LYNN H. LOOMIS and SHLOMO STERNBERG
Department of Mathematics, Harvard University
ADVANCED CALCULUS
REVISED EDITION
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Copyright © 1968 by Addison-Wesley Publishing Company, Inc.
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Library of Congress Cataloging-in-Publication Data
Loomis, Lynn H.
Advanced calculus / Lynn H. Loomis and Shlomo Sternberg. —Rev. ed.
p. cm.
Originally published: Reading, Mass. : Addison-Wesley Pub. Co., 1968.
ISBN 0-86720-122-3
1. Calculus. I. Sternberg, Shlomo. II. Title.
QA303.L87 1990
515-dc20 89-15620
CIP

PREFACE
This book is based on an honors course in advanced calculus that we gave in the
1960's. The foundational material, presented in the unstarred sections of
Chapters 1 through 11, was normally covered, but different applications of this basic
material were stressed from year to year, and the book therefore contains more
material than was covered in any one year. It can accordingly be used (with
omissions) as a text for a year's course in advanced calculus, or as a text for a
three-semester introduction to analysis.
These prerequisites are a good grounding in the calculus of one variable
from a mathematically rigorous point of view, together with some acquaintance
with linear algebra. The reader should be familiar with limit and continuity type
arguments and have a certain amount of mathematical sophistication. As
possible introductory texts, we mention Differential and Integral Calculus by R. Cou-
rant. Calculus by T. Apostol, Calculus by M. Spivak, and Pure Mathematics by
G. Hardy. The reader should also have some experience with partial derivatives.
In overall plan the book divides roughly into a first half which develops the
calculus (principally the differential calculus) in the setting of normed vector
spaces, and a second half which deals with the calculus of differentiable manifolds.
Vector space calculus is treated in two chapters, the differential calculus in
Chapter 3, and the basic theory of ordinary differential equations in Chapter 6.
The other early chapters are auxiliary. The first two chapters develop the
necessary purely algebraic theory of vector spaces. Chapter 4 presents the material
on compactness and completeness needed for the more substantive results of
the calculus, and Chapter 5 contains a brief account of the extra structure
encountered in scalar product spaces. Chapter 7 is devoted to multilinear (tensor)
algebra and is, in the main, a reference chapter for later use. Chapter 8 deals
with the theory of (Riemann) integration on Euclidean spaces and includes (in
exercise form) the fundamental facts about the Fourier transform. Chapters 9
and 10 develop the differential and integral calculus on manifolds, while Chapter
11 treats the exterior calculus of E. Cartan.
The first eleven chapters form a logical unit, each chapter depending on the
results of the preceding chapters. (Of course, many chapters contain material
that can be omitted on first reading; this is generally found in starred sections.)

Oil the other hand, Chapters 12^ 13, and the latter parts of Chapters 6 and 11
are independent of each other^ and are to be regarded m illustrative applications
of the methods developed in the earlier chapters. Presented here are elementary
Sturm-Liouville theory and Fourier series^ elementary differential geometry,
potential theory^ and classical mechanics. We usually covered only one or two
of these topics in our one-year course.
We have not hesitated to present the same material more than once from
different points of view. For example^ although we have selected the contraction
mapping fixed-point theorem as our basic approach to the implicit-function
theorem^ we have also outlined a "Kewton's method " proof in the text and have
sketched still a third proof in the exercises. Similarly^ the calculus of variations
is encountered twice—once in the context of the differential calculus of an
infinite-dimensional vector space and later in the context of classical mechanics.
The notion of a submanifold of a vector space is introduced in the early chapters^
while the invariant definition of a manifold is given later on.
In the introductory treatment of vector space theory, we are more careful
and precise than is customary. In fact, this level of precision of language is not
maintained in the later chapters. Our feeling is that in linear algebra, where the
concepts are so clear and the axioms so familiar, it is pedagogically sound to
illustrate various subtle points, such as distinguishing between spaces that are
normally identified, discussing the naturality of various maps, and so on. Later
on, when overly precise language would be more cumbersome, the reader should
be able to produce for himself a more precise version of any assertions that he
finds to be formulated too loosely. Similarly, the proofs in the first few chapters
are presented in more formal detail. Again, the philosophy is that once the
student has mastered the notion of what constitutes a formal mathematical
proof, it is safe and more convenient to present arguments in the usual
mathematical colloquialisms.
While the level of formality decreases, the level of mathematical
sophistication does not. Thus increasingly abstract and sophisticated mathematical
objects are introduced. It has been our experience that Chapter 9 contains the
concepts most difficult for students to absorb, especially the notions of the
tangent space to a manifold and the Lie derivative of various objects with
respect to a vector field.

There are exercises of many different kinds spread throughout the book.
Some are in the nature of routine applications. Others ask the reader to fill in
or extend various proofs of results presented in the text. Sometimes whole
topics, such as the Fourier transform or the residue calculus, are presented in
exercise form. Due to the rather abstract nature of the textual material, the
student is strongly advised to work out as many of the exercises as he possibly can.
Any enterprise of this nature owes much to many people besides the authors,
but we particularly wish to acknowledge the help of L. Ahlfors, A. Gleason,
R. Kulkami, R. Rasala, and G. Mackey and the general influence of the book by
Dieudonne. We also wish to thank the staff of Jones and Bartlett for their invaluable
help in preparing this revised edition.
Cambridge, Massachusetts L.H.L.
1968, 1989 S.S.

CONTENTS
Chapter 0 Introduction
1 Logic: quantifiers 1
2 The logical connectives 3
3 Negations of quantifiers 6
4 Sets 6
5 Restricted variables 8
6 Ordered pairs and relations 9
7 Functions and mappings 10
8 Product sets; index notation 12
9 Composition 14
10 Duality 15
11 The Boolean operations 17
12 Partitions and equivalence relations 19
Chapter 1 Vector Spaces
1 Fundamental notions 21
2 Vector spaces and geometry 36
3 Product spaces and Hom(y, W) 43
4 Affine subspaces and quotient spaces 52
5 Direct sums 56
6 Bilinearity 67
Chapter 2 Finite-Dimensional Vector Spaces
1 Bases 71
2 Dimension 77
3 The dual space 81
4 Matrices 88
5 Trace and determinant 99
6 Matrix computations 102
*7 The diagonalization of a quadratic form Ill
Chapter 3 The DifTerential Calculus
1 Reviewing 117
2 Norms 121
3 Continuity 126

4 Equivalent norms 132
5 Infinitesimals 136
6 The differential 140
7 Directional derivatives; the mean-value theorem 146
8 The differential and product spaces 152
9 The differential and R" 156
10 Elementary applications 161
11 The implicit-function theorem 164
12 Submanifolds and Lagrange multipliers 172
*13 Functional dependence 175
*14 Uniform continuity and function-valued mappings 179
*15 The calculus of variations 182
*16 The second differential and the classification of critical points . 186
*17 The Taylor formula 191
Chapter 4 Compactness and Completeness
1 Metric spaces; open and closed sets 195
*2 Topology 201
3 Sequential convergence 202
4 Sequential compactness 205
5 Compactness and uniformity 210
6 Equicontinuity 215
7 Completeness 216
8 A first look at Banach algebras 223
9 The contraction mapping fixed-point theorem 228
10 The integral of a parametrized arc 236
11 The complex number system 240
*12 Weak methods 245
Chapter 5 Scalar Product Spaces
1 Scalar products 248
2 Orthogonal projection 252
3 Self-adjoint transformations 257
4 Orthogonal transformations 262
5 Compact transformations 264

Chapter 6 Differential Equations
1 The fundamental theorem 266
2 Differentiate dependence on parameters 274
3 The linear equation 276
4 The nth-order linear equation 281
5 Solving the inhomogeneous equation 288
6 The boundary-value problem 294
7 Fourier series 301
Chapter 7 Multilinear Functionals
1 Bilinear functionals 305
2 Multilinear functionals 306
3 Permutations 308
4 The sign of a permutation 309
5 The subspace d"^ of alternating tensors 310
6 The determinant 312
7 The exterior algebra 316
8 Exterior powers of scalar product spaces 319
9 The star operator 320
Chapter 8 Integration
1 Introduction 321
2 Axioms 322
3 Rectangles and paved sets 324
4 The minimal theory 327
5 The minimal theory (continued) 328
6 Contented sets 331
7 When is a set contented? 333
8 Behavior under linear distortions 335
9 Axioms for integration . . 336
10 Integration of contented functions 338
11 The change of variables formula 342
12 Successive integration 346
13 Absolutely integrable functions 351
14 Problem set: The Fourier transform 355

Chapter 9 Differentiable Manifolds
1 Atlases 364
2 Functions, convergence 367
3 Differentiable manifolds 369
4 The tangent space 373
5 Flows and vector fields 376
6 Lie derivatives 383
7 Linear differential forms 390
8 Computations with coordinates 393
9 Riemann metrics 397
Chapter 10 The Integral Calculus on Manifolds
1 Compactness 403
2 Partitions of unity 405
3 Densities 408
4 Volume density of a Riemann metric 411
5 Fullback and Lie derivatives of densities 416
6 The divergence theorem 419
7 More complicated domains 424
Chapter 11 Exterior Calculus
1 Exterior differential forms 429
2 Oriented manifolds and the integration of exterior differential forms 433
3 The operator d 438
4 Stokes' theorem 442
5 Some illustrations of Stokes' theorem 449
6 The Lie derivative of a differential form 452
Appendix L "Vector analysis^^ 457
Appendix IL Elementary differential geometry of surfaces in E^ . 459
Chapter 12 Potential Theory in E'*
1 SoUd angle 474
2 Green's formulas 476
3 The maximum principle 477
4 Green's functions 479

5 The Poisson integral formula 482
6 Consequences of the Poisson integral formula 485
7 Harnack's theorem 487
8 Subharmonic functions 489
9 Dirichlet's problem 491
10 Behavior near the boundary 495
11 Dirichlet's principle 499
12 Physical applications 501
13 Problem set: The calculus of residues 503
Chapter 13 Classical Mechanics
1 The tangent and cotangent bundles 511
2 Equations of variation 513
3 The fundamental linear differential form on T*(M) . . . .515
4 The fundamental exterior two-form on T*(M) 517
5 Hamiltonian mechanics 520
6 The central-force problem 521
7 The two-body problem 528
8 Lagrange's equations 530
9 Variational principles 532
10 Geodesic coordinates 537
11 Euler's equations 541
12 Rigid-body motion 544
13 Small oscillations 551
14 Small oscillations (continued) 553
15 Canonical transformations 558
Selected References 569
Notation Index 572
Index 575

CHAPTER 0
INTRODUCTION
This preliminary chapter contains a short exposition of the set theory that
forms the substratum of mathematical thinking today. It begins with a brief
discussion of logic, so that set theory can be discussed with some precision, and
continues with a review of the way in which mathematical objects can be defined
as sets. The chapter ends with four sections which treat specific set-theoretic
topics.
It is intended that this material be used mainly for reference. Some of it
will be familiar to the reader and some of it will probably be new. We suggest
that he read the chapter through "lightly" at first, and then refer back to it
for details as needed.
1. LOGIC: QUANTIFIERS
A statement is a sentence which is true or false as it stands. Thus '1 < 2' and
'4 + 3 = 5' are, respectively, true and false mathematical statements. Many
sentences occurring in mathematics contain variables and are therefore not true
or false as they stand, but become statements when the variables are given
values. Simple examples are ^x < 4^ ^x < y\ ^x is an integer', '3a:^ + 2/^ = 10'.
Such sentences will be called statement frames. If P(x) is a frame containing the
one variable 'x\ then P(5) is the statement obtained by replacing ^x^ in P(x) by
the numeral '5\ For example, if P{x) is 'x < 4', then P(5) is '5 < 4', P{\/2)
is '\/2 < 4', and so on.
Another way to obtain a statement from the frame P(x) is to assert that P(x)
is always true. We do this by prefixing the phrase 'for every x\ Thus, 'for every
Xj X < 4:^ is Si false statement, and 'for every Xj x^ — 1 = (x — l)(x -{- 1)' is a
true statement. This prefixing phrase is called a universal quantifier.
Synonymous phrases are 'for each x^ and 'for all x\ and the symbol customarily
used is '(Vo:)', which can be read in any of these ways. One frequently presents
sentences containing variables as being always true without explicitly writing
the universal quantifiers. For instance, the associative law for the addition of
numbers is often written
x+ (y + z) = (x + y) + Zy
where it is understood that the equation is true for all Xy y and z. Thus the
1

2 INTRODUCTION 0.1
actual statement being made is
{^x){^y){^z)[x +{y + z)= {x + y)+ z].
Finally, we can convert the frame P{x) into a statement by asserting that
it is sometimes true, which we do by writing ^there exists an x such that P{xy.
This process is called existential quantification. Synonymous prefixing phrases
here are ^there is an x such that^ ^for some x\ and, symbolically, \^x)\
The statement ^(yx){x < 4)' still contains the variable ^x\ of course, but
^x^ is no longer free to be given values, and is now called a hound variable.
Roughly speaking, quantified variables are bound and unquantified variables
are free. The notation ^P{xy is used only when ^x^ is free in the sentence being
discussed.
Now suppose that we have a sentence P{x, y) containing two free variables.
Clearly, we need two quantifiers to obtain a statement from this sentence.
This brings us to a very important observation. // quantifiers of both types are
used, then the order in ivhich they are ivritten affects the meaning of the statement;
{^y)(Vx)P(x, y) and (Vx)(^y)P(Xy y) say different things. The first says that one y
can be found that works for all x: "there exists a y such that for all a:. . .".
The second says that for each x Oi y can be found that works: "for each x there
exists a y such that. . .". But in the second case, it may very well happen that
when X is changed, the y that can be found will also have to be changed. The
existence of a single y that serves for all x is thus the stronger statement. For
example, it is true that (yx){'^y){x < y) and false that {^y)(yx){x < y). The
reader must be absolutely clear on this point; his whole mathematical future is
at stake. The second statement says that there exists a y, call it 2/0, such that
(yx){x < 2/0), that is, such that every number is less than y^. This is false;
2/0 + 1, in particular, is not less than y^. The first statement says that for each x
we can find a corresponding y. And we can: take y = x -\- \.
On the other hand, among a group of quantifiers of the same type the order
does not affect the meaning. Thus ^(^x) (Vy)^ and '(Vy) (Vx)' have the same
meaning. We often abbreviate such clumps of similar quantifiers by using the
quantification symbol only once, as in \VXj y)\ which can be read ^for every x and y\
Thus the strictly correct ^(yx)(yy)(yz)[x + {y + z) = {x -^ y) -^ z\ receives the
slightly more idiomatic rendition ^(Vo:, y, z)[x + (2/ + 2;) = (^ + 2/) + A^- The
situation is clearly the same for a group of existential quantifiers.
The beginning student generally feels that the prefixing phrases ^for every x
there exists a y such that' and ^there exists a y such that for every x' sound
artificial and are unidiomatic. This is indeed the case, but this awkwardness is the
price that has to be paid for the order of the quantifiers to be fixed, so that the
meaning of the quantified statement is clear and unambiguous. Quantifiers do
occur in ordinary idiomatic discourse, but their idiomatic occurrences often
house ambiguity. The following two sentences are good examples of such
ambiguous idiomatic usage: "Every x is less than some y " and "Some y is greater
than every x". If a poll were taken, it would be found that most men on the

0.2 THE LOGICAL CONNECTIVES 3
street feel that these two sentences say the same thing, but half will feel that the
common assertion is false and half will think it true! The trouble here is that
the matrix is preceded by one quantifier and followed by anotlier, and the poor
reader doesn't know which to take as the inside, or first applied, quantifier. The
two possible symbolic renditions of our first sentence, ^[(Vx)(x < y)]{^y)^ and
\Vx)[(x < 2/)(3?/)]\ are respectively false and true. Mathematicians do use
hanging quantifiers in the interests of more idiomatic writing, but only if they
are sure the reader will understand their order of application, either from the
context or by comparison with standard usage. In general, a hanging quantifier
would probably be read as the inside, or first applied, quantifier, and with this
understanding our two ambiguous sentences become true and false in that order.
After this apology the reader should be able to tolerate the definition of
sequential convergence. It involves three quantifiers and runs as follows: The
sequence {xn} converges to x if (Ve) ('^N)(y7i) (if 7i > N then \xn — x\ < e).
In exactly the same format, we define a function / to be continuous at a if
(Ve)(3 5)(Vx)(if \x - a\ < 8 then \f{x) - f{a)\ < e). We often omit an inside
universal quantifier by displaying the final frame, so that the universal
quantification is understood. Thus we define / to be continuous at a if for every e
there is a 5 such that
if \x - a\ < 5, then \f{x) - f{a)\ < e.
We shall study these definitions later. We remark only that it is perfectly
possible to build up an intuitive understanding of what these and similar
quantified statements actually say.
2. THE LOGICAL CONNECTIVES
When the word 'and' is inserted between two sentences, the resulting sentence
is true if both constituent sentences are true and is false otherwise. That is, the
"truth value", T or F, of the compound sentence depends only on tlie truth
values of the constituent sentences. We can thus describe the way 'and' acts in
compounding sentences in the simple "truth table"
p
T
T
F
F
Q
T
F
T
F
P and Q
T
F
F
F
where 'P' and 'Q' stand for arbitrary statement frames. Words like 'and' are
called logical connectives. It is often convenient to use symbols for connectives,
and a standard symbol for 'and' is the ampersand '&'. Thus 'P & Q' is read
'P and Q\

4 INTRODUCTION 0.2
Another logical connective is the word 'or\ Unfortunately, this word is used
ambiguously in ordinary discourse. Sometimes it is used in the exclusive sense,
where ^P or Q^ means that one of P and Q is true, but not both, and sometimes
it is used in the inclusive sense that at least one is true, and possibly both are
true. Mathematics cannot tolerate any fundamental ambiguity, and in
mathematics 'or' is ahvays used in the latter way. We thus have the truth table
P Q P or Q
T T T
T F T
FT T
F F F
The above two connectives are binary, in the sense that they combine tivo
sentences to form one new sentence. The word 'not' applies to one sentence and
really shouldn't be considered a connective at all; nevertheless, it is called a
unary connective. A standard symbol for 'not' is ^^\ Its truth table is obviously
T F
F T
In idiomatic usage the word 'not' is generally buried in the interior of a
sentence. We write ^x is not equal to y' rather than 'not (x is equal to y)'.
However, for the purpose of logical manipulation, the negation sign (the word
'not' or a symbol like ''^') precedes the sentence being negated. We shall, of
course, continue to write ^x 9^ y\ but keep in mind that this is idiomatic for
'not {x = yY or ^^{x = yY.
We come now to the troublesome 'if ... , then . . .' connective, which we
write as either 'if P, then Q' or 'P =^ Q'. This is almost always applied in the
universally quantified context (yx){P{x) =^ Q{x))y and its meaning is best
unraveled by a study of this usage. We consider 'if a: < 3, then x < 5' to be a
true sentence. More exactly, it is true for all Xj so that the universal
quantification (Vx)(x < 3 =^ a: < 5) is a true statement. This conclusion forces us to
agree that, in particular, '2 < 3 =^ 2 < 5', '4 < 3 =^ 4 < 5', and '6 < 3 =^
6 < 5' are all true statements. The truth table for '=^' thus contains the
values entered below.
p
T
T
F
F
Q
T
F
T
F
P^Q
T
-
T
T

0.2 THE LOGICAL CONNECTIVES 5
On the other hand, we consider 'x < 7 =^ x < 5^ to he Si false sentence, and
therefore have to agree that '6 < 7 =^ 6 < 5Ms false. Thus the remaining row
in the table above gives the value T' for P =^ Q.
Combinations of frame variables and logical connectives such as we have
been considering are called truth-functional forms. We can further combine the
elementary forms such as ^P =^ Q^ and '^P^ by connectives to construct
composite forms such as ^^{P =^ QY and \P =^ Q) & (Q =^ Py. A sentence has a
given (truth-functional) form if it can be obtained from that form by substitution.
Thus ^x < y or ^{x < yY has the form 'P or ^P\ since it is obtained from this
form by substituting the sentence 'x < y^ for the sentence variable ^P\
Composite truth-functional forms have truth tables that can be worked out by
combining the elementary tables. For example, ^^{P =^ QY has the table below,
the truth value for the whole form being in the column under the connective
which is applied last (^^^ in this example).
p
T
T
F
F
Q
T
F
T
F
~iP ^ Q)
F
T
F
F
T
F
T
T
Thus ^(P =^ Q) is true only when P is true and Q is false.
A truth-functional form such as 'P or (^PY which is always true (i.e., has
only T' in the final column of its truth table) is called a tautology or a tautologous
form. The reader can check that
(P &(P^ Q)) ^ Q and ((P ^Q) &{Q^ R)) ^ {P ^ R)
are also tautologous. Indeed, any valid principle of reasoning that does not
involve quantifiers must be expressed by a tautologous form.
The 'if and only if' form T <^ Q\ or 'P if and only if Q\ or T iff Q\ is an
abbreviation for \P => Q) & {Q=> PY- Its truth table works out to be
P Q P^Q
T T T
T F F
F T F
F F T
That is, P <^ Q is true if P and Q have the same truth values, and is false
otherwise.
Two truth-functional forms A and B are said to be equivalent if (the final
columns of) their truth tables are the same, and, in view of the table for ^<f=^\
we see that A and B are equivalent if A ^^ B is tautologous^ and conversely.

6 INTRODUCTION 0.4
Replacing a sentence obtained by substitution in a form A by the equivalent
sentence obtained by the same substitutions in an equivalent form 5 is a device
much used in logical reasoning. Thus to prove a statement P true, it suffices to
prove the statement ^P false, since 'P' and ^^(^Py are equivalent forms.
Other important equivalences are
-(PorQ) ^ (-P) & (-Q),
(P=>Q) ^ Qor (-P),
^(P=>Q) <^ P & (^Q).
A bit of conventional sloppiness which we shall indulge in for smoother
idiom is the use of ^iV instead of the correct 'if and only if' in definitions. We
define/to be continuous at x if so-and-so, meaning, of course, that/is continuous
at X if and only if so-and-so. This causes no difficulty, since it is clear that 'if
and only if' is meant when a definition is being given.
3. NEGATIONS OF QUANTIFIERS
The combinations ^^(yxY and '{^x)^^ have the same meanings: something is
not always true if and only if it is sometimes false. Similarly, '^(^yY and '(Vy)'^'
have the same meanings. These equivalences can be applied to move a negation
sign past each quantifier in a string of quantifiers, giving the following important
practical rule:
In taking the negation of a statement beginning with a string of quantifiers^
we simply change each quantifier to the opposite kind and move the negation
sign to the end of the string.
Thus
^(Vx)(^y)(Vz)P(x,y,z) ^ (^x)(Vy)(^z)^P(x,y, z).
There are other principles of quantificational reasoning that can be isolated
and which we shall occasionally mention, but none seem worth formalizing here.
4. SETS
It is present-day practice to define every mathematical object as a set of some
kind or other, and we must examine this fundamental notion, however briefly.
A set is a collection of objects that is itself considered an entity. The objects
in the collection are called the elements or members of the set. The symbol for
'is a member of is 'g' (a sort of capital epsilon), so that 'x G A' is read "x is a
member of A", "x is an element of A", "x belongs to A", or "x is in A".
We use the equals sign '=' in mathematics to mean logical identity; A = B
means that A is B. Now a set A is considered to be the same object as a set B
if and only if A and B have exactly the same members. That is, 'A = B^ means
that (Vx)(x G A <^ X eB),

0.4 SETS 7
We say that a set A is a subset of a set B, or that A is included in B (or that
5 is a superset of A) if every element of A is an element of B. The symbol for
inclusion is f. Thus 'A C B' means that (Vx)(x G A => x G 5). Clearly,
(A = B) ^ (Ac B) and (5 C A).
This is a frequently used way of establishing set identity: we prove that A = B
by proving that A C B and that B C A. If the reader thinks about the above
equivalence, he will see that it depends first on the equivalence of the
truth-functional forms T<i=>Q' and '(P=>Q) & {Q=>P)\ and then on the obvious
quantificational equivalence between '(Vx)(R & Sy and '(yx)R & (yx)S\
We define a set by specifying its members. If the set is finite, the members
can actually be Hsted, and the notation used is braces surrounding a
membership list. For example {1, 4, 7} is the set containing the three numbers 1, 4, 7,
{x} is the unit set of x (the set having only the one object x as a member),
and {Xy y} is the paii' set of x and y. We can abuse this notation to name some
infinite sets. Thus {2, 4, 6, 8, . . .} would certainly be considered the set of all
even positive integers. But infinite sets are generally defined by statement
frames. If P(x) is a frame containing the free variable ^x\ then {x : P{x)} is the
set of all X such that P(x) is true. In other words, {x : P(x)} is that set A such
that
y gA^ P{y).
For example, {x : x^ < 9} is the set of all real numbers x such that x^ < 9,
that is, the open interval (—3, 3), and ?/ G {x : x^ < 9} <i=> y^ < 9. A statement
frame P{x) can be thought of as stating a property that an object x may or may
not have, and {x : P(x)} is the set of all objects having that property.
We need the empty set 0, in much the same way that we need zero in
arithmetic. If P(x) is never true, then {x : P{x)} = 0. For example,
{x :x ^ x} = 0.
When we said earlier that all mathematical objects are customarily
considered sets, it was taken for granted that the reader understands the distinction
between an object and a name of that object. To be on the safe side, we add a
few words. A chair is not the same thing as the word 'chair', and the number 4
is a mathematical object that is not the same thing as the numeral '4\ The
numeral '4' is a name of the number 4, as also are 'four\ '2 + 2\ and 'IV\
According to our present viewpoint, 4 itself is taken to be some specific set.
There is no need in this course to carry logical analysis this far, but some readers
may be interested to know that we usually define 4 as {0, 1, 2, 3}. Similarly,
2 = {0, 1}, 1 = {0}, and 0 is the empty set 0.
It should be clear from the above discussion and our exposition thus far
that we are using a symbol surrounded by single quotation marks as a name of
that symbol (the symbol itself being a name of something else). Thus ' '4' ' is a
name of '4' (which is itself a name of the number 4). This is strictly correct

8 INTRODUCTION 0.5
usage, but mathematicians almost universally mishandle it. It is accurate to
write: let x be the number; call this number 'x\ However, the latter is almost
always written: call this number x. This imprecision causes no difficulty to the
reading mathematician, and it often saves the printed page from a shower of
quotation marks. There is, however, a potential victim of such ambiguous
treatment of symbols. This is the person who has never realized that
mathematics is not about symbols but about objects to which the symbols refer. Since
by now the present reader has safely avoided this pitfall, we can relax and
occasionally omit the strictly necessary quotation marks.
In order to avoid overworking the word 'set', we use many synonyms,
such as 'class^ 'collection', 'family' and 'aggregate'. Thus we might say, "Let d
be a family of classes of sets". If a shoe store is a collection of pairs of shoes, then
a chain of shoe stores is such a three-level object.
5. RESTRICTED VARIABLES
A variable used in mathematics is not allowed to take all objects as values; it
can only take as values the members of a certain set, called the domain of the
variable. The domain is sometimes explicitly indicated, but is often only
implied. For example, the letter ' n' is customarily used to specify an integer, so
that' (yn)P{ny would automatically be read "for every integer n, P{n)".
However, sometimes n is taken to be a positive integer. In case of possible ambiguity
or doubt, we would indicate the restriction explicitly and write' (Vn G Z)P(n)\
where 'Z' is the standard symbol for the set of all integers. The quantifier is
read, literally, "for all n in Z", and more freely, "for every integer n". Similarly,
'(3n G Z)P(n)' is read "there exists an n in Z such that P{7i)" or "there exists
an integer n such that P{n)". Note that the symbol 'g' is here read as the
preposition 'in\ The above quantifiers are called restricted quantifiers.
In the same way, we have restricted set formation, both implicit and explicit,
as in '{n : P(n)}' and '{n G Z : P(n)}', both of which are read "the set of all
integers n such that P(n)".
Restricted variables can be defined as abbreviations of unrestricted variables
by
(Vx G A)P(x) <^ (yx) {xgA=> P(x)),
(^xgA)P(x) <^ (3x)(xG A &P(x)),
{xgA: P(x)} = {x:xeA& P(x)}.
Although there is never any ambiguity in sentences containing explicitly
restricted variables, it sometimes helps the eye to see the structure of the
sentence if the restricting phrases are written in superscript position, as in
(V€ ^^) (3n^^). Some restriction was implicit on page 1. If the reader agreed that
(yx){x^ — 1 = (x — l)(x -{- 1)) was true, he probably took x to be a real
number.

0.6 ORDERED PAIRS AND RELATIONS 9
6. ORDERED PAIRS AND RELATIONS
Ordered pairs are basic tools, as the reader knows from analytic geometry.
According to our general principle, the ordered pair < a, 6 > is taken to be a
certain set, but here again we don't care which particular set it is so long as it
guarantees the crucial characterizing property:
<Xyy> = <ayh> <^ X = a and y = h.
Thus <1, 3> 9^ <3, 1>.
The notion of a correspondence, or relation, and the special case of a
mapping, or function, is fundamental to mathematics. A correspondence is a pairing
of objects such that given any two objects x and ?/, the pair <Xy y> either does
or does not correspond. A particular correspondence (relation) is generally
presented by a statement frame P(x, y) having two free variables, with x and y
corresponding if any only if P{Xj y) is true. Given any relation (correspondence),
the set of all ordered pairs <x^y> of corresponding elements is called its graph.
Now a relation is a mathematical object, and, as we have said several times,
it is current practice to regard every mathematical object as a set of some sort
or other. Since the graph of a relation is a set (of ordered pairs), it is efficient and
customary to take the graph to he the relation. Thus a relation {correspondence)
is simply a set of ordered pairs. If 72 is a relation, then we say that x has the
relation R to y, and we write ^xRy\ if and only if <Xyy> G R. We also say
that X corresponds to y under R. The set of all first elements occurring in the
ordered pairs of a relation R is called the domain of R and is designated dom R
orT>(R). Thus
dom R = {x : (3?/) <x,y> G R},
The set of second elements is called the i^ange of R:
range R = {y : (3x) <Xy y> G R}.
The inverse, 72~^, of a relation R is the set of ordered pairs obtained by reversing
those of R:
R-^ = {^x, y> : <y, x> G R}.
A statement frame P(x, y) having two free variables actually determines a pair
of mutually inverse relations R & Sy called the graphs of P, as follows:
R = {<x, y> : P(x, y)}, S = {<y, x> : P(x, y)}.
A two-variable frame together with a choice of which variable is considered to
be first might be called a directed frame. Then a directed frame would have a
uniquely determined relation for its graph. The relation of strict inequality
on the real number system U would be considered the set { < x, y > : x < y},
since the variables in 'x < y^ have a natural order.
The set A X B = {<Xjy> : x G A & y G B} of all ordered pairs with
first element in A and second element in B is called the Cartesian product of the

10
INTKODUCTION
0.7
sets A and B, A relation R is always a subset of dom R X range R. If the two
"factor spaces" are the same, we can use exponential notation: A^ = A X A.
The Cartesian product IR^ = IR X i^ is the "analytic plane". Analytic
geometry rests upon the one-to-one coordinate correspondence between U^ and
the Euclidean plane E^ (determined by an axis system in the latter), which
enables us to tr»t geometric questions algebraically and algebraic questions
geometrically. In particular, since a relation between sets of real numbers is a
subset of U^, we can "picture" it by the corresponding subset of the Euclidean
plane, or of any model of the Euclidean plane, such as this page. A simple
Cartesian product is shown in Fig. 0.1 (A U -B is the union of the sets A and B).
1 .4 A
AXB when yl = [l, 2]u[2i 3] and i? = [l, li]u{2}
Fig. 0.1
Fig. 0.2
If jB is a relation and A is any set, then the restriction of R to A, R \ A,
is the subset of R consisting of those pairs with first element in A:
R \ A = {<x, y> : <x, y> G R and x G A},
Thus R \ A = R n (A X range R), where C n D is the intersection of the sets
C and D.
If E is a relation and A is any set, then the image of A under Rj R[A]^ is
the set of second elements of ordered pairs in R whose first elements are in A:
R[A] = {y : Ox)ix gA& <x,y> G R)}.
Thus R[A] = range (R t A), as shown in Fig. 0.2.
7. FUNCTIONS AND MAPPINGS
A function is a relation / such that each domain element x is paired with exactly
one range element y. This property can be expressed as follows:
<x, y> G/and <Xj z> G f =^ y =^ z.

0.7
FUNCTIONS AND MAPPINGS
11
The y which is thus uniquely determined by / and x is designated f{x):
y = fix) <=» <x,y> Gf.
One tends to think of a function as being active and a relation which is not
a function as being passive. A function / acts on an element x in its domain to
give/(a:). We take x and apply / to it; indeed we often call a function an operator.
On the other handj if ^ is a relation but not a function^ then there is in general
no particular y related to an element x in its domain^ and the pairing of x and y
is viewed more passively.
We often define a function / by specifying its value f{x) for each x in its
domain, and in this connection a stopped arrow notation is used to indicate the
pairing. Thus x^-^ x^ is the function assigning to each number x its square x^.
<-2, 4>
Fig. 0.3
If we want it to be understood that / is this function^ we can write "Consider
the function fix ^-^ x^". The domain of / must be understood for this notation
to be meaningful.
If / is a function, then /"^ is of course a relation^ but in general it is not a
function. For example^ if / is the function x *-» x^^ then f~^ contains the pairs
<4, 2> and <4; —2> and so is not a function (see Fig. 0.3). If/"^ is a
function, we say that/is one-to-one and that/is a one-to-one correspondence between
its domain and its range. Each x G dom/ corresponds to only one y G mnge/
(/is a function), and each y G range/ corresponds to only one x G dom/ (/""^ is
a function).
The notation
fiA^B
is rmd % (the) function f on A into B*' or "the function / from A to B". The
notation implies that / is a function, that dom/ = A, and that range f C B.
Many people feel that the very notion of function should include all these
ingredients; that is, a function should be considered an ordered triple < /, A, 5 >,
where / is a function according to our more limited definition, A is the domain

12 INTRODUCTION 0.8
of /j and 5 is a superset of the range of f, which we shall call the codomain of / in
this context. We shall use the terms 'map'^ 'mapping'^ and 'transformation'
for such a triple^ so that the notation/: A —> B in its totality presents a mapping.
Moreover^ when there is no question about which set is the codomain^ we shall
often call the function / itself a mapping^ since the triple </, A, B> is then
determined by /. The two arrow notations can be combined^ as in: "Define
/:IR-> Uhy x^x^'\
A mapping f: A —> B is said to be injedive if / is one-to-one^ surjedive if
range f = B^ and hijedive if it is both injective and surjective. A bijective
mapping f: A —> B is thus a one-to-one correspondence between its domain A
and its codomain B. Of course^ a function is always surjective onto its range R,
and the statement that / is surjective means that R = B, where B is the
understood codomain.
8. PRODUCT SETS; INDEX NOTATION
One of the characteristic habits of the modern mathematician is that as soon as
a new kind of object has been defined and discussed a little^ he immediately
looks at the set of all such objects. With the notion of a function from A to /S
well in handj we naturally consider the set of all functions from A to S, which we
designate S^. Thus R" is the set of all real-valued functions of one real variable^
and S^'^ is the set of all infinite sequences in S. (It is understood that an infinite
sequence is nothing but a function whose domain is the set Z"^ of all positive
integers.) Similarly^ if we set n = {1, . , , ^n}, then S^ is the set of all finite
sequences of length n in S.
If 5 is a subset of Sy then its charaderistic fundion (relative to S) is the
function on aSj usually designated Xbj which has the constant value 1 on 5 and the
constant value 0 off B. The set of all characteristic functions of subsets of S is
thus 2*^ (since 2 = {Oj 1}). But because this collection of functions is in a
natural one-to-one correspondence with the collection of all subsets of /S, Xb
corresponding to B^ we tend to identify the two collections. Thus 2*^ is also
interpreted as the set of all subsets of S. We shall spend most of the remainder
of this section discussing further similar definitional ambiguities which
mathematicians tolerate.
The ordered triple <XyyyZ> is usually defined to be the ordered pair
< <Xyy> y z>. The reason for this definition is probably that a function of
two variables x and y is ordinarily considered a function of the single ordered
pair variable <Xy y>^ so thatj for example^ a real-valued function of two real
variables is a subset of (R X R) X R. But we also consider such a function a
subset of Cartesian 3-space R^. Therefore, we define R^ as (R X R) X R;
that isj we define the ordered triple <XjyjZ> as <<Xjy> ,z>,
On the other hand^ the ordered triple <Xy y, z> could also be regarded as
the finite sequence {<lja:>j <2yy>y <d, z>}y which^ of course^ is a different
object. These two models for an ordered triple serve equally wellj and^ again,

0.8 PRODUCT sets; index notation 13
mathematicians tend to slur over the distinction. We shall have more to say
on this point later when we discuss natural isomorphisms (Section 1.6). For
the moment we shall simply regard U^ and U^ as being the same; an ordered
triple is something which can be "viewed" as being either an ordered pair of
which the first element is an ordered pair or as a sequence of length 3 (or^ for that
matter J as an ordered pair of which the second element is an ordered pair).
Similarly^ we pretend that Cartesian 4-space R^ is R^j U^ X U^y or
Ri X R^ = UX {(UXU) XU), etc. Clearly, we are in effect assuming an
associative law for ordered pair formation that we don^t really have.
This kind of ambiguity, where we tend to identify two objects that really are
distinct, is a necessary corollary of deciding exactly what things are. It is one
of the prices we pay for the precision of set theory; in days when mathematics
was vaguer, there would have been a single fuzzy notion.
The device of indices, which is used frequently in mathematics, also has
ambiguous implications which we should examine. An indexed collection, as a set,
is nothing but the range set of a function, the indexing function, and a particular
indexed object, say Xi^ is simply the value of that function at the domain element i.
If the set of indices is /, the indexed set is designated {xi: i G 1} or {xi] iei
(or {xi}Z=i in case 7 = Z"^). However, this notation suggests that we view the
indexed set as being obtained by letting the index run through the index set I
and collecting the indexed objects. That is, an indexed set is viewed as being
the set together with the indexing function. This ambivalence is reflected in the
fact that the same notation frequently designates the mapping. Thus we refer
to the sequence {Xr^n=\j where, of course, the sequence is the mapping n i—» x^.
We believe that if the reader examines his idea of a sequence he will find this
ambiguity present. He means neither just the set nor just the mapping, but the
mapping with emphasis on its range, or the range "together with" the mapping.
But since set theory cannot reflect these nuances in any simple and graceful way,
we shall take an indexed set to he the indexing function. Of course, the same
range object may be repeated with different indices; there is no implication that
an indexing is one-to-one. Note also that indexing imposes no restriction on the
set being indexed; any set can at least be self-indexed (by the identity function).
Except for the ambiguous ^{xi: i G /}', there is no universally used notation
for the indexing function. Since xi is the value of the function at z, we might
think of ^Xi as another way of writing ^x(i)\ in which case we designate the
function ^x^ or 'x'. We certainly do this in the case of ordered n-tuplets when
we say, "Consider the n-tuplet x = <Xi, . . . , Xn> ". On the other hand, there
is no compelling reason to use this notation. We can call the indexing function
anything we want; if it is/, then of course/(z) = xi for all i.
We come now to the general definition of Cartesian product. Earlier we
argued (in a special case) that the Cartesian product A X 5 X C is the set of
all ordered triples x = <Xi, X2j x^> such that xi G A, ^2 G 5, and X3 G C.
More generally, Ai X A2 X • • • X A^, or IILi ^h is the set of ordered n-
tuples X = <xij . . . J Xn> such that Xi G Ai for z = 1? . . . ? n. If we interpret

14 INTRODUCTION 0.9
an ordered n-tuplet as a function onn = {1^ . . . ^ n}^ we have
IIi=i ^i is the set of all functions x with domain n such that x^.G Ai
for all i G n.
This rephrasal generalizes almost verbatim to give us the notion of the
Cartesian product of an arbitrary indexed collection of sets.
Definition. The Cartesian product JlieiSi of the indexed collection of
sets {Si: i G /} is the set of all functions / with domain the index set I
such that f(i) G Si for all i G /.
We can also use the notation Y[{Si :iGl} for the product and fi for the
value/(z).
9. COMPOSITION
If we are given maps f:A—^B and g: B —> C, then the composition of g with f,
g ° fy is the map of A into C defined by
{9-f){x) = g{f{x)) for all x G A.
This is the function of a function operation of elementary calculus. If / and g are
the maps from R to R defined hy f{x) = x^'^ + 1 and g{x) = x^, then / © g{x) =
(^2)1/3 + 1 = ^2/3 _^ 1^ ^^^ ^ ,^(^) _ (^1/3 _^ 1)2 _ ^2/3 _^ 2^^/^ + 1. Note
that the codomain of / must be the domain of g in order forgof to be defined.
This operation is perhaps the basic binary operation of mathematics.
Lemma. Composition satisfies the associative law:
fo{goh)= {fog)oh.
Proof. {fo{goh)){x) = f{{goh){x)) = f{g{Hx))) = {fog){hix)) =
((/ ° q) ° h){x) for all X G dom h, D
If A is a set J the identity map 7^: A —> A is the mapping taking every
X G A to itself. Thus I a = {<x,x> : x G A}. If / maps A into B, then clearly
foI^=f= Is of.
If g:B —> A is such that g ° f = I Ay then we say that g isa left inverse of / and
that / is a right inverse of g.
Lemma. If the mapping f:A—>B has both a right inverse h and a left
inverse g, they must necessarily be equal.
Proof. This is just algebraic juggling and works for any associative operation.
We have
h = lAoh = (g o f) o h = g o (f o h) = g o Is = g, D

0.10 DUALITY 15
In this case we call the uniquely determined map g: B —^ A such that
f o g = J^ and g ° f = I a the inverse of /. We then have:
Theorem. A mapping/: A —^ B has an inverse if and only if it is bijective^
in which case its inverse is its relational inverse/~^
Proof. If / is bijectivej then the relational inverse/"^ is a function from B to A^
and the equations f ° f~^ = Ib and f~^ ° f = Ia are obvious. On the other
hand J if f o g = I^^ then / is surjective^ since then every y in B can be written
y = f{g{y))' And if g °f = I Ay then / is injective^ for then the equation
/W = fiy) implies that x = g{f{x)) = g{f{y)) = y. Thus / is bijective if it
has an inverse. D
Now let ®(A) be the set of all bisections /: A —^ A, Then ®(A) is closed
under the binary operation of composition and
1) f°{0°h)= {fog)oh for all /, g, h G ®;
2) there exists a unique I G ®(A) such that f o I = 7 o / = / for all / G ®;
3) for each / G ® there exists a unique {/ G ® such that f ° g = g ° f = I-
Any set G closed under a binary operation having these properties is called
a group with respect to that operation. Thus ®(A) is a group with respect to
composition.
Composition can also be defined for relations as follows. If 72 C A X 5 and
SCBXC, then S o R c A X C is defined by
<x,z> gSo R ^ {^y^^){<x, y> gR & <y,z> G S).
If R and S are mappings^ this definition agrees with our earlier one.
10. DUALITY
There is another elementary but important phenomenon called duality which
occurs in practically all branches of mathematics. Let F: A X B —^C he any
function of two variables. It is obvious that if x is held fixed^ then F{Xy y) is a
function of the one variable y. That iSj for each fixed x there is a function
h^:B —> C defined by h^{y) = F{x, y). Then x i-> /i"^ is a mapping <^ of A into
C^. Similarly^ each y gB yields a function gy G C^j where gy(x) = F{Xy y),
and y i-^ gy is a mapping 6 from B to C"^.
Now suppose conversely that we are given a mapping <^: A —> C^. For each
X G A we designate the corresponding value of (p in index notation as h"^^ so
that h"" is a function from B to C, and we define F: A X B —^ C hy F{Xy y) =
fv^{y). We are now back where we started. Thus the mappings <^: A —> C^^
F: A X B —^ Cj and 6: B —^ C^ are equivalent^ and can be thought of as three
different ways of viewing the same phenomenon. The extreme mappings (p and
0 will be said to be dual to each other.

16 INTRODUCTION 0.10
The mapping ip is the indexed family of functions {h^:x ^ A} c C^. Now
suppose that JF C C^ is an wmndexed collection of functions on B into C, and
define F: JF X 5 -> C by F{f, y) = f{y). Then d:B->C^ is defined by gy{f) =
f{y). What is happening here is simply that in the expression f{y) we regard both
symbols as variables^ so that f{y) is a function on JF X B. Then when we hold y
fixedj we have a function on JF mapping JF into C.
We shall see some important applications of this duality principle as our
subject develops. For example^ sm mX n matrix is a function t = {Uj} in
^mXn^ We picture the matrix as a rectangular array of numbers^ where 'i* is
the row index and 7' is the column indeXj so that Uj is the number at the
intersection of the ith row and the jth column. If we hold i fixedj we get the n-tuple
forming the ith rowj and the matrix can therefore be interpreted as an m-tuple
of row n-tuples. Similarly (dually) ^ it can be viewed as an n-tuple of column
m-tuples.
In the same vein^ an n-tuple <fu - - - ffn^ of functions from A to 5 can
be regarded as a single n-tuple-valued function from A to B^,
ai-> <fi{a),. . . Jn{a)>.
In a somewhat different application^ duality will allow us to regard a finite-
dimensional vector space V as being its own second conjugate space (F*)*.
It is instructive to look at elementary Euclidean geometry from this point
of view. Today we regard a straight line as being a set of geometric points.
An older and more neutral view is to take points and lines as being two different
kinds of primitive objects. Accordingly^ let A be the set of all points (so that A
is the Euclidean plane as we now view it)? and let B be the set of all straight lines.
Let F be the incidence function: F{p, I) = 1 if p and I are incident {p is "on" Z,
I is "on" p) and F{p, I) = 0 otherwise. Thus F maps Ax B into {0, 1}. Then
for each I G B the function gi{p) = F{p^ I) is the characteristic function of the
set of points that we think of as being the line I {gi{p) has the value 1 if /? is on Z
and 0 if 7? is not on I.) Thus each line determines the set of points that are on it.
But J dually^ each point p determines the set of lines I "on" it, through its
characteristic function h^{l). ThuSj in complete duality we can regard a line as being
a set of points and a point as being a set of lines. This duality aspect of geometry
is basic in projective geometry.
It is sometimes awkward to invent new notation for the "partial" function
obtained by holding a variable fixed in a function of several variables, as we did
above when we set gy{x) = F{x^ y)^ and there is another device that is frequently
useful in this situation. This is to put a dot in the position of the "varying
variable". Thus F(a, •) is the function of one variable obtained from F(x, y)
by holding x fixed at the value a, so that in our beginning discussion of duality
we have
h- = Fix,-), 9y = F{-,y).
If / is a function of one variable, we can then write f == f{-), and so express the

0.11 THE BOOLEAN OPERATIONS 17
above equations also ash^{') = F{x, ')y gy{') = F{' ^ y). The flaw in this notation
is that we can't indicate substitution without losing meaning. Thus the value
of the function F(Xf •) at 6 is F(x^ h)^ but from this evaluation we cannot read
backward and tell what function was evaluated. We are therefore forced to
some such cumbersome notation as F{x^ ')\b^ which can get out of hand.
Nevertheless, the dot device is often helpful when it can be used without evaluation
difl^iculties. In addition to eliminating the need for temporary notation^ as
mentioned above^ it can also be used^ in situations where it is strictly speaking
superfluous, to direct the eye at once to the position of the variable.
For example, later on D^F will designate the directional derivative of the
function F in the (fixed) direction f. This is a function whose value at a is
D^F{a)j and the notation D^F{') makes this implicitly understood fact explicit.
11. THE BOOLEAN OPERATIONS
Let aS be a fixed domain, and let JF be a family of subsets of S. The Mnion of JF,
or the union of all the sets in JF, is the set of all elements belonging to at least one
set in JF. We designate the union \J^ or [Jae^ ^, and thus we have
\J^= {x: (3A^^)(x G A)}, yG[j^^ {^A^^Xv G A).
We often consider the family JF to be indexed. That is, we assume given a set /
(the set of indices) and a surjective mapping z i-» Ai from I to 3^, so that 3^ =
{Ai :i E: I}. Then the union of the indexed collection is designated [jiei Ai or
\J{Ai:i G I}. The device of indices has both technical and psychological
advantages, and we shall generally use it.
If JF is finite, and either it or the index set is listed, then a different notation
is used for its union. If JF = {A, 5}, we designate the union A U 5, a notation
that displays the listed names. Note that here we have xGA\jB<=^xGAor
x gB. If JF == {Ai :i = 1, . . . , n}, we generally write 'Aj U Ag U • • • U A^
or Vi^i Ai for IJ^.
The intersection of the indexed family {A^j^e/j designated flze/ A^-, is the
set of all points that lie in every Ai. Thus
xG [\Ai ^ {\ffJ)(x^A,),
iei
For an unindexed family JF we use the notation f)^ or fl^ejF A, and if JF =
{A, B}, then f]^ = AnB.
The complement, A', of a subset of S is the set of elements x i' S not in
A: A^ = {x^^ : X ^ A}, The law of De Morgan states that the complement of
an intersection is the union of the complements:
(f) Ai)'=[j (A'i).
\iEI / lEI
This an immediate consequence of the rule for negating quantifiers. It is the

18 INTRODUCTION 0.11
equivalence between 'not always in' and 'sometimes not in': [^{Vi)(x G Ai) ^^
{^i){x S Ai)] says exactly that
xe(T\Aiy ^ xG\J{Al).
If we set Bi = A'l and take complements again^ we obtain the dual form:
Other principles of quantification yield the laws
Bc^({} aA = U {Bc^Ai)
from P & C^x)Q{x) <=> (3x)(P & Q{x)),
BufCl Ai) - n (BuAi),
BnfCi Ai) - n (BnAi),
Bu (D Ai) = U (BuAi),
In the case of two setSj these laws imply the following familiar laws of set algebra:
(A U By = A' n B% (A n By = A'uB' (De Morgan),
A n (5 u C) = (A n 5) u (A n C),
A u (5 n C) = (A u 5) n (A u C).
Even here, thinking in terms of indices makes the laws more intuitive. Thus
(Ai n A2)' = A'l U A'2
is obvious when thought of as the equivalence between 'not always in' and
'sometimes not in'.
The family ^ is disjoint if distinct sets in 3^ have no elements in common, i.e.,
if (VX, F^^)(X ^ Y=^ XnY = 0), For an indexed family {A,},ej the
condition becomes i 9^ j =^ Ai n Ay = 0. If 3^ = {A, 5}, we simply say that
A and B are disjoint.
Given f:U—^V and an indexed family {Bi} of subsets of F, we have the
following important identities:
and, for a single set B cVj
For example,
a; e/-' [ n 5i] ^/(a^) e n B,- ^ (Vz) (/(a;) e 5,)

0.12 PARTITIONS AND EQUIVALENCE RELATIONS 19
The first, but not the other two, of the three identities above remains valid
when / is replaced by any relation R. It follows from the commutative law,
(3x)(3^)A <^ (3^)(3x)A. The second identity fails for a general R because
\^x)(Vyy and '(V^)(3x)' have different meanings.
12. PARTITIONS AND EQUIVALENCE RELATIONS
A partition of a set A is a disjoint family ^ of sets whose union is A. We call the
elements of ^ 'fibers^ and we say that 'i^ fibers A or is afihering of A. For example,
the set of straight lines parallel to a given line in the Euclidean plane is a fibering
of the plane. If ^x^ designates the unique fiber containing the point x, then
X I—> ^ is a surjective mapping tt: A —> [F which we call the projection of A on [F.
Passing from a set A to a fibering [F of A is one of the principal ways of forming
new mathematical objects.
Any function / automatically fibers its domain into sets on which / is
constant. If A is the Euclidean plane and f{p) is the x-coordinate of the point p in
some coordinate system, then / is constant on each vertical line; more exactly,
f~^(x) is a vertical line for every x in U. Moreover, x ^-^ f~^(x) is a bijection
from U to the set of all fibers (vertical lines). In general, if f:A—^B is any
surjective mapping, and if for each value y in B we set
^y = r\y) = {x^A: fix) = y],
then 3^ = {^y '- y ^ ^} is a fibering of A and (p:y ^-^ Ay is a bijection from
B to 3^. Also (^ o / is the projection tt: A —> 3^, since (f o f{x) = (p{f{x)) is the
set X of all 2 in A such that f{z) = f{x).
The above process of generating a fibering of A from a function on A is
relatively trivial. A more important way of obtaining a fibering of A is from
an equality-like relation on A called an equivalence relation. An equivalence
relation '^ on A is a binary relation which is reflexive {x ^ x for every x E: A)^
symmetric (x ^ y =^ y ^ x)y and transitive (x ^ y and y ^ z =^ x ^ z). Every
fibering JF of A generates a relation ^ by the stipulation that x ^ y if and only if
x and y are in the same fiber, and obviously ^ is an equivalence relation. The
most important fact to be established in this section is the converse.
Theorem. Every equivalence relation '^ on A is the equivalence relation
of a fibering.
Proof. We obviously have to define x as the set of elements y equivalent to x,
X = {y '- y "^ x}y and our problem is to show that the family [F of all subsets of A
obtained this way is a fibering.
The reflexive, symmetric, and transitive laws become
X GXy X Gy =^ y GXf and x Gy and y Gz =^ x Gz.
Reflexivity thus implies that 3^ covers A. Transitivity says that if y GZy then
X Gy =^ X Gz; that is, if y Gz, then y Cz. But also, if y Gz, then z Gy by

20 INTRODUCTION 0.12
symmetry, and so z Cy- Thus y Gz implies y = z. Therefore, if two of our
sets a and h have a point x in common, then a = x = h. In other words, if a is
not the set 5, then a and h are disjoint, and we have a fibering. D
The fundamental role this argument plays in mathematics is due to the fact
that in many important situations equivalence relations occur as the primary
object, and then are used to define partitions and functions. We give two
examples.
Let Z be the integers (positive, negative, and zero). A fraction ^m/n^ can
be considered an ordered pair <m^n> of integers with n 9^ Q. The set of all
fractions is thus I.X{1— {0}). Two fractions <mjn> and <pyq> are
"equal" if and only if mq = np, and equality is checked to be an equivalence
relation. The equivalence class <m, n> is the object taken to be the rational
number m/n. Thus the rational number system Q is the set of fibers in a
partition of Z X (Z - {0}).
Next, we choose a fixed integer 7? G Z and define a relation jEJ on Z by
mEn <=^ p divides m — n. Then E is an equivalence relation, and the set Zp of
its equivalence classes is called the integers modulo p. It is easy to see that mEn
if and only if m and n have the same remainder when divided by 7?, so that in
this case there is an easily calculated function /, where f(m) is the remainder
after dividing m by 7?, which defines the fibering. The set of possible remainders
is {0, 1, . . . , 7? — 1}, so that Zp contains p elements.
A function on a set A can be "factored" through a fibering of A by the
following theorem.
Theorem. Let g he a function on A, and let JF be a fibering of A. Then g
is constant on each fiber of JF if and only if there exists a function g on 5
such that g = g o TT.
Proof. If g is constant on each fiber of JF, then the association of this unique
value with the fiber defines the function ^, and clearly g = "g o tt. The converse
is obvious. D

CHAPTER 1
VECTOR SPACES
The calculus of functions of more than one variable unites the calculus of one
variable, which the reader presumably knows, with the theory of vector spaces,
arid the adequacy of its treatment depends directly on the extent to which vector
space theory really is used. The theories of differential equations and differential
geometry are similarly based on a mixture of calculus and vector space theory.
Such "vector calculus" and its applications constitute the subject matter of this
book, and in order for our treatment to be completely satisfactory, we shall
have to spend considerable time at the beginning studying vector spaces
themselves. This we do principally in the first two chapters. The present chapter is
devoted to general vector spaces and the next chapter to finite-dimensional
spaces.
We begin this chapter by introducing the basic concepts of the subject—
vector spaces, vector subspaces, linear combinations, and linear
transformations—and then relate these notions to the lines and planes of geometry. Next
we establish the most elementary formal properties of linear transformations and
Cartesian product vector spaces, and take a brief look at quotient vector spaces.
This brings us to our first major objective, the study of direct sum
decompositions, which we undertake in the fifth section. The chapter concludes with a
])reliminary examination of bilinearity.
I. FUNDAMENTAL NOTIONS
Vector spaces and subspaces. The reader probably has already had some
('ontact with the notion of a vector space. Most beginning calculus texts discuss
f!;oometric vectors, which are represented by "arrows" drawn from a chosen
origin 0. These vectors are added geometrically by the parallelogram rule:
The sum of the vector OA (represented by the arrow from 0 to A) and the
vector OB is the vector OP, where P is the vertex opposite 0 in the parallelogram
liaving OA and OB as two sides (Fig. 1.1). Vectors can also be multiplied by
numbers: x(OA) is that vector OB such that B is on the line through 0 and
A, the distance from 0 to B is \x\ times the distance from 0 to A, and B and A
live on the same side of 0 if a: is positive, and on opposite sides if x is negative
21

22
VECTOR SPACES
1.1
OC=-iOA
Fig. 1.1
Fig. 1.2
(OA+OB)+OC = OP+OC = OX OA + iOB+OC)^OA+()Q = OX
Fig. 1.3
(Fig. 1.2). These two vector operations satisfy certain laws of algebra,
which we shall soon state in the definition. The geometric proofs of these laws
are generally sketchy, consisting more of plausibility arguments than of airtight
logic. For example, the geometric figure in Fig. 1.3 is the essence of the usual
proof that vector addition is associative. In each case the final vector OX is
represented by the diagonal starting from 0 in the parallelepiped constructed
from the three edges OA, OB, and OC. The set of all geometric vectors, together
with these two operations and the laws of algebra that they satisfy, constitutes
one example of a vector space. We shall return to this situation in Section 2.
The reader may also have seen coordinate triples treated as vectors. In this
system a three-dimensional vector is an ordered triple of numbers <xi, X2, xs>
which we think of geometrically as the coordinates of a point in space. Addition
is now algebraically defined,
<xuX2,x^> + <yuy2,y^> = <xi + yi, X2 + y2j x^ -{- y^y,
as is multiplication by numbers, t<xi^X2^ x^> = <txi,tx2jtx^>. The
vector laws are much easier to prove for these objects, since they are almost
algebraic formalities. The set IR^ of all ordered triples of numl)ers, together with
these two operations, is a second example of a vector space.

1.1 FUNDAMENTAL NOTIONS 23
If we think of an ordered triple <Xij X2, xs> as a function x with domain
the set of integers from 1 to 3, where Xi is the value of the function x at i (see
Section 0.8), then this vector space suggests a general type called a function
space, which we shall examine after the definition. For the moment we remark
only that we defined the sum of the triple x and the triple y as that triple z
such that Zi = xi + Vi for every i.
A vector space, then, is a collection of objects that can be added to each
other and multiplied by numbers, subject to certain laws of algebra. In this
context a number is often called a scalar.
Befinition* Let F be a set, and let there be given a mapping <ay j3> ^-^
a + ^ from F X F to F, called addition^ and a mapping <x, a> ^-^ xa
from il X F to F, called multiplication hy scalars. Then F is a vector space
with respect to these two operations if:
Al. a + (^ + 7) = (a + ^) + 7 for all a, ^, 7 G F.
A2. a + iS = ^ + a for all a.p^V.
A3. There exists an element 0 G F such that a + 0 = a for all a ^V.
A4. For every a ^V there exists a |S G F such that a-\- fi = Q.
51. {xy)a = x{ya) for all x,y ^R, a ^V.
52. {x + y)ci = xa-{- ya for all a:, ?/ G R, a ^V,
53. x{a -{- p) = xa + xfi for all x^U, a.fi^V.
54. la= a for all a^V.
In contexts where it is clear (as it generally is) which operations are intended,
we refer simply to the vector space F.
Certain further properties of a vector space follow directly from the axioms.
Thus the zero element postulated in A3 is unique, and for each a the ^ of A4
is unique, and is called —a. Also Oa = 0, a:0 = 0, and (—1)q:= —a. These
elementary consequences are considered in the exercises.
Our standard example of a vector space will be the set F = R"^ of all real-
valued functions on a set A under the natural operations of addition of two
functions and multiplication of a function by a number. This generalizes the
example [R^^'^.si ^ p|3 ^.j^^^ ^^ looked at above. Remember that a function /
in R^ is simply a mathematical object of a certain kind. We are saying that two
of these objects can be added together in a natural way to form a third such
object, and that the set of all such objects then satisfies the above laws for
addition. Of course, / + ^ is defined as the function whose value at a is /(a) +
(j{a), so that (/+ g){a) = f(a) + g(a) for ail a in A. For example, in U^ we
defined the sum x + y as that triple whose value at i is Xi + yi for all i. Similarly,
(•/ is the function defined by {cf){a) = c{f(a)) for all a. Laws Al through S4
follow at once from these definitions and the corresponding laws of algebra for
the real number system. For example, the equation (s + t)f = sf+tf means

24 VECTOR SPACES 1.1
that ((s + f)f) (a) = (s/ + tf){a) for all a e A. But
(is + t)f){a) = (s + 0 (/(«)) = s(/(a)) + t{f{a))
= (s/)(a) + (f/)(a) = (sf+tfXa),
where we have used the definition of scalar multiplication in R"^, the distributive
law in R, the definition of scalar multiplication in U^, and the definition of
addition in R^, in that order. Thus we have S2, and the other laws follow
similarly.
The set A can be anything at all. If A = R, then F = R"* is the vector
space of all real-valued functions of one real variable. If A = R X R, then
V = R"*^"* is the space of all real-valued functions of two real variables. If
A = {1, 2} = 2, then V = ^ = U^ is the Cartesian plane, and if A =
{I, . . . ,n} = n, then F = R"" is Cartesian n-space. If A contains a single
point, then R^ is a natural bijective image of R itself, and of course R is trivially
a vector space with respect to its own operations.
Now let V be any vector space, and suppose that TF is a nonempty subset of
V that is closed under the operations of V. That is, if a and 13 are in W, then so
is a + 13, and if a is in W, then so is xa for every scalar x. For example, let V be
the vector space R^^'^^ of all real-valued functions on the closed interval [a, h] C R,
and let W be the set e([a, h]) of all continuous real-valued functions on [a, h].
Then TF is a subset of V that is closed under the operations of V, since f + g
and cf are continuous whenever / and g are. Or let V be Cartesian 2-space R^,
and let W be the set of ordered pairs x = <a:i; 0:2^ such that 0:1 + 0:2 = 0.
Clearly, W is closed under the operations of V.
Such a subset W is always a vector space in its own right. The universally
quantified laws Al, A2, and Si through S4 hold in W because they hold in the
larger set V. And since there is some fi in W, it follows that 0 = 0(S is in W
because W is closed under multiplication by scalars. For the same reason, if a
is in Wj then so is ~a = (—1)q:. Therefore, A3 and A4 also hold, and we see
that TF is a vector space. We have proved the following lemma.
Lemma. If TF is a nonempty subset of a vector space V which is closed
under the operations of F, then TF is itself a vector space.
We call TF a suhspace of V. Thus e([a, h]) is a subspace of R^'''''^ and the
pairs <xi, X2> such that xi + ^2 = 0 form a subspace of R^. Subspaces will
be with us from now to the end.
A subspace of a vector space R^ is called a function space. In other words, a
function space is a collection of real-valued functions on a common domain
which is closed under addition and multiplication by scalars.
What we have defined so far ought to be called the notion of a real vector
space or a vector space over R. There is an analogous notion of a complex vector
space, for which the scalars are the complex numbers. Then laws Si through S4
refer to multiplication by complex numbers, and the space C^ of all complex-

1.1 FUNDAMENTAL NOTIONS 25
valued functions on A is the standard example. In fact, if the reader knew what
is meant by a field F, we could give a single general definition of a vector space
over F, where scalar multiplication is by the elements of F, and the standard
example is the space V = F"^ of all functions from A to F. Throughout this
book it will be understood that a vector space is a real vector space unless
explicitly stated otherwise. However, much of the analysis holds as well for complex
vector spaces, and most of the pure algebra is valid for any scalar field F,
EXERCISES
1.1 Sketch the geometric figure representing law S3,
x(dl + '0B) = x(OA) + x(pB),
for geometric vectors. Assume that x > 1.
1.2 Prove S3 for R^ using the explicit displayed form {o^i, X2j X'^] for ordered triples.
1.3 The vector 0 postulated in A3 is unique, as elementary algebraic fiddling will
show. For suppose that 0' also satisfies A3. Then
0' = 0' + 0 (A3 for 0)
= 0+0' (A2)
= 0 (A3 for 00.
Show by similar algebraic juggling that, given a, the fi postulated in A4 is unique.
This unique fi is designated —a.
1.4 Prove similarly that Oa = 0, a;0 = 0, and (—1)q: = —a.
1.5 Prove that if xa = 0, then either a; = 0 or a = 0.
1.6 Prove SI for a function space IR^. Prove S3.
1.7 Given that a is any vector in a vector space F, show that the set {xa -.xElW]
of all scalar multiples of a is a subspace of V.
1.8 Given that a and fi are any two vectors in F, show that the set of all vectors
xa + yfi, where x and y are any real numbers, is a subspace of V.
1.9 Show that the set of triples x in IR^ such that xi — 0:2 + 20:3 = 0 is a subspace
M. If A^ is the similar subspace {x: o^i + 0:2 + ^3 = 0}, find a nonzero vector a in
M n N. Show that if fl A' is the set {xa -.x^U] of all scalar multiples of a.
1.10 Let A be the open interval (0, 1), and let V be R^. Given a point x in (0, 1),
let Vx be the set of functions in V that have a derivative at x. Show that Vx is a sub-
space of V.
1.11 For any subsets A and 5 of a vector space V we define the set sum A -^ B hy
A + B = {a + ^ : a G ^1 and ^ G 5}. Show that {A + B) + C=A+{B+ C).
1.12 If ^ C F and ZC K, we similarly define XA = {xa'.x^X and a^A).
Show that a nonvoid set yl is a subspace if and only \i A-^ A = A and ^A = A,
1.13 Let V be IR^, and let M be the line through the origin with slope k. Let x be
any nonzero vector in M. Show that M is the subspace Rx = {^x: ^ G IR}.

26 VECTOR SPACES 1.1
1.14 Show that any other line L with the same slope k is of the form Af + a for some a.
1.15 Let M be a subspace of a vector space F, and let a and fi be any two vectors in V.
Given A = a+ M and B = fi+ M, show that either A=B or A r\B=0.
Show also that A + B = (a + P) + M.
1.16 State more carefully and prove what is meant by "a subspace of a subspace is
a subspace".
1.17 Prove that the intersection of two subspaces of a vector space is always itself
a subspace.
1.18 Prove more generally that the intersection W = flie/ ^i of ^^J family
{Wi :iG 1} of subspaces of F is a subspace of V.
1.19 Let V again be R^^.d^ ^nd let W be the set of all functions / in 7 such that/'(a;)
exists for every x in (0, 1). Show that W is the intersection of the collection of subspaces
of the form Vx that were considered in Exercise LIO.
1.20 Let 7 be a function space R"^, and for a point a in .1 let Wa be the set of functions
such that /(a) = 0. Wa is clearly a subspace. For a subset B C -I let Wb be the set
of functions / in 7 such that / = 0 on B. Show that Wb is the intersection f)aGB Wa-
1.21 Supposing again that X and Y are subspaces of V, show that if .Y + F = V and
X n y = {0}, then for every vector f in F there is a unique pair of vectors ^ G X
and rj G y such that f = ^ + r/.
1.22 Show that if .Y and F are subspaces of a vector space F, then the union A^ U F
can only be a subspace if either Y" C F or F C X.
Linear combinations and linear span. Because of the commutative and
associative laws for vector addition, the sum of a finite set of vectors is the same for all
possible ways of adding them. For example, the sum of the three vectors
aay abj ac can be calculated in 12 ways, all of which give the same result:
{aa + ab) -\- ac = ac-\- (^a + «&) = (^c + ^a) + a:6 = ^6 + (^c + ^a)? etc.
Therefore, if 7 = {a, 6, c} is the set of indices used, the notation Y^i^i ai^
which indicates the sum without telling us how we got it, is unambiguous. In
general, for any finite indexed set of vectors {ai: i G 1} there is a uniquely
determined sum vector Yliei cci which we can compute by ordering and
grouping the a/s in any way.
The index set I is often a block of integers n = {1, . . . , n}. In this case
the vectors ai form an n-tuple {a:z}ij and unless directed to do otherwise we
would add them in their natural order and write the sum as Xl?=i ^^ Note
that the way they are grouped is still left arbitrary.
Frequently, however, we have to use indexed sets that are not ordered.
For example, the general polynomial of degree at most 5 in the two variables
's' and 'V is
2_^ CijS t ,
0<z+i<5
and the finite set of monomials {sV}i+y<5 has no natural order.

1.1 FUNDAMENTAL NOTIONS 27
* The formal proof that the sum of a finite collection of vectors is
independent of how we add them is by induction. We give it only for the interested
reader.
In order to avoid looking silly, we begin the induction with two vectors,
in which case the commutative law 0:^ + ^6= ^6 + eta displays the identity of
all possible sums. Suppose then that the assertion is true for index sets having
fewer than n elements, and consider a collection {ai'.i G /} having n members.
Let (S and 7 be the sum of these vectors computed in two ways. In the
computation of fi there was a last addition performed, so that (S = (2Zze/i oci) +
{Hi^J2 ^i)y where {Ji, J2} partitions I and where we can write these two
partial sums without showing how they were formed, since by our inductive
hypothesis all possible ways of adding them give the same result.
Similarly, 7 = (HzeKi on) + (Lzei^2 «*)• Now set
Ljk = Jj n Kk and ^jk = J2 «^>
iELjk
where it is understood that ^jk = 0 if Ljk is empty (see Exercise 1.37). Then
^iEJi =^11 + ?i2 by the inductive hypothesis, and similarly for the other
three sums. Thus
^ = (?ii + ^12) + (?2i + ^22) = (?ii + ?2i) + (^2 + ^22) = -y,
which completes our proof, ^k
A vector 13 is called a linear combination of a subset A of the vector space V
if iS is a finite sum 2Z ^i^^h where the vectors ai are all in A and the scalars xi
are arbitrary. Thus, if A is the subset {t^}^ C U^ of all "monomials", then a
function / is a linear combination of the functions in A if and only if / is a
polynomial function f{t) = Xli ^z^^- If ^ is finite, it is often useful to take the
indexed set {ai} to be the whole of A, and to simply use a 0-coefKcient for any
vector missing from the sum. Thus, if A is the subset {sin ^, cos ^, e^} of IR°*,
then we can consider A an ordered triple in the listed ordering, and the function
3 sin ^ — 6* = 3 • sin ^ + 0 • cos t -j- (—l)e^ is the linear combination of the
triple A having the coefficient triple <3, 0, — 1 >.
Consider now the set L of all linear combinations of the two vectors
<1, 1, 1> and <0, 1, —1> in R^ It is the set of all vectors s<l, 1, 1> +
t<0, 1, —1 > = <Sj s + t, s — t> , where s and t are any real numbers. Thus
L = {<s^ s + t^ s — t> : <s^ t> G R^}. It will be clear on inspection that
L is closed under addition and scalar multiplication, and therefore is a subspace
of U^. Also, L contains each of the two given vectors, with coefficient pairs
<1,0> and <0, 1>, respectively. Finally, any subspace M of U^ which
contains each of the two given vectors will also contain all of their linear
combinations, and so will include L. That is, L is the smallest subspace of U^ containing
< 1, 1, 1 > and < 0, 1, — 1 > . It is called the linear span of the two vectors, or the
subspace generated by the two vectors. In general, we have the following theorem.

28 VECTOR SPACES 1.1
Theorem 1.1. If A is a nonempty subset of a vector space V^ then the set
L{A) of all linear combinations of the vectors in A is a subspace, and it is
the smallest subspace of V which includes the set A.
Proof, Suppose first that A is finite. We can assume that we have indexed A
in some way, so that A = {ai: i G 1} for some finite index set 7, and every
element of L{A) is of the form Yli^i ^i<^i' Then we have
(E xiai) + (E ym) = L (Xi + yi)ai
because the left-hand side becomes Ez (^z^z + Vio^i) when it is regrouped by
pairSj and then S2 gives the right-hand side. We also have
c{T.Xiai) = T.(cxi)ai
by S3 and mathematical induction. Thus L(A) is closed under addition and
multiplication by scalars and hence is a subspace. Moreover, L(A) contains
each ai (why?) and so includes A. Finally, if a subspace W includes A^ then it
contains each linear combination E ^z^z? so it includes L(A). Therefore, L{A)
can be directly characterized as the uniquely determined smallest subspace
which includes the set A.
If A is infinite, we obviously can't use a single finite listing. However, the
sum (Elf Xiai) + (Er Vjfij) of two linear combinations of elements of A is
clearly a finite sum of scalars times elements of A. If we wish, we can rewrite it
as 2^^+^ Xiaij where we have set fij = ^n+y and yj = Xn-{.j for J = 1, . . . , m.
In any case, L(A) is again closed under addition and multiplication by scalars
and so is a subspace. D
We call L{A) the linear span of A. If L(A) = F, we say that A spans V;
V is finite-dimensional if it has a finite spanning set.
If 7 = U^^ and if 5\ 5^, and 8^ are the "unit points on the axes", 8^ =
<1,0, 0>, 8^ = <0, 1,0>, and 8^ = <OjO, l>j then {8'}l spans F, since
x= <Xi^ X2y x^> = <a:i,0,0> + <0, a:2j0> + <0,0, a:3> = Xi8^ +
^25^ + xs8^ = E? Xi8' for every x in U^. More generally, if F = R'' and 8' is
the n-tuple having 1 in the jth place and 0 elsewhere, then we have similarly that
X = <xi, . . . ,Xn> = 117=1 Xi8\ so that {8'}i spans W". Thus W" is finite-
dimensional. In general, a function space on an infinite set A will not be finite-
dimensional. For example, it is true but not obvious that e([a, h]) has no finite
spanning set.
EXERCISES
1.23 Given a = <1,1,1>, 13 = <0, 1, —1 >, 7 = <2, 0, 1 >, compute the linear
combinations a + (3+7, Sa — 2(3 + 7, xa+ yl3 + z7. Find x, y, and z such that
xa+yl3+ z7 = < 0, 0, 1 > = 8^. Do the same for 8^ and 8^.
1.24 Given a = <1, 1, 1>, ^ = <0, 1,—1>, 7 = <1,0, 2>, show that each of
a, |S, 7 is a linear combination of the other two. Show that it is impossible to find
coefficients x, y, and z such that xa-\- yP-\- z7 = 8^,

1.1 FUNDAMENTAL NOTIONS 29
1.25 a) Find the linear combination of the set yl = <t,t^ — l,t^ -\- l> with
coefficient triple -<2, —1, 1>. Do the same for -<0, 1, 1>.
b) Find the coefficient triple for which the linear combination of the triple A
is (t-\- 1)^. Do the same for 1.
c) Show in fact that any polynomial of degree < 2 is a linear combination of .1.
1.26 Find the linear combination/of {e\ e'^} C U^ such that/(0) = 1 and/'(0) = 2.
1.27 Find a linear combination / of sin x, cos x, and e^ such that/(0) = 0,/'(O) = 1,
and/''(0) = 1.
1.28 Suppose that a sin a; + 6 cos x + ce^ is the zero function. Prove that a = b ~
c = 0.
1.29 Prove that < 1, 1 > and < 1, 2> span U^.
1.30 Show that the subspace M = {x : o^i + 0:2 = 0} C If^^ is spanned by one vector.
1.31 Let M be the subspace {x: o^i — X2-\- 2x3 = 0} in R^. Find two vectors a
and b in M neither of which is a scalar multiple of the other. Then show that M is
the linear span of a and b.
1.32 Find the intersection of the linear span of -<1, 1, 1> and -<0, 1, —1> in U^
with the coordinate subspace X2 = 0. Exhibit this intersection as a linear span.
1.33 Do the above exercise with the coordinate space replaced by
M = {x: 0^1+ 0:2 = 0}.
1.34 By Theorem 1.1 the linear span L(A) of an arbitrary subset A of a vector space
V has the following two properties:
i) L(A) is a subspace of V which includes A;
ii) If M is any subspace which includes A, then L(A) C M.
Using only (i) and (ii), show that
a) ACB=>L{A)CL{B);
b) L(L(A)) = L{A),
1.35 Show that
a) if M and A^ are subspaces of V, then so is ilf + A^;
b) for any subsets A.BCV, L(A U B) = L(A) + L{B).
1.36 Remembering (Exercise 1.18) that the intersection of any family of subspaces
is a subspace, show that the linear span L{A) of a subset yl of a vector space V is the
intersection of all the subspaces of V that include A, This alternative characterization
is sometimes taken as the definition of linear span.
1.37 By convention, the sum of an empty set of vectors is taken to be the zero vector.
This is necessary if Theorem 1.1 is to be strictly correct. Why? What about the
preceding problem?
Linear transformations. The general function space R"^ and the subspace
e([aj h\) of U^^'^^ both have the property that in addition to being closed under
the vector operations, they are also closed under the operation of multiplication
of two functions. That is, the pointwise product of two functions is again a
function [{fg){a) = f{a)g{a)]y and the product of two continuous functions is
continuous. With respect to these three operations, addition, multiplication.

30 VECTOR SPACES 1.1
and scalar multiplication, R^ and ©([a, h]) are examples of algebras. If the reader
noticed this extra operation, he may have wondered why, at least in the context
of function spaces, we bother with the notion of vector space. Why not study
all three operations? The answer is that the vector operations are exactly the
operations that are "preserved" by many of the most important mappings of
sets of functions. For example, define T:e([a, h]) -> R by T(f) = j^ f(t) dt.
Then the laws of the integral calculus say that T{f + g) = T{f) + T{g) and
T{cf) = cT{f). Thus T "preserves" the vector operations. Or we can say that T
"commutes" with the vector operations, since plus followed by T equals T
followed by plus. However, T does not preserve multiplication: it is not true in
general that TUg) = TU)T{g).
Another example is the mapping Trxi—»y from R^ to R^ defined by
?/i = 2a: 1 — ^^2 + ^3, 2/2 = ^1 + 30^2 — 5^:3, for which we can again verify
that T{x + y) = T{x) + T{j) and T{cx) = cT{x), The theory of the solvability
of systems of linear equations is essentially the theory of such mappings T] thus
we have another important type of mapping that preserves the vector operations
(but not products).
These remarks suggest that we study vector spaces in part so that we can
study mappings which preserve the vector operations. Such mappings are
called linear transformations.
Definition. If V and W are vector spaces, then a mapping T: F —> TF is a
linear transformation or a linear map if T(a + fi) = T{a) + T{fi) for all
a, ^ G F, and T{xa) = xT(a) for all a G F, a: G R.
These two conditions on T can be combined into the single equation
T(xa + yl3) = xT{a) + yT{ff) for all a,fi^V and all x, y G R.
Moreover, this equation can be extended to any finite sum by induction, so
that if T is linear, then
T f Z x,ai) = i: XiT{a,)
for any linear combination Y.Xiai. For example, j^ (E? cj,) = U ^/a U
EXERCISES
1.38 Show that the most general linear map from R to R is multiplication by a
constant.
1.39 For a fixed a in F the mapping x^—^xa from R to F is linear. Why?
1.40 Why is this true for a^—^xa when x is fixed?
1.41 Show that every linear mapping from R to F is of the form x^—^xa for a fixed
vector a in F.

1.1 FUNDAMENTAL NOTIONS 31
1.42 Show that every linear mapping from U^ to V is of the form -<a;i,a;2> •—*
xiai + X2a2 for a fixed pair of vectors ai and 0:2 in V. What is the range of this mapping?
1.43 Show that the map/i—» fa f(t) dt from C([a, b]) to R does not preserve products.
1.44 Let g be any fixed function in U^. Prove that the mapping T: R^ -^ R^
defined by T(f) = gf is linear.
1.45 Let (f be any mapping from a set A to a set B. Show that composition by (p is
a linear mapping from U^ to U^. That is, show that T:U^ -^ U^ defined by T(f) =
foipis linear.
In order to acquire a supply of examplesj we shall find all linear
transformations having R^ as domain space. It may be well to start by looking at one such
transformation. Suppose we choose some fixed triple of functions {fi} I in the
space R" of all real-valued functions on R, say fi(t) = sin t, f2(t) = cos t, and
/aCO = e^ = exp(t). Then for each triple of numbers x = {xi}l in R^ we have
the linear combination Xlf=i ^ifi with {xi} as coefficients. This is the element of
R" whose value at t is 2Zi ^ifiQ) = :^i sin ^ + ^2 cos t + x^eK Different coefficient
triples give different functions, and the mapping x 1—» 2Zf=i xifi = o^i sin +
X2 cos + Xs exp is thus a mapping from R^ to R". It is clearly linear. If we call
this mapping T, then we can recover the determining triple of functions from T
as the images of the "unit points" 8' in R^; T(8^') = J] djfi = fj, and so
T{8^) = sin, T{8^) = cos, and T(8^) = exp. We are going to see that every
linear mapping from R^ to R" is of this form.
In the following theorem {5*} 1 is the spanning set for R^ that we defined
earlier, so that x = ^i Xi8^ for every n-tuple x = <xij . . . , Xn> in R^.
Theorem 1.2. If {l3j}i is any fixed n-tuple of vectors in a vector space TF,
then the "linear combination mapping" x ^-^ ^i Xi^i is a linear
transformation T from R^ to W, and T(d^) = 13j forj= 1, . . . , n. Conversely,
if T is any linear mapping from R^ to Wy and if we set 13j = T(8^) ior j =
1, . . . , n, then T is the linear combination mapping x 1—> ^i xifii.
Proof. The linearity of the linear combination map T follows by exactly the
same argument that we used in Theorem 1.1 to show that L{A) is a subspace.
Thus
n n
T{X + y) = Z (^^• + yi)fii = Z (^^•^^ + ViPi)
1 1
= Z x,Pi + Z ViPi = T{x) + T(j),
1 1
and
n n n
T{sx) = Z (s^i)l^i = Z <^il^i) = s Z ^iPi = sT{x),
1 1 1
Also T(8') = ^^"=1 5|ft = 0j^ since 8J = 1 and 8{ = 0 for i 9^ j.

32 VECTOR SPACES 1.1
Conversely J if T-.W —^Wis linear, and if we set Pj = T{8^) for all j, then for
any x = <Xi, . , . , Xn> in R^ we have T(x) = T(Zi ^i 8') = E? XiT(8') =
X? Xil3i. Thus T is the mapping x i-» J^i xifii. D
This is a tremendously important theorem, simple though it may seem, and
the reader is urged to fix it in his mind. To this end we shall invent some
terminology that we shall stay with for the first three chapters. If a = {ai, . . . , an}
is an n-tuple of vectors in a vector space TF, let La be the corresponding linear
combination mapping x i-» Yl\ Xi<^i from W to W. Note that the n-tuple a
itself is an element of W^. If T is any linear mapping from W to TF, we shall call
the n-tuple {^(S')} J the skeleton of T. In these terms the theorem can be restated
as follows.
Theorem 1.2'. For each n-tuple a in TF^, the map La'. R^ —> TF is linear
and its skeleton is a. Conversely, if T is any linear map from W to TF, then
T = Lp where P is the skeleton of T.
Or again:
Theorem 1.2". The map a I—» La is a bisection from W^ to the set of all
linear maps T from U^ to TF, and T i—» skeleton (T) is its inverse.
A linear transformation from a vector space V to the scalar field U is called
a linear functional on F. Thus f ^-^ ja f{t) dt is a linear functional on F =
e([a, h]). The above theorem is particularly simple for a linear functional F:
since TF = R, each vector fii = 7^(5') in the skeleton of F is simply a number 6^,
and the skeleton {hi}^ is thus an element of U^. In this case we would write
F(x) = 2Z? hiXiy putting the numerical coefficient '6/ before the variable
^Xi. Thus F(x) = 3xi — ^2 + 4^3 is the linear functional on U^ with skeleton
<3, — 1, 4>. The set of all linear functionals on U^ is in a natural one-to-one
correspondence with U^ itself; we get b from F by hi = F(8^) for all ^, and we
get F from b by F(x) = E hiXi for all x in R^.
We next consider the case where the codomain space of 7" is a Cartesian
space R^, and in order to keep the two spaces clear in our minds, we shall, for
the moment, take the domain space to be R^. Each vector I3i = T(8^) in the
skeleton of T is now an m-tuple of numbers. If we picture this m-tuple as a
column of numbers, then the three m-tuples I3i can be pictured as a rectangular
array of numbers, consisting of three columns each of m numbers. Let tij be the
iih number in the jih column. Then the doubly indexed set of numbers {tij} is
called the matrix of the transformation T. We call it an m-by-3 (an m X 3)
matrix because the pictured rectangular array has m rows and three columns.
The matrix determines T uniquely, since its columns form the skeleton of T.
The identity ^(x) = E? XjT{8') = E? Xjl3j allows the m-tuple ^(x) to be
calculated explicitly from x and the matrix {tij}. Picture multiplying the
column m-tuple 13j by the scalar Xj and then adding across the three columns at

1.1 FUNDAMENTAL NOTIONS 33
the ^th row, as below :
Since Uj is the ^th number in the m-tuple fij^ the ^th number in the m-tuple
2Zy=i XjPj is Z^y=i Xjtij. That isj if we let y be the m-tuple T(x)^ then
3
and this set of 7n scalar equations is equivalent to the one-vector equation
y = Tix).
We can now replace three by 7i in the above discussion without changing
anything except the diagram, and thus obtain the following specialization of
Theorem 1.2.
Theorem 1.3. Every linear mapping T from R^ to W^ determines the
m X 71 matrix t = {tij} having the skeleton of T as its columns, and the
expression of the equation y = ^(x) in linear combination form is equivalent
to the m scalar equations
n
Vi = X tijXj for ^ = Ij . . . J m.
j=i
Conversely, each m X 7i matrix t determines the linear combination mapping
having the columns of t as its skeleton, and the mapping t •-> T is therefore
a bisection from the set of all m X 7i matrices to the set of all linear maps
from R^ to R^.
A linear functional F on U^ is a linear mapping from R^ to IR\ so it must
be expressed by a 1 X n matrix. That is, the 7i-tuple b in R^ which is the skeleton
of F is viewed as a matrix of one row and 7i columns.
As a final example of linear maps, we look at an important class of special
linear functionals defined on any function space, the so-called coordinate func-
tionals. If F = R^ and i G /, then the ith coordinate functional Ti is simply
evaluation at i, so that 7ri{f) = f{i). These functionals are obviously linear. In
factj the vector operations on functions were defined to make them linear; since
sf + tg is defined to be that function whose value at i is sf{i) + tg{i) for all i^
we see that sf + tg is by definition that function such that Tri{sf + tg) =
STTiif) + tTi{g) for all i\
If V is R^j then Wj is the mapping x = <xiy . . . ^ Xn> •-> xy. In this case
we know from the theorem that Tj must be of the form Tj(x) = 2Z^ hiXi for
some n-tuple b. What is b?

34: VECTOR SPACES 1.1
The general form of the Hnearity property, T(^Xiai) = J^XiT(ai)^ shows
that T and T~^ both carry subspaces into subspaces.
Theorem 1.4. If T:V —^ W is linear, then the T-image of the linear span
of any subset A C F is the linear span of the T-image of A: T[L(A)] =
L{T[A]). In particular, if A is a subspace, then so is T[A]. Furthermorej if Y
is a subspace of W^ then ^"^[F] is a subspace of V.
Proof. According to the formula T{J^Xiai) = JlxiT(ai)^ a vector in W is
the T-image of a linear combination on A if and only if it is a linear combination
on T[A]. That is, T[L(A)] = L(T[A]). If A is a subspace, then A = L(A) and
T[A] = L(T[A])j a subspace of W. Finally, if F is a subspace of W and {a^} C
T-'\Y], then T{Zx,ai) = Z x^T(a^) e L(Y) = F. Thus Zx^a^ G T-'[Y]
and T~^[Y] is its own linear span. D
The subspace ^"^(0) = {a e F : T(a) = 0} is called the 7iull space, or
kernel, of T, and is designated N{T) or 91(7"). The range of T is the subspace
T[V] of W. It is designated R{T) or 6{(T).
Lemma 1.1. A linear mapping T is infective if and only if its null space
is {0].
Proof. If T is injective, and if a ?^ 0, then T(a) ^ T(0) = 0 and the null space
accordingly contains only 0. On the other hand, if N(T) = {0}, then whenever
a 9^ fi, we have a - fi ^ 0, T{a) - T{ff) = T{a - ^) ^ 0, and T{a) ^ T(^);
this shows that T is injective. D
A linear map T:V —^ W which is bijective is called an isomorphism.
Two vector spaces F and W are isomorphic if and only if there exists an
isomorphism between them.
For example, the map <Ci^ . . . ^ Cn> —> 2Zo~^ c^+ix" is an isomorphism of
W^ with the vector space of all polynomials of degree < 7i.
Isomorphic spaces "have the same form", and are identical as abstract
vector spaces. That is, they cannot be distinguished from each other solely on
the basis of vector properties which they do or do not have.
When a linear transformation is from F to itself^ special things can happen.
One possibility is that T can map a vector a essentially to itself, T{a) = xa
for some x in R. In this case a is called an eigenvector (proper vector,
characteristic vector), and x is the corresponding eigenvalue.
EXERCISES
1.46 In the situation of Exercise 1.45, show that T is an isomorphism if ip is bijective
by showing that
a) if injective =^ T surjective,
b) <p surjective =^ T injective.

1.1 FUNDAMENTAL NOTIONS 35
1.47 Find the linear functional I on U^ such that l(<l,l» = 0andZ(<l,2>) = 1.
That is, find b = -<6i, 62^ in R^ such that I is the linear combination map
X -^ bixi + 62^2.
1.48 Do the samefor Z(<2, 1>) = —3andZ(<l, 2>) = 4.
1.49 Find the linear T:U^ -^ R" such that ^(^ 1, 1 >) = <2 ^nd ^(^ 1, 2>) = t^.
That is, find the functions/i (0 and/2(0 such that T is the linear combination map
X -^ a;i/i + 0:2/2.
1.50 Let T be the linear map from IR2 to IR3guch that ^(Si) = <2,—I, 1>, T{8^) =
-<1, 0, 3>. Write down the matrix of T in standard rectangular form. Determine
whether or not 5^ is in the range of T.
1.51 Let T be the linear map from U^ to U^ whose matrix is
1
2
3
2
0
—1
3
—1
1.
Find T(x) when x = <1, 1, 0>; do the same for x = <3, —2, 1>.
1.52 Let M be the linear span of -< 1, —1, 0> and -<0, 1, 1>. Find the subspace
T[M] by finding two vectors spanning it, where T is as in the above exercise.
1.53 Let T be the map <x, y> -^ <x + 2y, y> from U^ to itself. Show that T is a
linear combination mapping, and write down its matrix in standard form.
1.54 Do the same for T: <x,y, z> -^ <x — z, x -^ z, y> from U^ to itself.
1.55 Find a linear transformation T from U^ to itself whose range space is the span
of <1, —1,0> and <—1,0, 2>.
1.56 Find two linear functional on R^ the intersection of whose null spaces is the
linear span of -<1, 1, 1, 1> and -<1, 0, —1, 0>. You now have in hand a linear
transformation whose null space is the above span. What is it?
1.57 Let V = C([a, b]) be the space of continuous real-valued functions on [a, b],
also designated C^([a, 6]), and let W = CV[a, 6]) be those having continuous first
derivatives. Let D:W -^ V be differentiation (Df = f), and define T on F by
^(/) = ^7 where F(x) = fa f(t) dt. By stating appropriate theorems of the calculus,
show that D and T are linear, T maps into W, and D is a left inverse of T (D o T is
the identity on V).
1.58 In the above exercise, identify the range of T and the null space of D. We
know that D is surjective and that T is injective. Why?
1.59 Let V be the linear span of the functions sin x and cos a;. Then the operation
of differentiation D is a linear transformation from V to V. Prove that D is an
isomorphism from V to V. Show that D^ = —/ on V.
1.60 a) As the reader would guess, e^(IR) is the set of real-valued functions on R
having continuous derivatives up to and including the third. Show that / -^ /''' is a
surjective linear map T from e^{U) to e{U).
b) For any fixed a in R show that /-^ ^fia), fici), f\ci)^ is an isomorphism
from the null space N{T) to R^. [Hint: Apply Taylor's formula with remainder.]

36
VECTOR SPACES
1.2
1.61 An integral analogue of the matrix equations yi = ^jtijXjj i = 1, . . . , m, is
the equation
g(s) = f' K(s,t)mdt, se[0,l].
Jo
Assuming that K(s, t) is defined on the square [0, 1] X [0, 1] and is continuous as a
function of t for each s, check that / —> ^ is a linear mapping from C([0, 1]) to R^^'^^
1.62 For a finite set A = {«{], Theorem 1.1 is a corollary of Theorem 1.4. Why?
1.63 Show that the inverse of an isomorphism is linear (and hence is an isomorphism).
1.64 Find the eigenvectors and eigenvalues of T: R^ -^ R^ if the matrix of T is
1 —r
-2 0.
Since every scalar multiple xa of an eigenvector a is clearly also an eigenvector, it will
suffice to find one vector in each "eigendirection". This is a problem in elementary
algebra.
1.65 Find the eigenvectors and eigenvalues of the transformations T whose matrices
are
"2 r
-4 —2
1.66 The five transformations in the above two exercises exhibit four different kinds
of behavior according to the number of distinct eigendirections they have. What are
the possibilities?
1.67 Let V be the vector space of polynomials of degree < 3 and define T.V -^ V
by/ -^ tf{t). Find the eigenvectors and eigenvalues of T.
1
_—2
r
0_
}
'—1
_—1
—r
—1_
7
1
_—2
—r
2_
7
2. VECTOR SPACES AND GEOMETRY
The familiar coordinate systems of analytic geometry allow us to consider
geometric entities such as lines and planes in vector settings, and these geometric
notions give us valuable intuitions about vector spaces. Before looking at the
vector forms of these geometric ideas, we shall briefly review the construction of
the coordinate correspondence for three-dimensional Euclidean space. As usual,
the confident reader can skip it.
We start with the line. A coordinate correspondence between a line L and
the real number system U is determined by choosing arbitrarily on L a zero
point 0 and a unit point Q distinct from 0. Then to each point X on L is assigned
the number x such that \x\ is the distance from 0 to X^ measured in terms of
the segment OQ as unit, and x is positive or negative according as X and Q are
on the same side of 0 or on opposite sides. The mapping X i—» x is the coordinate
correspondence. Now consider three-dimensional Euclidean space E^. We want
to set up a coordinate correspondence between E^ and the Cartesian vector
space U^. We first choose arbitrarily a zero point 0 and three unit points
Qi7 Q27 and Q3 in such a way that the four points do not lie in a plane. Each of

1.2
VECTOR SPACES AND GEOMETRY
37
the unit points Qi determines a line Li through 0 and a coordinate
correspondence on this line, as defined above. The three lines Li, L2, and L3 are called
the coordinate axes. Consider now any point X in E^. The plane through X
parallel to L2 and L3 intersects Li at a point Xi, and therefore determines a
number xi, the coordinate of Xi on Li. In a similar way, X determines points
X2 on L2 and X3 on L3 which have
coordinates X2 and X3, respectively.
Altogether X determines a triple
X = <Xi, X2, X3>
in U^, and we have thus defined a mapping
(9:Xi->x from E'^ to
(see Fig. 1.4).
We call 6 the coordinate correspondence
defined by the axis system. The
convention implicit in our notation above is that
d(Y) is y, d(A) is a, etc. Note that the
unit point Qi on Li has the coordinate
triple 8^ = < 1, 0, 0>, and similarly, that
and
e(Q2) = 8^ = <o, i,o>
diQs) = 5'^- <0,0, 1>.
Fig. 1.4
There are certain basic facts about the coordinate correspondence that have
to be proved as theorems of geometry before the correspondence can be used to
treat geometric questions algebraically. These geometric theorems are quite
tricky, and are almost impossible to discuss adequately on the basis of the usual
secondary school treatment of geometry. We shall therefore simply assume
them. They are:
1) ^ is a bisection from E'^ to U^.
2) Two line segments AB and XY are equal in length and parallel, and the
direction from A to B is the same as that from X to Y if and only if b — a ==
y — X (in the vector space U^). This relationship between line segments is
important enough to formalize. A directed line segment is a geometric line
segment, together with a choice of one of the two directions along it. If we interpret
AB as the directed line segment from A to B, and if we define the directed line
segments AB and XF to be equivalent (and write AB '^ XY) if they are equal
in length, parallel, and similarly directed, then (2) can be restated:
AB ^ XY ^ h ~
y — X.
3) li X 9^ Oj then Y is on the line through 0 and X in E^ if and only if
y = tx for some t in R. Moreover, this t is the coordinate of Y with respect to X
as unit point on the line through 0 and X.

38
VECTOR SPACES
1.2
X3
S2=xf+a;^
|OX|2 = r2 = s2+xi
{y-x,y-x)^\XY\^
{y.y) = \OY\
Fig. 1.5
4) If the axis system in E^ is Cartesian^ that is, if the axes are mutually
perpendicular and a common unit of distance is used, then the length \0X\ of
the segment OX is given by the so-called Euclidean norm on R^, \0X\ =
(YAXi^y^. This follows directly from the Pythagorean theorem. Then this
formula and a second application of the Pythagorean theorem to the triangle
OXY imply that the segments OX and OY are perpendicular if and only if the
scalar product (x, y) = Z!f=i ^iVi h^s the value 0 (see Fig. 1.5).
In applying this result, it is useful to note that the scalar product (x, y) is
linear as a function of either vector variable when the other is held fixed. Thus
3 33
(ex + dj, z) = X (cXi + dyi)zi = c XI ^i^i + <i X Vi^i = ^(x, z) + d(j, z).
1 1 1
Exactly the same theorems hold for the coordinate correspondence between
the Euclidean plane E^ and the Cartesian 2-space U^, except that now, of course,
(x, y) = L? XiVi = xiVi + X22/2.
We can easily obtain the equations for lines and
planes in E^ from these basic theorems. First, we see
from (2) and (3) that if fixed points A and B are given,
with A 9^ 0^ then the line through B parallel to the
segment OA contains the point X if and only if there
exists a scalar t such that x — b = ^a (see Fig. 1.6).
Therefore, the equation of this line is
X = ^a + b. Fig. 1.6
This vector equation is equivalent to the three numerical equations Xi ~
ait + hij i = 1, 2, 3. These are customarily called the parametric equations of the
line, since they present the coordinate triple x of the varying point X on the line
as functions of the "parameter" t.

L2
VECTOR SPACES AND GEOMETRY
39
Nextj we know that the plane through B perpendicular to the direction of
the segment OA contains the point X if and only if BX ± OA^ and it therefore
follows from (2) and (4) that the plane contains X if and only if (x — b, a) = 0
(see Fig. 1.7). But (x — b, a) = (x, a) — (b, a) by the linearity of the scalar
product in its first variable, and if we set I = (b, a), we see that the equation of
the plane is
(x, a) = I
or
Zl (^i^
I
That iSj a point X is on the plane through B perpendicular to the direction of
OA if and only if this equation holds for its coordinate triple x. Converselyj
if a ?^ 0, then we can retrace the steps taken above to show that the set of points
X in E^ whose coordinate triples x satisfy (x, a) = Hs a plane.
V5^-..,
Fig. 1.7
The fact that U^ has the natural scalar product (x, y) is of course extremely
important, both algebraically and geometrically. However^ most vector spaces
do not have natural scalar products, and we shall deliberately neglect scalar
products in our early vector theory (but shall return to them in Chapter 5).
This leads us to seek a different interpretation of the equation J^l aiXi = I.
We saw in Section 1 that x i—> T^,? the most general linear functional / on
H^. Therefore, given any plane M m E^, there is a nonzero linear functional /
on M^ and a number I such that the equation of M is /(x) = I. And conversely,
given any nonzero linear functional /: R^ -^ H and any I E R, the locus of
/(x) = / is a plane M in E^. The reader will remember that we obtain the
coefficient triple a from / by ai = /(5'), since then /(x) = /(2Zi x^-5') =
Finally, we seek the vector form of the notion of parallel translation. In
plane geometry when we are considering two congruent figures that are parallel
and similarly oriented, we often think of obtaining one from the other by "sliding

40 VECTOR SPACES 1.2
the plane along itself" in such a way that all lines remain parallel to their original
positions. This description of a parallel translation of the plane can be more
elegantly stated as the condition that every directed line segment slides to an
equivalent one. If X slides to Y and 0 slides to B^ then OX slides to BYy so
that OX ^ BY and x = y — b by (2). Therefore, the coordinate form of such
a parallel sliding is the mapping x i—» y = x + b.
Conversely J for any b in R^ the plane mapping defined byxi—»y = x + b
is easily seen to be a parallel translation. These considerations hold equally well
for parallel translations of the Euclidean space E^.
It is geometrically clear that under a parallel translation planes map to
parallel planes and lines map to parallel lines, and now we can expect an easy
algebraic proof. Consider, for example, the plane M with equation /(x) = 1;
let us ask what happens to M under the translation x i—» y = x + b. Since
X = y — b, we see that a point x is on M if and only if its translate y satisfies
the equation /(y — h) = I or, since / is linear, the equation /(y) = l\ where
V = l-^/(b). But this is the equation of a plane N. Thus the translate of M
is the plane N.
It is natural to transfer all this geometric terminology from sets in E^
to the corresponding sets in R^ and therefore to speak of the set of ordered
triples X satisfying/(x) = Z as a set of points in U^ forming a plane in R^, and
to call the mapping x i-» x + b the (parallel) translation of U^ through b, etc.
Moreover, since R^ is a vector space, we would expect these geometric ideas to
interplay with vector notions. For instance, translation through b is simply the
operation of adding the constant vector b: x i—» x + b. Thus if M is a plane, then
the plane N obtained by translating M through b is just the vector set sum
M + b. If the equation of M is /(x) = Z, then the plane M goes through 0 if
and only if Z = 0, in which case M is a vector suhspace of U^ (the null space of /).
It is easy to see that any plane M is a translate of a plane through 0. Similarly,
the line {ta + h :t G U] is the translate through b of the line {^a : ^ G R}, and
this second line is a subspace, the linear span of the one vector a. Thus planes
and lines in R^ are translates of suhspaces.
These notions all carry over to an arbitrary real vector space in a perfectly
satisfactory way and with additional dimensional variety. A plane in R^
through 0 is a vector space which is two-dimensional in a strictly algebraic sense
which we shall discuss in the next chapter, and a line is similarly one-dimensional.
In R^ there are no proper subspaces other than planes and lines through 0,
but in a vector space V with dimension n > 3 proper subspaces occur with all
dimensions from 1 to n — 1. We shall therefore use the term "plane" loosely to
refer to any translate of a subspace, whatever its dimension. More properly,
translates of vector subspaces are called affine subspaces.
We shall see that if F is a finite-dimensional space with dimension n, then
the null space of a nonzero linear functional / is always {n — l)-dimensional, and
therefore it cannot be a Euclidean-like two-dimensional plane except when

1.2 VECTOR SPACES AND GEOMETRY 41
n = 3. We use the term hyperplane for such a null space or one of its translates.
Thus, in general, a hyperplane is a set with the equation /(x) = I, where / is
a nonzero linear functional. It is a proper affine subspace (plane) which is
maximal in the sense that the only affine subspace properly including it is the whole
of V. In U^ hyperplanes are ordinary geometric planes, and in U^ hyperplanes
are lines!
EXERCISES
2.1 Assuming the theorem AB ^ XY <^ h — a = y — x, show that OC is the sum
of OA and OB, as defined in the preliminary discussion of Section 1, if and only if
c = b+a. Considering also our assumed geometric theorem (3), show that the
mapping x i—> OX from R^ to the vector space of geometric vectors is linear and
hence an isomorphism.
2.2 Let L be the line in the Cartesian plane R^ w^ith equation X2 = Sxi. Express L
in parametric form as x = <a for a suitable ordered pair a.
2.3 Let V be any vector space, and let a and /? be distinct vectors. Show that the
line through a and ^ has the parametric equation
^ = tl3+ (I -t)a, tGR.
Show also that the segment from a to |S is the image of [0, 1] in the above mapping.
2.4 According to the Pythagorean theorem, a triangle with side lengths a, b, and c
has a right angle at the vertex "opposite c" if and only if c^ = a^ -\- b^.
Prove from this that in a Cartesian coordinate system in E^ the length \0X\ of a
segment OX is given by
\ox\^ = Z 4
1
where x = <xi, X2, xs^ is the coordinate triple of the point X. Next use our geometric
theorem (2) to conclude that
3
OX A. OY if and only if (x, y) = 0, where (x, y) = ^ Xiyi.
1
(Use the bilinearity of (x, y) to expand \X — Fp.)
2.5 More generally, the law of cosine says that in any triangle labeled as indicated,
c2 = a2 + 62 _ 2ab cos 6.

42 VECTOR SPACES 1.2
Apply this law to the diagram
0
to prove that
(x, y) = 2|x| |y| cos l9,
where (x, y) is the scalar product ^l Xiiji, |x| = (x, \)^^^ = \0X\, etc.
2.6 Given a nonzero linear functional /: U^ -^ R, and given A: E R, show that the
set of points X in E^ such that /(x) = A; is a plane. [Hint: Find a b in R^ such that
/(b) = k, and throw the equation/(x) = k into the form (x — b, a) = 0, etc.]
2.7 Show that for any b in U^ the mapping X i—» 1" from E^ to itself defined by
y = X + b is a parallel translation. That is, show that if .Y i—» 1^ and Z i—» W, then
XZ - YW.
2.8 Let M be the set in U^ with equation 3a;i — X2-h x^ = 2. Find triplets a and b
such that M is the plane through b perpendicular to the direction of a. What is the
equation of the plane P = 3/ + -< 1, 2, 1 > ?
2.9 Continuing the above exercise, what is the condition on the triplet b in order for
N = iVf + b to pass through the origin? What is the equation oi N?
2.10 Show that if the plane M in R^ has the equation/(x) = I, then M is a translate
of the null space N of the linear functional /. Show that any two translates M and P
of N are either identical or disjoint. What is the condition on the ordered triple b
in order that AT + b = M?
2.11 Generalize the above exercise to hyperplanes in R^.
2.12 Let N be the subspace (plane through the origin) in U^ with equation/(x) = 0.
Let M and F be any two planes obtained from N by parallel translation. Show that
Q = il/ + P is a third such plane. If M and P have the equations /(x) = h and
/(x) = h, find the equation for Q.
2.13 If M is the plane in R^ with equation/(x) = I, and if r is any nonzero number,
show that the set product rM is a plane parallel to M.
2.14 In view of the above two exercises, discuss how we might consider the set of all
parallel translates of the i)lane N with equation /(x) = 0 as forming a new vector
space.
2.15 Let L be the subspace (Hne through the origin) in U^ with parametric equation
X = ta. Discuss the set of all parallel translates of L in the spirit of the above three
exercises.
2.16 The best object to take as "being" the geometric vector AB is the equivalence
class of all directed line segments XY such that XY ^ AB. Assuming whatever you
need from properties (1) through (4), show that this is an equivalence relation on the
set of all directed line segments (Section 0.12).
2.17 Assuming that the geometric vector AB is defined as in the above exercise, show
that, strictly speaking, it is actually the mapping of the plane (or space) into itself that
we have called the parallel translation through AB. Show also that AB -\- CD is the
composition of the two translations.

1.3 PRODUCT SPACES AND HOM(F, TF) 43
3. PRODUCT SPACES AND HOM(F, W)
Product spaces. If TF is a vector space and A is an arbitrary set, then the set
V = W^ of all TF-valued functions on A is a vector space in exactly the same
way that W^ is. Addition is the natural addition of functions, (/+ g)(€i) =
f{a) + g(a)^ and, similarly, {xf){a) = x{f{a)) for every function/and scalar x.
Laws Al through S4 follow just as before and for exactly the same reasons. For
variety, let us check the associative law for addition. The equation/+ (Q + h) =
(f+9) + h means that {f + (g + h)) (a) = {(f + g) + h) (a) for all a G A.
But
{f+(g + h))(a)=f(a) + (g + hXa)
= m + {g{a) + h{a)) = (/(a) + g{a)) + h{a)
= {f+ g){a) + h{a) = ((/ +g) + h) (a),
where the middle equality in this chain of five holds by the associative law for W
and the other four are applications of the definition of addition. Thus the
associative law for addition holds in W^ because it holds in TF, and the other
laws follow in exactly the same way. As before, we let tti be evaluation at z,
so that Tri{f) = f(i). Now, however, tt^ is vector valued rather than scalar valued,
because it is a mapping from V to TF, and we call it the iih coordinate projection
rather than the iih coordinate functional. Again these maps are all linear.
In fact, as before, the natural vector operations on W^ are uniquely defined by
the requirement that the projections ti all be linear. We call the value/(j) =
7ry(/) theith coordinate of the vector /. Here the analogue of Cartesian n-space
is the set W^ of all n-tuples a = <ai, . . . , an^ of vectors in W; it is also
designated W^. Clearly, ay is the yth coordinate of the n-tuple a.
There is no reason why we must use the same space W at each index, as we
did above. In fact, if TFi, . . . , Wn are any n vector spaces, then the set of all
n-tuples a = <«!,..., a^^ such that ay E Wj forj= 1, . . . , n is a vector
space under the same definitions of the operations and for the same reasons.
That is, the Cartesian product W = Wi X W2 X • • • X Wn is also a vector
space of vector-valued functions. Such finite products will be very important
to us. Of course, W^ is the product Hi Wi with each Wi= R; but W^ can also
be considered W X R^"^, or more generally. Hi Wi, where Wi = Wi and
Y^l rrii = n. However, the most important use of finite product spaces arises
from the fact that the study of certain phenomena on a vector space V may
lead in a natural way to a collection {Fji of suhspaces of V such that V is
isomorphic to the product IJi Vi- Then the extra structure that V acquires
when we regard it as the product space Hi Vi is used to study the phenomena
in question. This is the theory of direct sums, and we shall investigate it in
Section 5.
Later in the course we shall need to consider a general Cartesian product of
vector spaces. We remind the reader that if {Wi :i G 1} is any indexed collection
of vector spaces, then the Cartesian product JliGi Wi of these vector spaces is

44
VECTOR SPACES
1.3
defined as the set of all functions / with domain I such that f{i) G Wi for all
i G / (see Section 0.8).
The following is a simple concrete example to keep in mind. Let aS be the
ordinary unit sphere in IR^j aS — {x : Yl\ ^f = 1}? and for each point x on aS
let TFx be the suhspace of U^ tangent to S at x. By this we mean the subspace
(plane through 0) parallel to the tangent plane to aS at x, so that the translate
Wj, + X is the tangent plane (see Fig. 1.8). A function / in the product space
W = YlnES Wj, is a function which assigns to each point x on aS a vector in W^,
that is, a vector parallel to the tangent plane to aS at x. Such a function is called
a vector field on aS. Thus the product set W is the set of all vector fields on aS,
and W itself is a vector space, as the next theorem states.
Fig. 1.8
Of course, the jih coordinate projection on IF = JJi^s Wi is evaluation
aty, Trj(f) = f{j), and the natural vector operations on W are uniquely defined
by the requirement that the coordinate projections all be linear. Thus f -]- g
must be that element of W whose value at j, 7ry(/+ g), is TTj{f) + TTj{g) =
fU) + qU) for all j G /, and similarly for multiplication by scalars.
Theorem 3.1. The Cartesian product of a collection of vector spaces can
be made into a vector space in exactly one way so that the coordinate
projections are all linear.
Proof. With the vector operations determined uniquely as above, the proofs of
Al through S4 that we sampled earlier hold verbatim. They did not require that
the functions being added have all their values in the same space, but only that
the values at a given domain element i all lie in the same space. D
Hoiii(F, W), Linear transformations have the simple but important properties
that the sum of two linear transformations is linear and the composition of two
linear transformations is linear. These imprecise statements are in essence the
theme of this section, although they need bolstering by conditions on domains
and codomains. Their proofs are simple formal algebraic arguments, but the
objects being discussed will increase in conceptual complexity.

1.3 PRODUCT SPACES AND HOM(F, TF) 45
If TF is a vector space and A is any set, we know that the space W^ of all
mappings /: A —> TF is a vector space of functions (now vector valued) in the
same way that U^ is. If A is itself a vector space V, we naturally single out for
special study the subset of W^ consisting of all linear mappings. We designate
this subset Hom(F, TF). The following elementary theorems summarize its
basic algebraic properties.
Theorem 3.2. Hom(Fj TF) is a vector subspace of TF^.
Proof. The theorem is an easy formality. If S and T are in Hom(Fj TF), then
{S + T){xa + yfi) = S{xa + y0) + T{xa + y0)
= xS(a) + yS(0) + xTia) + yT(ff) = x(S + T)(a) + y(S + T)0),
so S + T is linear and Hom(Fj TF) is closed under addition. The reader should
be sure he knows the justification for each step in the above continued equality.
The closure of Hom(F, TF) under multiplication by scalars follows similarly,
and since Hom(F, TF) contains the zero transformation, and so is nonempty,
it is a subspace. D
Theorem 3.3. The composition of linear maps is linear: if T g Hom(F, TF)
and aS g Hom(TF, X), then S o T G Hom(F, X). Moreover, composition
is distributive over addition, under the obvious hypotheses on domains and
codomains:
(Si + S2)oT = S10T + S20T and S o (Tx +T2) = S o Ti +S o T2.
Finally, composition commutes with scalar multiplication:
c{S o T) = (cS) oT = So (cT).
Proof. We have
S o T(xa + yfi) = S{T{xa + yfi)) = S{xT{a) + yT{fi))
= xS{T(a)) + yS(T(p)) = x{S o T)(a) + yiS o r)(^),
so aS o T is linear. The two distributive laws will be left to the reader. D
Corollary. If 7" G Hom(Fy TF) is fixed, then composition on the right by T
is a linear transformation from the vector space Hom(TF, X) to the vector
space Hom(F, X). It is an isomorphism if T is an isomorphism.
Proof. The algebraic properties of composition stated in the theorem can be
combined as follows:
icS, + C2S2) o T = c^{S, o T) + C2{S2 o T).
S o (ciTi + C2T2) = CiiS o Ti) + C2{S o T2).
The first equation says exactly that composition on the right by a fixed 7" is a
linear transformation. (Write aS o T as 3(aS) if the equations still don't look
right.) If T is an isomorphism, then composition by T~^ "undoes" composition
by Tj and so is its inverse. D

46 VECTOR SPACES 1.3
The second equation implies a similar corollary about composition on the
left by a fixed aS.
Theorem 3.4. If TF is a product vector space, W = Yli Wi, then a mapping
T from a vector space F to TF is linear if and only if tt^ o 7" is linear for
each coordinate projection tt^.
Proof. If T is linear, then tt^ o 7" is linear by the above theorem. Now suppose,
conversely, that all the maps tt^ q T are linear. Then
Ti{T{xa + yfi)) = TTi o T(xa + yfi) = x{ji o T){a) + 2/(7r, o T){fi)
= XTTiiTia)) + yWiiTifi)) = TTiixTia) + yT{^)),
But if TTiU) = Tiig) for all i, then f = g. Therefore, T(xa + yp) = xT{a) +
yT{0), and T is linear. D
If 7" is a linear mapping from W^ioW whose skeleton is {fij} ^, then tti q T
has skeleton {Tri{fij)}^=i, If W is R^, then in is the zth coordinate functional
y I—» yi^ and fij is theyth column in the matrix t = {Uj} of T, Thus Tr^dSy) = tijy
and TTi o Tm the linear functional whose skeleton is the zth row of the matrix of T.
In the discussion centering around Theorem 1.3, we replaced the vector
equation y = ^(x) by the equivalent set of m scalar equations yi = 2^y=i UjXj,
which we obtained by reading off the iih coordinate in the vector equation. But
in "reading off" the zth coordinate we were applying the coordinate mapping
TTi, or in more algebraic terms, we were replacing the linear map T by the set of
linear maps {wi © T}, which is equivalent to it by the above theorem.
Now consider in particular the space Hom(F, F), which we may as well
designate 'Hom(F)'. In addition to being a vector space, it is also closed under
composition, which we consider a multiplication operation. Since composition
of functions is always associative (see Section 0.9), we thus have for
multiplication the laws
Ao {BoC)= {AoB)oC,
Ao{B + C)= {AoB) + {Ao C),
{A+B)oC= {AoC) + {Bo C),
k{A o B) = (kA) oB = Ao (kB),
Any vector space which has in addition to the vector operations an operation
of multiplication related to the vector operations in the above ways is called
an algebra. Thus,
Theorem 3.5. Hom(F) is an algebra.
We noticed earlier that certain real-valued function spaces are also algebras.
Examples were W^ and e([0, 1]). In these cases multiplication is commutative,
but in the case of Hom(F) multiplication is not commutative unless F is a
trivial space (F = {0}) or V is isomorphic to U. We shall check this later when
we examine the finite-dimensional theory in greater detail.

1.3 PRODUCT SPACES AND HOM(F, TF) 47
Product projections and injections. In addition to the coordinate projections,
there is a second class of simple linear mappings that is of basic importance in
the handling of a Cartesian product space W = Yik^K Wk. These are, for each
y, the mapping dj taking a vector a G Wj to the function in the product space
having the value a at the index j and 0 elsewhere. For example, $2 for Wi X
W2 X W3 is the mapping a i-» <0, a, 0> from W2 to W, Or if we view U^ as
R X R^, then 62 is the mapping < 0:2, ^^3 >" •—* ^0, < 0:2, ^^3 >" >" = < 0, X2, X3 > .
We call dj the injection of Wj into Ufc ^A;- The linearity of 6j is probably obvious.
The mappings xy and Bj are clearly connected, and the following projection-
injection identities state their exact relationship. If Ij is the identity
transformation on Wj^ then
TTj o dj = Ij and TTj o di = 0 if i 9^ j.
If K is finite and I is the identity on the product space TF, then
Z h ° TTk = L
In the case II?=i TFj, we have 62 ^ 7r2{<oLi, a2, as>) = <0, a2, 0>, and
the identity simply says that <ai, 0, 0> + <0, 0:2? 0> + <0, 0, q:3> =
<ai, a2, (X3> for all ai, a2, (X3. These identities will probably be clear to the
reader, and we leave the formal proofs as an exercise.
The coordinate projections Wj are useful in the study of any product space,
but because of the limitation in the above identity, the injections 6j are of
interest principally in the case of finite products. Together they enable us to
decompose and reassemble linear maps whose domains or codomains are finite
product spaces.
For a simple example, consider the T in Hom([R^, U^) whose matrix is
2-1 1
1 1 4
Then tti q T i^ the linear functional whose skeleton <2, —1, 1 > is the first row
in the matrix of T, and we know that we can visualize its expression in equation
form, 2/1 == 2a:i — X2 + Xs, as being obtained from the vector equation y =
T{x) by "reading off the first row". Thus we "decompose" T into the two linear
functional k == tt^- o T. Then, speaking loosely, we have the reassembly
T = <lijl2> ; more exactly, ^(x) = <2a:i — X2 + Xs, Xi + X2 + 4.T3> ==
<li(x), l2(x)> for all X. However, we want to present this reassembly as the
action of the linear maps ^1 and 62. We have
<h{x), l2{x)> = ^iOl(x)) + d2{l2{x)) = (^1 o TTi + ^2 o 7r2)(7^(x)) = ^(x),
which shows that the decomposition and reassembly of T is an expression of the
identity E di o tt,- = 7. In general, if T G Hom(F, W) and W = Hz ^i, then
Ti = Ti o T is in Hom(F, Wj) for each z, and T^ can be considered "the part
of T going into Wi\ since Ti(a) is the ith coordinate of T(a) for each a. Then we

48 VECTOR SPACES 1.3
can reassemble the T/s to form T again hy T = Yl^i° Ti, for J^Oi o Ti =
(2Z 6i o TTi) o T = I o T = T, Moreover, any finite collection of T^s on a
common domain can be put together in this way to make a T. For example,
we can assemble an m-tuple {Ti}^ of linear maps on a common domain V to
form a single m-tuple-valued linear map T. Given a in F, we simply define
T{a) as that m-tuple whose ith coordinate is Ti{a) fori= 1, . . . , m, and then
check that T is linear. Thus without having to calculate, we see from this
assembly principle that T:xy-^ <2xi —0:2 +^35^1+^2 + ^x^> is a linear
mapping from U^ to R^, since we have formed T by assembling the two linear
functionals Zi(x) = 2a:i —0:2 + ^3 and Z2(x) = :^i + ^2 + 4a:3 to form a
single ordercd-pair-valued map. This very intuitive process has an equally
simple formal justification. We rigorize our discussion in the following theorem.
Theorem 3.6. If Ti is in Hom(F, Wi) for each i in a finite index set 7,
and if W is the product space IIzg/ ^ii then there is a uniquely determined
T in Hom(F, W) such that Ti = Wi o T for all i in 7.
Proof, If T exists such that Ti = Ti o T for each i, then T = Iw ° T =
(ZOiOTi) o T = Z Oi o {Ti o T) = Z^io Ti. Thus T is uniquely determined
as J^OiO Ti. ^loreover, this T does have the required property, since then
TjoT= Tj oiJ^OiO Ti) = E (Try o di) o T, = Ij o Tj = Tj. D
i
In the same way, we can decompose a linear T whose domain is a product
space F = IIy=i ^i ii^to the maps Tj = To Sj with domains Fy, and then
reassemble these maps to form T by the identity T = 2^7= 1 ^i ° ^i (check it
mentally!). Moreover, a finite collection of maps into a common codomain
space can be put together to form a single map on the product of the domain
spaces. Thus an n-tuple of maps {Ti}i into W defines a single map T into TF,
where the domain of T is the product of the domains of the T/s, by the equation
T(<ai,...,an>)=Zi Ti{ai) ov T = Zi TiOTi. For example, if Tii [R-> R^
is the map t ^-^ t<2, 1> = <2t, t>, and T2 and T^ are similarly the maps
11-» t<—l, 1> and ^ i-» ^<1, 4>, then T = Y.i Ti o Ti is the mapping from
U^ to U^ whose matrix is
2-1 1
.1 1 4
Again there is a simple formal argument, and we shall ask the reader to write
out the proof of the following theorem.
Theorem 3.7. If Tj is in Hom(Fy, W) for each j in a finite index set J,
and if F = Jlj^J Vji then there exists a unique T in Hom(F, W) such
that T o Sj = Tj for each j in J.
Finally we should mention that Theorem 3.6 holds for all product spaces,
finite or not, and states a property that characterizes product spaces. We shall

1.3
PRODUCT SPACES AND HOM(F, TF)
49
investigate this situation in the exercises. The proof of the general case of
Theorem 3.6 has to get along without the injections 6j; instead, it is an application
of Theorem 3.4.
The reader may feel that we are being overly formal in using the projections
Ti and the injections di to give algebraic formulations of processes that are
easily visualized directly, such as reading off the scalar "components" of a
vector equation. However, the mappings
and
<0,
0, .r„0, ...,0>
are clearly fundamental devices, and making their relationships explicit now
will be helpful to us later on when we have to handle their occurrences in more
complicated situations.
EXERCISES
3.1 Show that W^ X W' is isomorphic to W^'^
3.2 Show more generally that if Sf rn = n, then IJi^i ^""^ is isomorphic to W.
3.3 Show that if [B, C] is a partitioning of .1, then U-'^ and U^ X U^ are isomorphic.
3.4 Generalize the above to the case where {Ai][ partitions A.
3.5 Show that a mapping T from a vector space T^ to a vector space 11^ is linear if
and only if (the graph of) T is a subsi)ace of 7 X W.
3.6 Let S and T be nonzero linear maps from V to W. The definition of the map
S -\- T is not the same as the set sum of (the graphs of) S and T as subspaces of V X W.
Show that the set sum of (the graphs of) S and T cannot be a grai)h unless S = T.
3.7 Give the justification for each step of the calculation in Theorem 3.2.
3.8 Prove the distributive laws given in Theorem 3.3.
3.9 Let /;: Q^la, b]) -> e([a, b]) be differentiation, and let S:e{[a, b]) -> U be the
definite integral map/t-^ ja f. Compute the composition So J),
3.10 We know that the general linear functional F on R^ is the map x t-^ aixi -f a2i'2
determined by the i)air a in R^, and that the general linear maj) T in Hom(lR^) is
determined by a matrix
\j2l t22_\
Then F o T is another linear functional, and hence is of the form x h-> ^i^i + b2X2 for
some b in R^. Comi)ute b from t and a. Your comi)utation should -'lOw you that
a t-^ b is linear. What is its matrix?
3.11 Given S and T in Hom(lR^) whose matrices are
and
respectively, find the matrix oi S o T in Hom([R^).

50 VECTOR SPACES 1.3
3.12 Given S and T in Hom([R^) whose matrices are
and t =
Sll Sl2
.521 S22.
^11 h2
hi ^22.
find the matrix oi S o T.
3.13 With the above answer in mind, what would you guess the matrix of iS o T is
if S and T are in Hom(IR^) ? Verify your guess.
3.14 We know that if jT E Hom(F, W) is an isomorphism, t^en T~^ is an isomorphism
in Hom(]r, V). Prove that
S o T surjective => S surjective, S o T infective => T infective,
and, therefore, that if T G Hom(7, W),Sg Homdf, 7), and
So T = Iv, To S = I. ,
then T is an isomorphism.
3.15 Show that if S''^ and T'^ exist, then (So T)-'^ exists and equals T'^ o ^-i.
Give a more careful statement of this result.
3.16 Show that if S and T in Hom V commute with each other, then the null space of
T,N = N{T), and its range R = R{T) are invariant under S {S[N] C N and S[R] C R).
3.17 Show that if a is an eigenvector of T and S commutes with T, then S(a) is
an eigenvector of T and has the same eigenvalue.
3.18 Show that if S commutes with T and T~^ exists, then S commutes with T~^.
3.19 Given that a is an eigenvector of T with eigenvalue x, show that a is also an
eigenvector oi T^ = To T, of T^, and of T'^ (if T is invertible) and that the
corresponding eigenvalues are x^, x"", and l/x.
Given that p{t) is a polynomial in t, define the operator p(T), and under the above
hypotheses, show that a is an eigenvector of p(T) with eigenvalue p(x).
3.20 If S and T are in Hom V, we say that S doubly commutes with T (and write
S cc T) if S commutes with every .4 in Hom V which commutes with T. Fix T, and
set {T]'' = {S:Scc T}. Show that {T}'' is a commutative subalgebra of Hom V.
3.21 Given T in Hom V and a in V, let A^ be the linear span of the "trajectory of a
under T" (the set {T^a : n G Z+]). Show that N is invariant under T.
3.22 A transformation T in Hom V such that T^ = 0 for some n is said to be nilpotent.
Show that if T is nilpotent, then / — T is invertible. [Hint: The power series
1 °°
I — X 0
is a finite sum if x is replaced by T.]
3.23 Suppose that T is nilpotent, that iS commutes with T, and that S ^ exists, where
iS, T G Hom 7. Show that (iS — T)-^ exists.
3.24 Let <p be an isomorphism from a vector space F to a vector space W. Show that
T —^ (fo To (^-1 is an algebra isomorphism from the algebra Hom V to the algebra
Hom W.

1.3 PRODUCT SPACES AND HOM(F, TD 51
3.25 Show the tt/s and d/s explicitly for R^ = R X R X R using the stopped arrow
notation. Also write out the identity 2Z Oj o Try = / in explicit form.
3.26 Do the same for R^ = R^ X R^.
3.27 Show that the first two projection-injection identities (wi o Si = li and Try o ^^- = 0
if j lA i) are simply a restatement of the definition of 6i. Show that the linearity of 6i
follows formally from these identities and Theorem 3.4.
3.28 Prove the identity ^ Bi o wi = I by applying Try to the equation and remembering
that f = g ii wjif) = Tj{g) for all j (this being just the equation f{j) = g(j) for all j).
3.29 Prove the general case of Theorem 3.6. We are given an indexed collection of
linear maps {Ti: i E /} with common domain V and codomains {14\-: i G I]. The
first question is how to define T: V -^ W = II^• Wi. Do this by defining T{^) suitably
for each J E F and then applying Theorem 3.4 to conclude that T is linear.
3.30 Prove Theorem 3.7.
3.31 We know without calculation that the map
T:x -^ < 3a;i — X2 + xs, X2 + xs, xi — 5x's, 2a;i >
from R^ to R"* is linear. Why? (Cite relevant theorems from the text.)
3.32 Write down the matrix for the transformation T in the above example, and then
write down the mappings To Bi from R to R^ (for i = 1, 2, 3) in explicit ordered
quadruplet form.
3.33 Let W = U? Wi be a finite product vector space and set pi = Bi o tt^, so that
Pi is in Hom W for all i. Prove from the projection-injection identities that 23? pi = /
(the identity map on W), pio pj = 0 if i ?^ j, and pio pi = pi. Identify the range
Ri = R{pi).
3.34 In the context of the above exercise, define T in Hom W as
n
Show that a is an eigenvector of T if and only if a is in one of the subspaces Ri and that
then the eigenvalue of a is i.
3.35 In the same situation show that the polynomial
n
n {T-jD = (T-/)o...o(T-n/)
is the zero transformation.
3.36 Theorems 3.6 and 3.7 can be combined if T E Hom(F, If^), where both V and IT
are product spaces:
n m
V = YlVj and TF = n ^'i-
1 1
State and prove a theorem which says that such a T can be decomposed into a doubly
indexed family {T^•y} when TijE: Hom(Fi, Wj) and conversely that any such doubly
indexed family can be assembled to form a single T form V to W.

52 VECTOR SPACES 1.4
3.37 Apply your theorem to the special case where 7 = R"" and W = R"" (that is,
Vi = Wj = U for all i and j). Now Tij is from R to R and hence is simply
multiplication by a number Uj. Show that the indexed collection {Uj} of these numbers is the
matrix of T.
3.38 Given an m-tuple of vector spaces {Wi] f, suppose that there are a vector space
.Y and maps pi in Hom(X, Wi), i = 1, . . . , m, with the following property:
P. For any m-tuple of linear maps {Ti] from a common domain space V to the
above spaces Wi (so that TiG Hom(F, Wi), i = 1, . . . , m), there is a unique T
in Hom(F, A") such that Ti = pio T, i = I, . . . , rn.
Prove that there is a "canonical" isomorphism from
m
W = II Wi to X
1
under which the given maps pi become the projections in. [Remark: The product space
W itself has property P by Theorem 3.6, and this exercise therefore shows that P is an
abstract characterization of the product space.]
4. AFFINE SUBSPACES AND QUOTIENT SPACES
In this section we shall look at the "planes" in a vector space V and see what
happens to them when we translate them, intersect them with each other,
take their images under linear maps, and so on. Then we shall confine ourselves
to the set of all planes that are translates of a fixed subspace and discover that
this set itself is a vector space in the most obvious way. Some of this material
has been anticipated in Section 2.
Affine subspaces. If A^ is a subspace of a vector space V and a is any vector
of F, then the sot N + a = {^ + a : ^ E N} is called either the coset of N
containing a or the affine subspace of V through a and parallel to N. The set N + a
is also called the translate of N through a. We saw in Section 2 that affine sub-
spaces are the general objects that we want to call planes. If N is given and fixed
in a discussion, we shall use the notation a = N -\- a (see Section 0.12).
We begin with a list of some simple properties of affine subspaces. Some of
these will generalize observations already made in Section 2, and the proofs of
some will be left as exercises.
1) With a fixed subspace N assumed, if 7 G a, then 7 = a. For if 7 =
a + 770) then 7 + rj = a+ (r]o +rj) G a, SOJ Ca. Also a+rj = 7 + (rj — tjq) G7,
so a C 7. Thus a = 7.
2) With N fixed, for any a and (S, either a = p or a and /3 are disjoint.
For if a and /3 are not disjoint, then there exists a 7 in each, and a = 7 = ^
by (1). The reader may find it illuminating to compare these calculations with
the more general ones of Section 0.12. Here a ^ 13 ii and only li a — 13 G N.
3) Now let a be the collection of all affine subspaces of F; Q- is thus the set
of all cosets of all vector subspaces of V. Then the intersection of any sub-

1.4 AFFINE SUBSPACES AND QUOTIENT SP VCES 53
family of Q, is either empty or itself an affine subspace. In fact, if [Ai}isi is
an indexed collection of affine subspaces and A^- is a coset of the vector subspace
Wi for each i G /, then Hiei ^i is either empty or a coset of the vector subspace
rizG/ Wi. For if (S G rizG/ ^u then (1) implies that A,- = fi -\-Wi for all i, and
thenflA,- fi+V\Wi.
4) If A, 5 G Q-, then A + ^ G d. That is, the set sum of any two affine
subspaces is itself an affine subspace.
5) If A G a and T G Hom(F, W), then T[A] is an affine subspace of W.
In particular, if ^ G [R, then tA G Q-.
6) If B is an affine subspace of W and T G Hom(F, TF), then T'^B] is
either empty or an affine subspace of F.
7) For a fixed a E:V the translation of F through a is the mapping
Sa'-V —^ V defined by Sa(^) = ^ + « for all ^ gV. Translation is 7wt linear;
for example, ^^(0) = a. It is clear, however, that translation carries affine
subspaces into affine subspaces. Thus Sa(A) = A +« and Sa(l3 + W) =
(a + fi) + W,
8) An affine transformation from a vector space F to a vector space TF is a
linear mapping from F to TF followed by a translation in TF. Thus an affine
transformation is of the form J i-> T(^) +13, where T G Hom(F, TF) and 13 gW.
Note that J i—> T(^ + a) is affine, since
TU +cc)= T(0 + ^, where ^ - T(a).
It follows from (5) and (7) that an affine transformation carries affine
subspaces of F into affine subspaces of TF.
Quotient space. Now fix a subspace N of F, and consider the set TF of all
translates (cosets) of N. We are going to see that TF itself is a vector space in
the most natural way possible. Addition will be set addition, and scalar
multiplication will be set multiplication (except in one special case). For example, if N
is a line through the origin in R^, then TF consists of all lines in U^ parallel to A^.
We are saying that this set of parallel lines will automatically turn out to be a
vector space: the set sums of any two of the lines in TF turn out to be a line in TF!
And if L G TF and t 9^ 0, then the set product tL is a line in TF. The translates
of L fiber R^, and the set of fibers is a natural vector space.
During this discussion it will be helpful temporarily to indicate set sums by
'+s' and set products by '-5'. With A" fixed, it follows from (2) above that two
(^osets are disjoint or identical, so that the set TF of all cosets is a fibering of F
in the general case, just as it was in our example of the parallel lines. From (4)
or by a direct calculation we know that a +5 /3 = a + ^. Thus TF is closed
under set addition, and, naturally, we take this to be our operation of addition
on TF. That is, we define + on TF by a + /3 = a +5 /3. Then the natural map
tt.'q: I—> a from F to TF preserves addition, iri^a + |S) = 7r(Q:) + '7r(|S), since

54 VECTOR SPACES 1.4
this is just our equation a + (^ = « + /3 above. Similarly, if ^ G R, then the
set product t -s ais either ta or {0]. Hence if we define ta as the set product when
t 7^ 0 and as 0 = A^ when t = 0, then tt also preserves scalar multiplicationj
T(ta) = t7r(a).
We thus have two vectorlike operations on the set W of all cosets of N,
and we naturally expect W to turn out to be a vector space. We could prove this
by verifying all the laws, but it is more elegant to notice the general setting for
such a verification proof.
Theorem 4.1. Let V be a vector space, and let TF be a set having two
vectorlike operations, which we designate in the usual way. Suppose that
there exists a surjective mapping T:V —^ W which preserves the operations:
T(sa + tfi) = sT(a) + tT(l3). Then TF is a vector space.
Proof. We have to check laws Al through S4. However, one example should
make it clear to the reader how to proceed. We show that ^(O) satisfies A3 and
hence is the zero vector of W. Since every (3 gW is of the form T(a), we have
T(0) + fi= T(0) + T{a) = T(0 + a)= T(a) = fi,
which is A3. We shall ask the reader to check more of the laws in the exercises. D
Theorem 4.2. The set of cosets of a fixed subspace A^ of a vector space V
themselves form a vector space, called the quotient space V/N, under the
above natural operations, and the projection tt is a surjective linear map
from V to V/N.
Theorem 4.3. If T is in Hom(F, TF), and if the null space of T includes the
subspace M C F, then T has a unique factorization through V/M, That is,
there exists a unique transformation S in Hom(F/ii/, W) such that T =
Sot,
Proof. Since T is zero on 71/, it follows that T is constant on each coset A of il/,
so that T[A] contains only one vector. If we define S(A) to be the unique
vector in T[A], then S(a) = T{a)^ so S o w = T by definition. Conversely, if
T = R o TT, then R(a) = R o Tr(a) = T(a)^ and R is our above S. The linearity
of S is practically obvious. Thus
S(a + P) = S(a + fi)= T(a + fi) = T{a) + T{fi) = S{a) + S0),
and homogeneity follows similarly. This completes the proof. D
One more remark is of interest here. If N is invariant under a linear map
T in Horn F (that is, T[N] C A^), then for each a in F, T[a] is a subset of the
coset T(a), for
T[a] = T[a + N] = T(a) +, T[N] C T(a) +s N = T(a).

1.4 AFFINE SUBSPACES AND QUOTIENT SPACES 55
There is therefore a map S: V/N —> V/N defined by the requirement that
S{a) = T{a) {or S o TT = tt o T), and it is easy to check that S is linear.
Therefore,
Theorem 4.4. If a subspace N of Si vector space V is carried into itself by a
transformation T in Hom Fj then there is a unique transformation S in
Hom(F/iV) such that S o w = t o T,
EXERCISES
4.1 Prove properties (4), (5), and (6) of affine subspaces.
4.2 Choose an origin 0 in the Euchdean plane E^ (your sheet of paper), and let
Li and L2 be two parallel lines not containing 0. Let X and Y be distinct points on
Li and Z any point on L2. Draw the figure giving the geometric sums
OX+OZ and OY+CiZ
(parallelogram rule), and state the theorem from plane geometry that says that these
two sum points are on a third line L3 parallel to Li and L2.
4.3 a) Prove the associative law for addition for Theorem 4.1.
b) Prove also laws A4 and S2.
4.4 Return now to Exercise 2.1 and reexamine the situation in the light of Theorem
4.1. Show, finally, how we really know that the geometric vectors form a vector space.
4.5 Prove that the mapping S of Theorem 4.3 is infective if and only if A^ is the
null space of T.
4.6 We know from Exercise 4.5 that if T is a surjective element of Hom(Fj W) and
IV is the null space of T, then the S of Theorem 4.3 is an isomorphism from V/N to W,
Its inverse S"^ assigns a coset of A^ to each rj in W. Show that the process of "indefinite
integration" is an example of such a map S~^. This is the process of calculating an
integral and adding an arbitrary constant, as in
/
sin X dx = —cos x-\- c.
4.7 Suppose that A^ and M are subspaces of a vector space V and that N G M.
Show that then M/N is a subspace of V/N and that V/M is naturally isomorphic to the
(luotient space (V/N)/(M/N). [Hint: Every coset of A^ is a subset of some coset of M.]
4.8 Suppose that A^ and AI are any subspaces of a vector space V. Prove that
(M + A^)/A^ is naturally isomorphic to M/(M O N). (Start with the fact that each
coset of Af n A^ is included in a unique coset of A^.)
4.9 Prove that the map S of Theorem 4.4 is linear.
4.10 Given T G Hom V, show that T^ = 0 (T^ = T o T) if and only if R(T) C N(T).
4.11 Suppose that T G Hom V and the subspace A^ are such that T is the identity
on A^ and also on V/N. The latter assumption is that the S of Theorem 4.4 is the
identity on V/N. Set R = T — /, and use the above exercise to show that R'^ = 0.
Show that if T = I -\- R and 72^ = Oj then there is a subspace A^ such that T is the
identity on A^ and also on V/N.

56 VECTOI^ SPACES 1.5
4.12 We now view the above situation a little differently. Supposing that T is the
identity on N and on V/N, and setting R = I — T, show that there exists a
A' E liom(V/N, V) such that R = K o x. Show that for any coset A of N the action
of T on A can be viewed as translation through K(A). That is, if J E A and rj = K(A),
thenT(J) = ^+v-
4.13 Consider the map T: <xi, X2> ^-^ <a;i + 2x2, X2> in Horn U^, and let N be
the null space oi R = T — /. Identify N and show that T is the identity on N and
on R^/iV. Find the map K of the above exercise. Such a mapi)ing T is called a shear
transformation of T^ parallel to A^ Draw the unit square and its image under T.
4.14 If we remember that the linear span L{A) of a subset .1 of a vector space V can
be defined as the intersection of all the subspaces of V that include J, then the fact
that the intersection of any collection of affine subspaces of a vector space T^ is either
an affine subspace or empty suggests that we define the affine span M{A) of a nonempty
subset A(ZV as the intersection of all affine subspaces including A. Then we know
from (3) in our list of affine proi)erties that M{A) is an affine subspace, and by its
definition above that it is the smallest affine subspace including A. We now naturally
wonder whether M{A) can be directly described in terms of finear combinations.
Show first that if a E A, then M{A) = L(A — a) + a; then prove that MiA) is the
set of all Hnear combinations Yl ^i<^i ^^ -^ such that ^xi = 1.
4.15 Show that the finear span of a set B is the affine span of 5 U {0].
4.16 Show that yi{A + 7) = yi{A) + T for any 7 in T^ and that M{xA) = xM{A)
for any a; in R.
5. DIRECT SUMS
We come now to the heart of the chapter. It frequently happens that the study
of some phenomenon on a vector space V leads to a finite collection of subspaces
{FJ such that V is naturally isomorphic to the product space Yli ^i- Under
this isomorphism the maps Bi o in on the product space become certain maps
Pi in Horn F, and the projection-injection identities are reflected in the identities
YlPi = L Pj ° Pj = Pj ^or all ./', and Pi o Pj = 0 if i ^ j. Also, Vi = range
Pi. The product structure that V thus acquires is then used to study the
phenomenon that gave rise to it. For example, this is the way that we unravel the
structure of a linear transformation in Hom F, the study of which is one of the
central problems in linear algebra.
Direct sums. If Fi, . . . , F^ are subspaces of the vector space F, then the
mapping tt: <«!,..., a^^ "—^ 2Zi «z is a linear transformation from JYi Vi
to F, since it is the sum tt = 2Zi Wi of the coordinate projections.
Definition. We shall say that the F/s are independent if w is injective and
that F is the direct sum of the F/s if tt is an isomorphism. We express the
latter relationship by writing F = Fi 0 • • • 0 F^ = 0i F^
Thus F = 0r=i Vi if and only if tt is injective and surjective, i.e., if and
only if the subspaces {FJ ^ are both independent and span F. A useful restate-

1.5 DIRECT SUMS 57
ment of the direct sum condition is that each a G F is uniquely expressible as
a sum 2Z1 ai^ with ai G Vi for all i] a has some such expression because the F/s
span F, and the expression is unique by their independence.
For example, let F = e(IR) be the space of real-valued continuous functions
on R, let Ve be the subset of even functions (functions / such that f(—x) = f(x)
for all x), and let Vq be the subset of odd functions (functions such that f(—x) =
—f(x) for all x). It is clear that Ve and Vq are subspaces of F, and we claim that
7 = Fe e Fo. To see this, note that for any / in F, g(x) = {f{x) +f(-x))/2
is even, h(x) = {f{x) - f(-x))/2 is odd, Sindf= g + h. Thus V = Ve + Fo.
Moreover, this decomposition of / is unique, for if f = gi -{- hi also, where gi
is even and hi is odd, then g — gi = hi — h, and therefore ^ — ^i = 0 =
hi — /i, since the only function that is both even and odd is zero. The even-odd
components of e^ are the hyperbolic cosine and sine functions:
e^ = o + "^ o ^ ^^^^ ^ + ^"^^ ^*
Since tt is infective if and only if its null space is {0} (Lemma 1.1), we have:
Lemma 5.1. The independence of the subspaces [Vi}1 is equivalent to the
property that if ai G Vi for all i and XI i ^i = 0, then ai = 0 for all i.
Corollary. If the subspaces {F^jiare independent, ai G Vi for all z, and
2Z1 a^- is an element of Fy, then ai = 0 for i 9^ j.
We leave the proof to the reader.
The case of two subspaces is particularly simple.
Lemma 5.2. The subspaces M and A^ of F are independent if and only if
MnN= {0}.
Proof. If a G M, ^ G iV, and a + ^ = 0, then a= —l3GMnN. If M n N =
{0}, this will further imply that a = 13 = 0, so M and N are independent.
On the other hand, if 0 9^ fi G M D N, and if we set a = —fi, then a G M,
f-i G N, and a -{- ^ = 0, so M and N are not independent, D
Note that the first argument above is simply the general form of the
uniqueness argument we gave earlier for the even-odd decomposition of a function
on R.
Corollary. V = M ^ N if and only if V = M + N smd M D N = {0}.
Definition. If F = M 0 A^, then M and A^ are called complementary sub-
spaces, and each is a complement of the other.
Warning: A subspace M of F does not have a unique complementary subspace
unless M is trivial (that is, M = {0} or M = F). If we view R^ as coordinatized
lOuclidean 3-space, then M is a proper subspace if and only if M is a plane
containing the origin or M is a line through the origin (see Fig. 1.9). If M and N are

58 VECTOR SPACES 1.5
Fig. 1.9
proper subspaces one of which is a plane and the other a line not lying in that
plane, then M and N are complementary subspaces. Moreover, these are the
only nontrivial complementary pairs in 1^. The reader will be asked to prove
some of these facts in the exercises and they all will be clear by the middle of
the next chapter.
The following lemma is technically useful.
Lemma 5*S, If Vi and Vq are independent subspaces of V and {Vi}^ are
independent subspaces of Fq, then {Vijl are independent subspaces of V.
Proof, If m G Vi for all i and Ei a,- = 0, then, setting a^ = £^ ai, we have
"1 + ao = 0, with aQ G Vq, Therefore, ai = ao = 0 by the independence of
Vi and Vq. But then ag = ag = • • • = a^ = 0 by the independence of
{Vi}% and we are done (Lemma 5.1). D
Corollary. V = Vi ® Vq and Vq = ©t=2 Vi together imply that
V = e^i Vi,
Projections. If F = ®?=i F^, if w is the isomorphism <ai, , . , , an> ^-»
« = Si «ti and if Try is the jth projection map <ai^ , . . ^ an> ^-^ aj from
ffi=i Vi to Fi, then (tt,- o 7r""^)(a) = aj.
Definition. We call aj the jth component of a, and we call the linear map
Pj = TTy o 7r~^ the projection of V onto Vj (with respect to the given direct
sum decomposition of F). Since each a in F is uniquely expressible as a sum
« = ZIi «ii with ai in Vi for all f^ we can view Pj(a) = ay as "the part of
a in Vj'\
This use of the word "projection" is different from its use in the Cartesian
product situation^ and each is different from its use in the quotient space
context (Section 0.12). It is apparent that these three uses are related, and the
ambiguity causes little confusion since the proper meaning is always clear from
the context.
Theorem 5.1. If the maps Pi are the above projections, then range Pi = F^,
P. o p^. = 0 for i 9^ j, and E? Pi = /.
Proof, Since r is an isomoiphism and Pj = Try o 7r"~^ we have range Py =
range Try = Vj, Next, it follows directly from the corollary to Lemma 5.1 that

1.5 DIRECT SUMS 59
if a G Vj, then Pi{a) = 0 for i 9^ j, and so Pi o Py = 0 for i 9^ j. Finally^
Li Pi = El TTi ° TT"^ = (L" TTi) o TT"^ = 7r o tt"^ = I, and we are done. D
The above projection properties are clearly the reflection in V of the
projection-injection identities for the isomorphic space Hi Vi,
A converse theorem is also true.
Theorem 5.2. If {P,}^ C Hom F satisfy E^ Pi = I and Pi o Pj = 0 for
i ^ jj and if we set Vi = range Pi^ then V = ®iLi Vi, and Pi is the
corresponding projection on Vi.
Proof, The equation a — I (a) = Ei A'(«) shows that the subspaces {Vi}1
span F. Nextj if ^ G Vjy then Pi(fi) = 0 for i ^ j\ since fi G range Pj and
p. o Py =. 0 if i ^ J. Then also PyO) = (/ - Zi^j Pi)0) = 1(0) = 0-
Now consider a = E^ «i fo^ any choice of ai G Fj. Using the above two factSj
wehavePj(Q:) = PyCELi «i) = 2]?=i Pj{<^i) = «y- Therefore, a = 0 implies
that ay = Pi(0) = 0 for all j\ and the subspaces Vi are independent.
Consequently, F = ® 1 F^•. Finally, the fact that a = E Pi(a) and P.^a) G F^
for all i shows that Pj((x) is the jth component of a for every a and therefore that
Pj is the projection of F onto Fy. D
There is an intrinsic characterization of the kind of map that is a projection.
Lemma 5.4. The projections Pi are idempotent (Pf = Pi), or, equivalently,
each is the identity on its range. The null space of Pi is the sum of the spaces
Vj for J 9^ I.
Proof, Pj = Pj o (/ — EzVi Pi) = Pj ° I = Pj' Since this can be rewritten
as Pj{Pj{a)) = Pj(ci) for every a in F, it says exactly that Pj is the identity
on its range.
Now set Wi = Ejvi Fy, and note that if (S G Wi^ then Pi(p) = 0 since
p.[Vj] = 0 for J 9^ I, Thus Wi C N(Pi). Conversely, if P^a) = 0, then a =
I{a) = Z1 Pj{a) = Ey^,- Pj(a) G Wi, Thus N{Pi) C Wi, and the two spaces
are equal. D
Conversely:
Lemma 5.5. If P G Hom(F) is idempotent, then F is the direct sum of its
range and null space, and P is the corresponding projection on its range.
Proof, Setting Q = 7 — P, we have PQ = P — P^ = 0. Therefore, V is the
direct sum of the ranges of P and Q, and P is the corresponding projection on its
range, by the above theorem. Moreover, the range of Q is the null space of P,
by the corollary. D
If F = M 0 AT" and P is the corresponding projection on M, we call P the
projection on M along N, The projection P is not determined by M alone, since
M does not determine N. A pair P and Q in Hom F such that P -\- Q = I and
PQ = QP = 0 is called a pair of complementary projections.

60 VECTOR SPACES 1.5
In the above discussion we have neglected another fine point. Strictly
speaking, when we form the sum tt = 2Z^ Wi, we are treating each ttj as though
it were from JYi Vi to F, whereas actually the codomain of Try is Fy. And we
want Pj to be from F to F, whereas Try © tt"^ has codomain Vj, so the equation
Pj = TTj o 7r~^ can't quite be true either. To repair these flaws we have to
introduce the injection ty: Fy —> F, which is the identity map on Fy, but which
views Fy as a subspace of F and so takes F as its codomain. If our concept of a
mapping includes a codomain possibly larger than the range, then we have to
admit such identity injections. Then, setting Tfy = ty o Try, we have the correct
equations tt = 2Z^ iTi and Pj = Tfy o Tr~^
EXERCISES
5.1 Prove the corollary to Lemma 5.1.
5.2 Let a be the vector -<1, 1, 1> in U^, and let M = Ua be its one-dimensional
span. Show that each of the three coordinate planes is a complement of M.
5.3 Show that a finite product space V = YLi Fi has subspaces {ITiji such that
Wi is isomorphic to Vi and V = 0i Wi. Show how the corresponding projections
{Pi} are related to the tt/s and di's.
5.4 If r GHom(F, W), show that (the graph of) T is a complement of W =
{0} XWinVXW.
5.5 If Z is a linear functional on F (Z E Hom(F, U) = V*), and if a is a vector in V
such that l{a) j^ 0, show that V = N ® M, where N is the null space of I and M = Ua
is the linear span of a. What does this result say about complements in R^?
5.6 Show that any complement M of a subspace iV of a vector space V is isomorphic
to the quotient space V/N.
5.7 We suppose again that every subspace has a complement. Show that if
T E Hom V is not injective, then there is a nonzero S in Hom V such that 7" o ^ = 0.
Show that if 7" E Hom V is not surjective, then there is a nonzero S in Hom V such
that So T = 0.
5.8 Using the above exercise for half the arguments, show that T E Hom V is
injective if and only U T o S = 0=^ S = 0 and that T is surjective if and only ii S o T =
0=^ S = 0. We thus have characterizations of injectivity and surjectivity that are
formal, in the sense that they do not refer to the fact that S and T are transformations,
but refer only to the algebraic properties of S and T as elements of an algebra.
5.9 Let M and N be complementary subspaces of a vector space V, and let X be a
subspace such that X O N = {0). Show that there is a linear injection from A" to M.
[Hint: Consider the projection P of 7 onto M along N.] Show that any two
complements of a subspace N are isomorphic by showing that the above injection is surjective
if and only if Z is a complement of N.
5.10 Going back to the first point of the preceding exercise, let Y be a complement of
PIX] in M. Show that X nY = {0} and that X © F is a complement of N.
5.11 Let M be a proper subspace of V, and let {ai: i E /} be a finite set in V. Set
L = L({ai}), and suppose that M -[- L = V. Show that there is a subset J C I such

1.5 DIRECT SUMS 61
that {oii: i E J] spans a complement of M. [Hint: Consider a largest possible subset J
such that M 0 L {{a^] j) = {0}.]
5.12 Given T G Hom(F, W) and S G Hom(Tr, X), show that
a) So T'lii surjective <=^ S is surjective and R{T) + N{S) = W;
b) So T is injective <=^ T is injective and R{T) n N{S) = [0] ;
c) So T is an isomorphism <=> >S is surjective, T is injective, and ir = R{T) ® NiS).
5.13 Assuming that every subspace of T' has a complement, show that T G Horn V
satisfies T^ = OU and only if T^ has a direct sum decomposition V = M © A^ such that
r = 0 on A^ and T[M] C A^.
5.14 Suppose next that T^ = 0 but T- ^ 0. Show that T' can be written as V =
Vi © Vz © F:i, wliere ^[T^]] C V2, T[V2] C V-a, and T = 0 on V;^. (Assume again
that any subspace of a vector space has a complement.)
5.15 We now suppose that T'^ = 0 but T"-' 9^ 0. Set A\ = null space (T^) for
I = 1, . . . , n — 1, and let T^i be a complement of Nn-i in V. Show first that
T[Vi] n Nn-2 = {0}
and that T[Vi] C Nn-i- Extend T[Vi] to a complement V2 of Nn-2 in Nn-i, and
show that in this way we can construct subspaces Fi, . . . , F„ such that
and
© Vi, T[Vi] CVi^i for i < n,
1
TlVn] = {0}.
On solving a linear equation. Alany important problems in mathematics are
in the following general form. A linear operator 7": F -^ TF is given, and for a
given 77 G TF the equation T{^) = rj is to be solved for ^ ^V. In our terms, the
condition that there exist a solution is exactly the condition that rj be in the
range space of T. In special circumstances this condition can be given more or
less useful equivalent alternative formulations. Let us suppose that we know
how to recognize R{T), in which case we may as well make it the new codomain,
and so assume that T is surjective. There still remains the problem of
determining what we mean by solving the equation. The universal principle running
through all the important instances of the problem is that a solution process
calculates a right inverse to T, that is, a linear operator S:W-^ V such that
T o S = Iwj the identity on W. Thus a solution process picks one solution
vector J G F for each 77 G TF in such a way that the solving J varies linearly with
77. Taking this as our meaning of solving, we have the following fundamental
reformulation.
Theorem 5.3. Let 7" be a surjective linear map from the vector space F
to the vector space W, and let A^ be its null space. Then a subspace M is a
complement of A^ if and only if the restriction of T to M is an isomorphism
from M to W. The mapping M \-^ {T \ M)~^ is a bisection from the set
of all such complementary subspaces M to the set of all linear right inverses
of r.

62 VECTOR SPACES 1.5
Proof. It should be clear that a subspace M is the range of a linear right inverse
of T (a map S such that T o S = Iw) if and only if 7" \ M is an isomorphism to W,
in which case aS = {T \ M)~^. Strictly speaking, the right inverse must be from
TF to F and therefore must be R = lm ^ Sj where lm is the identity injection
from M to V, Then {R o T)^ = R o (T o R) o T = R o I^ o T = R o T, and
R o T is Si projection whose range is M and whose null space is N (since R is
injective). Thus F = M 0 iV. Conversely, if F == M 0 iV, then T \ M is
injective because M n N = {0} and surjective because M + N = V implies
that W = T[V] = T[M + N] = T[M] + T[N] = T[M] + {0} = T[M]. D
Polynomials in T. The material in this subsection will be used in our study of
differential equations with constant coefficients and in the proof of the diagonal-
izability of a symmetric matrix. In linear algebra it is basic in almost any
approach to the canonical forms of matrices.
If Pi{t) = 2Zo^ ciii^ a-nd 7?2(0 — 2Zo ^j^^ ^^^ ^^Y two polynomials, then their
product is the polynomial
m-{-n
pit) = Pi{t)p2{t) = E C/,
0
where Ck = Jli+j=k cabj = Jli=o ca^k-i- Now let T be any fixed element of
Hom(F), and for any polynomial q{f) let q{T) be the transformation obtained
by replacing t by T. That is, if q{t) = Lo ^/^ then qiT) = Eo CkT\ where, of
course, T^ is the composition product T o T o - - . o T with I factors. Then the
bilinearity of composition (Theorem 3.3) shows that if p{t) = Pi(0P2(0>
then p{T) = pi{T) o p2(T). In particular, any two polynomials in T commute
with each other under composition. More simply, the commutative law for
addition implies that
if p{t) = piit)+P2{t), then p{T) = p,{T)+p2{T).
The mapping p{t) ^-^ p{T) from the algebra of polynomials to the algebra
Hom(F) thus preserves addition, multiplication, and (obviously) scalar
multiplication. That is, it preserves all the operations of an algebra and is therefore
what is called an (algebra) homomorphism.
The word "homomorphism" is a general term describing a mapping 6
between two algebraic systems of the same kind such that 6 preserves the
operations of the system. Thus a homomorphism between vector spaces is
simply a linear transformation, and a homomorphism between groups is a
mapping preserving the one group operation. An accessible, but not really
typical, example of the latter is the logarithm function, which is a homomorphism
from the multiplicative group of positive real numbers to the additive group of U.
The logarithm function is actually a bijective homomorphism and is therefore
a group isomorphism.
If this were a course in algebra, we would show that the division algorithm
and the properties of the degree of a polynomial imply the following theorem.
(However, see Exercises 5.16 through 5.20.)

1.5 DIRECT SUMS 63
Theorem 5.4. If pi(^) and P2(0 ^^^ relatively prime polynomials, then there
exist polynomials ai{t) and a2{t) such that
ai{t)pi{t) + a2{t)p2{t) = 1.
By relatively prime we mean having no common factors except constants.
We shall assume this theorem and the results of the discussion preceding it in
proving our next theorem.
We say that a subspace M C F is invariant under T e Hom(F) if T[M] C M
[that is, r \ M G Hom(M)].
Theorem 5.5. Let T be any transformation in Hom F, and let q be any
polynomial. Then the null space N of q(T) is invariant under T, and if
Q. ^^ 0.10.2 is any factorization of q into relatively prime factors and A^i and
A'2 are the null space of qi{T) and q2{T), respectively, then A" = A^i 0 A'2-
Proof. Since T o q{T) = q{T) o T, we see that if q{T){a) = 0, then q{T){Ta) =
T{q{T)ia)) = 0, so ^[A^] C A". Note also that since q{T) = qi{T) o q^{T),
it follows that any a in A'2 is also in N, so A'2 C N. Similarly, A^i C A". We can
therefore replace F by A" and Thy T \ N; hence we can assume that T G Hom A"
and qiT) = q^{T) o q^{T) = 0.
Now choose polynomials ai and a2 so that aiqi + a2q2 = 1- Since p i-> p(r)
is an algebraic homomorphism, we then have
aiiT) o q,(T) + a2{T) o q2{T) = I.
Set Ai == ai{T)j etc., so that Ai oQ^ + A2 0Q2 = /, Qi ^ Q2 == 0, and all the
operators A^-, Qi commute with each other. Finally, set Pi = Ai o Q^ = Q^ o Ai
for i = 1, 2. Then Pi + P2 - 7 and P1P2 = P2P1 = 0. Thus Pi and P2 are
projections, and A" is the direct sum of their ranges: A" = Fi 0 F2. Since each
range is the null space of the other projection, we can rewrite this as A" =
A^i © ^2} where Ni = A"(Pi). It remains for us to show that A'(Pi) = N{Qi).
Note first that since Qi ^ P2 = Qi o Q2 o A2 = 0, we have Qi = Qi o I =
Qi ° {Pi + P2) = Qi ° Pi' Then the two identities Pi = Ai o Q^ and Qi =
Qi o P^ show that the null space of each of Pi and Qi is included in the other, and
so they are equal. This completes the proof of the theorem. D
Corollary. Let p(t) = TlT=i Pi(0 be a factorization of the polynomial
p{t) into relatively prime factors, let 7" be an element of Ilom(F), and set
Ni = N{pi{T)) iori= 1, . . . , m and A" - N{p{T)). Then A" and all the
Ni are invariant under T, and A" = @'iL ^ Ni.
Proof. The proof is by induction on m. The theorem is the case m = 2, and if
we set q = TV2 PiQ) and M = N{q{T))j then the theorem implies that
A" = A'l 0 M and that A'l and M are invariant under T. Restricting T to M,
we see that the inductive hypothesis implies that M = @T=2 Ni and that A^^ is
invariant under T for i = 2, . . . , m. The corollary to Lemma 5.3 then yields
our result. D

64 VECTOR SPACES 1.5
EXERCISES
5.16 Presumably the reader knows (or can see) that the degree d(P) of a polynomial
P satisfies the laws
d{P + Q) < mSix{diP),diQ)],
d(P • Q) = d{P) + d{Q) if both P and Q are nonzero.
The degree of the zero polynomial is undefined. (It would have to be — qc \) By
induction on the degree of P, prove that for any two polynomials P and D, with D 9^ 0,
there are polynomials Q and R such that P = DQ-\- R and d{R) < d{D) or R = 0.
[Hint: If d(P) < d{D), we can take Q and R as what? If d{P) > d{D), and if the
leading terms of P and D are ax*" and hx^^, respectively, with n > m, then the polynomial
has degree less than d{P), so P' = DQ' + R' by the Inductive hypothesis. Now finish
the proof.]
5.17 Assuming the above result, prove that R and Q are uniquely determined by
P and D. (Assume also that P = DQ' + R', and prove from the properties of degree
that R' = R and Q' = Q.) These two results together constitute the division algorithm
for polynomials.
5.18 If P is any polynomial
n
Pipe) = XI (^nX^,
0
and if t is any number, then of course P{t) is the number
n
0
Prove from the division algorithm that for any polynomial P and any number t there
is a polynomial Q such that
P{x) = {x-t)Q{x) + P{t),
and therefore that P{x) is divisible by {x — t) if and only if P{t) = 0.
5.19 Let P and Q be nonzero polynomials, and choose polynomials To and ^o such
that among all the polynomials of the form AP + BQ the polynomial
D = AoP+BoQ
is nonzero and has minimum degree. Prove that D is a factor of both P and Q.
(Suppose that D does not divide P and apply the division algorithm to get a contradiction
with the choice of .lo and ^o-)
5.20 Let P and Q be nonzero relatively prime polynomials. This means that if iiJ is a
common factor of P and Q {P = EP', Q = EQ'), then E' is a constant. Prove that
there are polynomials A and B such that A{x)P{x) -]- B{x)Q{x) = L (Apply the
above exercise.)

1.5 DIRECT SUMS 65
5.21 In the context of Theorem 5.5, show that the restriction of q2{T) = Q2 to Ni
is an isomorphism (from Ni to Ni).
5.22 An involution on 7 is a mapping T G Hom V such that T^ = I. Show that if
T is an involution, then F is a direct sum V = Fi © F2, where T{^) = J for every
i^Vi {T = I on Vi) and r(© = —f for every J G V2 {T = -I on V2). (Apply
Theorem 5.5.)
5.23 We noticed earlier (in an exercise) that if (p is any mapping from a set .1 to a
set B, then /i—» /o <^ is a linear map T^ from U^ to W^. Show now that if xp: B -^ C,
then
(This should turn out to he a direct consequence of the associativity of composition.)
5.24 Let .1 be any set, and let <^: .1 —> A be such that ipo (p(a) = a for every a.
Then T^i/i—»/o <^ is an involution on 7 = U^^ (since T^^rf^ = T^o T^). Show that
the decomposition of U^ as the direct sum of the subspace of even functions and the
subspace of odd functions arises from an involution on R"* defined by such a map
5.25 Let 7 be a subspace of U^ consisting of differentiable functions, and suppose
that V is invariant under differentiation (/E V =^ Df E^ V). Suppose also that on V
the linear operator D E Hom V satisfies D^ — 2D — 3/ = 0. Prove that V is the
direct sum of two subspaces M and A^ such that D = 3/ on M and D = —lonN.
Actually, it follows that M is the linear span of a single vector, and similarly for N.
Find these two functions, if you can. (/' = ^f=^f= ?)
*Block decompositions of linear maps. Given T in Hom V and a direct sum
decomposition V = 0i Vi, with corresponding projections {Pi}1, we can
consider the maps Tij = P^o T o P-, Although Tij is from V to F, we may also
want to consider it as being from Vj to Vi (in which case, strictly speaking, what
is it?). We picture the T^/s arranged schematically in a rectangular array
similar to a matrix, as indicated below for n = 2.
^11 ^12
T21 ^22
Furthermore, since T = J^ij Tij, we call the doubly indexed family the block
decomposition of T associated with the given direct sum decomposition of V.
More generally, if 7" G Hom(F, W) and W also has a direct sum
decomposition W — @T=i ^u with corresponding projections {Qij^j then the family
{Tij) defined by Tij = Qi o T o Pj and pictured as an m X n rectangular array
is the block decomposition of T with respect to the two direct sum decompositions.
Whenever T in Hom V has a special relationship to a particular direct sum
decomposition of V, the corresponding block diagram may have features that
display these special properties in a vivid way; this then helps us to understand
the nature of T better and to calculate with it more easily.

66
VECTOR SPACES
1.5
Forexample, if F=- Vi 0 F2, then Fi is invariant under T (i.e., T[Fi] C Fi)
if and only if the block diagram is upper triangulaVy as shown in the following
diagram.
^11 ^12
0
22
Suppose, next, that T^ = 0. Letting Vi be the range of T, and supposing that
Vi has a complement V2, the reader should clearly see that the corresponding
block diagram is ^
0.
0
Tu
^0.
This form is called strictly upper triangular; it is upper triangular and also zero
on the main diagonal. Conversely, if T has some strictly upper-triangular
2X2 block diagram, then T^ = 0.
If 72 is a composition product, 72 = ST, then its block components can be
computed in terms of those of S and T. Thus
72a =- PiRPic = PiSTPk
'•<,?/')
TPk = E SijT,
ijl jlc
y=i
We have used the identities / = E7=i ^j ^^^ ^j = ^1- The 2x2 case is
pictured below.
^11^11 + ^12^21
^21^11 + S22T2I
S11T12 + S12T22
S21T12 + S22T22
From this we can read off a fact that will be useful to us later: If 7" is 2 X 2
upper triangular {T21 = 0), and if Ta is invertible as a map from Vi to
Vi (i = 1, 2), then T is invertible and its inverse is
Tu
rn —1 /Trr n^ —1
"^11 ^ 12^ 22
0
We find this solution by simply setting the product diagram equal to
h
0
0
and solving; but of course with the diagram in hand it can simply be checked to
be correct.
EXERCISES
5.26 Show that if T G Horn 7, if V = ®i Vi, and if {P^] i are the corresponding
projections, then the sum of the transformation Tij = P^o To Py is T.

1.6 BILINEARITY 67
5.27 If S and T are in Horn V and [Sij], {Tij} are their block components with
respect to some direct sum decomposition of V, show that Sij o Ta = 0 if j 9^ I.
5.28 Verify that if T has an upper-triangular block diagram with respect to the
direct sum decomposition V = Vi @ V2, then Vi is invariant under T.
5.29 Verify that if the diagram is strictly upper triangular, then T^ = 0.
5.30 Show that if 7 = 7i © ^2 © F3 and T G Hom V, then the subspaces Vi are
all invariant under T if and only if the block diagram for T is
7^11
0
0
0
T22
0
0
0
^33
Show that T is invertible if and only if Ta is invertible (as an element of Hom Vi)
for each i.
5.31 Supposing that T has an upper-triangular 2X2 block diagram and that Ta
is invertible as an element of Hom Vi for i = 1,2, verify that T is invertible by
forming the 2X2 block diagram that is the product of the diagram for T and the diagram
given in the text as the inverse of T.
5.32 Supposing that T is as in the preceding exercise, show that S = T~^ must have
the given block diagram by considering the two equations T o S = I and S o T = I
in their block form.
5.33 What would strictly upper triangular mean for a 3 X 3 block diagram? What
is the corresponding property of T? Show that T has this property if and only if it has
a strictly upper-triangular block diagram. (See Exercise 5.14.)
5.34 Suppose that T in Hom V satisfies T"" = 0 (but T""^ ^ 0). Show that T has
a strictly upper-triangular nX n block decomposition. (Apply Exercise 5.15.)
6. BILINEARITY
Bilinear mappings. The notion of a bilinear mapping is important to the
understanding of linear algebra because it is the vector setting for the duality
principle (Section 0.10).
Definition. If f7, F, and W are vector spaces, then a mapping
co: < J, 77> '->co(J, 7])
from f7 X F to TF is bilinear if it is linear in each variable lohen the other
variable is held fixed.
That is, if we hold J fixed, then 17 •—> co(J, rj) is linear [and so belongs to
Hom(F, TF)]; if we hold rj fixed, then similarly co(J, 77) is in Hom(l7, TF) as a
function of J. This is not the same notion as linearity on the product vector
space U X V. For example, <x,y> —^x + y is a linear mapping from U^
to R, but it is not bilinear. If y is held fixed, then the mapping x ^-^ x + y is
affine (translation through y), but it is not linear unless y is 0. On the other
hand, <x, y> 1—> 0:7/ is a bilinear mapping from R^ to R, but it is not linear. If y

68 VECTOR SPACES 1.6
is held fixed, then the mapping x ^-^ yx is linear. But the sum of two ordered
couples does not map to the sum of their images:
<x^y> + <u^v> = <x -]- u^y -]- v> i-»(a: + u){y + y),
which is not the sum of the images, xy + uv. Similarly, the scalar product
(x, y) = 2Zi xiyi is bilinear from U^ X W to R, as we observed in Section 2.
The linear meaning of bilinearity is partially explained in the following
theorem.
Theorem 6.1. If co: 17 X F —> TF is bilinear, then, by duality, co is
equivalent to a linear mapping from U to Hom(F, W) and also to a linear mapping
from F to Hom(l7, TF).
Proof. For each fixed 77 G F let co^ be the mapping J •-> co(J, 77). That is,
^rj(^) = w(^j f])' Then co^ G Hom(l7, TF) by the bilinear hypothesis. The
mapping 77 1—> co^ is thus from F to Hom(f/, TF), and its linearity is due to the
linearity of co in 77 when J is held fixed:
COcr,+rfr(^) = ^(^5 ^^7 + CU) = CC0(J, 77) + CZC0(J, f) = CCO^(^) + cZcOf(^),
so that , ,
Similarly, if we define co^ by co^(rj) = co( J, 77), then J 1—» co^ is a linear mapping
from U to Hom(F, TF). Conversely, if cp: U —^ Hom(F, TF) is linear, then the
function co defined by co(J, 77) = (p(0(v) is bilinear. Moreover, co^ = (p(Oj so
that (f is the mapping J 1—» co^. D
We shall see that bilinearity occurs frequently. Sometimes the reinterpreta-
tion provided by the above theorem provides new insights; at other times it
seems less helpful.
For example, the composition map <S,T>i-^SoTis bilinear, and the
corollary of Theorem 3.3, which in effect states that composition on the right by
a fixed 7" is a linear map, is simply part of an explicit statement of the bilinearity.
But the linear map T 1—» composition by 7" is a complicated object that we have
no need for except in the case W = U.
On the other hand, the linear combination formula XI1 ^i^i and Theorem 1.2
do receive new illumination.
Theorem 6.2. The mapping co(x, a) = Xli ^i^^i is bilinear from W X V
to F. The mapping a 1—» coa is therefore a linear mapping from F^ to
Hom([R^, F), and, in fact, is an isomorphism.
Proof. The linearity of co in x for a fixed a was proved in Theorem 1.2, and its
linearity in a for a fixed x is seen in the same way. Then a 1—» coa is linear by
Theorem 6.1. Its bijectivity was implicit in Theorem 1.2. D
It should be remarked that we can use any finite index set I just as well as
the special set n and conclude that co(x, a.) = Yli^i ^i^i is bilinear from U^ X F^

1.6 BILINEARITY 69
to V and that a i-» coa is an isomorphism from V^ to Hom([R^, V). Also note
that coa = La in the terminology of Section 1.
Corollary. The scalar product (x, a) = X!i bilinear from R^ x R^
to R; therefore, a i-» co^ = L^ is an isomorphism from U^ to Hom([R^, U).
Natural isomorphisms. We often find two vector spaces related to each other
in such a way that a particular isomorphism between them is singled out. This
phenomenon is hard to pin down in general terms but easy to describe by
examples.
Duality is one source of such "natural" isomorphisms. For example, an
m X n matrix {tij} is a real-valued function of the two variables <i,j>j and
as such it is an element of the Cartesian space R^^^. We can also view {tij} as
a sequence of n column vectors in U^. This is the dual point of view where we
hold j fixed and obtain a function of i for each j. From this point of view {tij}
is an element of (W^)^. This correspondence between R^x^ and (R"^)^ is clearly
an isomorphism, and is an example of a natural isomorphism.
We review next the various ways of looking at Cartesian n-space itself.
One standard way of defining an ordered n-tuplet is by induction. The ordered
triplet <x, y, z> is defined as the ordered pair < <x, y>, z> , and the ordered
n-tuplet <xi, . . . , Xn> is defined as < <xij . . . , Xn-i>, Xn>- Thus we
define R"" inductively by setting R^ = R and R'' = R''"^ X R.
The ordered n-tuplet can also be defined as the function onn = {1, . . . , n}
which assigns Xi to i. Then
and Cartesian n-space is R^' = R^^'--"^L
Finally, we often wish to view Cartesian (n + m)-space as the Cartesian
product of Cartesian n-space with Cartesian m-space, so we now take
and R^+^ as R^ X R^.
Here again if we pair two different models for the same n-tuplet, we have an
obvious natural isomorphism between the corresponding models for Cartesian
n-space.
Finally, the characteristic properties of Cartesian product spaces given in
Theorems 3.6 and 3.7 yield natural isomorphisms. Theorem 3.6 says that an
71-tuple of linear maps {Ti}^ on a common domain V is equivalent to a single
/i-tuple-valued map T, where T{^) = <7^i(J), . . . , Tn{0> for all ^gV.
(This is duality again! Ti(^) is a function of the two variables i and J.) And it is
not hard to see that this identification of T with {Ti}^ is an isomorphism from
Ui Hom(F, Wi) to Hom(F, Ui Wi).
Similarly, Theorem 3.7 identifies an n-tuple of linear maps {Ti}'l into a
common codomain V with a single linear map T of an n-tuple variable, and this
identification is a natural isomorphism from Hi Hom(TFj, V) to Hom(n^ Wi, F).

70 VECTOR SPACES 1.6
An arbitrary isomorphism between two vector spaces identifies them in a
transient way. For the moment we think of the vector spaces as representing the
same abstract space, but only so long as the isomorphism is before us. If we
shift to a different isomorphism between them, we obtain a new temporary
identification. Natural isomorphisms, on the other hand, effect permanent
identifications, and we think of paired objects as being two aspects of the same
object in a deeper sense. Thus we think of a matrix as "being" either a sequence
of row vectors, a sequence of column vectors, or a single function of two integer
indices. We shall take a final look at this question at the end of Section 3 in the
next chapter.
*We can now make the ultimate dissection of the theorems centering
around the linear combination formula. Laws Si through S3 state exactly that
the scalar product xa is bilinear. More precisely, they state that the mapping
S: <x^ a> ^-^ xa from R X TF to TF is bilinear. In the language of Theorem 6.1,
xa = coai^): and from that theorem we conclude that the mapping c^: i—» co^ is
an isomorphism from W to Hom([R, W).
This isomorphism between W and Hom([R, W) extends to an
isomorphism from W^ to (Hom([R, TF))^, which in turn is naturally isomorphic to
Hom([R^,TF) by the second Cartesian product isomorphism. Thus W^ is
naturally isomorphic to Hom([R^, TF); the mapping is a i—> La, where La(x) =
Z1 Xiai.
In particular, U^ is naturally isomorphic to the space Hom([R^, U) of all
linear functionals on U^, the n-tuple a corresponding to the functional co.^
defined by cOa(x) = X!i (^i^i-
Also, (W^y is naturally isomorphic to Hom([R'', Wy ^nd since 1R^><^ is
naturally isomorphic to (W^y, it follows that the spaces R^^^ and Hom([R^, R^)
are naturally isomorphic. This is simply our natural association of a
transformation T in Hom([R^, U^) to an m X n matrix {tij}.

CHAPTER 2
FINITE-DIMENSIONAL VECTOR SPACES
We have defined a vector space to be finite-dimensional if it has a finite spanning
set. In this chapter we shall focus our attention on such spaces, although this
restriction is unnecessary for some of our discussion. We shall see that we can
assign to each finite-dimensional space V a unique integer, called the dimension
of F, which satisfies our intuitive requirements about dimensionality and which
becomes a principal tool in the deeper explorations into the nature of such
spaces. A number of "dimensional identities" are crucial in these further
investigations. We shall find that the dual space of all linear functional on V,
V* = Hom(F, R), plays a more satisfactory role in finite-dimensional theory
than in the context of general vector spaces. (However, we shall see later in
the book that when we add limit theory to our algebra, there are certain special
infinite-dimensional vector spaces for which the dual space plays an equally
important role.) A finite-dimensional space can be characterized as a vector
space isomorphic to some Cartesian space U^, and such an isomorphism allows a
transformation T in Hom V to be "transferred" to R^, whereupon it acquires a
matrix. The theory of linear transformations on such spaces is therefore mirrored
completely by the theory of matrices. In this chapter we shall push much
deeper into the nature of this relationship than we did in Chapter 1. We also
include a section on matrix computations, a brief section describing the trace
and determinant functions, and a short discussion of the diagonalization of a
([uadratic form.
I. BASES
(yonsider again a fixed finite indexed set of vectors a = {ai: i G 1} inV and the
corresponding linear combination map La: x i—» JZ ^ic^i from U^ to V having a
lis skeleton.
Definition. The finite indexed set {ai: i G 1} is independent if the above
mapping La is infective, and {ai} is a basis for V if La is an isomorphism
(onto F). In this situation we call {ai: i G 1} an ordered basis or frame if
I = n = {Ij . , , J n} ior some positive integer n.
Thus {ai :i G 1} is a basis if and only if for each ^ gV there exists a unique
indexed "coefficient" set x = {xi: i G 1} G U^ such that J = Yi Xiai. The
71

72 FINITE-DIMENSIONAL VECTOR SPACES 2.1
numbers Xi always exist because {ai: i G /} spans F, and x is unique because
La is injective.
For example, we can check directly that b^ = < 2, 1 > and b^= <1, —3>
form a basis for R^. The problem is to show that for each y G IR^ there is a
unique x such that
2
J =Y1, ^*b* = ''^'i"^^, 1> + 0:2<1, —3> = <2a:i + X2, xi — 3x'2>.
1
Since this vector equation is equivalent to the two scalar equations 2/1 =
2x\ + X2 and 2/2 = ^1 ~ ^0:25 we can find the unique solution x\ = (3yi + 2/2)/^,
^2 = (2/1 "~ 22/2)/7 by the usual elimination method of secondary school
algebra.
The form of these definitions is dictated by our interpretation of the linear
combination formula as a linear mapping. The more usual definition of
independence is a corollary.
Lemma 1.1. The independence of the finite indexed set {ai : i G 1} is
equivalent to the property that 2]/ Xiai = 0 only if all the coefficients Xi
are 0.
Proof. This is the property that the null space of La consist only of 0, and is
thus equivalent to the injectivity of La, that is, to the independence of {ai}, by
Lemma LI of Chapter L D
If {aiji is an ordered basis (frame) for F, the unique n-tuple x such that
J = XI1 ^io^i is called the coordinate n-tuple of J (with respect to the basis {ai}),
and Xi is the ith coordinate of J. We call Xiai (and sometimes Xi) the ith component
of J. The mapping La will be called a basis isomorphism, and its inverse La^,
which assigns to each vector J G F its unique coordinate n-tuple x, is a coordinate
isomorphism. The linear functional J 1—» Xj is the yth coordinate functional;
it is the composition of the coordinate isomorphism J 1—» x with the yth
coordinate projection x 1—» Xj on U^. We shall see in Section 3 that the n coordinate
functionals form a basis for F* = Hom(F, U).
In the above paragraph we took the index set / to be n = {1, . . , , n} and
used the language of n-tuples. The only difference for an arbitrary finite index
set is that we speak of a coordinate function x = {xi: i G /} instead of a
coordinate n-tuple.
Our first concern will be to show that every finite-dimensional (finitely
spanned) vector space has a basis. We start with some remarks about indices.
We note first that a finite indexed set {ai: i G 1} can be independent only
if the indexing is injective as a mapping into F, for if ak = ai, then 2Z ^io^i = 0,
where o^A: = 1, ^z = —1, and Xi = 0 for the remaining indices. Also, if {ai : i G /}
is independent and J C I, then {ai :i G J} is independent, since if Zl j Xiai = 0,
and if we set Xi =0 for z G / — J, then 2Z/ ^i<^i = 0? and so each Xj is 0.
A finite unindexed set is said to be independent if it is independent in some

2.1 BASES 73
(necessarily bijective) indexing. It will of course then be independent with
respect to any bijective indexing. An arbitrary set is independent if every finite
subset is independent. It follows that a set A is dependent (not independent) if
and only if there exist distinct elements ai, . . . , q:,^ in A and scalars Xi^ . . . ^ Xn
not all zero such that Yl\ ^i(^i = 0. An unindexed basis would be defined in the
obvious way. However, a set can always be regarded as being indexed, by itself
if necessary!
Lemma 1.2. If B is an independent subset of a vector space V and 13 is any
vector not in the linear span L(B), then B U [13] is independent.
Proof. Otherwise tliere is a zero linear combination, xjS + X^i •^'//^z = 0, where
^1, . . . , p,i are distinct elements of B and tlie coefficients are not all 0. But then
X cannot be zero; if it were, the equation would contradict the independence of
B. We can therefore divide by x and solve for 13, so that /3 G L(B), a
contradiction. D
The reader will remember that we call a vector space V finite-dimensional
if it has a finite spanning set [ai] \. We can use the above lemma to construct a
basis for such a F by choosing some of the aiS. We simply run through the
sequence [ai] i and choose those members that increase the linear span of the
preceding choices. We end up with a spanning set since [ai] \ spans, and our
subsequence is independent at each step, by the lemma. In the same way we
can extend an independent set {/3/} i to a basis by choosing some members of a
spanning set {apy^l. This procedure is intuitive, but it is messy to set up
rigorously. We shall therefore proceed differently.
Theorem 1.1. Any minimal finite spanning set is a basis, and therefore any
finite-dimensional vector space F has a basis. More generally, if {0j : j G J}
is a finite independent set and [ai : i G 1} is a finite spanning set, and if K
is a smallest subset of / such that {0j} j U {ai] k spans, then this collection is
independent and a basis. Therefore, any finite independent subset of a
finite-dimensional space can be extended to a basis.
Proof. It is sufficient to prove the second assertion, since it includes the first
as a special case. If {fij} j U {ai}K is not independent, then there is a nontrivial
zero linear combination X!/ Vj^j + 2Za' Xiai = 0. If every Xi were zero, this
(equation would contradict the independence of {fij] j. Therefore, some Xk is
not zero, and we can solve the equation for ak. That is, if we set L = K — {A;},
I hen the linear span of {0j} j U {a^} l contains ak. It therefore includes the whole
original spanning set and hence is F. But this contradicts the minimal nature of
/v, since L is a proper subset of K. Consequently, {0j} j U {ai} k is independent. D
We next note that IR^ itself has a very special basis. In the indexing map
/ H-> ai the vector aj corresponds to the index j, but under the linear
combination map X t-^ 2Z ^i<^i the vector aj corresponds to the function b-' which has
(he value 1 at j and the value 0 elsewhere, so that Yli K(^i = «y- This function

74 FINITE-DIMENSIONAL VECTOR SPACES 2.1
8^' is called a Kronecker delta function. It is clearly the characteristic function Xb
of the one-point set B = {j}, and the symbol '5-^' is ambiguous, just as 'X^'
is ambiguous; in each case the meaning depends on what domain is implicit from
the context. We have already used the delta functions on R^ in proving Theorem
1.2 of Chapter 1.
Theorem 1.2. The Kronecker functions {d^}J=i form a basis for W\
Proof. Since Yli ^i^XJ) = ^j by the definition of 8\ we see that 2Zi ^i^* is
the n-tuple x itself, so the linear combination mapping Lg: x ^^ 2Zi ^i^* is the
identity mapping x ^^ x, a trivial isomorphism. D
Among all possible indexed bases for R^, the Kronecker basis is thus singled
out by the fact that its basis isomorphism is the identity; for this reason it is
called the standard basis or the natural basis for U^. The same holds for U^ for
any finite set /.
Finally, we shall draw some elementary conclusions from the existence of
a basis.
Theorem 1.3. If 7" G Hom(F, W) is an isomorphism and a = {ai : i G 1}
is a basis for F, then {T(ai) : i G 1} is a basis for W.
Proof. By hypothesis La is an isomorphism in Hom(IR'^, F), and so T o La is
an isomorphism in Hom([R^, W). Its skeleton {T(ai)} is therefore a basis for W. D
We can view any basis {ai} as the image of the standard basis {5*} under
the basis isomorphism. Conversely, any isomorphism d:U^ —> B becomes a basis
isomorphism for the basis ay = 6(8^).
Theorem 1.4. If X and Y are complementary subspaces of a vector space F,
then the union of a basis for X and a basis for F is a basis for F. Conversely,
if a basis for F is partitioned into two sets, with linear spans X and F,
respectively, then X and F are complementary subspaces of F.
Proof. We prove only the first statement. If {ai: i G J} is a basis for X and
{ai: i G K} is a basis for F, then it is clear that {ai : i G J U K} spans F, since
its span includes both X and F, and so X + F = F. Suppose then that
Hjuk Xiai = 0. Setting J = Xl/ ^i<^i and t) = ^k Xiai, we see that ^ G X,
77 G F, and J + 77 = 0. But then f = 77 = 0, since X and F are complementary.
And then Xi = 0 for i G J because {ai} j is independent, and Xi = 0 for i G K
because {ai}K is independent. Therefore, {ai} j\jk is a basis for F. We leave the
converse argument as an exercise. D
Corollary. If F = 0i Vi and Bi is a basis for Vi, then B = IJi ^z is a
basis for F.
Proof. We see from the theorem that Bi u ^2 is a basis for Fi 0 ¥2-
Proceeding inductively we see that Ui=i Bi is a basis for 0^=1 Vi for j = 2, . . . , n,
and the corollary is the case j = n. D

2.1 BASES 75
If we follow a coordinate isomorphism by a linear combination map, we get
the mapping of the following existence theorem, which we state only in n-tuple
form.
Theorem 1.5. If /3 = {0i}\ is an ordered basis for the vector space F, and
if {ai}\ is any n-tuple of vectors in a vector space W, then there exists a
unique S E Hom(F, W) such that S(Pi) = ai for i = 1, ... , n.
Proof. By hypothesis Lp is an isomorphism in Hom(IR'^, F), and so
S = La^ (^p)~^ is an element of Hom(F, W) such that S(Pi) = La(5') = ai.
Conversely, if S G Hom(F, W) is such that S{0i) = ai for all z, then S o Lp(5*) =
ai for all z, so that aS o Lp = La. Thus S is uniquely determined as La o (Lp)~\ D
It is natural to ask how the unique S above varies with the 7?-tuple {ai}.
The answer is: linearly and ^^isomorphically".
Theorem 1.6. Let {Pi}\ be a fixed ordered basis for the vector space
F, and for each 7i-tuple a = {aji chosen from the vector space W let
Sa E Hom(F, W) be the unique transformation defined above. Then the
map a ^-^ /Sa is an isomorphism from W^ to Hom(F, W).
Proof. As above, Sa = La o d~^, where 6 is the basis isomorphism Lp. Now we
know from Theorem 6.2 of Chapter 1 that a ^-^ La is an isomorphism from W^
to Hom([R^, W), and composition on the right by the fixed coordinate
isomorphism d~^ is an isomorphism from Hom([R^, W) to Hom(F, W) by the corollary
to Theorem 3.3 of Chapter 1. Composing these two isomorphisms gives us the
theorem. D
*Infinite bases. Most vector spaces do not have finite bases, and it is natural to
try to extend the above discussion to index sets I that may be infinite. The
Kronecker functions {5*: z E /} have the same definitions, but they no longer
span U^. By definition / is a linear combination of the functions 5* if and only
if/ is of the form Ylieii Ci8\ where 7i is a finite subset of I. But then / = 0
outside of 7i. Conversely, if / E R^ is 0 except on a finite set /i, then / =
12ieiif(i)8^' The linear span of {5* : z E /} is thus exactly the set of all
functions of R^ that are zero except on a finite set. We shall designate this sub-
space Ui.
If {ai: i G 1} is an indexed set of vectors in F and / E R/, then the sum
Y^i^i f{i)oLi becomes meaningful if we adopt the reasonable convention that the
sum of an arbitrary number of O's is 0. Then XiiG/ = Szg/oj where Iq is any
finite subset of I outside of which / is zero.
With this convention, La'.f^-^ Y^if{i)<^i is a linear map from R/ to F, as in
Theorem 1.2 of Chapter 1. And with the same convention, Y^i^i f{i)ai is an
elegant expression for the general linear combination of the vectors ai. Instead
of choosing a finite subset 7i and numbers Ci for just those indices z in 7i, we
define Ci for all i E 7, but with the stipulation that Cj = 0 for all but a finite
number of indices. That is, we take c = {ci: i G 1} as a function in Uj.

76 FINITE-DIMENSIONAL VECTOR SPACES 2.1
We make the same definitions of independence and basis as before. Then
{ai : i G 1} is a basis for F if and only if La: H^/ —> F is an isomorphism, i.e., if
and only if for each ^ gV there exists a unique x G R/ such that ^ = 2Zz Xiai.
By using an axiom of set theory called the axiom of choice, it can be shown
that every vector space has a basis in this sense and that any independent set
can be extended to a basis. Then Theorems 1.4 and 1.5 hold with only minor
changes in notation. In particular, if a basis for a subspace M of F is extended
to a basis for F, then the linear span of the added part is a subspace A^
complementary to M. Thus, in a purely algebraic sense, every subspace has
complementary subspaces. We assume this fact in some of our exercises.
The above sums are always finite (despite appearances), and the above
notion of basis is purely algebraic. However, infinite bases in this sense are not
very useful in analysis, and we shall therefore concentrate for the present on
spaces that have finite bases (i.e., are finite-dimensional). Then in one
important context later on we shall discuss infinite bases where the sums are genuinely
infinite by virtue of limit theory.
EXERCISES
1.1 Show by a direct computation that {-<1, —l^, <0, 1>} is a basis for U^.
1.2 The student must realize that the ith coordinate of a vector depends on the whole
basis and not just on the ith basis vector. Prove this for the second coordinate of
vectors in U^ using the standard basis and the basis of the above exercise.
1.3 Show that {-<1, 1>, -<1, 2>} is a basis for V = U^. The basis isomorphism
from U^ to V is now from U^ to U^. Find its matrix. Find the matrix of the coordinate
isomorphism. Compute the coordinates, with respect to this basis, of -< —1, 1 >, -< 0, 1 >,
<2, 3>.
1.4 Show that {h% where b^ = <1, 0, 0>, b^ = <1, 1, 0>, and b^ =
<1, 1, 1>, is a basis for U^.
1.5 In the above exercise find the three linear functional k that are the coordinate
functional with respect to the given basis. Since
3
1
finding the h is equivalent to solving x = 2Zi Vi^^ for the yi's in terms of
X = <Xl, X2, X3>.
1.6 Show that any set of polynomials no two of which have the same degree is
independent.
1.7 Show that if {ai}^ is an independent subset of V and T in Hom(F, W) is injec-
tive, then {T{ai)]^ is an independent subset of W.
1.8 Show that if T is any element of Hom(F, W) and {T(ai)] ^ is independent in TF,
then [ai] ^ is independent in V.

2.2 DIMENSION 77
1.9 Later on we are going to call a vector space V n-dimensional if ever}^ basis for
V contains exactly n elements. If T^ is the span of a single vector a, so that T^ = Ua,
then V is elearl}^ one-dimensional.
Let {Vi]^ be a collection of one-dimensional subspaces of a vector space V, and
choose a nonzero vector ai in ]\ for each i. Prove that [ai]\ is independent if and
only if the subspaces {V,]\ are inde])endent and that [oii]^ is a basis if and only if
V = 0"; Vi.
1.10 Finish the proof of Theorem 1.4.
1.11 Give a proof of Theorem 1.4 based on the existence of isomorphisms.
1.12 The reader would guess, and we shall prove in the next section, that every
subspace of a finite-dimensional s})ace is finite-dimensional. Prove now that a sub-
space A^ of a finite-dimensional vector space T is finite-dimensional if and only if it has
a complement M. (Work from a combination of Theorems LI and 1.4 and direct sum
})rojections.)
L13 Since {h^]\ = {<1,0, 0>, <1, ],0>, <\, 1, 1 >] is a basis for U^, there is a
unique T in Hom([R^ R^) ^^^.^ that T{h^) = <1,0>, ^(b^) = <0, 1>, and
T(h^) = < 1, 1 >. Find the matrix of T. (Find 7^(50 for i = 1,2, 3.)
1.14 Find, similarly, the S in Hom U'^ such that S(h') = 8' for i = 1, 2, 3.
1.15 Show that the infinite sequence [r'] q i« a basis for the vector space of all
polynomials.
2. DIMENSION
The concept of dimension rests on the fact that two different bases for the same
space always contain the same number of elements. This number, which is
then the number of elements in every basis for F, is called the dimension of F.
It tells all there is to know about F to within isomorphism: There exists an
isomorphism between two spaces if and only if they have the same dimension.
We shall consider only finite dimensions. If F is not finite-dimensional, its
dimension is an infinite cardinal number, a concept with which the reader is
probably unfamiliar.
Lemma 2.1. If F is finite-dimensional and T in Hom V is surjective, then T
is an isomorphism.
Proof, Let n be the smallest number of elements that can span F. That is, there
is some spanning set {oLi}\ and none with fewer than n elements. Then {ai]1 is a
basis, by Theorem LI, and the linear combination map ^: x i—> Yl\ ^i<^i is
accordingly a basis isomorphism. But {fii]\ = {T(ai)}1 also spans, since T is
surjective, and so 7" o ^ is also a basis isomorphism, for the same reason. Then
T = (T o e) o d~^ is an isomorphism. D
Theorem 2.1. If F is finite-dimensional, then all bases for V contain the
same number of elements.
Proof. Two bases with ii and m elements determine basis isomorphisms
6: [R^ -> F and <p: W -> F. Suppose that m < n and, viewing R^ as R^ X R^"^,

78 FINITE-DIMENSIONAL VECTOR SPACES 2.2
let TT be the projection of R"" onto W,
TT^^ "S ^Ij . . . J X^n: ' ' ' ■) Xn ^ ) ^ "S ^ij . . . j Xm ^ •
Since T = d~^ o ,p is an isomorphism from W to W and T o tt: W -> W is
therefore surjective, it follows from the lemma that 7" © tt is an isomorphism.
Then tt = T~^ o (7" o tt) is an isomorphism. But it isn't, because 7r(5'^) = 0,
and we have a contradiction. Therefore no basis can be smaller than any other
basis, n
The integer that is the number of elements in every basis for F is of course
called the dimensioii of F, and we designate it (1{V). Since the standard basis
[b^] ^ for R'^ has n elements, we see that W is n-dimensional in this precise sense.
Corollary. Two finite-dimensional vector spaces are isomorphic if and only
if they have the same dimension.
Proof. If T is an isomorphism from F to TF and 5 is a basis for F, then T[B] is a
basis for W by Theorem 1.3. Therefore d{V) = #B = #T[B] = d(W), where
#A is the number of elements in A. Conversely, if d(V) = d(W) = n, then F
and W are each isomorphic to R'^ and so to each other. D
Theorem 2.2. Every subspace M of a finite-dimensional vector space F is
finite-dimensional.
Proof. Let d be the family of finite independent subsets of M. By Theorem 1.1,
if A G d, then A can be extended to a basis for F, and so #A < d(V). Thus
{#A : A G a} is a finite set of integers, and we can choose B G d such that
71 = jj^B is the maximum of this finite set. But then L{B) = M, because
otherwise for any a G M ~ L{B) we have B \J {a} G Ct, by Lemma 1.2, and
#(5uM) = n+l,
contradicting the maximal nature of n. Thus M is finitely spanned. D
Corollary. Every subspace M of a finite-dimensional space F has a
complement.
Proof. Use Theorem 1.1 to extend a basis for M to a basis for F, and let N be
the linear span of the added vectors. Then apply Theorem 1.4. D
Dimensional identities. We now prove two basic dimensional identities.
We will always assume F finite-dimensional.
Lemma 2.2. If Fi and F2 are complementary subspaces of F, then d{V) =
d(Vi) + d(V2)^ More generally, if F = ©^ Vi then d(V) = Ei d(Vi).
Proof. This follows at once from Theorem 1.4 and its corollary. D
Theorem 2.3. If U and W are subspaces of a finite-dimensional vector space,
then d(U +W) + d(U nW) = d(U) + d(W).

2.2 DIMENSION 79
Proof, Let F be a complement of 17 n TF in 17. We start by showing that then F
is also a complement of TF in 17 + TF. First
F + TF = F + ((17 n TF) + TF) = (F + (17 n TF)) + TF = 17 + TF.
We have used the obvious fact that the sum of a vector space and a subspace
is the vector space. Next,
V nw = {V nu)nw =v n{u nW)= {o},
because F is a complement of 17 fi TF in C/. We thus have both F + TF =
17 + TF and F fi TF = {0}, and so F is a complement of TF in 17 + TF by
the corollary of Lemma 5.2 of Chapter L
The theorem is now a corollary of the above lemma. We have
d{U) + d{W) = {d{U n TF) + d{V)) + d{W) = d{U n W) +{d{V) + d{W))
= d{U n TF) + d{U + TF). D
Theorem 2.4. Let F be finite-dimensional, and let TF be any vector space.
Let T G Hom(F, TF) have null space N (in F) and range R (in TF). Then R
is finite-dimensional and d{V) = d{N) + d(R).
Proof. Let 17 be a complement of N in F. Then we know that 7" f 17 is an
isomorphism onto R, (See Theorem 5.3 of Chapter L) Therefore, R is finite-
dimensional and d{R) + d{N) = d{U) + d{N) = d{V) by our first identity. D
Corollary. If TF is finite-dimensional and d(W) = ^(F), then T is injective
if and only if it is surjective, so that in this case injectivity, surjectivity, and
bijectivity are all equivalent.
Proof, T is surjective if and only if 72 = TF. But this is equivalent to d{R) =
d(W), and if d(W) = d(V), then.the theorem shows this is turn to be equivalent
to d(N) = 0, that is, to N = {0}. D
Theorem 2.5. If d(V) = n and d{W) = m, then Hom(F, TF) is finite-
dimensional and its dimension is mn.
Proof, By Theorem L6, Hom(F, TF) is isomorphic to TF^ which is the direct
sum of the n subspaces isomorphic to TF under the injections ^^ for z = 1, . . . , n.
The dimension of TF"^ is therefore 2Zi ^ = ^^ by Lemma 2.2. D
Another proof of Theorem 2.5 will be available in Section 4.
EXERCISES
2.1 Prove that if d(V) = n, then any spanning subset of n elements is a basis.
2.2 Prove that if d(V) = n, then any independent subset of n elements is a basis.
2.3 Show that if d(V) = n and TF is a subspace of the same dimension, then W = V,

80 FINITE-DIMENSIONAL VECTOR SPACES 2.2
2.4 Prove by using dimensional identities that if / is a nonzero linear functional on
an n-dimensional space V, then its null space has dimension n — 1.
2.5 Prove by using dimensional identities that if / is a linear functional on a finite-
dimensional space V, and if a is a vector not in its null space N, then V = N ® Ua.
2.6 Given that A^ is an (n — l)-dimensional subspace of an n-dimensional vector
space V, show that A^ is the null space of a linear functional.
2.7 Let A' and Y be subspaces of a finite-dimensional vector s})ace V, and suppose
that T in Hom(7, W) has null space N = Xn Y. Show that T[X + Y] = T[X] ©
T(Y), and then deduce Theorem 2.3 from Lemma 2.2 and Theorem 2.4. This proof
still depends on the existence of a T having N = X f) Y as its null space. Do we know
of any such T?
2.8 Show that if T^ is finite-dimensional and S, T E Horn V, then
So T = I =^ T is invertible.
Show also that T o S = I =^ T is invertible.
2.9 A subspace A of a vector space V has finite codimension n if the quotient space
V/N is finite-dimensional, with dimension n. Show that a subspace A has finite
codimension n if and only if A has a complementary subspace M of dimension n.
(Move a basis for V/N back into V.) Bo not assume V to be finite-dimensional.
2.10 Show that if Ai and A2 are subspaces of a vector space V with finite codimen-
sions, then A = Ai H A2 has finite codimension and
cod(A) < cod(Ai) + cod(A2).
(Consider the mapping Ji—> < ^1, ^2^^ when ^^ is the coset of A^ containing J.)
2.11 In the above exercise, suppose that cod(Ai) = cod(A2), that is, d{V/Ni) =
d(V/N2). Prove that d(Ai/A) = d{N2/N).
2.12 Given nonzero vectors /3 in 7 and / in 7* such that/O) 9^ 0, show that some
scalar multiple of the mapping ?'-^/(J)/3 is a projection. Prove that any projection
having a one-dimensional range arises in this way.
2.13 We know that the choice of an origin 0 in Euclidean 3-space E^ induces a
vector space structure in E^ (under the correspondence Xt-^ OX) and that this vector
space is three-dimensional. Show that a geometric plane through 0 becomes a two-
dimensional subspace.
2.14 An m-dimensional plane M is a translate A + ao of an m-dimensional subspace N.
Let [^i] f be any basis of A, and set ai = ^i + ao. Show that M is exactly the set of
linear combinations
m m
/ , Xiai such that zL ^i — 1-
1=0 0
2.15 Show that Exercise 2.14 is a corollary of Exercise 4.14 of Chapter 1.
2.16 Show, conversely, that if a plane M is the affine span of m + 1 elements, then
its dimension is < m.
2.17 From the above two exercises concoct a direct definition of the dimension of an
affine subspace.

2.3 THE DUAL SPACE 81
2.18 Write a small essay suggested by the following definition. An {m + l)-tuple
{ai}^ is affinely independent if the conditions
together imply that
y^. Xiai = 0 and 7 . o^i = 0
0 0
Xi = 0 for all I.
2.19 A polynomial on a vector space 7 is a real-valued function on V which can be
represented as a finite sum of finite products of linear functional. Define the degree
of a polynomial; define a homogeneous polynomial of degree k. Show that the set of
homogeneous polynomials of degree A: is a vector space Xk.
2.20 Continuing the above exercise, show that if A:i < A:2 < • • • < ^n, then the
vector spaces {Xk^^ are independent subspaces of the vector space of all polynomials.
[Assume that a polynomial p{t) of a real variable can be the zero function only if all
its coefficients are 0. For any polynomial P on F consider the polynomials Pa{t) =
Pita).]
2.21 Let -Ka, fi> be a basis for the two-dimensional space 7, and let <\, fi> be the
corresponding coordinate projections (dual basis in V*), Show that every polynomial
on V "is a polynomial in the two variables X and /x".
2.22 Let -<q:, ^> be a basis for a two-dimensional vector space 7, and let -<X,/x>
be the corresponding coordinate projections (dual basis for V*), Show that
<X2,X/x,/x'>
is a basis for the vector space of homogeneous polynomials on V of degree 2. Similarly,
compute the dimension of the space of homogeneous polynomials of degree 3 on a
two-dimensional vector space.
2.23 Let V and W be two-dimensional vector spaces, and let F be a mapping from
V to W, Using coordinate systems, define the notion of F being quadratic and then
show that it is independent of coordinate systems. Generalize the above exercise to
higher dimensions and also to higher degrees.
2.24 Now lei F: V -^ W he Si mapping between two-dimensional spaces such that
for any u, v E F and any I E W*, l(F(tu + v)) is a quadratic function of t, that is, of
the form at^ + 6< + c. Show that F is quadratic according to your definition in the
above exercises.
3. THE DUAL SPACE
Although throughout this section all spaces will be assumed finite-dimensional,
many of the definitions and properties are valid for infinite-dimensional spaces
as well. But for such spaces there is a difference between pTirely algebraic
situations and situations in which algebra is mixed with hypotheses of continuity.
One of the blessings of finite dimensionality is the absence of this complication.
As the reader has probably surmised from the number of special linear functionals
we have met, particularly the coordinate functionals, the space Hom(F, U)
of all linear functionals on V plays a special role.

82 FINITE-DIMENSIONAL VECTOR SPACES 2.3
Definition. The dual space (or conjugate space) F* of the vector space V is
the vector space Hom(F, R) of all linear mappings from V to R. Its elements
are called linear functionals.
We are going to see that in a certain sense V is in turn the dual space of
V* (V and (F*)* are naturally isomorphic), so that the two spaces are
symmetrically related. We shall briefly study the notion of aiiiiihilation
(orthogonality) which has its origins in this setting, and then see that there is a natural
isomorphism between Hom(F, W) and Hom(TF*, F*). This gives the
mathematician a new tool to use in studying a linear transformation T in Hom(F, W)]
the relationship between T and its image T* exposes new properties of T itself.
Dual bases. At the outset one naturally wonders how big a space F* is, and we
settle the question immediately.
Theorem 3.1. Let {Pi}\ be an ordered basis for F, and let 8y be the
corresponding jth coordinate functional on F: 8y(J) = Xj, where J = Z!i ^t/^i-
Then {8y} i is an ordered basis for F*.
Proof. Let us first make the proof by a direct elementary calculation.
a) Independence. Suppose that Yl\ Cj8>j = 0, that is, J^i Cj&j(^) = 0 for
all ^ gV. Taking J = Pi and remembering that the coordinate n-tuple of /3^•
is d\ we see that the above equation reduces to Ci = 0, and this for all i.
Therefore, {Sy} 1 is independent.
b) Spanning. First note that the basis expansion ^ = Yl ^i&i can be
rewritten J = L Si(0/3f. Then for any \gV we have X(0 = Li h^i{i)y
where we have set U = \(l3i). That is, X = XI h^i- This shows that {Sy} i spans
F*, and, together with (a), that it is a basis. D
Definition. The basis {8y} for F* is called the dual of the basis {^i} for F.
As usual, one of our fundamental isomorphisms is lurking behind all this,
but we shall leave its exposure to an exercise.
Corollary. 6^(F*) = d{V).
The three equations
^ = E UOfii, X = L x(^,)8,, x(0 = L x(^,). &i(0
are worth looking at. The first two are symmetrically related, each presenting
the basis expansion of a vector with its coefficients computed by applying the
corresponding element of the dual basis to the vector. The third is symmetric
itself between J and X.
Since a finite-dimensional space F and its dual space F* have the same
dimension, they are of course isomorphic. In fact, each basis for F defines an
isomorphism, for we have the associated coordinate isomorphism from F to R^,
the dual basis isomorphism from R^ to F*, and therefore the composite isomor-

2.3 THE DUAL SPACE 83
phism from F to F*. This isomorphism varies with the basis, however, and there
is in general no natural isomorphism between F and F*.
It is another matter with Cartesian space U^ because it has a standard
basis, and therefore a standard isomorphism with its dual space (R"")*. It is
not hard to see that this is the isomorphism a i—» L^, where La(x) = 2Zi a^•a:^•,
that we discussed in Section 1.6. We can therefore feel free to identify U^ with
(R"")*, only keeping in mind that when we think of an 7?-tuple a as a linear
functional, we mean the functional La(x) = 2^i a^•a:^.
The second conjugate space. Despite the fact that F and F* are not naturally
isomorphic in general, we shall now see that V is naturally isomorphic to F** =
(F*)*.
Theorem 3.2. The function co: F X F* -> R defined by co(J,/) = /(J) is
bilinear, and the mapping J ^^ co^ from F to F** is a natural isomorphism.
Proof. In this context we generally set J** = co^, so that J** is defined by
j**(/) = /(J) for all / e F*. The bilinearity of co should be clear, and Theorem
G.l of Chapter 1 therefore applies. The reader might like to run through a
direct check of the linearity of J i-» J** starting with (ci Ji + C2J2)**(/)•
There still is the question of the injectivity of this mapping. If a ?^ 0, we
can find / G F* so that f(a) 9^ 0. One way is to make a the first vector of an
ordered basis and to take/as the first functional in the dual basis; then/(a) = 1.
Since «**(/) = /(a) ^ 0, we see in particular that a** ?^ 0. The mapping
J —> J** is thus injective, and it is then bijective by the corollary of Theorem
2.4. D
If we think of F** as being naturally identified with F in this way, the two
spaces V and F* are symmetrically related to each other. Each is the dual of
t,he other. In the expression y(J)' we think of both symbols as variables and
then hold one or the other fixed for the two interpretations. In such a situation
we often use a more symmetric symbolism, such as (^,/), to indicate our inten-
t,ion to treat both symbols as variables.
Lemma 3.1. If {X^•} is the basis in F* dual to the basis {ai} in F, then
{a**} is the basis in F** dual to the basis {X^•} in F*.
Proof. We have a**(Xj) = \j(ai) = 5), which shows that af* is the iih
coordinate projection. In case the reader has forgotten, the basis expansion/ = X! ^y^y
implies that (xf*{f) = f{ai) = {Jlcj\j)(ai) = Ci, so that af* is the mapping
f^ci. D
Annihilator subspaces. It is in this dual situation that orthogonality first
naturally appears. However, we shall save the term 'orthogonal' for the latter
context in which V and F* have been identified through a scalar product, and
shall speak here of the annihilator of a set rather than its orthogonal
complement.

84 FINITE-DIMENSIONAL VECTOR SPACES 2.3
Definition. If A C F, the annihilator of .4, A°, is the set of all/ in F* such
that/(a) = 0 for all a in A. Similarly, if A C F*, then
A° = {a G F:/(a) = 0 for all / G A).
If we view F as (F*)*, the second definition is included in the first.
The following properties are easily established and will be left as exercises:
1) A° is always a subspace.
2) ACB=^ B"" CA\
3) {L{A)Y = A°.
4) (A uBy == .rn/^°.
5) A C A°°.
We now add one more crucial dimensional identity to those of the last
section.
Theorem 3.3. If TF is a subspace of F, then d(V) = cl(W) + d(TF°).
Proof. Let {I3i]'i be a basis for TF, and extend it to a basis {tSiYl for F. Let
{X^} 1 be the dual basis in F*. We claim that then {XjJJ^+i is a basis for TF°.
I'irst, if,/ > 7^1, then \j{l3i) = 0 for i = 1, . . . , m, and so \j is in W° by (3)
above. Thus {X^^.!, . . . , X,J C TF°. Now suppose that / G TF°, and let / =
2]y=i cy^i ^e its (dual) basis expansion. Then for each i < m we have c^• =
/(^.) = 0, since ^,- G W and / G TF°; therefore, / = Em+i cyXy. Thus every / in
TF° is in the span of {Xi}^_|_i. Altogether, we have shown that TF° is the span of
{Xj^+i, as claimed. Then cZ(TF°) + d(W) = {71 - m) + m = n = cl(V), and
we are done. D
Corollary. A°° = L(A) for every subset A CV.
Proof. Since (L(A))° = A°, we have d(L(A)) + d(A°) = cZ(F), by the
theorem. Also d{A°) + fZ(A°°) = d(F*) = fZ(F). Thus d(A°°) = d{L(A)),
and since L(A) C A°°, by (5) above, we have L{A) = A°°. 0
The adjoint of T. We shall now see that with every T in Hom(F, TF) there is
naturally associated an element of Hom(TF*, F*) which we call the adjoint of
T and designate T*. One consequence of the intimate relationship between T
and T* is that the range of T* is exactly the annihilator of the null space of
T, Combined with our dimensional identities, this implies that the ranges of T
and T* have the same dimension. And later on, after we have established the
connection between matrix representations of T and 7"*, this turns into the very
mysterious fact that the dimension of the linear span of the row vectors of an m-
hy-n matrix is the same as the dimension of the linear span of its column vectors,
which gives us our notion of the rank of a matrix. In Chapter 5 we shall study a
situation (Hilbert space) in which we are given a fixed fundamental isomorphism
between F and F*. If T is in Hom F, then of course T* is in Hom F*, and we
can use this isomorphism to "transfer" T* into Hom F. But now T can be com-

2.3 THE DUAL SPACE 85
pared with its (transferred) adjoint T*, and they may be equal. That is, T may
be self-adjoint. It turns out that the self-adjoint transformations are "nice" ones,
as we shall see for ourselves in simple cases, and also, fortunately, that many
important linear maps arising from theoretical physics are self-adjoint.
If r G Hom(F, W) and I G TF*, then of course Z o T g F*. Moreover, the
mapping I y-^ I o T (T fixed) is a linear mapping from TF* to F* by the corollary
to Theorem 3.3 of Chapter 1. This mapping is called the adjoint of T and is
designated T*. Thus T* G Hom(T7*, F*) and r*(0 = I o T for Silll G TF*.
Theorem 3.4. The mapping T i-^ T* is an isomorphism from the vector
space Hom(F, W) to the vector space Hom(TF*, F*). Also (T o S)* =
aS* o T* under the relevant hypotheses on domains and codomains.
Proof. Everything we have said above through the linearity of 7" i—» T* is a
consequence of the bilinearity of co(l, T) = I o T. The map we have called T*
is simply cot, and the linearity of 7" i—» T* thus follows from Theorem 6.1 of
Chapter 1. Again the reader might benefit from a direct linearity check,
beginning with (ci^i + C2T2)*©.
To see that T i-» T* is injective, we take any T ^ 0 and choose a G F so
that T{a) 9^ 0. We then choose Z G TF* so that l{T{a)) ^ 0. Since l{T{a)) =
(r*(0)(a), we have verified that T* ^ 0.
Next, if d(V) = m and d{W) = n, then also d{V*) = m and (i(TF*) = n
by the corollary of Theorem 3.1, and 6^(Hom(F, TF)) = mn= 6^(Hom(TF*, F*))
by Theorem 2.5. The injective map 7" 1—» T* is thus an isomorphism (by the
corollary of Theorem 2.4).
Finally, {T o S)n = I o (T o S) = {I o T) o S = aS*(Z o T) = >S*(r*(0) =
(S* o r*)Z, so that {ToSy = S*o T*. D
The reader would probably guess that T** becomes identified with T under
the identification of F with F**. This is so, and it is actually the reason for
calling the isomorphism J i-» J** natural. We shall return to this question at the
end of the section. Meanwhile, we record an important elementary identity.
Theorem 3.5. (72(r*))° = N(T) and iV(r*) = {R(T)y.
Proof. The following statements are definitionally equivalent in pairs as they
occur: I G iV(r*), r*(0 = OJ o T = 0, l{T(0) = 0 for alU G F, Z G {R(T)y,
Therefore, N(T*) = {R{T)y. The other proof is similar and will be left to the
reader. [Start with a G N{T) and end with a G (72(r*))°.] D
The rank of a linear transformation is the dimension of its range space.
Corollary. The rank of T* is equal to the rank of T,
Proof. The dimensions of R{T) and {N{T)y are each d{V) - d{N(T)) by
Theorems 2.4 and 3.3, and the second is d{R{T*)) by the above theorem.
Therefore, d{R{T)) = d{R{T*)). D

86 FINITE-DIMENSIONAL VECTOR SPACES 2.3
Dyads. Consider any T in Hom(F, W) whose range M is one-dimensional. If fi
is a nonzero vector in ilf, then x i—> X|S is a basis isomorphism d:R —^ M and
(9-^ o T: F -> R is a linear functional X G F*. Then T = 6> o x and 7^(0 =
X( J)(S for all ^ We write this as T = X(-)(S, and call any such T a dyad.
Lemma 3.2. If T is the dyad X(-)^, then T* is the dyad ^**(-)X.
Proof, (r*(0)(^) = {I o TXO = KnO) = KHm) = mnO, so that
r*(Z) = l(p)\ = ^**(0X, and r* = ^**(-)X. D
*Natural isomorphisms again. We are now in a position to illustrate more
precisely the notion of a natural isomorphism. We saw above that among all the
isomorphisms from a finite-dimensional vector space F to its second dual, we
could single one out naturally, namely, the map J i-> J**, where J**(/) = f(^)
for all /in F*. Let us call this isomorphism (pv- The technical meaning of the
word 'natural' pertains to the collection {(pv} of all these isomorphisms; we
found a way to choose one isomorphism (pv for each space F, and the proof that
this is a "natural" choice lies in the smooth way the various <^f's relate to each
other. To see what we mean by this, consider two finite-dimensional spaces F
and W and a map T in Hom(F, W). Then T* is in Hom(TF*, F*) and T** =
(T*)* is in Hom(F**, TF**). The setting for the four maps T, T**, <pv, and <pw
can be displayed in a diagram as follows:
The diagram indicates two maps, (pw ° T and T** o (py^ from F to TF**, and we
define the collection of isomorphisms {ipv} to be natural if these two maps are
always equal for any F, W and T. This is the condition that the two ways of
going around the diagram give the same result, i.e., that the diagram he
commutative.
Put another way, it is the condition that T "become" T** when F is
identified with F** (by (pv) and W is identified with TF** (by (pw)- We leave its proof
as an exercise.
EXERCISES
3.1 Let 6 be an isomorphism from a vector space V to U^. Show that the functional
Wi °6}i form a basis for V*.
3.2 Show that the standard isomorphism from R^ to (R")* that we get by composing
the coordinate isomorphism for the standard basis for R" (the identity) with the dual
basis isomorphism for (R^)* is just our friend a i—» /«, where Za(x) = 2i (^i^i- (Show
that the dual basis isomorphism is a i—» J^^ atWi.)

2.3 THE DUAL SPACE 87
3.3 We know from Theorem 1.6 that a choice of a basis {^i} for V defines an
isomorphism from W^ to Hom(F, W) for any vector space W. Apply this fact and Theorem
1.3 to obtain a basis in F*, and show that this basis is the dual basis of {Pi}.
3.4 Prove the properties of A° that are listed in the text.
3.5 Find (a basis for) the annihilator of <1, 1, 1> in U^. (Use the isomorphism
of (R^)* with U^ to express the basis vectors as triples.)
3.6 Find (a basis for) the annihilator of {< 1, 1, 1 >, < 1, 2, 3>} in R^.
3.7 Find (a basis for) the annihilator of {< 1, 1, 1, 1 >, < 1, 2, 3, 4>} in R'*.
3.8 Show that if V = AI @ N, then 7* = M° 0 iV°.
3.9 Show that if M is any subspace of an n-dimensional vector space V and d(M) =
m, then M can be viewed as being the linear span of an independent subset of m
elements of V or as being the annihilator of (the intersection of the null spaces of) an
independent subset of n — m elements of F*.
3.10 If B = {fi}^ is a finite collection of linear functionals on V (B C V*), then its
annihilator B° is simply the intersection A^ = Cl^Ni of the null spaces Ni = N(fi)
of the functionals/t. State the dual of Theorem 3.3 in this context. That is, take W
as the linear span of the functionals /», so that W C V* and W° C V. State the dual
of the corollary.
3.11 Show that the following theorem is a consequence of the corollary of Theorem 3.3.
Theorem. Let A^ be the intersection fli ^i of the null spaces of a set {fi}^ of
linear functionals on V, and suppose that g in V* is zero on A^. Then ^ is a linear
combination of the set {fi}^.
3.12 A corollary of Theorem 3.3 is that if IF is a proper subspace of V, then there is
at least one nonzero linear functional / in V* such that / = 0 on W. Prove this fact
directly by elementary means. (You are allowed to construct a suitable basis.)
3.13 An m-tuple of linear functionals {/t}f on a vector space V defines a linear
mapping a i—» <fi(a),... ,fm{ci) > from V to W^. What theorem is being applied
here? Prove that the range of this linear mapping is the whole of W^ if and only if
(/i}i* is an independent set of functionals. [Hint: If the range is a proper subspace W,
there is a nonzero m-tuple a such that 23i (^i^i = 0 for ^-H x E W.]
3.14 Continuing the above exercise, what is the null space A^" of the linear mapping
(X i—^ < fi(a), . . . ,fm(oL) > ? If ^ is a linear functional which is zero on A^, show that g
is a linear combination of the/t, now as a corollary of the above exercise and Theorem
4.3 of Chapter 1. (Assume the set {fi}^ independent.)
3.15 Write out from scratch the proof that T* is linear [for a given T in Hom(F, W)].
Also prove directly that T i—» T* is linear.
3.16 Prove the other half of Theorem 3.5.
3.17 Let 6{ be the isomorphism a i—» a** from Vi to 7i** for i = 1, 2, and suppose
^iven T in Hom(Fi, ¥2)- The loose statement T = T** means exactly that
T** = 620 To ^fi or T** o Oi = 620 T,
Prove this identity. As usual, do this by proving that it holds for each a in Vi.

88 FINITE-DIMENSIONAL VECTOR SPACES 2.4
3.18 Let ^: R^ —>7 be a basis isomorphism. Prove that the adjoint 6* is the
coordinate isomorphism for the dual basis if (R^)* is identified with R^ in the natural way.
3.19 Let CO be any bilinear functional on F X W. Then the two associated linear
transformations are T: V-^ W* defined by (Ti^))ir}) = cc(^, rj) and S:W-^ 7*
defined by iSir})){^) = co(J, 77). Prove that S = T* if W is identified with M'**.
3.20 Suppose that / in (R"^)* has coordinate m-tuple a [/(y) = ^f aiyi] and that T
in Hom(IR^, U^) has matrix t = {Uj}. Write out the explicit expression of the number
f(T(\)) in terms of all these coordinates. Rearrange the sum so that it appears in
the form
n
1
and then read off the formula for b in terms of a.
4. MATRICES
Matrices and linear transformations. The reader has already learned something
about matrices and their relationship to linear transformations from Chapter 1 ;
we shall begin our more systematic discussion by reviewing this earlier material.
By popular conception a matrix is a rectangular array of numbers such as
^11 ^12 • • • hn
^21 ^22 • • • ^2n
^ml ^m2 ' ' • ^mn
Note that the first index numbers the rows and the second index numbers the
columns. If there are m rows and n columns in the array, it is called an m-by-n
(m X n) matrix. This notion is inexact. A rectangular array is a way of picturing
a matrix, but a matrix is really a function, just as a sequence is a function. With
the notation m = {1, . . . , m}, the above matrix is a function assigning a
number to every pair of integers < i, j> inm X n. It is thus an element of the set
U^mxn ^j^g addition of two m X n matrices is performed in the obvious place-
by-place way, and is merely the addition of two functions in [R^x^^ the same is
true for scalar multiplication. The set of all m X n matrices is thus the vector
space U^^'^j a Cartesian space with a rather fancy finite index set. We shall use
the customary index notation Uj for the value t(z, j) of the function t at <z, j>,
and we shall also write {tij} for t, just as we do for sequences and other indexed
collections.
The additional properties of matrices stem from the correspondence between
m X n matrices {tij} and transformations T G Hom(IR'', U^),
The following theorem restates results from the first chapter. See Theorems
1.2, 1.3, and 6.2 of Chapter 1 and the discussion of the linear combination map
at the end of Section 1.6.

2.4 MATRICES 89
Theorem 4.1. Let {Uj} be an m-by-n matrix, and let t-^ be the m-tuple that
is itsyth column forj= 1, . . . , n. Then there is a unique T in Hom(IR'', U^)
such that skeleton T = {t^}, i.e., such that T(d^) = t^ for ally. T is defined
as the linear combination mapping x ^-^ y = 2Z7=i ^j^^\ ^^^ ^^ equivalent
presentation of T is the collection of scalar equations
n
i=i
Each T in Hom(IR^, U^) arises this way, and the bijection {tij} ^-^ T from
U^mxn ^Q Hom(IR'^, R^) is a natural isomorphism.
The only additional remark called for here is that in identifying an m X n
matrix with an n-tuple of m-tuples, we are making use of one of the standard
identifications of duality (Section 0.10). We are treating the natural isomorphism
between the really distinct spaces R^x^ and (R'^)'' as though it were the identity.
We can also relate T to {tij} by way of the rows of {1^}. As above, taking
iih coordinates in the m-tuple equation y = 2Z7=i ^i*"^? we get the equivalent
and familiar system of numerical (scalar) equations yi = 2^y=i UjXj for
i = 1, . . . , m. Now the mapping x ^-^ 2Zy=i CjXj from R"" to R is the most
general linear functional on R^. In the above numerical equations, therefore, we
have simply used the m rows of the matrix {tij} to present the m-tuple of linear
functionals on R^ which is equivalent to the single m-tuple-valued linear
mapping T in Hom(R^, R^) by Theorem 3.6 of Chapter 1.
The choice of ordered bases for arbitrary finite-dimensional spaces F and W
allows us to transfer the above theorem to Hom(F, W). Since we are now going
to correlate a matrix t in R^x^ with a transformation T in Hom(F, W), we shall
designate the transformation in Hom(R'', R"^) discussed above by T.
Theorem 4.2. Let {ay} i and {Pi}"^ be ordered bases for the vector spaces F
and W, respectively. For each matrix {tij} in R^Xn j^^ rp ^^ ^^iq unique
element of Hom(F, W) such that T{aj) = T.T=i Ujfii for j = 1, . . . , n.
Then the mapping {tij} i—» 7" is an isomorphism from R^Xn ^^ Hom(F, W).
Proof. We simply combine the isomorphism {tij} i—» T of the above theorem
with the isomorphism T^T = ypoTo^-^ from Hom(R^, R^) to Hom(F, TF),
where (p and \p are the two given basis isomorphisms. Then T is the transforma-
(ion described in the theorem, for T(aj) = yp{T{<p-\aj))) = yp{T(d')) =
\p{t^) = JlT=i iijPi' The map {tij} ^-^ T is the composition of two isomorphisms
and so is an isomorphism. D
It is instructive to look at what we have just done in a slightly different way.
(liven the matrix {tij}, let Tj be the vector in TF whose coordinate m-tuple is the
,/(h column t-^ of the matrix, so that Tj = YlT=i kjfii- Then let T be the unique
clement of Hom(F, TF) such that T{aj) = Tj forj= 1, . . . , n. Now we have
obtained T from {tij} in the following two steps: T corresponds to the n-tuple

90 FINITE-DIMENSIONAL VECTOR SPACES 2.4
{ry} 1 under the isomorphism from Hom(F, W) to W given by Theorem 1.6,
and {Tj} 1 corresponds to the matrix {tij} by extension of the coordinate
isomorphism between W and U^ to its product isomorphism from W to (R'^)''.
Corollary. If y is the coordinate m-tuple of the vector rj in W and x is the
coordinate n-tuple of J in F (with respect to the given bases), then rj = T{^)
if and only if yi = 2^7= i ^u^i ^ov i = 1, . . . , m.
Proof. We know that the scalar equations are equivalent to y = T(x), which is
the equation y = ^~^o7'o(^(x). The isomorphism ^ converts this to the
equation rj = T(^). D
Our problem now is to discover the matrix analogues of relationship between
linear transformations. For transformations between the Cartesian spaces U^
this is a fairly direct, uncomplicated business, because, as we know, the matrix
here is a natural alter ego for the transformation. But when we leave the
Cartesian spaces, a transformation T no longer has a matrix in any natural way, and
only acquires one when bases are chosen and a corresponding T on Cartesian
spaces is thereby obtained. All matrices now are determined with respect to
chosen bases, and all calculations are complicated by the necessary presence of
the basis and coordinate isomorphisms. There are two ways of handling this
situation. The first, which we shall follow in general, is to describe things
directly for the general space F and simply to accept the necessarily more
complicated statements involving bases and dual bases and the corresponding loss in
transparency. The other possibility is first to read off the answers for the
Cartesian spaces and then to transcribe them via coordinate isomorphisms.
Lemma 4.1. The matrix element tkj can be obtained from T by the formula
tkj = lXk{T{aj)),
where p,k is the A:th element of the dual basis in TF*.
Proof. fJik{T{aj)) = fXk(ZT=i Ujfii) = HiUM^i) = Hikj b\ = t^j. D
In terms of Cartesian spaces, T{d^), is the yth column m-tuple t^ in the
matrix {tij} of T, and tkj is the A:th coordinate of t-^. From the point of view of
linear maps, the A:th coordinate is obtained by applying the A:th coordinate
projection Tk, so that % = Trk{T{8^)). Under the basis isomorphisms, Tk
becomes jujt, T becomes T, 8^ becomes aj, and the Cartesian identity becomes
the identity of the lemma.
The transpose. The transpose of the m X n matrix {tij} is the n X m matrix
{tfj} defined by t* = tji for all z, j. The rows of t* are of course the columns of t,
and conversely.
Theorem 4.3. The matrix of T* with respect to the dual bases in TF* and
F* is the transpose of the matrix of T.

2.4 MATRICES 91
Proof, If s is the matrix of T*, then Lemmas 3.1 and 4.1 imply that
s^,= aT{T\ix,)) = aT{iXioT)
= {iXioT){aj) = iXi{T{aj)) = Uj. D
Definition. The row space of the matrix {tij} G W^^^ is the subspace of IR^
spanned by the m row vectors. The column space is similarly the span of
the n column vectors in U^.
Corollary. The row and column spaces of a matrix have the same dimension.
Proof. If T is the element of Hom(lR^, IR^) defined by T(8') = t', then the
set {t-'} 1 of column vectors in the matrix {tij} is the image under T of the
standard basis of IR^, and so its span, which we have called the column space of the
matrix, is exactly the range of T. In particular, the dimension of the column
space is d{R(T)) = rank T,
Since the matrix of T* is the transpose t* of the matrix t, we have, similarly,
that rank T* is the dimension of the column space of t*. But the column space
of t* is the row space of t, and the assertion of the corollary is thus reduced to
the identity rank T* = rank T, which is the corollary of Theorem 3.5. Q
This common dimension is called the rank of the matrix.
Matrix products. If T G Hom(lR^, W^) and S G Hom(lR^, IRO, then of course
R = S o T G Hom(lR^, IR^), and it certainly should be possible to calculate the
matrix r of 72 from the matrices s and t of aS and T, respectively. To make this
computation, we set y = T(x) and z = S(y)f so that z = (S o T)(x) = R(x).
The equivalent scalar equations in terms of the matrices t and s are
Vi = 23 tihXh and Zk = ^ SkiVi,
h=l z = l
so that
m n n / m \
Zk = ^ Ski S tihXh = S ( X) Skitihj Xh.
i = \ h=\ h=\ \i=\ '
But Zk = Y.h=i '^khXh for fc = 1, . . . , Z. Taking x as 8\ we have
m
Vkj = 23 ^kitij for all k and j,
z = l
We thus have found the formula for the matrix r of the map R = So Tix—>z.
Of course, r is defined to be the product of the matrices s and t, and we write
r = s • t or r = St.
Note that in order for the product st to be defined, the number of columns
in the left factor must equal the number of rows in the right factor. We get the
element Vkj by going across the fcth row of s and simultaneously down the yth

92
FINITE-DIMENSIONAL VECTOR SPACES
2.4
column of t, multiplying corresponding elements as we go, and adding the
resulting products. This process is illustrated in Fig. 2.1. In terms of the scalar
product (x, y) = ^i xiyi on IR^, we see that the element Vkj in r = st is the
scalar product of the fcth row of s and the yth column of t.
{I by m)
I
kih. row|---o e3-
m
(m by n)
n 1
d)
1
1
1
1
1
1
jth column
t
Fig. 2.1
{I l)y n)
Since we have defined the product of two matrices as the matrix of the
product of the corresponding transformations, i.e., so that the mapping T i-^ {tij}
preserves products (So T i-^ st), it follows from the general principle of
Theorem 4.1 of Chapter 1 that the algebraic laws satisfied by composition of
transformations will automatically hold for the product of matrices. For
example, we know without making an explicit computation that matrix
multiplication is associative. Then for square matrices we have the following theorem.
Theorem 4.4. The set Mn of square n X n matrices is an algebra naturally
isomorphic to the algebra Hom(lR^).
Proof, We already know that T \-^ {tij} is a natural linear isomorphism from
Hom(lR^) to Mn (Theorem 4.1), and we have defined the product of matrices
so that the mapping also preserves multiplication. The laws of algebra (for an
algebra) therefore follow for Mn from our observation in Theorem 3.5 of Chapter
1 that they hold for Hom(lR^). D
The identity / in Hom(lR^) takes the basis vector 8^ into itself, and therefore
its matrix e has 8^ for its yth column: e^ = 8^, Thus eij = 81 = 1 if i = j
and eij =8^ = 0 if i 9^ J. That is, the matrix e is 1 along the main diagonal
(from upper left to lower right) and 0 elsewhere. Since / i-^ e under the algebra
isomorphism T i-^ t, we know that e is the identity for matrix multiplication.
Of course, we can check this directly: J^]=i Ujejk = Uk, and similarly for
multiplying by e on the left. The symbol 'e^ is ambiguous in that we have used it
to denote the identity in the space IR^^^ of square 7i X 7i matrices for any n.
Corollary. A square 71 X n matrix t has a multiplicative inverse if and only
if its rank is 7i.

2.4 MATRICES 93
Proof. By the theorem there exists an s G M^ such that st = ts = e if and
only if there exists an S G Hom(lR'') such that SoT=ToS=I. But such
an S exists if and only if T is an isomorphism, and by the corollary to Theorem 2.4
this is equivalent to the dimension of the range of T being n. But this dimension
is the rank of t, and the argument is complete. D
A square matrix (or a transformation in Hom V) is said to be nonsingular
if it is invertible.
Theorem 4.5. If {az}i, {/5j}T> ^^^ {^/c}i are ordered bases for the
vector spaces (7, F, and TF, respectively, and if T G Hom ((7, V) and
S G Hom(F, W), then the matrix of aS o T is the product of the matrices
of S and T (with respect to the given bases).
Proof. By definition the matrix of aS o T is the matrix oi S o T = X~^ o {S o T) o ip
in Hom(lR^, W), where <^ and X are the given basis isomorphisms for U and W.
But if yj/ is the basis isomorphism for F, we have
and therefore its matrix is the product of the matrices of S and T by the defini-
tion of matrix multiplication. The latter are the matrices of S and T with respect
to the given bases. Putting these observations together, we have the theorem. □
There is a simple relationship between matrix products and transposition.
Theorem 4.6. If the matrix product st is defined, then so is t*s*, and
t*s* = (st)*.
Proof. A direct calculation is easy. We have
m m
ist)jk = (St)kj = X Skitij = X tjiSik = (t S )jk.
z = l z = l
Thus (st)* = t*s*, as asserted. D
This identity is clearly the matrix form of the transformation identity
{S o T)"^ = T* o aS*, and it can be deduced from the latter identity if desired.
Cartesian vectors as matrices. We can view an n-tuple x^ -Kxij . . . , Xn^
as being alternatively either an n X 1 matrix, in which case we call it a column
vector, or a 1 X n matrix, in which case we call it a row vector. Of course, these
identifications are natural isomorphisms. The point of doing this is, in part, that
then the equations yi = l^y=i tijXj say exactly that the column vector y is the
matrix product of t and the column vector x, that is, y = t • x. The linear map
T:U^ —^ W^ becomes left multiplication by the fixed matrix t when U^ is viewed as
the space of n X 1 column vectors. For this reason we shall take the column
vector as the standard matrix interpretation of an n-tuple x; then x* is the
corresponding row vector.

94 FINITE-DIMENSIONAL VECTOR SPACES 2.4
In particular, a linear functional F G (IR^)* becomes left multiplication by
iU matrix, which is of course 1 X n (F being from W^ to IR^), and therefore is
simply the row matrix interpretation of an n-tuple in R^. That is, in the natural
isomorphism a i-^ La from W^ to (IR^)*, where L^(x) = YJi (^i^i^ the functional
La can now be interpreted as left matrix multiplication by the 7i-tuple a viewed
as the row vector a*. The matrix product of the row vector (1 X n matrix) a*
and the column vector (rz X 1 matrix) x is a 1 X 1 matrix a* • x, that is, a
number.
Let us now see what these observations say about T*. The number L^{T{x))
is the 1 X 1 matrix a*tx. Since L^{T(x)) = (r*(La))(x) by the definition of
T*, we see that the functional T*(L^) is left multiplication by the row vector
a*t. Since the row vector form of La is a* and the row vector form of T'*(La) is
a*t, this shows that when the functionals on IR^ are interpreted as roio vectors,
T* becomes right multiplication by t. This only repeats something we already
know. If we take transposes to throw the row vectors into the standard column
vector form for 7?-tuples, it shows that T* is left multiplication by t*, and so
gives another proof that the matrix of T* is t*.
Change of basis. If<^:xi-^ ^ = J2i ^iPi and d:y ^-^ ^ = l^i yifii are two basis
isomorphisms for F, then A = ^~^ o <^ is the isomorphism in Hom(lR^) which
takes the coordinate 7i-tuple x of a vector ^ with respect to the basis {/? J into the
coordinate n-tuple y of the same vector with respect to the basis {jSi}. The
isomorphism A is called the "change of coordinates" isomorphism. In terms
of the matrix a of A, we have y = ax, as above.
The change of coordinate map A = d~^ o ip should not be confused with the
similar looking T = 6 o (p~^. The latter is a mapping on F, and is the element
of Hom(y) which takes each Pi to /?'.
Fig. 2.2
We now want to see what happens to the matrix of a transformation
T G Hom(F, W) when we change bases in its domain and codomain spaces.
Suppose then that cpi and <^2 are basis isomorphisms from IR^ to F, that i^i and \l/2
are basis isomorphisms from IR^ to W, and that t' and t" are the matrices of T
with respect to the first and second bases, respectively. That is, t' is the matrix
of r = (^Pl)-^ o T o <^i G Hom(lR^, IR^), and similarly for t'\ The mapping
^ = <^2~^ ° <^i ^ Hom(lR^) is the change of coordinates transformation for
V: if X is the coordinate n-tuple of a vector { with respect to the first basis
[that is, ^ = <pi(x)], then A(x) is its coordinate n-tuple with respect to the second
basis. Similarly, let B be the change of coordinates map rj/^^ o xj/^ for W. The
diagram in Fig. 2.2 will help keep the various relationships of these spaces and

2.4 MATRICES 95
mappings straight. We say that the diagram is commutative, which means that
any two paths between two points represent the same map. By selecting various
pairs of paths, we can read off all the identities which hold for the nine maps
T, T", T"', (fly (P2, A,i^i,i^2f B. For example, T"' can be obtained by going
backward along A J forward along T", and then forward along B. That is, T"' =
B o T^ o A~^. Since these "outside maps" are all maps of Cartesian spaces, we
can then read off the corresponding matrix identity
t" = bt'a-S
showing how the matrix of T with respect to the second pair of bases is obtained
from its matrix with respect to the first pair.
What we have actually done in reading off the above identity from the
diagram is to eliminate certain retraced steps in the longer path which the
definitions would give us. Thus from the definitions we get
Bo r o A~^ = (yl/2^ o yp{) o (yPY^ o T o ^^) o (<pf ^ o ^^) = yfy^^ o T o ^^ = T".
In the above situation the domain and codomain spaces were different, and
the two basis changes were independent of each other. If TF = F, so that
T G Hom(F), then of course we consider only one basis change and the formula
becomes
t" = a • t' • a-^
Now consider a linear functional F G F*. If i" and f are its coordinate
n-tuples considered as column vectors (n X 1 matrices), then the matrices of F
with respect to the two bases are the row vectors (f)* and (f")*> as we saw
earlier. Also, there is no change of basis in the range space since here IF = IR,
with its permanent natural basis vector 1. Therefore, b = e in the formula
t" = bt'a~\ and we have (f")* = (fO*a~^ or
f"= (a-"i)*f'.
We want to compare this with the change of coordinates of a vector { G F,
which, as we saw earlier, is given by
These changes go in the oppositive directions (with a transposition thrown in).
For reasons largely historical, functionals F in V* are called covariant vectors,
and since the matrix for a change of coordinates in F is the transpose of the
inverse of the matrix for the corresponding change of coordinates in F*, the
vectors { in F are called contravariant vectors. These terms are used in classical
tensor analysis and differential geometry.
The isomorphism {tij} i-> T, being from a Cartesian space IR'^^^, is
automatically a basis isomorphism. Its basis in Hom(F, W) is the image under the
isomorphism of the standard basis in IR'^^^, where the latter is the set of
Kronecker functions 8^^ defined by d^\i,j) = 0 if <k,l> 7^ <hj> and
S^\k, l) = 1. (Remember that in IR^, S"" is that function such that d'^ib) = 0

90 FIXITE-DIMEXSIOXAL VECTOR SPACES 2.4
if 6 7^ a and 5"*(a) — 1. Here A = m X n and the elements a of A are ordered
pairs a = </v, />.) The function 8^^ is that matrix whose columns are all 0
except for the /th, and the lih column is the m-tuple 8^. The corresponding
transformation D^i thus takes every basis vector aj to 0 except ai and takes ai
to Pfc' That is, Dki(aj) = Oiij 9^ I, and Dki(ai) = I3k. Again, Dki takes the lih
basis vector in V to the /ith basis vector in W and takes the other basis vectors
in V to 0.
If ? = E ^'io^h it follows that Z>a-KO = xih-
Since {D^z} is the basis defined by the isomorphism {tij} i-^ T, it follows
that [tij] is the coordhiate set of T with respect to this basis; it is the image of T
under the coordinate isomorphism. It is interesting to see how this basis
expansion of T automatically appears. We have
TiO = T (Z xja\ = i: xjTiaj) = ^ UjXjfii = ^ kjO^jU),
so that
1 / J tijUij.
Our original discussion of the dual basis in F* was a special case of the
present situation. There we had Hom(F, R) = 7*, with the permanent
standard basis 1 for R. The basis for F* corresponding to the basis {aJ for T'"
therefore consists of those maps Di taking ai to 1 and aj to 0 for j 9^ I. Then
Di{^) — Di(Y, Xjctj) = ^h ^ii<^l ^i '^^ the /th coordinate functional 8^
Finally, we note that the matrix expression of T G Hom(lR^, W^') is very
suggestive of the block decompositions of T that we discussed earlier in Section
1.5. In the exercises we shall ask the reader to show that in fact Tj^i = tuDki.
EXERCISES
4.1 Prove that if co: T'X ]'^ IR is a bilinear functional on T' and T: T^ ^ T'*
is the corresponding linear transformation defined by {T{ri)){X) = 60(^,77), then for
any basis [ai] for T' the matrix tij = o)(ai, aj) is the matrix of T.
4.2 Verify that the row and column rank of the following matrix are both 1:
-5 2 3
-10 4 6
4.3 Show by a direct calculation that if the row rank of a 2 X 3 matrix is 1, then so
is its column rank.
4.4 Let [fi] ^ be a linearly' dependent set of C^-functions (twice continuously^ differ-
entiable real-valued functions) on R. Show that the three triples <fi{x),fi(x),fi\x) >
are dependent for any x. Prove therefore that sin t, cos t, and e^ are linearly-
independent. (Compute the derivative triples for a well-chosen x.)

2.4
MATRICES
97
4.5 Compute
4.6 Compute
] ^ U
From your answer give a necessary and sufficient condition for
I a b~\
\_c d I
to exist.
4.7 A matrix a is idempotent if a^ = a. Find a basis for the vector space R^^^ of
all 2 X 2 matrices consisting entirely of idempotents.
4.8 By a direct calculation show that
1 2
L—2 3.
is invertible and find its inverse.
4.9 Show by explicitly solving the equation
fa ^1. r^ ^1 = r^
[c d\ Iz w\ [o
that the matrix on the left is invertible if and only if (the determinant) ad
zero.
4.10 Find a nonzero 2X2 matrix
be is not
[: 3
whose square is zero.
4.11 Find all 2X 2 matrices whose squares are zero.
4.12 Prove by computing matrix products that matrix multiplication is associative.
4.13 Similarly, prove directly the distributive law, (r + s) • t = r • t + s • t.
4.14 Show that left matrix multiplication by a fixed r in U.^^^ is a linear
transformation from R'^^p to R'^^p, What theorem in Chapter 1 does this mirror?
4.15 Show that the rank of a product of two matrices is at most the minimum of their
ranks. (Remember that the rank of a matrix is the dimension of the range space of its
associated T.)
4.16 Let a be an m X n matrix, and let b be n X m. If m > w, show that a • b cannot
be the identity e {mX m).

98
FINITE-DIMENSIONAL VECTOR SPACES
2.4
4.17 Let Z be the subset of 2 X 2 matrices of the form
fa h\
a
Prove that Z is a subalgebra of IR2X2 (that is, Z is closed under addition, scalar
multiplication, and matrix multiplication). Show that in fact Z is isomorphic to the complex
number system.
4.18 A matrix (necessarily square) which is equal to its transpose is said to be
symmetric. As a square array it is symmetric about the main diagonal. Show that for any
mX n matrix t the product t • t* is meaningful and symmetric.
4.19 Show that if s and t are symmetric nX n matrices, and if they commute, then
s • t is symmetric. (Do not try to answer this by writing out matrix products.) Show
conversely that if s, t, and s • t are all symmetric, then s and t commute.
4.20 Suppose that T in Hom R^ has a symmetric matrix and that T is not of the
form cl. Show that T has exactly two eigenvectors (up to scalar multiples). What
does the matrix of T become with respect to the "eigenbasis" for IR^ consisting of these
two eigenvectors?
4.21 Show that the symmetric 2X2 matrix t has a symmetric square root s (s^ = t)
if and only if its eigenvalues are nonnegative. (Assume the above exercise.)
4.22 Suppose that t is a 2 X 2 matrix such that t* = t~^. Show that t has one of
the forms
where a^ -\- b^ = 1.
4.23 Prove that multipHcation by the above t is a EucHdean isometry. That is,
show that if y = t • x, wherex and y E IR^, then ||a:|| = ||?/||, where ||a:|| = (x\-\- x^)^^^.
4.24 Let {Dki] be the basis for Hom(y, W) defined in the text. Taking W = V,
show that these operators satisfy the very important multiplication rules
Dij o D,i = 0 if j 9^ k,
Diko Dki = Da.
4.25 Keeping the above identities in mind, show that if / 9^ ??i, then there are
transformations *S and T in Hom V such that
So T — To S
Du
Also find *S and T such that
So T — To S = Du
Drr,
4.26 Given T in Hom U"", we know from Chapter 1 that T = J^ij Tij, where Tij =
PiTPj and Pi = BiWi. Now we also have
T = X) ikiDki.
kl
Show from the definition of D^ in the text that PiDijPj = Dij and that PiDkiPj = 0
if either i 9^ k or j 9^ I. Conclude that Ta = tijDij.

2.5 TRACE AND DETERMINANT 99
5. TRACE AND DETERMINANT
Our aim in this short section is to acquaint the reader with two very special
real-valued functions on Horn V and to describe some of their properties.
Theorem 5.1. If F is an 7i-dimensional vector space, there is exactly one
linear functional X on the vector space Hom(F) with the property that
\(S o T) = \(T o S) for all S, T in Hom(F) and normalized so that X(/) = n.
If a basis is chosen for F and the corresponding matrix of T is {Uj}, then
X(T') = Y^i=i tiij the sum of the elements on the main diagonal.
Proof. If we choose a basis and define \(T) as X)i Ui^ then it is clear that X is a
linear functional on Hom(F) and that X(/) = n. Moreover,
n / n \ n
XC-S o T) = E ( E Sijtji) = E Sijtji = £ tjiSij = \{T o S).
That is, each basis for F gives us a functional X in (Hom F)* such that \(S o T) =
\{T o aS), X(/) = ??, and \(T) = X) Ui for the matrix representation of that basis.
Now suppose that jjl is any element of (Hom(F))* such that ijl(S o T) =
Ijl(T o S) and /x(/) = n. If we choose a basis for F and use the isomorphism
d: {tij} ^ T from IR^^^ to Hom F, we have a functional v = fx o 6 on IR^^^
(p = d^fji) such that v(st) = v(ts) and v(e) = n. By Theorem 4.1 (or 3.1) v is
given by a matrix c, v{i) = HXj=i CijUj, and the equation p(st — ts) = 0
becomes T.ij,k=i Cij(siktkj — SjAi) = 0.
We are going to leave it as an exercise for the reader to show that if Z f^ m,
then very simple special matrices s and t can be chosen so that this sum reduces
to cim = 0, and, by a different choice, to c^ — Cmm = 0.
Together with the requirement that p(e) = n, this implies that cim = 0 for
I 7^ m and Cmm = 1 for m = 1, . . . -, n. That is, v{t) = YJi imm, and v is the
X of the basis being used. Altogether this shows that there is a unique X in
(Hom F)* such that \(S o T) = \{T o S) for all S and T and X(/) = n, and that
\(T) has the diagonal evaluation as X) Ui in every basis. □
This unique X is called the trace functional^ and \{T) is the trace of T. It is
usually designated tr(T').
The determinant function A(T) on Hom F is much more complicated, and
we shall not prove that it exists until Chapter 7. Its geometric meaning is as
follows. First, |A(T')| is the factor by which T multiplies volumes. More
precisely, if we define a "volume" v for subsets of V by choosing a basis and using
the coordinate correspondence to transfer to F the "natural" volume on IR^,
then, for any figure A C F, v(T[A]) = \A(T)\ • v(A), This will be spelled out in
Chapter 8. Second, A(T) is positive or negative according as T preserves or
reverses orientation^ which again is a sophisticated notion to be explained later.
For the moment we shall list properties of A(T') that are related to this geometric
interpretation, and we give a sufficient number to show the uniqueness of A.

100 FINITE-DIMENSIONAL VECTOR SPACES
2.5
T[A]
Fig. 2.3
We assume that for each finite-dimensional vector space V there is a
function A (or Av when there is any question about domain) from Hom(F) to U such
that the following are true:
a) A(S o T) = AiS) AiT) for any S, T in Hom(y).
b) If a subspace N of V is invariant under T and T is the identity on N and
on V/N (that is, T[a] = a for each coset a = a + N oi N), then
A(T) = 1. Such a T is a shearing of V along the planes parallel to N.
In two dimensions it can be pictured as in Fig. 2.3.
c) If F is a direct sum F = ilf + ^ of T-invariant subspaces M and N, and
if R = T \ Mand S = T \ N, then A(T) = A(R) A(S). More exactly,
Av(T) = Am(R) An(S).
d) If F is one-dimensional, so that any T in Hom(F) is simply multiplication
by a constant ct, then A(T) is that constant ct-
e) If F is two-dimensional and T interchanges a pair of independent vectors,
then A(T) = — 1. This is clearly a pure orientation-changing property.
The fact that A is uniquely determined by these properties will follow from
our discussion in the next section, which will also give us a process for calculating
A. This process is efficient for dimensions greater than two, but for T in Hom(IR^)
there is a simple formula for A(T) which every student should know by heart.
Theorem 5.2. If T is in Hom(lR2) and {Uj} is its 2x2 matrix, then
A(T) = tiit22 ~ ^12^21-
This is a special case of a general formula, which we shall derive in Chapter 7,
that expresses A(T') as a sum of n! terms, each term being a product of n numbers
from the matrix of T, This formula is too complicated to be useful in
computations for large n, but for n = 3 it is about as easy to use as our row-reduction
calculation in the next section, and for n = 2 it becomes the above simple
expression. There are a few more properties of A with which every student
should be familiar. They will all be proved in Chapter 7.
Theorem 5.3. If T is in Hom F, then A(T*) = A{T). If d is an isomorphism
from F to IF and aS = ^ o T o ^-i, then A{S) = A{T).

2.5 TRACE AND DETERMINANT 101
Theorem 5.4. The transformation T is nonsingular (invertible) if and only
if ^{T) 9^ 0.
In the next theorem we consider T in Hom IR^, and we want to think of A(T')
as a function of the matrix t of T. To emphasize this we shall use the notation
D{t) = h{T).
Theorem 5.5 {Cramer^s rule). Given an n X n matrix t and an n-tuple y,
let t \j y be the matrix obtained by replacing theyth column of t by y. Then
y = t • X =4> D{t)Xj = D(t \j y)
for all j.
If t is nonsingular [D(t) 9^ 0], this becomes an explicit formala for the
solution X of the equation y = t • x; it is theoretically important even in those
cases when it is not useful in practice (large n).
EXERCISES
5.1 Finish Theorem 5.1 by applying Exercise 4.25.
5.2 It follows from our discussion of trace that tr(T) = ^ ta is independent of the
basis. Show that this fact follows directly from
tr(t • s) = tr(s • t)
and the change of basis formula in the preceding section.
5.3 Show by direct computation that the function d{t) = tiit22 — ^12^21 satisfies
d(s • t) = d(s) d(t) (where s and t are 2X2 matrices). Conclude that if V is two-
dimensional and d{T) is defined for T in Hom V by choosing a basis and setting
d(T) = d(t), then d(T) is actually independent of the basis.
5.4 Continuing the above exercise, show that d(T) = A(T) in any of the following
cases:
1) T interchanges two independent vectors.
2) T has two eigenvectors.
3) T has a matrix of the form
ri al
[0 ij
Show next that if T has none of the above forms, then T = R o S, where *S is of type
(1) and R is of type (2) or (3). [Hint: Suppose T(a) = fi, with a and ^ independent,
l^et S interchange a and jS, and consider R = T o S.] Show finally that d{T) = A(T)
for all T in Hom V. (F is two-dimensional.)
5.5 If t is symmetric and 2X2, show that there is a 2 X 2 matrix s such that
H* = s~^, A(s) = 1, and sts~^ is diagonal.
5.6 Assuming Theorem 5.2, verify Theorem 5.4 for the 2X2 case.
5.7 Assuming Theorem 5.2, verify Theorem 5.5 for the 2X2 case.

102 FINITE-DIMENSIONAL VECTOR SPACES 2.G
5.8 In this exercise we suppose that the reader remembers what a continuous
function of a real variable is. Suppose that the 2X2 matrix function
a(0 =
aiiit) ai2(0l
a2i(t) a22(0j
has continuous components aij(t) for t E (0, 1), and suppose that a(t) is nonsingular
for every t. Show that the solution y{t) to the Unear equation a{t) • y(t) = x(t) has
continuous components yi{t) and y2{i) if the functions xi{t) and X2{t) are continuous.
5.9 A homogeneous second-order linear differential equation is an equation of the
form
y'' + aiy' + aoy = 0,
where ai = ai(t) and ao = ao(t) are continuous functions. A solution is a C^-function
/ (i.e., a twice continuously differentiate function) such that f'{t) + ai{t)f{t) +
cio{t)f{t) = 0. Suppose that / and g are C^-functions [on (0, 1), say] such that the
2X2 matrix
\m g{t)l
Ifit) g'{t)\
is always nonsingular. Show that there is a homogeneous second-order differential
equation of which they are both solutions.
5.10 In the above exercise show that the space of all solutions is a two-dimensional
vector space. That is, show that if h(t) is any third solution, then h is a. Hnear
combination of / and g.
5.11 By a "linear motion" of the Cartesian plane R^ into itself we shall mean a
continuous maj) X ^-^ t(x) from [0, 1] to the set of 2 X 2 nonsingular matrices such that
t(0) = e. Show that A(t(l)) > 0.
5.12 Show that if A(s) = 1, then there is a linear motion whose final matrix t(l) is s.
6. MATRIX COMPUTATIONS
The computational process by which the reader learned to solve systems of
linear equations in secondary school algebra was undoubtedly "elimination by-
successive substitutions". The first equation is solved for the first unknown, and
the solution expression is substituted for the first unknown in the remaining
equations, thereby eliminating the first unknown from the remaining equations.
Next, the second equation is solved for the second unknown, and this unknown is
then eliminated from the remaining equations. In this way the unknowns are
eliminated one at a time, and a solution is obtained.
This same procedure also solves the following additional problems:
1) to obtain an explicit basis for the linear span of a set of m vectors in IR^;
therefore, in particular,
2) to find the dimension of such a subspace;
3) to compute the determinant of an m X m matrix;
4) to compute the inverse of an invertible m X m matrix.

2.6 MATRIX COMPUTATIONS 103
In this section we shall briefly study this process and the solutions to these
problems.
We start by noting that the kinds of changes we are going to make on a
finite sequence of vectors do not alter its span.
Lemma 6.1. Let {ai}^ be any m-tuple of vectors in a vector space, and let
{Pi}7 be obtained from {ai}^ by any one of the following elementary
operations:
1) interchanging two vectors;
2) multiplying some ai by a nonzero scalar;
3) replacing ai by ai — xaj for some./ 9^ i and some a: G K.
Then
L({^,}T) = L({a,}T).
Proof. If a'i = ai — xaj, then ai = a- + xaj. Thus if {^i}^ is obtained from
{ai}^ by one operation of type (3), then {af}^ can be obtained from {^i}"^ by
one operation of type (3). In particular, each sequence is in the linear span of
the other, and the two linear spans are therefore the same.
Similarly, each of the other operations can be undone by one of the same
type, and the linear spans are unchanged. □
When we perform these operations on the sequence of roiv vectors in a
matrix, we call them elementary row operations.
We define the order of an n-tuple x = <xi, . . . ,Xn> as the index of the
first nonzero entry. Thus if Xi = 0 for i < j and Xj 9^ 0, then the order of x
kj. The order of <0, 0, 0, 2, — 1, 0> is 4.
Let {aij} be an m X n matrix, let V be its row space, and let rii < 712 <
• • • < rifc be the integers that occur as orders of nonzero vectors in V. We are
going to construct a basis for V consisting of k elements having exactly the
above set of orders.
If every nonzero row in {aij} has order > p, then every nonzero vector x in
V has order >p, since x is a linear combination of these row vectors. Since some
vector in V has the minimal order rii, it follows that some row in {aij} has order
?ii. We move such a row to the top by interchanging two rows. We then multiply
this row X by a constant, so that its first nonzero entry Xm is 1. Let a\ . . . , a^
be the row vectors that we now have, so that a^ has order rii and a^j = 1. We
next subtract multiples of a^ from each of the other rows in such a way that the
new ith row has 0 as its ni-coordinate. Specifically, we replace a* by a* — aj^j • a^
for i > 1. The matrix that we thus obtain has the property that its yth column
is the zero m-tuple for eachy < rii and its riith column is 8^ in U^. Its first row
has order rii, and every other row has order >ni. Its row space is still V. We
again call it a.
Now let X = ^7 Cz-a* be a vector in V with order 712- Then ci = 0, for if
Ci 7^ Oj then the order of x is ni. Thus x is a linear combination of the second

104
FINITE-DIMENSIONAL VECTOR SPACES
2.6
to the mth rows, and, just as in the first case, one of these rows must therefore
have order 712.
We now repeat the above process all over again, keying now on this vector.
We bring it to the second row, make its n2-coordinate 1, and subtract multiples
of it from all the other rows (including the first), so that the resulting matrix
has 8^ for its n2th column. Next we find a row with order 713, bring it to the
third row, and make the 713th column 5^, etc.
We exhibit this process below for one 3X4 matrix. This example is
dishonest in that it has been chosen so that fractions will not occur through the
application of (2). The reader will not be that lucky when he tries his hand.
Our defense is that by keeping the matrices simple we make the process itself
more apparent.
0
2
2
-1
2
4
2
4
0
3
-2
3
(1)
(3)
(3)
(3)
2
0
2
1
0
0
"1
0
0
1
0
0
2
-1
4
1
-1
2
0
1
0
0
1
0
4
2
0
2
2
-4
4
-2
0
4
-2
0
-2
3
3
-1
3
5.
2
-3
11
0"
0
1
(2)
(2)
(1)
1
0
2
"1
0
P
1
0
0
1
-1
4
1
1
2
0
1
0
2
2
0
2
-2
-4
4
-2
0
-1
3
3
-1
-3
5.
2
-3
1
Note that from the final matrix we can tell that the orders in the row space
are 1, 2, and 4, whereas the original matrix only displays the orders 1 and 2.
We end up with an m X n matrix having the same row space V and the
following special structure:
1) For 1 < j < k the jth row has order Uj,
2) If /c < m, the remaining m — k rows are zero (since a nonzero row would
have order >nk, a contradiction).
3) The Ujih column is 8^.
It follows that any linear combination of the first k rows with coefficients
Ci, . . . , Cfc has Cj in the riyth place, and hence cannot be zero unless all the
c/s are zero. These k rows thus form a basis for F, solving problems (1) and (2).
Our final matrix is said to be in row-reduced echelon form. It can be shown to
be uniquely determined by the space V and the above requirements relating its
rows to the orders of the elements of V, Its rows form the canonical basis of V.

2.6
MATRIX COMPUTATIONS
105
A typical row-reduced echelon matrix is shown in Fig. 2.4. This matrix is 8 X 11,
its orders are 1, 4, 5, 7, 10, and its row space has dimension 5. It is entirely 0
below the broken line. The dashes in the first five lines represent arbitrary
numbers, but any change in these remaining entries changes the spanned space V.
We shall now look for the significance of the row-reduction operations from
the point of view of general linear theory. In this discussion it will be convenient
to use the fact from Section 4 that if an n-tuplet in U^ is viewed as an n X 1
matrix (i.e., as a column vector), then the system of linear equations yk =
1^7=1 (^ij^jf ^* = 1, . . . , m, expresses exactly the single matrix equation y = a • x.
Thus the associated linear transformation A G Hom(IR^, W^) is now viewed as
being simply multiplication by the matrix a; y = A(x) if and only if y = a • x.
1 -
Fig. 2.4
We first note that each of our elementary row operations on an m X n
matrix a is equivalent to premultiplication by a corresponding m X m elementary
matrix u. Supposing for the moment that this is so, we can find out what u
is by using the m X m identity matrix e. Since u • a = (u • e) • a, we see that
the result of performing the operation on the matrix a can also be obtained by
premultiplying a by the matrix u • e. That is, if the elementary operation can
be obtained as matrix multiplication by u, then the multiplier is u • e. This
argument suggests that we should perform the operation on e and then see if
premultiplying a by the resulting matrix performs the operation on a.
If the elementary operation is interchanging the z'oth and joth rows, then
performing it on e gives the matrix a with Ukk = 1 for /c f^ Iq and k 7^ jo,
= 1 and Uki = 0 for all other indices. Moreover, examination of
^*o^o
= U
;o»o
the sums defining the elements of the product matrix u • a will show that
premultiplying by this u does just interchange the z'oth and joth rows of any
771 X n matrix a.
In the same way, multiplying the z'oth row of a by c is equivalent to
premultiplying by the matrix u which is the same as e except that Ui^i^ = c.
Finally, multiplying the ^o^h row by x and adding it to the z'oth row is equivalent
to premultiplying by the matrix u which is the identity e except that Ui^j^ is x
instead of 0.

106
FINITE-DIMENSIONAL VECTOR SPACES
2.6
These three elementary matrices are indicated schematically in Fig. 2.5.
Each has the value 1 on the main diagonal and 0 off the main diagonal except as
indicated.
Jo
Fig. 2.5
These elementary matrices u are all nonsingular (invertible). The row
interchange matrix is its own inverse. The inverse of multiplying the yth row by c
is multiplying the same row by 1/c. And the inverse of adding c times thejth
row to the zth row is adding — c times the yth row to the zth row.
If u\ u^, . . . , u^ is a sequence of elementary matrices, and if
b = u^
.p-i .
then b • a is the matrix obtained from a by performing the corresponding
sequence of elementary row operations on a. If u\ . . . , u^ is a sequence which
row reduces a, then r = b • a is the resulting row-reduced echelon matrix.
Now suppose that a is a square m X m matrix and is nonsingular (invertible).
Thus the dimension of the row space is m, and hence there are m different orders
Til, ... , Uk. That is, k = m, and since 1 < rii < 712 < • • • < n^ = m, we
must also have n^ = z, z = 1, . . . , m. Remembering that the n^th column in r is
5*, we see that now the iih column in r is 5* and therefore that r is simply the
identity matrix e. Thus b • a = e and b is the inverse of a.
Let us find the inverse of
'1 21
13 41
[3
by this procedure. The row-reducing sequence is
1
3
i\ (3) |_0 -2J (2) [c
The corresponding elementary matrices are
[-1 ?]' [J -i]
?]
(3)
1
0
3
[I -i\

2.6
MATRIX COMPUTATIONS
107
The inverse is therefore the product
[J
0] _ [i i] r 1 0] _ [-2 1
iJ~Lo -iJL-3 iJ-L f -i
Check it if you are in doubt.
Finally, since b • e = b, we see that we get b from e by applying the same
row operations (gathered together as premultiplication by b) that we used to
reduce a to echelon form. This is probably the best way of computing the inverse
of a matrix. To keep track of the operations, we can place e to the right of a to
form a single m X 2m matrix a | e, and then row reduce it. In echelon form it
will then be the m X 2m matrix e | b, and we can read off the inverse b of the
original matrix a.
Let us recompute the inverse of
by this method. We row reduce
getting
1
0
0"
1
w
w
1
3
[l
ri
2
4
2
-2
0
1
1
0
1
-3
-2
3
2
0]
ij
0"
1_
11
1
2J
(2)
[J
3 1.
2 2.
from which we read off the inverse to be
Finally we consider the problem of computing the determinant of a square
mX m matrix. We use two elementary operations (one modified) as follows:
1') interchanging two rows and simultaneously changing the sign of one of
them;
2) as before, replacing some row a^ by ai — xaj for some j 9^ i.
When applied to the rows of a square matrix, these operations leave the
determinant unchanged. This follows from the properties of determinants listed in
Section 5, and its proof will be left as an exercise. Moreover, these properties
will be trivial consequences of our definition of a determinant in Chapter 7.
Consider, then, a square mX m matrix {aij}. We interchange the first
and pth rows to bring a row of minimal order rii to the top, and change the sign
of the row being moved down (the first row here). We do not make the leading

108
FINITE-DIMENSIONAL VECTOR SPACES
2.6
coefficient of the new first row 1; this elementary operation is not being used
now. We do subtract multiples of the first row from the remaining rows, in order
to make all the remaining entries in the riith column 0. The riith column is now
Ci^S where Ci is the leading coefficient in the first row. And the new matrix has
the same determinant as the original matrix.
We continue as before, subject to the above modifications. We change the
sign of a row moved downward in an interchange, we do not make leading
coefficients 1, and we do clear out the Ujth column so that it becomes Cjb^j,
where Cj is the leading coefficient of the ^th row {I < j < k). As before, the
remaining m — k rows are 0 (if /c < m). Let us call this resulting matrix
semzreduced. Note that we can find the corresponding reduced echelon matrix
from it by k applications of (2); we simply multiply the jth row by 1/cy for
y = 1, . . . , /c. If s is the semireduced matrix which we obtained from a using
(r) and (3), then we shall show below that its determinant, and therefore the
determinant of a also, is the product of the entries on the main diagonal: Y[a=\ ^a-
Recapitulating, we can compute the determinant of a square matrix a by using
the operations (1') and (3) to change a to a semireduced matrix s, and then
taking the product of the numbers on the main diagonal of s.
If we apply this process to
ri 21
3 4|
we get
(3)
[J
(3)
and the determinant is 1 • (-
to
-2) = —2. Our 2 X 2 determinant formula, appHed
gives 1.4-2-3=4-6= -2.
If the original matrix {aij} is nonsingular, so that k = m and n^ = i for
i = 1, . . . , m, then the ^th column in the semireduced matrix is Cj8\ so that
Sjj = Cj, and we are claiming that the determinant is the product YLT= i ^z of the
leading coefficients.
To see this, note that if T is the transformation in Hom(IR^) corresponding
to our semireduced matrix, then T(8^) = Cjb\ so that K^ is the direct sum of n
T-invariant, one-dimensional subspaces, on the ji\\ of which T is multiplication
by Cj. It follows from (c) and (d) of our list of determinant properties that
A(T) = III Cj = III Sjj. This is nonzero.
On the other hand, if {aij} is singular, so that k = d(V) < m, then the mth
row in the semireduced matrix is 0 and, in particular, Smm = 0. The product
IIz Sii is thus zero. Now, without altering the main diagonal, we can subtract
multiples of the columns containing the leading row entries (the columns with

2.6
MATRIX COMPUTATIONS
109
indices Uj) to make the mth column a zero column. This process is equivalent
to postoultiplying by elementary matrices of type (2) and, therefore, again
leaves the determinant unchanged. But now the transformation S of this matrix
leaves W^~^ invariant (as the span of 5S . . . , 5"^"^ in IR^) and takes b"^ to 0,
so that A(aS) = 0 by (c) in the list of determinant properties. So again the
determinant is the product of the entries on the main diagonal of the semi-
reduced matrix, zero in this case.
We have also found that a matrix is nonsingular (invertible) if and only if its
determinant is nonzero.
EXERCISES
6.1 Compute the canonical basis of the row space of
1
3
1
0
1
1
1
2
2
-3
4
2
2
-2
1
3
0
— 1
4
3
0
2
2
4
-3.
5
4
2
6.2 Do the same for
6.3 Do the same for the above matrix but with a different first choice.
6.4 Calculate the inverse of
"l 2 3]
12 3 4
[3 4 7J
by row reduction. Check your answer by multiplication.
6.5 Row reduce
yi
2/2
ys.
How does the fourth column in the row-reduced matrix compare with the inverse of
computed in the above exercise? Explain.
6.6 Check whether or not <1, 1, 1, 1>, <1, 2, 3, 4>, <0, 1, 0, 1>, and
<4, 3, 2, 1 > are linearly independent by row reducing. Part of one of the
row-reducing operations is unnecessary for this check. What is it?

110
FINITE-DIMENSIONAL VECTOR SPACES
2.6
6.7 Let us call a A:-tuple of vectors {ai}i in U^ canonical if the kX n matrix a with
ai as its iih row for all i is in row-reduced echelon form. Supposing that an n-tuple {
is in the row space of a, we can read off what its coordinates are with respect to the
above canonical basis. What are they? How then can we check whether or not an
arbitrary n-tuple { is in the row space?
6.8 Use the device of row reducing, as suggested in the above exercise, to determine
whether or not 8^ = < 1, 0, 0, 0> is in the span of -<1,1, 1,1>, -<1,2, 3, 4>-, and
<2, 0, 1, —1>. Do the same for <1,2, 1,2>, and also for <1, 1,0, 4>.
6.9 Supposing that a f^ 0, show that
[: :]
is invertible if and only U ad — be 9^ 0 by reducing the matrix to echelon form.
6.10 Let a be an m X n matrix, and let u be the nonsingular matrix that row reduces
a, so that r = u • a is the row-reduced echelon matrix obtained from a. Suppose that r
has m — k > 0 zero rows at the bottom (the kth row being nonzero). Show that the
bottom m — k rows of u span the annihilator (range A)° of the range of A. That is,
y = ax for some x if and only if
m
S CiVi = 0
for each m-tuple c in the bottom m — k rows of u. [Hint: The bottom row of r is
obtained by applying the bottom row of u to the columns of a.]
6.11 Remember that we find the row-reducing matrix u by applying to the mX m
identity matrix e the row operations that reduce a to r. That is, we row reduce the
mX (n-\~ m) juxtaposition matrix a ( e to r ( u. Assuming the result stated in the
above exercise, find the range oi A G Hom(IR^) as the null space of a functional if the
matrix of A is
1 2 31
12 3 4
1_3 5 7J
6.12 Similarly, find the range of A if the matrix of A is
1
2
0
4
1
0
2
2
1
1
1
3
6.13 Let a be an ?n X n matrix, and let a be row reduced to r. Let A and R be the
corresponding operators in Hom(IR^ W^) [so that A(x) = a • x]. Show that A and R
have the same null space and that A* and R* have the same range space.
6.14 Show that solving a system of m linear equations in n unknowns is equivalent
to solving a matrix equation
k = tx
for the n-tuple x, given the mX n matrix t and the m-tuple k. Let T G Hom(IR", W^)
be multiplication by t. Review the possibilities for a solution from our general linear
theory for T (range, null space, affine subspace).

2.7 THE DIAGONALIZATION OF A QUADRATIC FORM 111
6.15 Let b = c I d be the mX (n-{- p) matrix obtained by juxtaposing the mX n
matrix c and the mX p matrix d. If a is an / X m matrix, show that
a • b = ac I ad.
State the similar result concerning the expression of b as the juxtaposition of n column
m-tuples. State the corresponding theorem for the "distributivity" of right
multiplication over juxtaposition.
6.16 Let a be an m X n matrix and k a column m-tuple. Let b | 1 be the niX (n + 1)
matrix obtained from the m X (n+ 1) juxtaposition matrix a | k by row reduction.
Show that a • X = k if and only if b • x = L Show that there is a solution x if and only
if every row that is zero in b is zero in L Restate this condition in terms of the notion
of row rank.
6.17 Let b be the row-reduced echelon matrix obtained from an m X n matrix a.
Thus b = u • a, where u is nonsingular, and B and A have the same null space (where
B G Hom(lR'*, R^) is multiplication by b). We can read off from b a basis for a sub-
space W C^^ such that B \ W is slu isomorphism onto range B. What is this basis?
We then know that the null space A^" of jB is a complement of W. One complement of W,
call it M, can be read off from W. What is M?
6.18 Continuing the above exercise, show that for each standard basis vector 8i in M
we can read off from the matrix b a vector ai in W such that 5' — ai G A^. Show that
these vectors {8^ — ai} form a basis for A^".
6.19 We still have to show that the modified elementary row operations leave the
determinant of a square matrix unchanged, assuming the properties (a) through (e)
from Section 5. First, show from (a), (c), (d), and (e) that if T in Hom R^ is defined
by T{8^) = 52 and T(8^) = —8\ then A(T) = 1. Do this by a very simple
factorization, T = R ° S, where (e) can be applied to S. Conclude that a type (1') elementary
matrix has determinant 1.
6.20 Show from the determinant property (b) that an elementary matrix of type (2)
has determinant 1. Show, therefore, that the modified elementary row operations on a
square matrix leave its determinant unchanged.
*7. THE DIAGONALIZATION OF A QUADRATIC FORM
As we mentioned earlier, one of the crucial problems of linear algebra is the
analysis of the "structure" of a linear transformation T in Hom V. From the
point of view of bases, every theorem in this area asserts that with the choice
of a special basis for V the matrix of T can be given the such-and-such simple
form. This is a very difficult part of the subject, and we are only making
contact with it in this book, although Theorem 5.5 of Chapter 1 and its corollary
form a cornerstone of the structural results.
In this section we are going to solve a simpler problem. In the above
language it is the problem of choosing a basis for V making simple the matrix of a
transformation T in Hom(F, F*). Such a transformation is equivalent to a
bilinear functional on V (by Theorem 6.1 of Chapter 1 and Theorem 3.2 of this
chapter); we shall tackle the problem in this setting.

112 FINITE-DIMENSIONAL VECTOR SPACES 2.7
Let F be a finite-dimensional real vector space, and let co: F X F —> IR be
a bilinear functional. If {az}i is a basis for V, then co determines a matrix
tij = co(aij aj). We know that if corf(^) = co(f, 77), then co^, G F* and 77 i-^ co^, is a
linear mapping T from F to F*. We leave it as an exercise for the reader to
show that {tij} is the matrix of T with respect to the basis {a^} for F and its
dual basis for F* (Exercise 4.1).
If f = L^ XiOLi and 77 = Li 2/i«i> then
w(f> ^) = Z) XiyjO)(ai, aj) = X) ^zi^zl/i-
In particular, if we set g(Q = co(f, f), then g(Q = X!^,^ UjXiXj is a homogeneous
quadratic polynomial in the coordinates Xi.
For the rest of this section we assume that co is symmetric: co(f, 77) =
co(77, f). Then we can recover co from the quadratic form q by
as the reader can easily check. In particular, if the bilinear form co is not
identically zero, then there are vectors f such that g(Q = co(f, f) 9^ 0.
What we want to do is to show that we can find a basis {aJ 1 for F such that
co(ai, aj) ^ 0 if i 7^ j and co(ai, ai) has one of the three values 0, ± 1.
Borrowing from the standard usage of scalar product theory (see Chapter 5), we say
that such a basis is orthonormal. Our proof that an orthonormal basis exists will
be an induction on n = dim F. If n = 1, then any nonzero vector /? is a basis,
and if co(/3,13) 9^ 0, then we can choose a = xjS so that a:^co(/3,13) = o)(a, a) =
±1, the required value of x obviously being x = |co(/3,13)\~^'^. In the general
case, if co is the zero functional, then any basis will trivially be orthonormal, and
we can therefore suppose that co is not identically 0. Then there exists a fi such
that co(/3,13) 7^ 0, as we noted earlier. We set an = xfi^ where x is chosen to
make qian) = o)(an, an) = ±1. The nonzero linear functional/(f) = co(f, an)
has an (n — 1)-dimensional null space N^ and if we let co' be the restriction of
ootoNxNf then co' has an orthonormal basis {aJ i~^ by the inductive
hypothesis. Also co(ai, an) = o)(an, oti) = 0 if z < n, because ai is in the null space of/.
Therefore, {az} 1 is an orthonormal basis for co, and we have reached our goal:
Theorem 7.1. If co is a symmetric bilinear functional on a finite-dimensional
real vector space F, then F has an co-orthonormal basis.
For an co-orthonormal basis the expansion co(f, 77) = 2] ^iVj^if^ii «>) reduces to
n
w(f, v) = Yl XiyiQ(oLi),
z = l
where qiai) = ±1 or 0. If we let Vi be the span of those basis vectors ai for
which q(ai) = 1, and similarly for F_i and Fq, then we see that q(^) > 0 for
every nonzero f in Fi, g(f) < 0 for every nonzero vector f in F_i, and q = 0

2.7 THE DIAGONALIZATION OF A QUADRATIC FORM 113
on Vq. Furthermore, F = Fi 0 F_i 0 Fq, and the three subspaces are
co-orthonormal to each other (which means that co.(f, 77) = 0 if ^ G Vi and
7} G F_i, etc.). Finally, g(f) < 0 for every f in F_i 0 Vq.
If we choose another orthonormal basis {I3i} and let TFi, TF_i, and TFq be
its corresponding subspaces, then Wi may be different from Vi, but their
dimensions must be the same. For Wi n (F_i 0 Fq) = {0}, since any nonzero f
in this intersection would yield the contradictory inequalities q(^) > 0 and
q(0 ^ 0- Thus Wi can be extended to a complement of F_i 0 Fq, and since
Fi zs a complement, we have (i(TFi) < d(Vi). Similarly, d(Vi) < d(TFi),
and the dimensions therefore are equal. Incidentally, this shows that TFi is a
complement of F_i 0 Vq. In exactly the same way, we find that d(TF_i) =
d(F_i) and finally, by subtraction, that d(Wo) = d(Vo). It is conventional to
reorder an co-orthonormal basis {aj 1 so that all the a/s with q(ai) = 1 come first,
then those with g(a/) = — 1, and finally those with g(a^) = 0. Our results
above can then be stated as follows:
Theorem 7.2. If co is a symmetric bilinear functional on a finite-dimensional
space F, then there are integers n and p such that if {aiJT is any co-ortho-
normal basis in conventional order, and if f = Hi* Xiai, then
q(0 = xi^ \- Xp — Xp^i — . • . — Xp^n
p p-j-n
= / J Xi / J X{.
1 P + 1
The integer p — n is called the signature of the form q (or its associated
symmetric bilinear functional co), and p + n is its rank. Note that p + n is the
dimension of the column space of the above matrix of g, and hence equals the
dimension of the range of the related linear map T. Therefore, p + n is the
rank of every matrix of q.
An inductive proof that an orthonormal basis exists doesn^t show us how to
find one in practice. Let us suppose that we have the matrix {tij} of co with
respect to some basis {oLi}\ before us, so that co(f, 77) = Hxiyjtij, where
f = Hi Xiai, r] = Hi Vi^if and tij = co(a^, a^), and we want to know how to go
about actually finding an orthonormal basis {/3j i- The main problem is to find
an orthogonal basis; normalization is then trivial. The first objective is to find
a vector 13 such that co(/3, /?) 9^ 0. If some ta = co(ai, a^) is not zero, we can take
13 = ai. If all tii = 0 and the form co is not the zero form, there must be some
lij 9^ 0, say ti2 9^ 0. If we set Ti = ai + a2 and 7^- = ai for i > 1, then {7^} 1
is a basis, and the matrix s = {s^J of co with respect to the basis {7^} has
sii = w(^i, ^1) = w(«i + «2, «i + 0L2) = til + 2^12 + ^22 = 2^12 ^ 0.
Similarly, Sij = tij if either i or j is greater than 1.
For example, if co is the bilinear form on U^ defined by co(x, y) = 0:12/2 +
X22/1, then its matrix tij = o)(8\ 8^) is
fO ll
1 oh

114 FINITE-DIMENSIONAL VECTOR SPACES 2.7
and we must change the basis to get ^n f^ 0. According to the above scheme,
we set Ti = 5^ + 8^ and 72 = 8^ and get the new matrix Sij = co(7^, 7^),
which works out to
Li oj
The next step is to find a basis for the null space of the functional co(f, 7i) =
^ XiSu. We do this by modifying 72, ... , 7^; we replace Jj by Jj + c7i and
calculate c so that this vector is in the null space. Therefore, we want 0 =
co(7y + c7i, 7i) = Sij + csii, and so c = —SiyAii. Note that we cannot take
this orthogonalizing step until we have made Sn f^ 0. The new set still spans
and thus is a basis, and the new matrix {vij} has rn f^ 0 and Vij = Vji = 0 for
j > 1. We now simply repeat the whole procedure for the restriction of co to this
(n — l)-dimensional null space, with matrix {rij : 2 < i, j < n} j and so on.
This is a long process, but until we normalize, it consists only of rational
operations on the original matrix. We add, subtract, multiply, and divide, but we
do not have to find roots of polynomial equations.
Continuing our above example, we set /3i = 7i, but we have to replace 72
by /32 = ^2 — (si2Aii)^i = ^2 — i^i- The final matrix Uj = o)(l3i, pj)
has
^11 = Sll = 2, ri2 = C0(/3i, 132) = C0(7i, 72 - i7i) = Si2 - iSll = 0,
and r22 = ^(^2 — i^i? ^2 — 2^1) = ^22 — ^12 + Sii/4 = —J, so that
Ol
{rij}
2J
The final basis is/3i = 7^= 8^ + 8^2indp2 = ^2 - i^i = 8^ - ^8^ + 8^) =
(52 - 5i)/2.
The steps we had to take above are reminiscent of row reduction, but since
we are changing bases simultaneously in the domain and range spaces of the
transformation T:V—^V* associated with co, each step involves simultaneously
premultiplying and postmultiplying by an elementary matrix. That is, we are
simultaneously row and column reducing. It should be intuitively clear that this
has to be the case if we are to operate on a symmetric matrix in such a way as
to keep it symmetric.
For additional information about quadratic forms, we go back to the change
of basis formula for the matrix of a transformation: t" = b • t' • a~^ Here the
transformation T associated with the form co is from F to F*, and sob = (a*)~\
according to our calculations in Section 4. Now one of the properties of the
determinant function is that A(T*) = A(T'), and so A(a*) = A(a). Therefore,
if t and s are the matrices of a quadratic form with respect to a first and second
basis in V, and if a is the change of basis matrix, then s = (a*)""^ • t • a"'^ and
A(s) = (A(a~^))^ A(<). Therefore, a quadratic form has parity. If it is non-
singular, then its determinant is either always positive or always negative, and

2.7 THE DIAGONALIZATION OF A QUADRATIC FORM 115
we can call it even or odd. In our continuing example, the beginning and final
matrices
0 1
1 0
and
both have determinant —1.
In the two-dimensional case, the determinant of a form with respect to an
orthonormalized basis is +1 if the diagonal elements are both +1 or both —1,
and —1 if they are of opposite sign. We can therefore read off the signature of a
nonsingular form over a two-dimensional space ivithout orthonormalizing. If the
determinant ^11^22 ~ (^12)^ is positive, the signature is d=2, and we can
determine which by looking at ^n (since ^n is then unchanged by our orthogonalizing
procedure). Thus the signature is +2 or —2 depending on whether ^n > 0 or
^11 < 0. If the determinant is negative, then the signature is 0. Thus the
signature of the form co(x, y) = ^12/2 + ^22/1 > with matrix
ro r
Li 0.
is known to be 0, without any calculation.
Theorems 7.1 and 7.2 are important for the classification of critical points
of real-valued functions on vector spaces. We shall see in Section 3.16 that the
second differential of such a function F is a symmetric bilinear functional, and
that the signature of its form has the same significance in determining the
behavior of F near a point at which its first differential is zero that the sign of
the second derivative has in the elementary calculus.
A quadratic form q is said to be definite if q(^) is never zero except for ^ = 0.
Then g(f) must always have the same sign, and q is accordingly called positive
definite or negative definite. Looking back to Theorem 7.2, it should be obvious
that q is positive definite only \i p = d{V) and n = 0, and negative definite
only if n = d{V) and p = 0. A symmetric bilinear functional whose associated
(quadratic form is positive definite is called a scalar product. This is a very
important notion on general vector spaces, and the whole of Chapter 5 is
devoted to developing some of its implications.

CHAPTER 3
THE DIFFERENTIAL CALCULUS
Our algebraic background is now adequate for the differential calculus, but we
still need some multidimensional limit theory. Roughly speaking, the
differential calculus is the theory of linear approximations to nonlinear mappings,
and we have to know what we mean by approximation in general vector settings.
We shall therefore start this chapter by studying the notion of a measure of
length, called a nornif for the vectors in a vector space V. We can then study
the phenomenon suggested by the way in which a tangent plane to a surface
approximates the surface near the point of tangency. This is the general theory
of unique local linear approximations of mappings, called differentials. The
collection of rules for computing differentials includes all the familiar laws of
the differential calculus, and achieves the same goal of allowing complicated
calculations to be performed in a routine way. However, the theory is richer
in the multidimensional setting, and one new aspect which we must master is
the interplay between the linear transformations which are differentials and their
evaluations at given vectors, which are directional derivatives in general and
partial derivatives when the vectors belong to a basis. In particular, when the
spaces in question are finite-dimensional and are replaced by Cartesian spaces
through a choice of bases, then the differential is entirely equivalent to its matrix,
which is a certain matrix of partial derivatives called the Jacobian matrix of the
mapping. Then the rules of the differential calculus are expressed in terms of
matrix operations.
Maximum and minimum points of real-valued functions are found exactly
as before, by computing the differential and setting it equal to zero. However,
we shall neglect this subject, except in starred sections. It also is much richer
than its one-variable counterpart, and in certain infinite-dimensional situations
it becomes the subject called the calculus of variations.
Finally, we shall begin our study of the inverse-mapping theorem and the
implicit-function theorem. The inverse-mapping theorem states that if a mapping
between vector spaces is continuously differentiable, and if its differential at a
point a is invertible (as a linear transformation), then the mapping itself is
invertible in the neighborhood of a. The implicit-function theorem states that if
a continuously differentiable vector-valued function G of two vector variables
is set equal to zero, and if the second partial differential of G is invertible (as a
linear mapping) at a point <a, P> where G(a, /?) = 0, then the equation
116

3.1
REVIEW IN
117
^(^v) = 0 can be solved for rj in terms of f near this point. That is, there is a
uniquely determined mapping tj = F(^) defined near a such that 13 = F(a) and
such that G(f, ^(O) = 0 in the neighborhood of a. These two theorems are
fundamental to the further development of analysis. They are deeper results
than our work up to this point in that they depend on a special property of
vector spaces called completeness; we shall have to put off part of their proofs to
the next chapter, where we shall study completeness in a fairly systematic way.
In a number of starred sections at the end of the chapter we present some
liarder material that we do not expect the reader to master. However, he should
try to get a rough idea of what is going on.
I. REVIEW IN U
l^]very student of the calculus is presumed to be familiar with the properties of
the real number system and the theory of limits. But we shall need more than
familiarity at this point. It will be absolutely essential that the student
understand the €-definitions and be able to work with them.
To be on the safe side, we shall review some of this material in the setting of
limits of functions; the confident reader can skip it. We suppose that all the
functions we consider are defined at least on an open interval containing a,
(except possibly at a itself. The need for this exception is shown by the difference
(juotients of the calculus, which are not defined at the point near which their
i)ehavior is crucial.
Definition. f(x) approaches I as x approaches a (in symbols, f(x)
X -^ a) U for every positive e there exists a positive 8 such that
0 <\x- a\ < 8 =^ \f(x) -l\<€.
I as
We also say that I is the limit of f(x) as x approaches a and write
\imx-^af(x) = l. The displayed statement in the definition is understood to be
universally quantified in a:, so that the definition really begins with the three
(juantifiers (V€^^)(35^^)(Va:). These prefixing quantifiers make the definition
sound artificial and unidiomatic when read as
ordinary prose, but the reader will remember from
our introductory discussion of quantification that
(his artificiality is absolutely necessary in order
for the meaning of the sentence to be clear and
unambiguous. Any change in the order of the
quantifiers (Ve) (3 8) (Vx) changes the meaning of the
statement.
The meaning of the inner universal
quantification
(Va:)(0 <\x~ a\ < 8 =^ \f{x) - l\ < e)
is intuitive and easily pictured (see Fig. 3.1).

118 THE DIFFERENTIAL CALCULUS 3.1
For all X closer to a than 8 the value of / at a: is closer to I than e. The
definition begins by stating that such a positive 8 can be found for each positive e.
Of course, 8 will vary with e; if € is made smaller, we will generally have to
go closer to a, that is, we will have to take 8 smaller, before all the values of /
on (a — 5, a + 5) — {a} become e close to I.
The variables ^e^ and '8^ are almost always restricted to positive real
numbers, and from now on we shall let this restriction be implicit unless there seems
to be some special call for explicitness. Thus we shall write simply (V€)(35) . . .
The definition of convergence is used in various ways. In the simplest
situations we are given one or more functions having limits at a, say, f(x) —> u
and g(x) -^ v, and we want to prove that some other function h has a limit iv
at a. In such cases we always try to find an inequality expressing the quantity we
wish to make small, \h(x) — iv], in terms of the quantities which we know can be
made small, \f(x) — u\ and \g(x) — v\.
For example, suppose that h = f + g. Since f(x) is close to u and g(x) is
close to Vy clearly h(x) is close toio = u + v. But how close? Since h(x) — iv =
{f{x) — u) + (g(x) — v), we have
\Hx) - iv\ < \f(x) -u\ + \g{x) - v\.
From this it is clear that in order to make \h(x) — iv\ less than e it is sufficient
to make each of \f(x) — u\ and \g(x) — v\ less than e/2. Therefore, given any €,
we can take 8i so that 0 < \x — a\ < 8i =^ \f(x) — u\ < e/2, and ^2 so that
0 < \x — a\ < 82 =^ \g(x) — v\ < e/2, and we can then take 8 as the smaller
of these two numbers, so that if 0 < |a: — a| < 5, then both inequalities hold.
Thus
0 <\x- a\ < 8 ^ \hix) -w\< \f(x) -u\ + \g(x) -v\ <^ + ^= e,
and we have found the desired 8 for the function h.
Suppose next that u 9^ 0 and that h = 1//. Clearly, h(x) is close to to = 1/u
when/(a:) is close to li, and so we try to express h(x) — iv in terms oi f{x) — u.
Thus
1 I _u- fix)
h(x) — w =
f{x) u f(x)u
and so \h(x) — io\ < \f(x) — u\/\f(x)u\. The trouble here is that the
denominator is variable, and if it should happen to be very small, it might cancel the
smallness of \f(x) — u\ and not force a small quotient. But the answer to this
problem is easy. Since f(x) is close to u and u is not zero, f(x) cannot be close to
zero. For instance, if f(x) is closer to u than \u\/2, then f(x) must be farther
from 0 than \u\/2. We therefore choose 81 so that 0 < \x — a\ < 81 =>
\fix) — u\ < \u\/2, from which it follows that \f(x)\ > \u\/2. Then
\h(x) -w\ < 2\f(x) - u\/\u\^

3.1 REVIEW IN IR 119
and now, given any €, we take ^2 so that
0 < |a: — a| < 52 => l/W — u\ < e\u\^/2.
Again taking 8 as the smaller of 5i and ^2, so that both inequalities will hold
simultaneously when 0 < |a: — a| < 5, we have
0 <\x - a\ < 8 =^ \hix) -w\< 2\f(x) - u\/\u\^ < 2e\u\y2\u\^ = e,
and again we have found our 8 for the function h.
We have tried to show how one would think about these situations. The
actual proof that would be written down would only show the choice of 8. Thus,
Lemma 1.1. If/(a:) -^ u and g(x) -^ v as x -^ a, then f(x) + g(x) -^ u + v
as X -^ a.
Proof. Given €, choose 8i so that 0<|a: — a|<5i=^ \f(x) — a\ < e/2
(by the assumed convergence of f to u at a), and, similarly, choose ^2 so that
0 < \x — a\ < 82 =^ loi^) — v\ < €/2. Take 8 as the smaller of 81 and 52-
Then
0<\x- a\ < 8 ^ \ (fix) + g(x)) - (u + v)\
< \f(x) -u\ + \g{x) -v\ < e/2 + e/2 = e.
Thus we have proved that for every e there is a 5 such that
0 <\x-a\ < 8 ^ \{f(x) + g(x)) - (u + v)\ < €,
and we are done. D
In addition to understanding e-techniques in limit theory, it is necessary to
understand and to be able to use the fundamental property of the real number
system called the least upper hound property. In the following statement of the
property the semi-infinite interval (— 00, a] is of course the subset {xEl^'.x <a].
If i4 is a nonempty subset of R such that ^4 C (—00, a] for some a, then
there exists a uniquely determined smallest number h such that ^4 C (— 00, 6].
A number a such that ^4 C (— 00, a] is called an upper hound of A; clearly, a
is an upper bound of A if and only if every a: in ^4 is less than or equal to a.
A set having an upper bound is said to be bounded above. The property says that
u nonempty set A which is bounded above has a least upper bound (lub). If
we reverse the order relation by multiplying everything by —1, then we have the
alternative formulation which asserts that a nonempty subset of R that is
i)ounded below has a greatest lower hound (gib). The least upper bound of the
interval (0, 1) is 1. The least upper bound of [0, 1] is also 1. The greatest lower
hound of {1/n : n a positive integer} is 0. Furthermore, lub {x : a: is a positive
rational number and x"^ < 2} = \/2, gib {e^ :x E:R} =0, and lub {e^ : x is
rational and x < \/2} = e^^

120 THE DIFFERENTIAL CALCULUS 3.1
EXERCISES
1.1 Prove that if f(x) —> / and f(x) —> ?n as x —> a, then / = m. We can therefore
talk about the limit of / as x —> a.
1.2 Prove that if/(x) —> / and g(x) —> m (as a; —> a), then /(x) gf(x) —> //?t as a; —> a.
1.3 Prove that |x — a| < \a\/2=^ \x\ > \a\/2.
1.4 Prove (in detail) the greatest lower bound property from the least upper bound
l)roperty.
1.5 Show that lub A = a; if and only if x is an upper bound of .1 and, for every
positive e, X — e is not an upper bound of .1.
1.6 Let A and B be subsets of R that are nonempty and bounded above. Show that
A + J5 is nonempty and bounded above and that lub (A + B) = lub A + lub B.
1.7 Formulate and prove a correct theorem about the least upper bound of the
product of two sets.
1.8 Define the notion of a one-sided limit for a function whose domain is a subset of IR.
For example, we want to be able to discuss the limit of f(x) as x approaches a from
below, which we might designate
lim/(x).
xt a
1.9 If the domain of a real-valued function / is an interval, say [a, b], we say that / is
an increasing function if
X < y =^ fix) <f{y).
Prove that an increasing function has one-sided limits everywhere.
1.10 Let [a, b] be a closed interval in R, and let/: [a, 6] —> IR be increasing. Show that
\imx-^yf{x) = f(y) for all y in [a, b] (/ is continuous on [a, b]) if and only if the range
of / does not omit an3^ subinterval (c, d) C lf(ci),f(b)]. [Hint: Suppose the range omits
(c, d), and set y = lub {x :f(x) < c}. Then/(x) -{^ f{y) as x —> y.]
1.11 A set that intersects every open subinterval of an interval [s, t] is said to be
dense in [s, t]. Show that if/: [a, 6] —> IR is increasing and range/is dense in [f{a),f(b)],
then range/ = lf(a),f(b)]. (For an}^ 2 between/(a) and/(6) set 2/ = lub {x:f(x) < z],
etc.)
1.12 Assuming the results of the above two exercises, show that if / is a continuous
strictly increasing function from [a, b] to R, and if r = /(a) and s = f(b), then f~^ is a
continuous strictly increasing function from [r, s] to R. [A function / is continuous if
f(x) —^f{y) as X —> 2/ for every y in its domain; it is strictly increasing U x < y =>
fix) <f{y).]
1.13 Argue somewhat as in Exercise Lll above to prove that if /: [a, b] —^ R is
continuous on [a, 6], then the range of / includes lf(a),f(b)]. This is the intermediate-
value theorem.
1.14 Suppose the function (/: IR —> IR satisfies q{xy) = q(x) q(y) for all x^yEiR.
Note that q(x) = x"* (n a positive integer) and q(x) = \x\'' (r any real number) satisfy
this "functional equation". So does q(x) = 0 (r = —00 ?). Show that if q satisfies the
functional equation and q{x) > 1 for x > 1, then there is a real number r > 1 such
that q(x) = x^ for all positive x.

3.2 NORMS 121
1.15 Show that if q is continuous and satisfies the functional equation q{xy) =
^l{^) Q(y) for Q-ll ^, 2/ E f^, and if there is at least one point a where q(a) 9^ 0, I, then
<l(x) = x" for all positive x. Conclude that if also q is nonnegative, then q{x) = Ixl"" on IR.
1.16 Show that if q{x) = \x\% and if q{x + y) < q{x) + q{y), then r < 1. (Try y = I
and X large; what is q'{x) like if r > 1?)
2. NORMS
Fn the limit theory of IR, as reviewed briefly above, the absolute-value function
is used prominently in expressions like ^\x — y^ to designate the distance
i)etween two numbers, here between x and y. The definition of the convergence
of/(a:) to u is simply a careful statement of what it means to say that the distance
\f(x) — u\ tends to zero as the distance \x — a\ tends to zero. The properties of
|.r| which we have used in our proofs are
1) \x\ > Oif a: F^ 0, and |0| = 0;
2) \xy\ = \x\ \y\]
3) \x + y\< \x\ + \y\.
The limit theory of vector spaces is studied in terms of functions called
norms, which serve as multidimensional analogues of the absolute-value function
on U. Thus, if p: V —^ IR is a norm, then we want to interpret p(a) as the "size"
of a and p{a — ^) as the "distance" between a and ^. However, if V is not
one-dimensional, there is no one notion of size that is most natural. For example,
if / is a positive continuous function on [a, 6], and if we ask the reader for a
number which could be used as a measure of how "large" / is, there are two
possibilities that will probably occur to him: the maximum value of/and the area
under the graph of/. Certainly, / must be considered small if max / is small.
Hut also, we would have to agree that / is small in a different sense if its area is
small. These are two examples of norms on the vector space V = 6 ([a, h]) of all
continuous functions on [a, h]:
p{f) = max {\m\ :tG[a, b]} and q{f) = [' |/(0| dt.
Ja
Note that / can be small in the second sense and not in the first.
In order to be useful, a notion of size for a vector must have properties
analogous to those of the absolute-value function on R.
Definition. A norm is a real-valued function p on a vector space V such that
nl. p(a) > 0 if a F^ 0 (positivity);
n2. p{xa) = \x\p{a) for all a G F, a: G IR (homogeneity);
n3. p{a -\- P) < p(a) + j){fi) for all a, j^ G F (triangle inequality).
A normed linear space (nls), or normed vector space, is a vector space V
t()^j;ether with a norm p on V. A normed linear space is thus really a pair

122 THE DIFFERENTIAL CALCULUS 3.2
<V, p> f but generally we speak simply of the normed linear space V, a definite
norm on V then being understood.
It has been customary to designate the norm of a by ||a||, presumably to
suggest the analogy with absolute value. The triangle inequality n3 then
becomes ||a + i3|| < ||a|| + ||i3||, which is almost identical in form with the basic
absolute-value inequality |a: + 2/| < \x\ + \y\. Similarly, n2 becomes \\xa\\ =
\x\ ||a||, analogous to \xy\ = \x\ \y\ in U. Furthermore, \\a — ^\\ is similarly
interpreted as the distance between a and p. This is reasonable since if we set
a = ^ — 7] and ^ = v — ^y then n3 becomes the usual triangle inequality of
geometry:
U- fll < U- v\\ + \\v- fl|.
We shall use both the double bar notation and the "p"-notation for norms; each
is on occasion superior to the other.
The most commonly used norms on IR^ are ||x||i = JZi \xi\f the Euclidean
norm ||x||2 = (Uli^fy^) ai^d ||x||oo = max {|a:e|}e. Similar norms on the
infinite-dimensional vector space e([a, h]) of all continuous real-valued functions
on [a, h] are
11/111= C \m\dt,
Ja
\\f\\2 = (^[\m\u^"\
ll/IU = max {|/(0| : a < ^ < 6}.
It should be easy for the reader to check that || ||i is a norm in both cases
above, and we shall take up the so-called uniform norms || ||oo in the next
paragraph. The Euclidean norms || II2 are trickier; their properties depend on
scalar product considerations. These will be discussed in Chapter 5. Meanwhile,
so that the reader can use the Euclidean norm || II2 on IR^, we shall ask him to
prove the triangle inequality for it (the other axioms being obvious) by brute
force in an exercise. On U itself the absolute value is a norm, and it is the only
norm to within a constant multiple.
We can transfer the above norms on IR^ to arbitrary finite-dimensional
spaces by the following general remark.
Lemma 2.1. If p is a norm on a vector space W and T is an infective linear
map from a vector space V to TT, then p o 7" is a norm on V.
Proof. The proof is left to the reader.
Uniform norms. The two norms || ||oo considered above are special cases of a
very general situation. Let A be an arbitrary nonempty set, and let (B(i4, IR)
be the set of all bounded functions /: ^4 -^ IR. That is, / G (B(i4, IR) if and only if
/ G IR^ and range / C [—b, h] for some 6 G IR. This is the same as saying that
range |/| C [0, 6], and we call any such h a hound of |/|. The set (B(i4, IR) is a

3.2 NORMS 123
vector space V, since if |/| and |^| are bounded by h and c, respectively, then
l^/+ vol is bounded by \x\h + \y\c. The uniform norm \\f\\oo is defined as the
smallest bound of |/|. That is,
||/|U=lub{|/(p)|:pe4}.
Of course, it has to be checked that || ||oo is a norm. For any p in A,
\Kp) + g(p)\ < \Kp)\ + \9(P)\ < ll/IU + \\g\U.
Thus ll/lloo + ll^lloo is a bound of |/ + g\ and is therefore greater than or equal to
the smallest such bound, which is ||/ + ^||x. This gives the triangle inequality.
Next we note that if a: f^ 0, then h bounds |/| if and only if \x\h bounds |a:/|, and
it follows that ||a:/||oo = |a:| ||/||oo. Finally, ||/|U > 0, and ||/|U = 0 only if / is
the zero function.
We can replace R by any normed linear space W in the above discussion.
A function f: A -^ W is bounded by h if and only if ||/(7))|| < h for all p in Aj
and we define the corresponding uniform norm on (^(A, W) by
||/|U = Iub{||/(p)||:pe4}.
If / G e([0, 1]), then we know that the continuous function / assumes the
least upper bound of its range as a value (that is, / "assumes its maximum value"),
so that then ||/||oo is the maximum value of |/|. In general, however, the definition
must be given in terms of lub.
Balls. Remembering that \\a — f|| is interpreted as the distance from a to f, it is
natural to define the open hall of radius r about the center a as {f : ||a — f || < r}.
We designate this ball Br(a). Translation through 13 preserves distance,
\\TpU) - ^^('?)ll = ll(^ + ^) - ('? + ^)ll = U-vl
and therefore f G Br(a.) if and only if ^ + 13 G Br(a + 13), That is, translation
through ^ carries Br(a) into Br(a + 13): Tp[Br(a)] = Br(a + ^). Also, scalar
multiplication by c multiplies all distances by c, and it follows in a similar way
that cBr(a) = Bcr(ca).
Although Br (a) behaves like a ball, the actual set being defined is different
for different norms, and some of them "look unspherelike". The unit balls about
the origin in U^ for the three norms || ||i, || II2, and || ||oo are shown in Fig. 3.2.
A subset i4 of a nls V is hounded if it lies in some ball, say Br (a). Then it
also lies in a ball about the origin, namely Br+llall(0). This is simply the fact that
if II f — a|| < r, then ||f|| < r+ ||a||, which we get from the triangle inequality
upon rewriting ||f|| as ||(f — a) + a\\.
The radius of the largest ball about a vector ^ which does not touch a set A
is naturally called the distance from 13 to A, It is clearly gib {|| ^ — ^\\ : ^ G A}
(see Fig. 3.3).

124 THE DIFFERENTIAL CALCULUS
3.2
p{^,A)=r
Fig. 3.2
Fig. 3.3
Fig. 3.4
A point a is an interior point of a set A if some ball about a is included in A.
This is equivalent to saying that the distance from a to the complement of A is
positive (supposing that A is not the whole of F), and should coincide with
the reader^s intuitive notion of what an "inside" point should be. A subset A
of a normed linear space is said to be open if every point of A is an interior
point.
If our language is to be consistent, an open ball should be an open set. It is:
if a G Br(^), then \\a — I3\\ < r, and then B§(a) C Br(l3), provided that 8 < r —
\\a — i^ll, by virtue of the triangle inequality (see Fig. 3.4). The reader should
write down the detailed proof. He has to show that if f G B§(a)f then f G Br{fi).
Our intuitions about distances are quite trustworthy, but they should always be
checked by a computation. The reader probably can see by a mental argument
that the union of any collection of open sets is open. In particular, the union of
any collection of open balls is open (Fig. 3.5), and this is probably the most
intuitive way of visualizing an open set. (See Exercise 2.9.)
Fig. 3.5
Fig. 3.6
A subset C is said to be closed if its complement C is open.
Our discussion above shows that a nonempty set C is closed if and only if
every point not in it is at a positive distance from it: a ^ C =^ p(aj C) > 0.
The so-called closed ball of radius r about 13, B = {^ :\\^ — I3\\ < r}, is a closed
set. As Fig. 3.6 suggests, the proof is another application of the triangle
inequality.

3.2 NORMS 125
EXERCISES
2.1 Show that if ||f - a|| < ||a||/2, then U\\ > \\a\\/2.
2.2 Prove in detail that
11x111 = i: N
1
is a norm on W^. Also prove that
ll/lli = [' 1/(01 rf<
is a norm on 6([a, 6]).
2.3 For X in W let |a:| be the Euclidean length
|x| = [E x^J \
and let (x, y) be the scalar product
n
(xi y) = Zl ^t2/t-
The Schwarz inequality says that
l(x,y)| < H|y|
and that the inequality is strict if x and y are independent.
a) Prove the Schwarz inequality for the case n = 2 by squaring and canceling.
b) Now prove it for the general n in the same way.
2.4 Continuing the above exercise, prove that the Euclidean length |x| is a norm.
The crucial step is the triangle inequality, |x + y| < |x| + |y|. Reduce it to the
Schwarz inequaUty by squaring and canceHng. This is of course our two-norm ||x||2.
2.5 Prove that the unit balls for the norms || ||i and || ||oo on M^ are as shown in
Fig. 3.2.
2.6 Prove that an open ball is an open set.
2.7 Prove that a closed ball is a closed set.
2.8 Give an example of a subset of R^ that is neither open nor closed.
2.9 Show from the definition of an open set that any open set is the union of a
family (perhaps very large!) of open balls. Show that any union of open sets is open.
Conclude, therefore, that a set is open if and only if it is a union of open balls.
2.10 A subset ^ of a normed hnear space V is said to be convex if A includes the line
segment joining any two of its points. We know that the line segment from a to jS is
the image of [0, 1] under the mapping t —^ t^ -\- {I — t)a. Thus A is convex if and
only li a,l3G A and t G [0, l]=^tl3+ (I — t)a G A. Prove that every ball jBr(T) in
a normed hnear space V is convex.
2.11 A seminorm is the same as a norm except that the positivity condition nl is
relaxed to nonnegativity:
nl'. p{a) > 0 for all a.

126 THE DIFFERENTIAL CALCULUS 3.3
Thus p{a) may be 0 for some nonzero a. Every norm is in particular a seminorm.
Prove:
a) If p is a seminorm on a vector space W and T is a linear mapping from V to W,
then J) o T is a seminorm on V.
b) p o T' is a norm if and only if T is infective and p is a norm on range T.
2.12 Show that the sum of two seminorms is a seminorm.
2.13 Prove from the above two exercises (and not by a direct calculation) that
qif) = ll/'IU + l/«o)|
is a seminorm on the space 6^ ([a, h]) of all continuously differentiable real-valued
functions on [a, h], where ^o is a fixed point in [a, h]. Prove that g is a norm.
2.14 Show that the sum of two bounded sets is bounded.
2.15 Prove that the sum Brio) + Bs{fi) is exactly the ball jBr+s(a + P).
3. CONTINUITY
Let V and W be any two normed linear spaces. We shall designate both norms
by II II. This ambiguous usage does not cause confusion. It is like the ambiguous
use of "0" for the zero elements of all the vector spaces under consideration. If we
replace the absolute value sign | | by the general norm symbol || || in the
definition we gave earlier for the limit of a real-valued function of a real variable,
it becomes verbatim the corresponding definition of convergence in the general
setting. However, we shall repeat the definition and take the occasion to relax
the hypothesis on the domain of/. Accordingly, let A by any subset of F, and let
/ be any mapping from A to W.
Definition. We say that /(f) approaches 13 as ^ approaches a, and write
/(f) -^ i^ as f -^ a, if for every e there is a 5 such that
^eAandO<U-a\\<8=^ ||/(f) - I3\\ < e.
If a G A and /(f) -^ f(a) as f -^ a, then we say that / is continuous at a.
We can then drop the requirement that f f^ a and have the direct €,5-
characterization of continuity: / is continuous at a if for every e there exists a 8
such that llf — ajl < 8=^ ||/(f) — /(a)|| < €. It is understood here that f is
universally quantified over the domain A of/. We say that/is continuous if / is
continuous at every point a in its domain. If the absolute value of a number is
replaced by the norm of a vector, the limit theorems that we sampled in Section 1
hold verbatim for normed linear spaces. We shall ask the reader to write out a
few of these transcriptions in the exercises.
There is a property stronger than continuity at a which is much simpler to
use when it is available. We say that / is Lipschitz continuous at a if there is a
constant c such that ||/(f) — /(a)|| < c||f — a|| for all f sufficiently close to a.

3.3 CONTINUITY 127
That is, there are constants c and r such that
U-a\\ <r ^ \\m) -f{a)\\ <c||f-a||.
The point is that now we can take b simply as e/c (provided e is small enough so
that this makes 5 < r; otherwise we have to set b = min {e/c, r}). We say
that / is a Lipschitz function (on its domain A) if there is a constant c such that
WfiO - f(v)\\ < c||f - v\\ for all f, T] in A. For a /mear map T:V -^W the
Lipschitz inequality is more simply written as
mm < ciifii
for all f G F; we just use the fact that now T(^) — T{ri) = T{^ — rj) and set
f = ^ — -q. In this context it is conventional to call T a hounded linear mapping
rather than a Lipschitz linear mapping, and any such c is called a hound of T,
We know from the beginning calculus that if / is a continuous real-valued
function on [a, h] (that is, if/G e([a, 6])), then |Jj'/(x) (ix| < m(6 — a), where
m is the maximum value of \f{x)\. But this is just the uniform norm of/, so that
the inequality can be rewritten as \^a f\ ^ (h — a)||/||oo. This shows that if the
uniform norm is used on e([a, 6]), then /»—^ Jj'/ is a bounded linear functional,
with bound h — a.
It should immediately be pointed out that this is not the same notion of
boundedness we discussed earlier. There we called a real-valued function
bounded if its range was a bounded subset of U. The analogue here would be to
call a vector-valued function bounded if its range is norm bounded. But a
nonzero linear transformation cannot be bounded in this sense, because
\\T{xa)\\ = \x\ \\T{a)l
The present definition amounts to the boundedness in the earlier sense of the
quotient T(a)/\\a\\ (on V — {0}). It turns out that for a linear map T, being
continuous and being Lipschitz are the same thing.
Theorem 3.1. Let T be a linear mapping from a normed linear space F to a
normed linear space W. Then the following conditions are equivalent:
1) T is continuous at one point;
2) T is continuous;
3) T is bounded.
Proof. (1) => (3). Suppose T is continuous at ao- Then, taking e = 1, there
exists 8 such that \\a — aoll < 5 => \\T(a) — T(ao)\\ < 1. Setting ^ = a ~ ao
iand using the additivity of T, we have ||f|| < 5 => ||T(Q|| < 1. Now for any
nonzero v^ ^ = ^v/^vW has norm 8/2. Therefore, \\T(^)\\ < 1. But
mm = 8\\Tir})\\/2\\vl giving ||r(^)|| < 2\\r}\\/8. Thus T is bounded by
C = '2/5.

128 THE DIFFERENTIAL CALCULUS 3.3
(3) => (2). Suppose ||T(Q|| < CUW for all ^ Then for any ao and any e
we can take 8 = e/C and have
\\a - aoW < 8 ^ \\Tia) - T(ao)|| = \\Tia - ao)\\ < C\\a - ao\\ <Cd=€.
(2) => (1). Trivial. □
In the lemma below we prove that the norm function is a Lipschitz function
from V to U.
Lemma 3.1. For all a,l3eV,\ \\a\\ - \\l3\\ | < ||a - I3\\.
Proof. We have ||a|| = \\ia - 13) + I3\\ < \\a - p\\ + p||, so that ||a|| - \\p\\ <
\\a - p\\. Similarly, \\I3\\ - \\a\\ < \\^ - a\\ = \\a - I3\\. This pair of
inequalities is equivalent to the lemma. □
Other Lipschitz mappings will appear when we study mappings with
continuous differentials. Roughly speaking, the Lipschitz property lies between
continuity and continuous differentiability, and it is frequently the condition
that we actually apply under the hypothesis of continuous differentiability.
The smallest bound of a bounded linear transformation T is called its norm.
That ic;
n| = Iub{||r(a)||/||a||:ap^O}.
For example, let T: e(fa, b]) -^ IR be the Riemann integral, T(f) = ja fix) dx.
We saw earlier that if we use the uniform norm ||/||oo on e([a, 6]), then T is
bounded by 6 — a : | T{f)\ < (b — a) ||/||oo. On the other hand, there is no smaller
bound, because j^ 1 = b - a = (b - a)||l||oc. Thus ||T|| = b - a. Other
formulations of the above definition are useful. Since
\\T(a)\\/\\a\\ = ||r(a/||a||)|l
by homogeneity, and since 13 = oi/\\a\\ has norm 1, we have
||r|| = iub{r(^)||: 11^11 = 1}.
Finally, if ||7|| < 1, then 7 = x^, where ||/3|| = 1 and |a;| < 1, and
\\F{y)\\ = \x\ \\Fm\ < \\Fm-
We therefore have an inefficient but still useful characterization:
liril = lub {||r(7)||: IMI <i}.
These last two formulations are uniform norms. Thus, if ^i is the closed unit
ball {f : llfll < 1}, we see that a linear T is bounded if and only if T \ Bi is
bounded in the old sense, and then
iiT|| = iiTr^iiioo.
AlinearmapT: F-^ Wisbouncledbeloiv by b if \\Ti^)\\ > bU\\ forall ^nF.
If T has a bounded inverse and m = HT"^!!, then T is bounded below by 1/m,
for 11^-1(77)11 < rn\\v\\ for all 77 G TF if and only if ||f|| < m\\T{0\\ forall ^gV.

3.3 CONTINUITY 129
If V is finite-dimensional, then it is true, conversely, that if T is bounded below,
then it is invertible (why?), but in general this does not follow.
If V and W are normed linear spaces, then Hom(F, W) is defined to be the
set of all hounded linear maps T: V -^ W. The results of Section 2.3 all remain
true, but require some additional arguments.
Theorem 3.2. Hom(F, W) is itself a normed linear space if \\T\\ is defined
as above, as the smallest bound for T.
Proof. This follows from the uniform norm discussion of Section 2 by virtue
of the identity ||T|| = \\T \ Bi\\^. D
Theorem 3.3. If C/, F, and W are normed linear spaces, and if
T G Hom(C/, V) and S G Hom(F, TT), then SoTe Hom(C/, W) and
\\S o T\\ < \\S\\ \\T\\. It follows that composition on the right by a fixed T
is a bounded linear transformation from Hom(F, W) to Hom(C/, TT), and
similarly for composition on the left by a fixed S.
Proof
||(.So T)(a)|| = \\S(T(a))\\ < \\S\\ ma)\\ < ||.S||(||T|| ||a||) = (||.S|| • ||T||)(||a||).
Thus S o T is bounded by ||aS|| • \\T\\ and everything else follows at once. D
As before, the conjugate space F* is Hom(F, IR), now the space of all hounded
linear functionals.
EXERCISES
3.1 Write out the e,5-proofs of the following limit theorems.
1) Let V and W be normed linear spaces, and let F and G be mappings from V to W.
If \im^^aF(0 = fJi and lim^^a G(f) = v, then lim^^a {F + G)(f) = (jl+v.
2) Given F: F -> TF and g: V -^ U, if F(0 -> M and g(^) -> 6 as f -> a, then
3.2 Prove that if F(^) —> /x as f —> a and G{r]) —> X as 77 —> /x, then G o F{^) —> X as
f —> a. Give a careful, complete statement of the theorem you have proved.
3.3 Suppose that A is an open subset of a nls V and that ao G A. Suppose that
F: A —> IR is such that lima_^ao F(a) = b 9^ 0. Prove that l/F{a) —> 1/6 as a —> ao
(e,5-proof).
3.4 The function/(a:) = |a:|^ is continuous at a: = 0 for any positive r. Prove that/is
not Lipschitz continuous at a: = 0 if r < 1. Prove, however, that / is Lipschitz
continuous at a: = a if a > 0. (Use the mean-value theorem.)
3.5 Use the mean-value theorem of the calculus and the definition of the derivative
to show that if / is a real-valued function on an interval /, and if/' exists everywhere,
then / is a Lipschitz mapping if and only if /' is a bounded function. Show also that
then [|/'||oo is the smallest Lipschitz constant C.

130 THE DIFFERENTIAL CALCULUS 3.3
3.6 The "working rules" for ||r|| are
1) \\Tm< lirilllJIlforalU;
2) 117(^)11 < 611^11, alU ^ \\T\\<b.
Prove these rules.
3.7 Prove that if we use the one-norm ||x||i = ^i \xi\ on IR'', then the norm of the
linear functional
n
^a^ X —> 2^ aiXi
1
is ||a||oo.
3.8 Prove similarly that if ||x|| = ||x||oo, then ||La|| = ||a||i.
3.9 Use the above exercises to show that if ||x|| on R^ is the one-norm, then
||x|| =lub{|/(x)|:/G(ff^'^)* and 11/11 < 1}.
3.10 Show that if T in Hom(lR'', IR'") has matrix t = {Uj], and if we use the one-
norm ||x||i on W and the uniform norm ||y||oo on W^, then ||T|1 = ||t||oo.
3.11 Show that the meaning of 'Hom(F, TF)' has changed by giving an example of a
linear mapping that fails to be bounded. There is one in the text.
3.12 For a fixed f in F define the mapping ev^: Hom(F, W) -> W by ev^(T) = T(f).
Prove that ev^ is a bounded linear mapping.
3.13 In the above exercise it is in fact true that llev^|| = \\^\\, but to prove this we
need a new theorem.
Theorem. Given f in the normed linear space V, there exists a functional/in F*
such that 11/11 = land 1/(^1 = U\l
Assuming this theorem, prove that ||ev^|| = ||f||. [Hint: Presumably you have already
shown that ||ev^|| < ||f||. You now need a T in Hom(F, W) such that ||T|| = 1 and
\\T(0\\ = Ull Consider a suitable dyad.]
3.14 Let t = {tij] be a square matrix, and define ||t|| as max^ (^y \tij\). Prove that
this is a norm on the space IR''^^'' of all n X n matrices. Prove that ||st|| < ||s|| • ||t||.
Compute the norm of the identity matrix.
3.15 Let V be the normed linear space IR'' under the uniform norm ||x||oo = max {\xi\].
If T G Hom V, prove that ||T|| is the norm of its matrix ||t|| as defined in the above
exercise. That is, show that
Ly=i J
(Show first that ||t|| is an upper bound of T, and then show that ||T(x)|| = ||t|| ||x|| for
a specially chosen x.) Does part of the previous exercise now become superfluous?
3.16 Assume the following fact: If/G ©([0, 1]) and ||/||i = a, then given e, there is a
function g G ©([0, 1]) such that
||gf||oo = 1 and / fg > a — e.

3.3 CONTINUITY 131
Let K(s, t) be continuous on [0, 1] X [0, 1] and bounded by h. Define T\ e([0, 1]) ->
^([0, 1]) by Th = k, where
k(s) = f K(s, t) hit) dt.
Jo
If V and IF are the normal linear spaces 6 and (B under the uniform norms, prove that
liril =luh I\K{s,t)\clt.
[Hint: Proceed as in the above exercise.]
3.17 Let V and W be normed linear spaces, and let A be any subset of V containing
more than one point. Let £(^1, TL) be the set of all Lipschitz mappings from A to IF.
For/in £(.i, TF), let p(/) be the smallest Lipschitz constant for/. That is,
^^n II? — ^1!
Prove that £(-.1, TF) is a vector space V and that p is a seminorm on V.
3.18 Continuing the above exercise, show that if a is any fixed point of A, then
pU) + ll/(«)ll is a norm on V.
3.19 Let K be a mapping from a subset .1 of a normed Unear space F to F which
differs from the identity by a Lipschitz mapping with constant c less than 1. We may
as well take c = J, and then our hypothesis is that
\\K{i) - K{,r,) - {^ - ■nn < m - ril
Prove that K is injective and that its inverse is a Lipschitz mapping with constant 2.
3.20 Continuing the above exercise, suppose in addition that the domain .4 of K is
an open subset of F and that K[C] is a closed set whenever (7 is a closed ball lying in A.
Prove that if C = Cr(a), the closed ball of radius r about a, is a subset of .1, then
K[C] includes the ball B = Br/iiy), where 7 = K(a). This proof is elementary but
tricky. If there is a point u of jB not in K[C], then since K[C] is closed, there is a largest
hall B' about v disjoint from K[C] and a point 77 = K(^) in K[C] as close to B' as we
wish. Now if we change f by adding v — rj, the change in the value of K will
approximate V — 7] closely enough to force the new value of K to be in B\ If we can also show
that the new value f + (u — rj) is in C, then this new value of K is in K[C], and we
have our contradiction.
Draw a picture. Obviously, the radius p of B' is at most r/7. Show that if
77 = K(^) is chosen so that ||u — r]\\ < 3/2p, then the above assertions follow from the
triangle inequality, and the Lipschitz inequafity displayed in Exercise 3.19. You have
to prove that
\\K(^+iv-v})-v\\ <p
and
\\{^+iv-v))-c^\\<r.
.'1.21 Assume the result of the above exercise and show that
Br,8(y) C K[Br{a)].

132 THE DIFFERENTIAL CALCULUS 3.4
Show, therefore, that iv[.l] is an open subset of T^. State a theorem about the Lipschitz
invertibility of K, including all the hypotheses on K that were used in the above
exercises.
3.22 We shall see in the next chapter that if V and W are finite-dimensional spaces,
then any continuous map from V to IT takes bounded closed sets into bounded closed
sets. Assuming this and the results of the above exercises, prove the following theorem.
Theorem. Let F he Si mapping from an open subset .1 of a finite-dimensional
normed linear space F to a finite-dimensional normed linear space W. Suppose
that there is a T in Hom(F, W) such that T"^ exists and such that F — T is
Lipschitz on .1, with constant l/2m, where m = ||T~^||. Then F is injective, its
range R = F[.l] is an open subset of W, and its inverse F~^ is Lij^schitz
continuous, with constant 2?n.
4. EQUIVALENT NORMS
Two normed linear spaces V and W are 7mrm isomorphic if there is a bijection T
from V toW such that T G Hom(F, W) and T'^ G Hom(Tr, V). That is, an
isomorphism is a linear isomorphism T such that both T and T~^ are continuous
(bounded). As usual, we regard isomorphic spaces as being essentially the same.
For two different norms on the same space we are led to the following definition.
Definition. Two norms p and q on the same vector space V are equivalent
if there exist constants a and h such that p < aq and q < hp.
Then (l/h)q < p < aq and (l/a)p < q < hp, so that two norms are
equivalent if and only if either can be bracketed by two multiples of the other.
The above definition simply says that the identity map i^-^ ^ from V to F,
considered as a map from the normed linear space <V, p> to the normed
linear space <V, q>, is bounded in both directions, and hence that these two
normed linear spaces are isomorphic.
If F is infinite-dimensional, two norms will in general not be equivalent.
For example, if F = e([0, 1]) and fn{t) = T, then ||/^||i = l/(n+ 1) and
11/^ 1100 = L Therefore, there is no constant a such that ||/||oo < ^||/||i for all
/G e[0, 1], and the norms || ||oo and || ||i are not equivalent on F = e[a, h].
This is why the very notion of a normed linear space depends on the assumption
of a given norm.
However, we have the following theorem, which we shall prove in the next
chapter by more sophisticated methods than we are using at present.
Theorem 4.1. On a finite-dimensional vector space F all norms are
equivalent.
We shall need this theorem and also the following consequence of it
occasionally in the present chapter.
Theorem 4.2. If F and W are finite-dimensional normed linear spaces, then
every linear mapping T from F to IF is necessarily bounded.

3.4 EQUIVALENT NORMS 133
Proof. Because of the above theorem, it is sufficient to prove T bounded with
respect to some pair of norms. Let 6: W^ -^ V and (p: W -^ W he any basis
isomorphisms, and let {Uj} be the matrix oi T = (p~^ o 7" o ^ in Hom(IR'', W^).
Then
\Tx\\ao = max
< (max|^
^,J
ij\) (t h)
where h = max \tij\. Now q(r}) = \\(p ^(^)||oo and p(^) = ||^ ^(?)||i are norms
on W and V respectively, by Lemma 2.1, and since
q(.m)) = md-H)\U < b\\d-'^h = bp(0,
we see that T is bounded by h with respect to the norms p and g on F and W. D
If we change to an equivalent norm, we are merely passing through an
isomorphism, and all continuous linear properties remain unchanged. For
example :
Theorem 4.3. The vector space Hom(F, W) remains the same if either the
domain norm or the range norm is replaced by an equivalent norm, and the
two induced norms on Hom(F, W) are equivalent.
Proof. The proof is left to the reader.
We now ask what kind of a norm we might want on the Cartesian product
V X W of two normed linear spaces. It is natural to try to choose the product
norm so that the fundamental mappings relating the product space to the two
factor spaces, the two projections Wi and the two injections d{, should be
continuous. It turns out that these requirements determine the product norm
uniquely to within equivalence. For if \\<a, ^>\\ has these properties, then
\\<a, ^>\\ = ||<a:,0> + <0, ?> || < ||<a,0>|| + ||<0, ?>||
<k,\\a\\ + k2U\\ <A;(||a|| + 11^11),
where A;^ is a bound of the injection ${ and A; is the larger of A;i and A;2- Also,
Ikll ^ Ci\\<a, ^>||and||^|| < C2\\<a, ^> ||, by the boundedness of the
projections TTi, and so ||a:|! + ||?|| ^ c||<a:, ^>\\, where c = Ci-\~ C2- Now ||a:|| +
II ^11 is clearly a norm || ||i on F X W, and our argument above shows that
II <a, ^> II will satisfy our requirements if and only if it is equivalent to || ||i.
Any such norm will be called a product norm for V X W. The product norms
most frequently used are the uniform (product) norm
\\l<oi, ?>||oo = max {||a||, ll^ll),
tlie Euclidean (product) norm \\<a, ^>\\2 = (||«IP + l!?lP)^^^j and the above
sum (product) norm \\<a, ^> ||i. We shall leave the verification that the
uniform and Euclidean norms actually are norms as exercises.
Each of these three product norms can be defined as well for n factor spaces
jis for two, and we gather the facts for this general case into a theorem.

134 THE DIFFERENTIAL CALCULUS 3.4
Theorem 4.4. If {<Vi, Pi>}1 is a finite set of normed linear spaces, then
II 111, II II2, and II U defined on V=IlUVi by Mi = ZlPi(ad,
II^IU = (El Pi{oii)^)^'^, and Halloo = max {pi{ai) :i = 1, . . . ,n}, are
equivalent norms on V, and each is a product norm in the sense that the
projections Wi and the injections di are all continuous.
* It looks above as though all we are doing is taking any norm || || on W^ and
then defining a norm jjj jjj on the product space V by
IHII = \\<Pl{0il), . . . ,Pn{oin)>\\-
This is almost correct. The interested reader will discover, however, that
II II on W^ must have the property that if \xi\ < \yi\ for i = 1, . . . , n, then
||x|i < ||y|| for the triangle inequality to follow for jjj jjj in V. If we call such a
norm on [R^ an increasing norm, then the following is true.
If II II is any increasing norm on W^, then jjjajjj = || <pi{ai), . . . , Pnio^n) > \\
is a product norm on V = Hi
billow ever, we shall use only the 1-, 2-, go-product norms in this book.*
The triangle inequality, the continuity of addition, and our requirements on
a product norm form a set of nearly equivalent conditions. In particular, we
make the following observation.
Lemma 4.1. If F is a normed linear space, then the operation of addition
is a bounded linear map from F X F to F.
Proof. The triangle inequality for the norm on F says exactly that addition is
bounded by 1 when the sum norm is used on F X F. D
A normed linear space F is a (norm) direct sum 0^ Vi if the mapping
<Xi, . . . , Xn> i-^ El ^i is a norm isomorphism from JJi ^i to F. That is, the
given norm on F must be equivalent to the product norm it acquires when it is
viewed as IJi ^i- If ^ is algebraically the direct sum 0^ Vi, we always have
Y^xi
1
< EII^'
by the triangle inequality for the norm on F, and the sum on the right is the one-
norm for Hi Vi- Therefore, F will be the norm direct sum 0^ Vi if, conversely,
there is an 7i-tuple of constants {hi} such that \\xi\\ < hi\\x\\ for all x. This is
the same as saying that the projections Pi'.x \-^ xi are all bounded. Thus,
Theorem 4.5. If F is a normed linear space and F is algebraically the direct
sum F = 01 Vi, then F = 0^ Vi as normed linear spaces if and only if
the associated projections [Pi] are all bounded.

3.4 EQUIVALENT NORMS 135
i:XERCISES
4.1 The fact that Hom(F, W) is unchanged when norms are replaced by equivalent
norms can be viewed as a corollary of Theorem 3.3. Show that this is so.
4.2 Write down a string of quite obvious inequalities showing that the norms
II 111, II II2, and II ||oo on W are equivalent. Discuss what happens as n —> 00.
4.3 Let V be an n-dimensional vector space, and consider the collection of all norms
on V of the form p ^ 6, where ^: F —> IR'* is a coordinate isomorphism and p is one of
the norms || ||i, || II2, || ||oo on IR'*. Show that all of these norms are equivalent. (Use
the above exercise and the reasoning in Theorem 4.2.)
4.4 Prove that \\<a, ^>\\ = max {||a||, ||^||} is a norm on F X W.
4.5 Prove that ||<a, ^>|| = ||a|| + ||^|| is a norm on F X W.
4.6 Prove that II < a, ?> II = (||af + ||g||2)i/2 is a norm on F X TF.
4.7 Assuming Exercises 4.4 through 4.6, prove by induction the corresponding part
of Theorem 4.4.
4.8 Prove that if A is an open subset of F X TF, 1>hen Tri[A] is an open subset of F.
4.9 Prove (e, 8) that <T, S> —>*So Tisa continuous map from
Hom(Fi, F2) X Hom(F2, F3) to Hom(Fi, Vs),
where the Vi are all normed hnear spaces.
[.10 Let II II be any increasing norm on IR""; that is, ||x|| < ||y|| if Xi < yi for all i.
Lot Pi be a norm on the vector space Vi for i = 1, . . . , n. Show that
llklll = \\<pi{ai),...,pn{an)>\\
is a norm on F = IJi ^i-
1.11 Suppose that p: F —> IR is a nonnegative function such that p{xa) = |a:|p(a:)
for all x, a. This is surely a minimum requirement for any function purporting to be a
measure of length of a vector.
a) Define continuity with respect to p and show that Theorem 3.1 is valid.
b) Our next requirement is that addition be continuous as a map from F X F to F,
and we decide that continuity at 0 means that for every e there is a 5 such that
p{a) < 5 and p(l3) < 8 =^ p(a + 13) < e.
Argue again as in Theorem 3.1 to show that there is a constant c such that
p(a + p) < c{p(a) + p(p)) for all a,pGV.
1.12 Let F and TF be normed Hnear spaces, and let /: F X TF —> IR be bounded and
bilinear. Let T be the corresponding linear map from F to TF*. Prove that T is bounded
iind that ||T|| is the smallest bound to/, that is, the smallest b such that
|/(a,^)| < 6||a||||^|| for all a, fi.
1.13 Let the normed linear space F be a norm direct sum M ® N. Prove that the
^uhspaces M and A^ are closed sets in F. (The converse theorem is false.)

136 THE DIFFERENTIAL CALCULUS 3.5
4.14 Let A^ be a closed subspace of the normed linear space V. If A is a coset N -\- a,
define |||J||| as gib {||^|| : ^G ^1}. Prove that |||.4||| is a norm on the quotient space V/N.
Prove also that if | is the coset containing ^, then the mapping $ i-^ | (the natural
projection tt of F onto V/N) is bounded by 1.
4.15 Let V and W be normed linear spaces, and let T in Hom(F, 11') have a null space
which includes the closed subspace A^. Prove that the unique linear S from V/N to W
defined hy T = S o tt (Theorem 4.3 of Chapter 1) is bounded and that \\S\\ = \\T\\.
4.16 Let N he 8l closed subspace of a normed linear space, and suppose that A^ has a
finite-dimensional complement in the purely algebraic sense. Prove that then V is the
norm direct sum M 0 A^. (Use the above exercise and Theorem 4.2 to prove that if P
is the projection of V onto A^ along ilf, then P is bounded.)
4.17 Let A^i and A^2 be closed subspaces of the normed linear space V, and suppose
that they have the same finite codimension. Prove that A^i and A^2 are norm
isomorphic. (Assume the results of the above exercise and Exercise 2.11 of Chapter 2.)
4.18 Prove that if p is a seminorm on a vector space F, then its null set is a subspace
A^, p is constant on the cosets of A^, and p factors: p = g o tt, where g is a norm on V/N
and TV is the natural projection ^ i-^ ^ of V onto V/N. Note that ^ i-^ | is thus an
isometric surjection from the seminormed space V to the normed space V/N. An
isometry is a distance-preserving map.
5. INFINITESIMALS
The notion of an infinitesimal was abused in the early literature of the calculus,
its treatment generally amounting to logical nonsense, and the term fell into
such disrepute that many modern books avoid it completely. Nevertheless, it
is a very useful idea, and we shall base our development of the differential upon
the properties of two special classes of infinitesimals which we shall call ''big oh^^
and ''little oh^^ (and designate'0^ and ^e\ respectively).
Originally an infinitesimal was considered to be a number that "is infinitely
small but not zero". Of course, there is no such number. Later, an infinitesimal
was considered to be a variable that approaches zero as its limit. However, we
know that it is functions that have limits, and a variable can be considered to
have a limit only if it is somehow considered to be a function. We end up looking
at functions (p such that (p(t) -^ 0 as t -^ 0. The definition of derivative involves
several such infinitesimals. If f\x) exists and has the value a, then the
fundamental difference quotient {f(x + t) — f{x))/t is the quotient of two
infinitesimals, and, furthermore, {{f{x + 0 ~ /(^))/0 "~ ^ ^^so approaches 0 as ^ -^ 0.
This last function is not defined at 0, but we can get around this if we wish by
multiplying through by ^, obtaining
{Kx + t) - fix)) - at= cp(t),
where/(a: -\- t) — f(x) is the "change in/" infinitesimal, at is a linear infinitesimal,
and (p(t) is an infinitesimal that approaches 0 faster than t (i.e., (p(t)/t -^ 0 as ^ -^ 0).
If we divide the last equation by t again, we see that this property of the infin-

:i5
INFINITESIMALS
137
itesimal <^, that it converges to 0 faster than ^ as ^ -^ 0, is exactly equivalent to
the fact that the difference quotient of / converges to a. This makes it clear that
the study of derivatives is included in the study of the rate at which
infinitesimals get small, and the usefulness of this paraphrase will shortly become clear.
Definition. A subset A of a normed linear space F is a neighborhood of a
point a if A includes some open ball about a. A deleted neighborhood of a is a
neighborhood of a minus the point a itself.
We define special sets of functions ^, 0, and o as follows. It will be assumed
in these definitions that each function is from a neighborhood of 0 in a normed
linear space F to a normed linear space W.
/ E £r if /(O) = 0 and / is continuous at 0. These functions are the infi-
nitesimals.
/ E 0 if /(O) = 0 and / is Lipschitz continuous at 0. That is, there exist
positive constants r and c such that ||/(?)|| < c||^|| on ^^.(O).
/ e e if/(O) = 0 and ||/(a||/||^|| ^ 0 as ? ^ 0.
When the spaces V and W are not understood, we specify them by writing
c)(F, TF), etc.
A simple set of functions from IR to IR makes the qualitative difference
between these classes apparent. The function/(a:) = \x\ ^'^ is in ^(U, U) but not
in 0, ^(a:) = a: is in 0 and therefore in ^ but not in o, and h(x)
three classes (Fig. 3.7).
X is in all
Fig. 3.7
It is clear that ^, 0, and o are unchanged when the norms on F and W are
replaced by equivalent norms.
Our previous notion of the sum of two functions does not apply to a pair
<»f functions fy g G ^{V^ W) because their domains may be different. However,
/ I ^ is defined on the intersection dom / D dom g, which is still a neighborhood
<»f 0. Moreover, addition remains commutative and associative when extended
ifi this way. It is clear that then ^(F, W) is almost a vector space. The only
trouble occurs in connection with the equation /+ (—/) = 0; the domain of
On\ function on the left is dom /, whereas we naturally take 0 to be the zero
function on the whole of F.

138 THE DIFFERENTIAL CALCULUS 3.5
* The way out of this difficulty is to identify two functions / and ^ in ^ if
they are the same on some ball about 0. We define / and g to be equivalent
(/ '^ O) if ^i^d only if there exists a neighborhood of 0 on which f = g. We then
check (in our minds) that this is an equivalence relation and that we now do
have a vector space. Its elements are called germs of functions at 0. Strictly
speaking, a germ is thus an equivalence class of functions, but in practice one
tends to think of germs in terms of their representing functions, only keeping in
mind that two functions are the same as germs when they agree on a
neighborhood of 0.*
As one might guess from our introductory discussion, the algebraic
properties of the three classes ^, 0, and o are crucial for the differential calculus.
We gather them together in the following theorem.
Theorem 5.1
1) 6(V,W)C0(V,W)C^(V,W)y and each of the three classes is closed
under addition and multiplication by scalars.
2) If fGeiV,W), and if (/G 0(17, Z), then ^o/G0(F,Z), where
dom g o f = /~^[dom g].
3) If either f or g above is in o, then so is ^ o /.
4) If / G 0(7, W) and g G ^(F, U), then fg G ©(F, W), and similarly if
/ G ^ and g GO.
5) In (4) if either / or ^ is in o and the other is merely bounded on a
neighborhood of 0, then fg ee(V,W).
6) Hom(F, W) C 0(F, W).
7) Hom(F, W) n e(V, W) = {0}.
Proof. Let £.,(¥, W) be the set of infinitesimals / such that ||/(f)|| < eU\\ on
some ball about 0. Then / G 0 if and only if / is in some £e, and / G o if and only
if / is in every £e- Obviously, o C 0 C ^.
1) If 11/(011 < «ll^ll on BM and ||<;(0|| < 6||^|| on 5„(0), then
\\m) + gm < (a + b)U\\
on Br(0)y where r = min {^, u}. Thus 0 is closed under addition. The
closure of o under addition follows similarly, or simply from the limit of a
sum being the sum of the limits.
2) If 11/(011 < all^ll when ||^|| < t and \\g{v)\\ < bM when ||„|| < u, then
himm < Hfm < abuw
when llfll < t and ||/(f)|| < u, and so when ||f|| < r == min {^, u/a}.

3.5 INFINITESIMALS 139
3) Now suppose that / G o in (2). Then, given e, we can take a = e/h and have
lls(/(^))ll < eUW
when II f II < r. Thus ^ o / G o. The argument when g Ge and / G 0 is
essentially the same.
4) Given ||/(f)|| < c||f|| on Br(0) and given €, we choose 8 such that \g(^)\ <
e/c on ^5(0) and have
\\m)gm<4^\\
when llfll < min (5, r). The other result follows similarly, as also does (5).
G) A bounded linear transformation is in 0 by definition.
7) Suppose that / G Hom(F, W) D ©(F, W). Take any a 9^ 0. Given e,
choose r so that ||/(f)|| < e\\^\\ on Br(0). Then write a as a = a:f, where
llfll < r. (Find f and a:.) Then
\\fM\\ = WKxm = \x\' WmW < N.€.||f|| = €|H|.
Thus ||/(a)|| < €||a|| for every positive e, and so f(a) = 0. Thus / = 0,
proving (7). D
Remark. The additivity of/was not used in this argument, only its homogeneity.
It follows therefore that there is no homogeneous function (of degree 1) in 0
except 0.
Sometimes when more than one variable is present it is necessary to indicate
with respect to which variable a function is in 0 or 0. We then write "/(f) = 6(f)"
for "/ G 0", where "0(f)" is used to designate an arbitrary element of 0.
The following rather curious lemma will be useful later in our proof of the
differentiability of an implicitly defined function. It is understood that rj = f(^),
where / is the function we are studying.
Lemma 5.1. U rj = 0(f) + o(< ^, v>) and also rj = ^(f), then rj = 0(f).
Proof. The hypotheses imply that there are numbers h, ri and p such that
ll^ll < ?>||f|| + idlfll + h\\) if llfll < r, and ||f|| + ||^|| < p, and then that
ll^ll ^ p/2 if llfll is smaller than some r2. If ||f|| < r = min {ri, r2, p/2}, then
all the conditions are met and \\7]\\ < 6||f|| + i(||f|| + ||^||)- But this is the
inequality \\7]\\ < (26 + l)||f||, and so 77 = 0(f). D
We shall also need the following straightforward result.
Lemma 5.2. If / G 0(F, X) and g G 0(7, F), then <f,g> G 0(7, X X Y).
That is, <0(f), 0(f) > = 0(f).
Proof. The proof is left to the reader.

140 THE DIFFERENTIAL CALCULUS 3.6
EXERCISES
5.1 Prove in detail that the class ^{V, W) is unchanged if the norms on V and W
are replaced by equivalent norms.
5.2 Do the same for 0 and 0.
5.3 Prove (5) of the 00-theorem (Theorem 5.1).
5.4 Prove also that if in (4) either/ or gf is in 0 and the other is merely bounded on a
neighborhood of 0, then fg G 0{V, W).
5.5 Prove Lemma 5.2. (Remember that F = •<Fi,F2^ is loose language for
F = di o Fi-\- 62 ° F2-) State the generalization to n functions. State the e-form of
the theorem.
5.6 Given Fi G 0(Fi, W) and F2 G 0(F2, W), define F from (a subset of) V =
Vi X V2 to M' by F(ai, 0:2) = Fi(ai) + F2{ol2). Prove that F G 0(F, M'). (First
state the defining equation as an identity involving the projections tti and 7r2 and not
involving explicit mention of the domain vectors ai and 0:2-)
5.7 Given Fi G 0(Fi, W) and F2 G 0(^2, ^), define precisely what you mean by
F1F2 and show that it is in 0(Fi X F2, W).
5.8 Define the class 0" as follows:/ G 0" if / G ^ and ||/(f) ||/||f ||" is bounded in some
deleted ball about 0. (A deleted neighborhood of a is a neighborhood minus a.) State
and prove a theorem about /+ g when / G 0"^ and g G 0^.
5.9 State and prove a theorem about fog when f G 0"^ and g G 0^.
5.10 State and prove a theorem about fg when / G 0"^ and g G 0^.
5.11 Define a similar class Q"^. State and prove a theorem about/o g when/G 0"^
and g G o"^.
6. THE DIFFERENTIAL
Before considering the notion of the differential, we shall review some geometric
material from the elementary calculus. We do this for motivation only; our
subsequent theory is independent of the preliminary discussion.
In the elementary one-variable calculus the derivative/'(a) of a function/
at the point a has geometric meaning as the slope of the tangent line to the graph
of / at the point a. (Of course, according to our notion of a function, the graph
of / is /.) The tangent line thus has the (point-slope) equation y — /(a) =
f\a){x — a), and is the graph of the affine map x i-^ f'{a){x — a) + /(^)-
We ordinarily examine the nature of the curve / near the point < a, /(a) >
by using new variables which are zero at this point. That is, we express
everything in terms oi s = y — f(a) and t = x — a. This change of variables is
simply the translation <x, y> i-^ <ty s> = <x — a, y — f(a) > in the
Cartesian plane U^ which brings the point of interest <a^f{a) > to the origin.
If we picture the situation in a Euclidean plane, of which the next page is a
satisfactory local model, then this translation in IR^ is represented by a choice of new
axes, the t- and s-axes, with origin at the point of tangency. Since y = f(x)

3.6
THE DIFFERENTIAL
141
if and only if s = f(a + t) ~ f{a), we see that the image of / under this
translation is the function Afa defined by Afa(t) = f(a + t) — f(a). (See Fig. 3.8.)
Of course, Afa is simply our old friend the change in / brought about by changing
X from a to a + ^.
A/aW
dfa{t)-Afa{t)=Qit)
f{a + t) I
^ T" ^^ ' '>tf\a)=dja{t)
Fig. 3.8
a + t
Similarly, the equation y — f{a) = f'{a){x — a) becomes s = f'{o)t, and
the tangent line accordingly translates to the line that is (the graph of) the
linear functional l\ t \-^ f'{a)t having the number/'(ot) as its skeleton (matrix).
Remember that from the point of view of the geometric configuration (curve and
tangent line) in the Euclidean plane, all that we are doing is choosing the natural
axis system, with origin at the point of tangency. Then the curve is (the graph
of) the function Afa, and the tangent line is (the graph of) the linear map I.
Now it follows from the definition off'(a) that I can also be characterized as
llie linear function that approximates Afa most closely. For, by definition.
A/a(0
►/'(a) as ^-^0,
and this is exactly the same as saying that
Afajt) - m
t
0
or
A/a - I
Q.
But we know from the Oe-theorem that the expression of the function Afa as the
sum Z + 0 is unique. This unique linear approximation I is called the differential
of/at a and is designated elf a- Again, the differential of/at a is the linear function
/: IR 1-^ IR that approximates the actual change in /, Afa, in the sense that
Afa — I E o; we saw above that if the derivative/'(c^) exists, then the differential
of/at a exists and hsisf(a) as its skeleton (1x1 matrix).
Similarly, if / is a function of two variables, then (the graph of) / is a surface
ill Cartesian 3-space IR^ = IR^ X IR, and the tangent plane to this surface at
<a, h, f(a, h) > has the equation z — f(a, h) = fi{a, h)(x — a) + /2(ot, h)(x — 6),

142
THE DIFFERENTIAL CALCULUS
3.6
where /i = df/dx and /2 = df/dy. If, as above, we set
and Z(s, t) = s/i(a, b) + tf2{a, 6), then A/<a,6> is the change in / around a, b
and Z is the linear functional on IR^ with matrix (skeleton) <fi(a, 6), /2(a, b) >.
Moreover, it is a theorem of the standard calculus that if the partial derivatives
of / are continuous, then again I approximates Af^a,b>, with error in ©. Here
also I is called the differential of/at < a, 6 > and is designated df^a,b> (Fig. 3.9).
The notation in the figure has been changed to show the value at t = < ^i, ^2 ^
of the differential dfg, of / at a = < ai, a2 >.
Fig. 3.9
The following definition should now be clear. As above, the local function
AFa is defined by AF«(f) = F(a + f) - F{a).
Definition. Let V and W be normed linear spaces, and let 4 be a
neighborhood of a in F. A mapping F: A -^ W is differentiable at a if there is a T
in Hom(F, W) such that AFa(^) = r(f) + ©(f).
The 00-theorem implies then that T is uniquely determined, for if also AF« =
S + Oj then T — Sgo, and soT — S = Ohy {7) of the theorem. This imiquely
determined T is called the differential of F at a and is designated dFa. Thus
AF« = dFa + 0,
where dFa is the unique (bounded) linear approximation to AF«.

3.6 THE DIFFERENTIAL 143
* Our preliminary discussion should make it clear that this definition of the
differential agrees with standard usage when the domain space is IR". However,
in certain cases when the domain space is an infinite-dimensional function space,
dFa is called the first variation of F at a. This is due to the fact that although the
early writers on the calculus of variations saw its analogy with the differential
calculus, they did not realize that it was the same subject.*
We gather together in the next two theorems the familiar rules for
differentiation. They follow immediately from the definition and the 00-theorem.
It will be convenient to use the notation a)«(F, W) for the set of all mappings
from neighborhoods of a in F to TF that are differentiable at a.
Theorem 6.1
1) UFg a)«(F, Tf), then AF« G 0(F, W),
2) If F, G G a)«(F, Tf), then F+G G a)«(F, W) and d{F + G)« =
dF^ + dGa.
3) UF G a)«(F, IR) and G G a)«(F, Tf), thenFG G 3D«(F, W) and d(FG)a =
F{a) dGa + dFaG{a), the second term being a dyad.
4) If -F is a constant function on F, then F is differentiable and dFa = 0.
5) If F G Hom(F, Tf), then F is differentiable at every a G F and dFa == F,
Proof
1) AFa = dFa + 0 = e + 0 = e by (I) and (6) of the 00-theorem.
2) It is clear that A(F +(?)« = AF« + AG«. Therefore, A(F + G)a =
(dFa + e) + (dGa + ©) = (dFa + dGa) + 0 by (1) of the 00-theorem. Since
dFa + dGa G Hom(F, Tf), we have (2).
3) A(FG)a(0 = F(a + i)G(a + f) -~ F(a)G(a)
= AF«(f)G(a) + F(a) AG«(|) + AF,(f) AG«(f),
as the reader will see upon expanding and canceling. This is just the usual
device of adding and subtracting middle terms in order to arrive at the form
involving the A's. Thus
A(FG)a = (dFa + Q)G(a) + F(a)(dGa + o) + 00 = dFaG(a) + F(a) dGa + e
by the 00-theorem.
4) If AFa = 0, then dFa = 0 by (7) of the 00-theorem.
5) AFa(^) == F(a + f) - F(a) = F(f). Thus AFa = F G Hom(F, W). D
The composite-function rule is somewhat more complicated.
Theorem 6.2. If F Ga)«(F, W) and GG^Fia)(W, Z), then GoFga)a(F, X)
and
d(G o F)a = dGp^a, o dFa.

144 THE DIFFERENTIAL CALCULUS 3.6
Proof. We have
A(G o FUQ = G{F(a + 0) - G{F(a))
= GiFia) + AF^(O) - G(Fia))
= AG^(«)(AF«(f))
= dGr.aMF^iO) + o{AFa(^))
= dGF^a){dFa{0) + dGFiaMO) + « ° ©
= (dGF(a) ° dFa)(0 + e o 0 + 0 o 0.
Thus A(G o F)a = dGF(a) ° dFc, + 0, and since dGF(a) ° dF^ G Hom(F, Tf),
this proves the theorem. The reader should be able to justify each step taken in
this chain of equalities. D
EXERCISES
6.1 The coordinate mapping <x^y> i—> x from IR^ to IR is differentiable. Why?
What is its differential?
6.2 Prove that differentiation commutes with the application of bounded linear
maps. That is, show that H F:V —^ W is differentiable at a and if T E Hom(TF, X),
then T o F is differentiable at a smd d(T o F) a = To dF^.
6.3 Prove that F G T>a{V, U) and F{a) ^ 0=^ G = l/FG 3D«(7, IR) and
6.4 Let F: V —^ R be differentiable at a, and let /; IR —» IR be a function whose
derivative exists at a == F(a). Prove that/o F is differentiable at a and that
d(foF). =r(a)dF..
[Remember that the differential of / at a is simply multiplication by its derivative:
dfa(h) = hf'(a).] Show that the preceding problem is a special case.
6.5 Let V and W be normed linear spaces, and let F:V —^ W and G:W—^ V he
continuous maps such that G o F = Iv and F o G = Iw- Suppose that F is
differentiable at a and that G is differentiable at jS = F(a). Prove that
dG^ = (dFa)
-1
6.6 Let/: F —» IR be differentiable at a. Show that g = f^ is differentiable at a and
that
dga = np-Ha) dfa.
(Prove this both by an induction on the product rule and by the composite-function
rule, assuming in the second case that D^x'' = nx''~^.)

3.6 THE DIFFERENTIAL 145
6.7 Prove from the product rule by induction that if the n functions fi'.V -^ IR,
i = 1, . . . , n, are all differentiable at a, then so is/ = Ili/i) and that
6.8 A monomial of degree n on the normed linear space F is a product XI? U of
linear functionals {U E F*). A homogeneous polynomial of degree n is a finite sum of
monomials of degree n. A polynomial of degree n is a sum of homogeneous polynomials
Pt, 1 = 0, . . . , n, where Pq is a constant. Show from the above exercise and other
known facts that a polynomial is differentiable everywhere.
6.9 Show that \i Fi:V -^ Wi and F2''V —^ W2 are both differentiable at a, then
so is /^ = <Fij F2>' from 7 to IF = TFi X W2 (use the injections ^1 and 62).
6.10 Show without using explicit computations, but using the results of earlier
exercises instead, that the mapping F = U^ —^ U^ defined by
<x,y>y-^ <{x — y)^, {x+y)^>
is everywhere differentiable. Now compute its differential at <a,b>.
6.11 Let F:V—^X and G:W—^X be differentiable at a and ^ respectively, and
defined: 7 X W -^ X by
K{^,V) = Fa)+G{v).
Show that K is differentiable at <a, I3>
a) by a direct A-calculation;
b) by using the projections tti and 7r2 to express K in terms of F and G without
explicit reference to the variable, and then applying the differentiation rules.
6.12 Now suppose given F:V—^R and GiW—^X^ and define K by
Ka,v) = F(^)G(v).
Show that if F and G are differentiable at a and (3 respectively, then K is differentiable
at < a, jS > in the manner of (b) in the above exercise.
6.13 Let V and W be normed linear spaces. Prove that the map <a^fi> 1—^ ||a|| ||i3||
from F X IF to IR is in ©(F X TF, IR). Use the maximum norm on the product space.
Let/: F X IF -^ IR be bounded and bilinear. Here boundedness means that there
is some h such that \f(a^fi)\ < 6||a|| \\fi\\ for all a, jS. Prove that / is differentiable
everywhere and find its differential.
6.14 Let/ and g be differentiable functions from IR to IR. We know from the composite-
function rule of the ordinary calculus that
(/°ff)'(a) =/'(ff(a)V(«).
Our composite-function rule says that
d{f o g)a = dfg^a) ° dga,
where dfx is the linear mapping t -^ f\x)t. Show that these two statements are
equivalent.

146 THE DIFFERENTIAL CALCULUS 3.7
6.15 Prove that f{x^y) = ||<^, 2/>"||i = |x| + |2/| is differentiable except on the
coordinate axes (that is, df^a,b> exists if a and h are both nonzero).
6.16 Comparing the shapes of the unit balls for || ||i and || ||oo on IR^^ guess from the
above the theorem about the differentiabiHty of || ||oo. Prove it.
6.17 Let V and W be fixed normed linear spaces, let Xd be the set of all maps from
V to W that are differentiable at 0, let Xo be the set of all maps from V to W that
belong to o(V, W), and let Xi be Hom(F, W). Prove that Xd and Xo are vector spaces
and that Xd = Xo 0 Xi.
6.18 Let Fhea, Lipschitz function with constant C which is differentiable at a point a.
Prove that \\dFa\\ < C.
7. DIRECTIONAL DERIVATIVES; THE MEAN-VALUE THEOREM
Directional derivatives form the connecting link between differentials and the
derivatives of the elementary calculus, and, although they add one more concept
that has to be fitted into the scheme of things, the reader should find them
intuitively satisfying and technically useful.
A continuous function / from an interval / C S^ to a normed linear space W
can have a derivative/'(^) at a point a: G / in exactly the sense of the elementary
calculus:
/'(.) = limfct^V^^i^^-
The range of such a function / is a curve or arc in TT, and it is conventional to
call / itself a parametrized arc when we want to keep this geometric notion in
mind. We shall also call /'(x), if it exists, the tangent vector to the arc / at x.
This terminology fits our geometric intuition, as Fig. 3.10 suggests. For
simplicity we have set a: = 0 and/(a:) = 0. If/'(a:) exists, we say that the
parametrized arc / is smooth at x. We also say that / is smooth at a = /(a:), but this
terminology is ambiguous if / is not injective (i.e., if the arc crosses itself). An
arc is smooth if it is smooth at every value of the parameter.
We naturally wonder about the relationship between the existence of the
tangent vector/'(x) and the differentiability of/at x. If dfx exists, then, being a
linear map on IR, it is simply multiplication "by" the fixed vector a that is its
skeleton, dfxQi) = h dfxil) = ha, and we expect a to be the tangent vector
./(Q+0-/(Q)
t
Fig. 3.10

3.7 DIRECTIONAL DERIVATIVES; THE MEAN-VALUE THEOREM 147
f'(x). We showed this and also the converse result for the ordinary calculus in
our preliminary discussion in Section 6. Actually, our argument was valid for
vector-valued functions, but we shall repeat it anyway.
When we think of a vector-valued function of a real variable as being an
arc, we often use Greek letters like 'X' and '7' for the function, as we do below.
This of course does not in any way change what is being proved, but is slightly
suggestive of a geometric interpretation.
Theorem 7.1. A parametrized arc 7: [a, 6] -^ F is differentiable at a: G (a, h)
if and only if the tangent vector (derivative) a = y\x) exists, in which case
the tangent vector is the skeleton of the differential, dJ^ih) = hy'{x) = ha.
Proof. If the parametrized arc 7: [a, 6] -^ V is differentiable at a: G (a, 6), then
dJxih) = hdlxil) = ha, where a = dlxW- Since A7a; — dy^ G o, this gives
||A7a;(/i) — /ia||/|/i| -^ 0, and so A7a;(/i)//i -^ a as /i -^ 0. Thus a is the derivative
y\x) in the ordinary sense. By reversing the above steps we see that the
existence of y\x) implies the differentiability of 7 at a:. D
Now let F be a function from an open set ^4 in a normed linear space F to a
normed linear space W. One way to study the behavior of F in the neighborhood
of a point a in ^4 is to consider how it behaves on each straight line through a.
That is, we study F by temporarily restricting it to a one-dimensional domain.
The advantage gained in doing this is that the restricted F is then simply a
parametrized arc, and its differential is simply multiplication by its ordinary
derivative.
For any nonzero ^ G V the straight line through a in the direction f has the
parametric representation t ^-^ a + t^. The restriction of F to this line is the
parametrized arc 7: 7(^) = F(a + t^). Its tangent vector (derivative) at the
origin ^ = 0, if it exists, is called the derivative of F in the direction f at a, or the
derivative of F with respect to f at a, and is designated D^F(a). Clearly,
D,Fia) = lim ^(- + '^] - ^(") •
Comparing this with our original definition of /', we see that the tangent vector
y\x) to a parametrized arc 7 is the directional derivative Diy(x) with respect
to the standard basis vector 1 in IR.
Strictly speaking, we are misusing the word "direction", because different
vectors can have the same direction. Thus, if r; = cf with c > 0, then r; and f
point in the same direction, but, because D^F(a) is linear in f (as we shall see in a
moment), their associated derivatives are different: Dy^F{a) = cD^F(a).
We now want to establish the relationship between directional derivatives,
which are vectors, and differentials, which are linear maps. We saw above that
for an arc 7 differentiability is equivalent to the existence of y\x) = Di7(a:).
In the general case the relationship is not as simple as it is for arcs, but in one
direction everything goes smoothly.

148 THE DIFFERENTIAL CALCULUS 3.7
Theorem 7.2. If F is differentiable at a, and if X is any smooth arc through a,
with a = \(x), then y = F o \ is smooth at x, and y'(x) = dFa{y(x)).
In particular, if F is differentiable at a, then every directional derivative
D^F(a) exists, and D^F(a) = dF^i^).
Proof. The smoothness of 7 is equivalent to its differentiability at x and
therefore follows from the composite-function theorem. Moreover, y'{x) = dyx{l) =
d{F o\)^{l) = dF^{d\:,{l)) = dFc,{y{x)). If X is the parametrized line
\{t) = a-\- t^^ then it has the constant derivative f, and since a = X(0) here,
the above formula becomes 7'(0) = cZF«(Q. That is, D^F{a) = 7'(0) =
dF^{^). D
It is not true, conversely, that the existence of all the directional derivatives
D^F(a) of a function F at a point a implies the differentiability of F at a. The
easiest counterexample involves the notion of a homogenous function. We say
that a function F: V -^ W is homogeneous if F(xi) = xF{^) for all x and f.
For such a function the directional derivative D^F(0) exists because the arc
y(t) = F(0 + t^) = tF(^) is linear, and 7'(0) = F(f). Thus, all of the directional
derivatives of a homogeneous function F exist at 0 and D^F{0) = F(^). If F is
also differentiable at 0, then dFo(0 = D^F(0) = F(0 and F = dFo. Thus a
differentiable homogeneous function must he linear. Therefore, any nonlinear
homogeneous function F will be a function such that D^F(0) exists for all f but
dFo does not exist. Taking the simplest possible situation, define F: IR^ -^ IR by
F(x, y) = x^/{x^ + y^)ii <x,y> 9^ <0, 0> and F(0, 0) = 0. Then
F{tx, ty) = tF(x, y),
so that F is homogeneous, but F is not linear.
However, if V is finite-dimensional, and if for each f in a spanning set of
vectors the directional derivative D^F{a) exists and is a continuous function of a
on an open set A, then F is continuously differentiable on A. The proof of this
fact depends on the mean-value theorem, which we take up next, but we shall
not complete it until Section 9 (Theorem 9.3).
The reader will remember the mean-value theorem as a cornerstone of the
calculus, and this is just as true in our general theory. We shall apply it in the
next section to give the proof of the general form of the above-mentioned
theorem, and practically all of our more advanced work will depend on it. The
ordinary mean-value theorem does not have an exact analogue here. Instead we
shall prove a theorem that in the one-variable calculus is an easy consequence of
the mean-value theorem.
Theorem 7.3. Let / be a continuous function (parametrized arc) from a
closed interval [a, h] to a normed linear space, and suppose that/'(0 exists
and that ||/'(0|| < m for all t G (a, h). Then ||/(6) - f{a)\\ < m{h - a).
Proof. Fix € > 0, and let A be the set of points x G [a, h] such that
\\Kx)-f{a)\\ < (m + 6)(x-a) + 6.

3.7 DIRECTIONAL DERIVATIVES; THE MEAN-VALUE THEOREM 149
A includes at least a small interval [a, c], because / is continuous at a. Set
I = lub A. Then ||/(0 - f(a)\\ < (m + €)(/ - a) + ehy the continuity of/at I.
Thus le A, and a < I < h. We claim that I = h. For if / < 6, then f\l)
exists and ||/'(Oll ^ ^- Therefore, there is a 5 such that
\mx)-fm{x-l)\\ <m + e
when \x — l\ < 8. It follows that
ll/a + 5) - /(a)II < ii/(? + 5) - /(Oil + |i/(0 - /(a)II
< {m + e)5+ {m + e){l - a) + e
= (m+ €)(/+ S-a)+e,
so that Z + 5 e. A, a, contradiction. Therefore, I = b. We thus have
11/(6) -/(a)II < (m + e)(h-a) + e,
and, since € is arbitrary, ||/(6) — /(a)|| < m(b — a). 0
The following more general version of the mean-value theorem is the form in
which it is ordinarily applied. As usual, F and G are from a subset of V to W.
Theorem 7.4. If F is differentiable in the ball 5r(«), and if \\dF^\\ < e for
every (3 in this ball, then ||AF^(Q|| < ejj f || whenever (3 and (3 + ^ are in the
ball. More generally, the same result holds if the ball Br (a) is replaced by
any convex set C.
Proof. The segment from 13 to ^ + ^ is the range of the parametrized arc
\(t) = i3 + t^ from [0, 1] to V. If ^ Siud ^ + ^ are in the ball E,(a), then this
segment is a subset of the ball. Setting 7(^) = F{I3 + ^Q, we then have y'(x) =
(lF^^a:^(\'(x)) = dF^^a^^i^), from Theorem 7.2. Therefore, ||T'(a:)|| < €||f|| on
[0, 1], and the mean-value theorem then implies that
iiA^^(^)ii = i!^(^ +0-nm = pw-T(o)ii <€|i^ii(i-0) = eiuii,
which is the desired inequality. The only property of Br(a) that we have used is
that it includes the line segment joining any two of its points. This is the
definition of convexity, and the theorem is therefore true for any convex set. D
Corollary. If G is differentiable on the convex set C, if T G Hom(F, IF),
and if \\dG^ - ^|| < € for all 13 in C, then \\AG^(0 - T{0\\ < eU\\
whenever p and P + ^ are in C.
Proof, SetF = G - T, and note that cZF^ = dG^ - TsmdAF^ = AG^ - T. U
We end this section with a few words about notation. Notice the reversal
of the positions of the variables in the identity {D^F)(a) = dFaiO- This differ-
i^nce has practical importance. We have a function of the two variables ^a
and ^ f' which we can convert to a function of one variable by holding the other
variable fixed; it is convenient technically to put the fixed variable in subscript

150 THE DIFFERENTIAL CALCULUS 3.7
position. Thus we think of dFaiO with a held fixed and have the function dFa
in Hom(F, TF), whereas in (D^F)(a) we hold f fixed and have the directional
derivative D^F: A -^ W in the fixed direction f as a function of a, generalizing
the notation for any ordinary partial derivative dF/dxi(a) as a function of a.
We can also express this implication of the subscript position of a variable in the
dot notation (Section 0.10): when we write D^F(a), we are thinking of the value
at a of the function D^F(').
Still a third notation that we shall use in later chapters puts the function
symbol in subscript position. We write
Jria) = dFa.
This notation implies that the mapping F is going to be fixed through a discussion
and gets it "out of the way" by putting it in subscript position.
If F is differentiable at each point of the open set A^ then we naturally
consider dF to be the map a i-^ dFa from A to Hom(F, W). In the V'-notation,
dF = J p. Later in this chapter we are going to consider the differentiability
of this map at a. This notion of the second differential d^F^ = d(dF)a is probably
confusing at first sight, and a preliminary look at it now may ease the later
discussion. We simply have a new map G = dF from an open set ^4 in a normed
linear space F to a normed linear space X = Hom(F, W), and we consider its
differentiability at a. If dGa = d(dF)a exists, it is a linear map from V to
Hom(F, W)j and there is something special now. Referring back to Theorem 6.1
of Chapter 1, we know that dOa = d^F^ is equivalent by duality to a bilinear
mapping co from V X V to W: since dGaiO is itself a transformation in
Hom(F, W), we can evaluate it at tj, and we define co by
0,(1 V)= idGM))(r,).
The dot notation may be helpful here. The mapping a \-^ dFa is simply
(iF(.), and we have defined G by (?(•) = dF(^.y Later, the fact that dGa(0 is a
mapping can be emphasized by writing it as dGdOi')- ^^ ^^^h case here we
have a function of one variable, and the dot only reminds us of that fact and
shows us where we shall put the variable when indicating an evaluation. In the
case of CO we have the original use of the dot, as in co(f, •) = dGaiO-
EXERCISES
7.1 Given /: IR —» IR such that /'(a) exists, show that the "directional derivative"
Dbf(a) has the value hf\a)^ by a direct evaluation of the limit of the difference quotient.
7.2 Let/ be a real-valued function on an n-dimensional space F, and suppose that/ is
differentiable at a E F. Show that the directions ^ in which the derivative D^F(a) is
zero make up an (n — l)-dimensional subspace of V (or the whole of V). What similar
conclusions can be drawn if/ maps F to a two-dimensional space W?

3.7 DIRECTIONAL DERIVATIVES; THE MEAN-VALUE THEOREM 151
7.3 a) Show by a direct argument on limits that if / and g are two functions from an
interval / C f^ to a normed linear space F, and if f'{x) and g'(x) both exist, then
(/+ gYix) exists and (/+ gYix) = f{x) + g\x).
b) Prove the same result as a corollary of Theorems 7.1 and 7.2 and the
differentiation rules of Section 6.
7.4 a) Given f:I—^V and g:I -^W^ show by a direct limit argument that if
j'{x) and g\x) both exist, and \i F = <fyg>: I -^ VX W^ then F'(x) exists and
F'(x) = <f(x),g\x)>.
b) Prove the same result from Theorems 7.1 and 7.2 and the differentiation rules of
Section 6, using the exact relation F = di o f-^ $2 o g,
7.5 In the spirit of the above two exercises, state a product law for derivatives of
arcs and prove it as in the (b) proofs above.
7.6 Find the tangent vector to the arc -<e', sin O at < = 0; at < = t/2. [Apply
Mxercise 7.4(a).] What is the differential of the above parametrized arc at these two
points? That is, if/(O = "<^S sin O, what are dfo and cjy^/2?
7.7 Let /^: 1R2-» (}^2 ^^ ^j^g mapping <x, ?/>»—^ <3x^yjX^y^>. Compute the
directional derivative Z)<i,2>/^(3, —1)
a) as the tangent vector at <3, —1 >- to the arc /o X, where X is the straight line
through < 3, — 1 > in the direction < 1, 2> ;
b) by first computing d/^<3,_i> and then evaluating at < 1, 2>.
7.8 Let X and n be any two linear functionals on a vector space V. Evaluate the
product /(f) = X(f))Lt(f) along the line f = ta, and hence compute /)«/(«). Now
rvaluate/along the general line f = <a + jS, and from it compute Daf(l3).
7.9 Work the above exercise by computing differentials.
7.10 If /: IR'' —» IR is differentiable at a, we know that its differential d/a, being a
linear functional on R^, is given by its skeleton n-tuple L according to the formula
n
^^(x) = (L,x) = 22 liXi.
1
In this context we call the n-tuple Lthe gradient of / at a. Show from the Schwarz
ine(|uality (Exercise 2.3) that if we use vectors y of Euclidean length 1, then the
directional derivative Dyf(a) is maximum when y points in the direction of the gradient
nif,
7.11 Let IT be a normed linear space, and let V be the set of parametrized arcs
X; 1 — 1, 1] —» W such that X(0) = 0 and X'(0) exists. Show that F is a vector space and
that X —» X'(0) is a surjective linear mapping from V to W. Describe in words the
i'li'iiients of the quotient space V/N, where A^ is the null space of the above map.
7.12 Find another homogeneous nonlinear function. Evaluate its directional deriva-
ftves D^F(0)y and show again that they do not make up a linear map.
7.13 Prove that if /^ is a differentiable mapping from an open ball jB of a normed
linear space F to a normed linear space W such that dFa = 0 for every a in 5, then F
in a constant function.
7.14 Generalize the above exercise to the case where the domain of F is an open set A
wjth the property that any two points of A can be joined by a smooth arc lying in A.

152 THE DIFFERENTIAL CALCULUS 3.8
Show by a counterexample that the result does not generalize to arbitrary open sets A
as the domain of F.
7.15 Prove the following generalization of the mean-value theorem. Let / be a
continuous mapping from the closed interval [a, b] to a normed linear space V, and let g
be a continuous real-valued function on [a, b]. Suppose that/'(0 and g\t) both exist
at all points of the open interval (a, b) and that ||/'(0|| ^ 9^0 on (a, b). Then
11/(6) -/(a)II <^(6) -g(a).
[Consider the points x such that \\f(x) — /(a)|| ^ g(x) — g(a) + e(x — a) + e.]
8. THE DIFFERENTIAL AND PRODUCT SPACES
In this section we shall relate the differentiation rules to the special configurations
resulting from the expression of a vector space as a finite Cartesian product.
When dealing with the range, this is a trivial consideration, but when the domain
is a product space, we become involved with a deeper theorem. These general
product considerations will be specialized to the IR ^-spaces in the next section,
but they also have a more general usefulness, as we shall see in the later sections
of this chapter and in later chapters.
We know that an m-tuple of functions on a common domain, F^: A -^ Wi,
i = 1, . . . , m, is equivalent to a single m-tuple-valued function
m
F:A-^W=Yl Wi,
1
F(a) being the m-tuple {F\a)}^ for each a G A. We now check the obviously
necessary fact that F is differentiable at a if and only if each F^ is differentiate
at a.
Theorem 8.1. Given F': A-^Wi,i= 1, . . . , m, and F = <F\ . . . , F^>,
then F is differentiable at a if and only if all the functions F^ are, in which
csisedFa = <dFl. . . ,dF^>.
Proof. Strictly speaking, F = 117 ^i ° F\ where dj is the injection of Wj into
the product space W = HT Wi (see Section 1.3). Since each di is linear and
hence differentiable, with d(di)a = Oi, we see that if each F^ is differentiable at a,
then so is F, and dF^ = 117 ^i ° <^^L- Less exactly, this is the statement
dFa = < dF\, . . . , dF"^ >. The converse follows similarly from F* = in o F,
where ttj is the projection of HT Wi onto Wj. D
Theorems 7.1 and 8.1 have the following obvious corollary (which can also
be proved as easily by a direct inspection of the limits involved).
Lemma 8.1. If fi is an arc from [a, h] to Wi, fori= 1, . . . , n, and if / is
the n-tuple-valued arc / = </i, . . . ,fn>, then f\x) exists if and only if
fi(x) exists for each i, in which case/'(a:) = <fi(^), • • • ,fn(^) ^•
When the domain space F is a product space IIi Vj the situation is more
complicated. A function F(f i, . . . , fn) of n vector variables does not decompose*

8.8 THE DIFFERENTIAL AND PRODUCT SPACES 153
into an equivalent n-tuple of functions. Moreover, although its differential
(IFa does decompose into an equivalent n-tuple of partial differentials {cZFi},
we do not have the simple theorem that dFa exists if and only if the partial
differentials dFa all exist.
Of course, we regard a function /^(f i, . . . , fn) of n vector variables as being
a function of the single n-tuple variable f = < f i, . . . , fn ^ > so that in principle
there is nothing new when we consider the differentiability of F. However, when
we consider a composition F o G, the inner function G must now be an n-tuple-
valued function G = <^\...,^^>", where g^ is from an open subset A of some
iiormed linear space X to Vi, and we naturally try to express the differential
of F o G in terms of the differentials dg\ To accomplish this we need the partial
differentials dFa of F. For the moment we shall define the jth partial differential
of F at Qj = <ai, . . . , a^^ as the restriction of the differential dFa to Fy,
(U)nsidered as a subspace of F = Hi Vi- As usual, this really involves the
injection dj of Vj into Hi Vi, and our formal (temporary) definition, accordingly,
is
dFL = dFa o dj.
Then, since f = < h, • • • , fn^ = Li ^z(fz)> we have
dFa{& = E dFi{^i),
1
Similarly, since G = <^\ . . . , ^^^ = Hi 6i o ^*, we have
d{F o G)t = E dFho) ° dg\,
1
which we shall call the general chain rule. There is ambiguity in the "z^'-super-
H(!ripts in this formula: to be more proper we should write (dF)a and d(g^)y.
We shall now work around to the real definition of a partial differential.
Since
AFa o dj = (dFa + e) o dj = dFa o dj + e= dF^ + e,
w'v, see that dFa can be directly characterized, independently of dFa, as follows:
dFa is the unique element Ti of Hom(F^, W) such that AFa o di = Ti + o.
That is, dF\ is the differential at ai of the function of the one variable ^i
obtained by holding the other variables in F(fi, . . . , fn) fixed at the values
ii = aj. This is important because in practice it is often such partial differen-
liubility that we come upon as the primary phenomenon. We shall therefore
luke this direct characterization as our definition of dFa, after which our
motivating calculation above is the proof of the following lemma.
Lemma 8.2. If A is an open subset of a product space V = Hi Vi, and if
F: A -^ W is differentiable at a, then all the partial differentials dFa exist
and dFi = dFa o di.

154 THE DIFFERENTIAL CALCULUS 3.8
The question then occurs as to whether the existence of all the partial
differentials dFa implies the existence of dFa. The answer in general is negative,
as we shall see in the next section, but if all the partial differentials dFa exist for
each Qj in an open set A and are continuous functions of qj, then F is continuously
differentiable on A. Note that Lemma 8.2 and the projection-injection identities
show us what dFa must be if it exists: dFa = dFa o di and ^ di o tt^ = / together
imply that dFa = H dF\^ o tt-.
Theorem 8.2. Let A be an open subset of the normed linear space
V = V\X F2, and suppose that F: A -^ W has continuous partial
differentials dF\a,^> and dF%a,0> on A. Then dF^a,0> exists and is continuous
on A, and dF<a,^>(f, v) = ^^<a.^>(f) + dF%cc,Mv)'
Proof. We shall use the sum norm on F = Fi X F2. Given €, we choose 8 so
that ||dF<^,^> — dF\a,^>\\ < € for every </x, ^> in the 5-ball about <a^ fi>
and for i = 1, 2. Setting
we have
WdG^W = ||dFi,«+f.^+,> - dFl,«.^>|| < € if ||<f, r;>|| < 5,
and the corollary of Theorem 7.4 implies that
\\Fia +^,p+rj)- Fia, 0 + v) - ^^<a.^>(0|| < €|| f ||
when II < f, r? > II < 8. Arguing similarly with
H(v) = F(a,0+v) -dF^^Mv).
we find that
\\F{a, 0 + v)- F(a, fi) - dF%„,^^(r,)\\ < e||,||
when II <0, r7> II < 8. Combining the two inequalities, we have
||AF<«.^>(f, r;) - T«^,v>)\\ < e\\<^,v>\\
when II < f, r7>|| < 8, where T = dF\cc,^> ° tti + dF%a,^> ° '7r2. That is,
AF^a,^> — T = 0, and so dF^a,^> exists and equals T. D
The theorem for more than two factor spaces is a corollary.
Theorem 8.3. If A is an open subset of JTiVi and F -^ W is such that for
each i = 1, . . . , n the partial differential dFa exists for all qj G A and is
continuous as a function of qj = <ai, . . . , a^^j then dFa exists and is
continuous on .4. If € = < fi, . . . , fn>, then dFa(^) = L^ dFi(^i),
Proof. The existence and continuity of dFa and dFa imply by the theorem that
dFi 0x1 + dFa o 7r2 is the differential of F considered as a function of the first
two variables when the others are held fixed. Since it is the sum of continuous

3.8 THE DIFFERENTIAL AND PRODUCT SPACES 155
functions, it is itself continuous in qj, and we can now apply the theorem again
to add dFa to this sum partial differential, concluding that Y.\ dFaOTr^is the
partial differential of F on the factor space Vi X V2X F3, and so on (which is
colloquial for induction). D
As an illustration of the use of these two theorems, we shall deduce the
general product rule (although a direct proof based on A-estimates is perfectly
feasible). A general product is simply a bounded bilinear mapping co:X X Y-^W,
where X, F, and W are all normed linear spaces. The boundedness inequality
hereis||co(f, r;)|| < 6||f|| ||r;||.^
We first show that co is differentiable.
Lemma 8.3. A bounded bilinear mapping co: X X Y -^ W is everywhere
differentiable and <ico<a,^>(f, tj) = co(a, rj) + co(f, P).
Proof. With 13 held fixed, g^(^) = co(f, 13) is in Hom(Z, W) and therefore is
everywhere differentiable and equal to its own differential. That is, dco^ exists
and dco\a,^>(0 = ^(f>i^)- Since jS 1-^ ^^ is a bounded linear mapping,
^w<a,^> = ^^ is a continuous function of <aj^>. Similarly, do)%a,0>(v) =
w(a, ry), and doo^ is continuous. The lemma is now a direct corollary of Theorem
8.2. D
If co(f, Tj) is thought of as a product of f and tj, then the product of two
functions g(^) and /i(r) is co(^(f), /i(f)), where g is from an open subset ^4 of a
normed linear space F to X and h is from A to F. The product rule is now just
what would be expected: the differential of the product is the first times the
differential of the second plus the second times the differential of the first.
Theorem 8.4. If g: A -^ X and h: A -^ Y are differentiable at jS, then so
is the product F(^) = co(^(f), /i(r)) and
dF^U) = co(^(^), dh^U)) + co(d^^(f), HP)),
Proof. This is a direct corollary of Theorem 8.1, Lemma 8.3, and the chain
rule. D
KXERCISES
8.1 Find the tangent vector to the arc <sint,cost,t^>- at < = 0; at < = ir/2.
What is the differential of the above parametrized arc at the two given points? That is,
>^ fiO = < sin t, cos t,t^>, what are dfo and dI/^/2?
8.2 Give the detailed proof of Lemma 8.1.
8.3 The formula
n
dFa{0 = E dFii^i)
1

156 THE DIFFERENTIAL CALCULUS 3.9
is probably obvious in view of the identity
n
1
and the definition of partial differentials, but write out an explicit, detailed proof
anyway.
8.4 Let F he 8l differentiable mapping from an n-dimensional vector space F to a
finite-dimensional vector space W, and define G: VX W -^ W by G(^, tj) = tj — F(^).
Thus the graph of /^ in V X W is the null set of G. Show that the null space of
dG^a,p> has dimension n for every <a, I3> E V X W.
8.5 Let F(^, tj) be a continuously differentiable function defined on a product
A X B, where jB is a ball and A is an open set. Suppose that dF\a,^> = 0 for all
<a, I3> in A X B. Prove that F is independent of tj. That is, show that there is a
continuously differentiable function G(^) defined on A such that /^(f, tj) = G(^) on
AX B.
8.6 By considering a domain in R^ as indicated at the right, show
that there exists a function f(x, y) on an open set A in IR^ such that
dy
everywhere and such that /(x, y) is not a function of x alone.
8.7 Let F{^, TJ, f) be any function of three vector variables, and for fixed 7 set
^(Jj v) = ^(S, V,^)- Prove that the partial differential dF]<a,fi,y> exists if and only
if dG\a,^> exists, in which case they are equal.
8.8 Give a more careful proof of Theorem 8.3. That is, state the inductive hypothesis
and show that the theorem follows from it and Theorem 8.2. If you are meticulous in
your argument, you will need a form of the above exercise.
8.9 Let/be a differentiable mapping from R^ to R. Regarding IR^ as IR X R, show
that the two partial differentials of/are simply multiplication by its partial derivatives.
Generalize to n dimensions. Show that the above is still true for a map F from R^ to a
general vector space V, the partial derivatives now being vectors.
8.10 Give the details of the proof of Theorem 8.4.
9. THE DIFFERENTIAL AND R"-
We shall now apply the results of the last two sections to mappings involving the
Cartesian spaces IR^, the bread and butter spaces of finite-dimensional theory.
We start with the domain.
Theorem 9.1. If F is a mapping from (an open subset of) IR^ to a normed
linear space TF, then the directional derivative of F in the direction of the jth
standard basis vector 8^ is just the partial derivative dF/dxj, and the jth
partial differential is multiplication by dF/dxj: dF^JJi) = h(dF/dxj)(a).
More exactly, if any one of the above three objects exists at a, then they
all do, with the above relationships.

3.9 THE DIFFERENTIAL AND W 157
Proof. We have
dF F(ai, ..., ay + /,..., an) — F(ai,. . . , ay, . . . , an)
-— (a) = lim
dxj ^ ' t-.o t
Moreover, since the restriction of F to a + U8^' is a parametrized arc whose
differential at 0 is by definition the jth partial differential of F at a and whose
tangent vector at 0 we have just computed to be (dF/dxj)(a)j the remainder of
the theorem follows from Theorem 7.1. D
Combining this theorem and Theorem 7.2, we obtain the following result.
Theorem 9.2. U V = W^ and F is differentiable at a, then the partial
derivatives (dF/dxj)(a) all exist and the n-tuple of partial derivatives at
a, {(^^/^a:y)(a)} 1, is the skeleton of dF^. In particular,
n -JET
DyF{a)= Z2/>|f (a).
Proof. Since dF^(8^) = D^^F^si) = {dF/dxi){a), as we noted above, we have
DyF{a) = dF^iy) = dF^ (Z Vi^') = E Vi dF.(8') = E y, g (a).
All that we have done here is to display dF^ as the linear combination mapping
defined by its skeleton {dF^(8')} (see Theorem 1.2 of Chapter 1), where T(8') =
(lFa(8^) is now recognized as the partial derivative (dF/dxi)(a.). D
The above formula shows the barbarism of the classical notation for partial
derivatives: note how it comes out if we try to evaluate dFa(x). The notation
Ds^F is precise but cumbersome. Other notations are Fj and DjF. Each has its
problems, but the second probably minimizes the difficulties. Using it, our
formula reads dF^(y) = Y.7=i yjDjF(a).
In the opposite direction we have the corresponding specialization of
Theorem 8.3.
Theorem 9.3. If A is an open subset of IR^, and if F is a mapping from A
to a normed linear space W such that all of the \ partial derivatives
(dF/dxj)(a) exist and are continuous on A, then F is continuously
differentiable on A.
Proof. Since the jth partial differential of F is simply multiplication by dF/dxj,
we are (by Theorem 9.1) assuming the existence and continuity of all the partial
differentials dF{ on A. Theorem 9.3 thus becomes a special case of Theorem 8.3. D
Now suppose that the range space of F is also a Cartesian space, so that F
is a mapping from an open subset A of W" to W^. Then dF^ is in Hom(lR'^, W^).

158 THE DIFFERENTIAL CALCULUS 3.9
For computational purposes we want to represent linear maps from IR^ to W"^
by their matrices, and it is therefore of the utmost importance to find the matrix
t of the differential T = dF^. This matrix is called the Jacobian matrix of F
at a.
The columns of t form the skeleton of dF^^ and we saw above that this
skeleton is the n-tuple of partial derivatives (dF/dxj)(a). If we write the m-
tuple-valued F loosely as an m-tuple of functions, F = </i, . . . ffm>, then
according to Lemma 8.1, thejth column of t is the m-tuple
^^■)HI<" i<4
dxj
Thus,
Theorem 9.4. Let F be a mapping from an open subset of W^ to IR^, and
suppose that F is differentiable at a. Then the matrix of dF^ (the Jacobian
matrix of F at a) is given by
dxj
If we use the notation yi = fi(x), we have
'.■,=&(»)■
If we also have a differentiable map z = G(y) = <gi(y), . . . , giiy) > from
an open set B C^'^ into U\ then tZGb has, similarly, the matrix
Also, if B contains b = F(a)y then the composite-function rule
d(G o F)a = dGb o dF^
has the matrix form
d^j (") = 5 ^- (*») d^j ("^'
or simply
dzk ^ y> dzk dyi
dxj ^i dyi dxj
This is the usual form of the chain rule in the calculus. We see that it is merely
the expression of the composition of linear maps as matrix multiplication.
We saw in Section 8 that the ordinary derivative/'(a) of a function/ of one
real variable is the skeleton of the differential d/a, and it is perfectly reasonable to
generalize this relationship and define the derivative ^'(a) of a function F of
n real variables to be the skeleton of dF^, so that ^'(a) is the n-tuple of partial
derivatives {idF/dxi){a)}iy as we saw above. In particular, if F is from an open
subset of W^ to R^y then F\a) is the Jacobian matrix of F at a. This gives the

3.9 THE DIFFERENTIAL AND U^ 159
n>atrix chain rule the standard form
(Go FY {a) = G'(Fia))F\a).
Some authors use the word ^derivative' for what we have called the
differential, but this is a change from the traditional meaning in the one-variable case,
and we prefer to maintain the distinction as discussed above: the differential dFa
is the linear map approximating AFa and the derivative ^'(a) must be the
matrix of this linear map when the domain and range spaces are Cartesian.
However, we shall stay with the language of Jacobians.
Suppose now that A is an open subset of a finite-dimensional vector space V
and that H: A -^ W is differentiate Sit a G A. Suppose that W is also finite-
dimensional and that <p:V-^U^ and xpiW -^ W^ are any coordinate
isomorphisms. If A = <p[A]j then A is an open subset of W^ and H = xp o H o <^~i is a
mapping from A to W^ which is differentiate at a = <^(«), with dH^ =
x// o dHa o (p~~^. Then dH^ is given by its Jacobian matrix {idh^/dxj)(a)}y which
we now call the Jacobian matrix of H with respect to the chosen bases in V and
W, Change of bases in V and W changes the Jacobian matrix according to the
rule given in Section 2.4.
If F is a mapping from IR^ to itself, then the determinant of the Jacobian
matrix (df/dxj) (a) is called the Jacobian of F at a. It is designated
y'---'^"ha), or y^'---'^")(a)
d{Xi, . . . ,Xn) d(Xi, . . . ,Xn)
if it is understood that yi = f(x). Another notation is J/r(a) (or simply J(a) if
F is understood). However, this is sometimes used to indicate the differential
dF^f and we shall write det Jf{^) instead.
If F{x) = <xi — xl, 2a:ia:2 >", then its Jacobian matrix is
2xi —2x2
2a:2 2xi
and det J(x) = 4(a:f + 4i = 4(||x||2)
2
EXERCISES
9.1 By analogy with the notion of a parametrized arc, we define a smooth
parametrized two-dimensional surface in a normed linear space W to be a continuously
differentiable map F from a rectangle / X J in IR^ to W. Suppose that I X J =
[-^1, 1] X [—1, 1], and invent a definition of the tangent space to the range of F in W
at the point F(0, 0). Show that the two vectors
are a basis for this tangent space. (This should not have been your definition.)

160 THE DIFFERENTIAL CALCULUS 3.9
9.2 Generalize the above exercise to a smooth parametrized n-dimensional surface
in a normed linear space W.
9.3 Compute the Jacobian matrix of the mapping •<x,y> i-^ -Kx^^y^, (x-j- y)^>.
Show that its rank is two except at the origin.
9.4 Let F = <f,P,f > from U^ to U^ be defined by
fi{x, y,z) = x + y + z, f2{x, y, z) = x^ + ^/^ + z^,
and
fs(x,y,z) = x^ + y^ + z^.
Compute the Jacobian of /^ at <a,h, c>. Show that it is nonsingular unless two of the
three coordinates are equal. Describe the locus of its singularities.
9.5 Compute the Jacobian of the mapping F: <x,y> \-^ -< (x-{-y)^, y^ > from
IR^tolR^at <1, —l>;at <l,0>;at <a, 6>. Compute the Jacobian of (?: <s,t>^
<s — t, s -{- t> at <w, t;>.
9.6 In the above exercise compute the compositions F o G and G o F. Compute the
Jacobian oi F o (7 at <yyV>. Compute the corresponding product of the Jacobians
of F and G.
9.7 Compute the Jacobian matrix and determinant of the mapping T defined by
X = r cos 6, y = r sm 6, z = z. Composing a function f(x, y, z) with this mapping
gives a new function:
gf(r, ^, z) = /(r cos ^, r sin ^, z).
That is, gf = / o T. This composition (substitution) is called the change to cylindrical
coordinates in W*,
9.8 Compute the Jacobian determinant of the polar coordinate transformation
<r^Q> \-^ <x^y> ^ where x = r cos d^y — r sin d,
9.9 The transformation to spherical coordinates is given by x = r sin <^ cos ^,
y = r sin (^ sin d^z = r cos d. Compute the Jacobian
^(^, V, z)
d(r,<p,d)'
9.10 Write out the chain rule for the following special cases:
dw/dt = ?, where w = F(Xyy)y x = g(t), y = h(t).
Find dw/dt when w = F{xi, . . . , xj and Xi = gi(t), i = 1, . . . , n. Find dw/du when
w = F{x,y), X = g{Ujv), y = h(u,v). The special case where g{u,v) = w can be
rewritten
£F(a;,Mx,t;)).
Compute it.
9.11 li w = f(x, y), X = r cos 6, and y = r sin 6, show that
[^lyl , \dw\ \dw\ , r^iy I

3.10 ELEMENTARY APPLICATIONS 161
10. ELEMENTARY APPLICATIONS
The elementary max-min theory from the standard calculus generalizes with
little change, and we include a brief discussion of it at this point.
Theorem 10.1. Let F be a real-valued function defined on an open subset A
of a normed linear space F, and suppose that F assumes a relative maximum
value at a point a in A where dFa exists. Then dFa = 0.
Proof. By definition D^F(a) is the derivative 7'(0) of the function 7(^) =
F(a -\~ t^)f and the domain of 7 is a neighborhood of 0 in U. Since 7 has a relative
maximum value at 0, we have 7'(0) = 0 by the elementary calculus. Thus
clFaiO = D^F(a) = 0 for all f, and so dFa = 0. D
A point a such that dFa = 0 is called a critical point. The theorem states
that a differentiable real-valued function can have an interior extremal value
only at a critical point.
If F is R^, then the above argument shows that a real-valued function F
can have a relative maximum (or minimum) at a only if the partial derivatives
(dF/dxi)(a) are all zero, and, as in the elementary calculus, this often provides
a way of calculating maximum (or minimum) values. Suppose, for example,
that we want to show that the cube is the most efficient rectangular parallelepiped
from the point of view of minimizing surface area for a given volume V. If
the edges are a:, y and z, we have V = xyz and A = 2(xy -\- xz ^ yz) =
2(xy + V/y + V/x). Then from 0 = dA/dx = 2(y - V/x^), we see that
V = yx^y and, similarly, dA/dy = 0 impHes that V = xy^. Therefore, yx^ =
xy^f and since neither x nor y can be 0, it follows that x = y. Then V = yx^ =
x^f and X = V^^^ = y. Finally, substituting in F = xyz shows that z = V^^^.
Our critical configuration is thus a cube, with minimum area A = QV^'^.
It was assumed above that A has an absolute minimum at some point
<x^y^z>. The reader might enjoy showing that ^4 -^ oo if any of a:, y^ z tends
to 0 or 00, which implies that the minimum does indeed exist.
We shall return to the problem of determining critical points in Sections
12, 15, and 16.
The condition dFa = 0 is necessary but not sufficient for an interior
maximum or minimum. The reader will remember a sufficient condition from
beginning calculus: If f'{x) = 0 and f"{x) < 0 (>0), then x is a relative
maximum (minimimi) point for /. We shall prove the corresponding general theorem
in Section 16. There are more possibilities now; among them we have the
analogous sufficient condition that if dFa = 0 and d^Fa is negative (positive)
definite as a quadratic form on V, then a is a relative maximum (minimum)
point of F,
We consider next the notion of a tangent plane to a graph. The calculation
of tangent lines to curves and tangent planes to surfaces is ordinarily considered
a geometric application of the derivative, and we take this as sufficient
justification for considering the general question here.

162 THE DIFFERENTIAL CALCULUS 3.10
Let F be a mapping from an open subset ^4 of a normed linear space V to
a normed linear space W. When we view F as a graph in F X TF, we think of it
as a "surface" S lying "over" the domain A, generalizing the geometric
interpretation of the graph of a real-valued function of two real variables inIR^ = IR^XlR.
The projection tti'.V X W -^ V projects S "down" onto A^
and the mapping ^ \~^ <^,F(^)> gives the point of S lying "over" f. Our
geometric imagery views V as the plane (subspace) F X {0} in F X TF, just as
we customarily visualize U as the real axis U X {0} in IR^.
We now assume that F is differentiable at a. Our preliminary discussion in
Section 6 suggested that (the graph of) the linear function dFa is the tangent
plane to (the graph of) the function AFa in F X TF, and that its translate M
through -< a, F{a) > is the tangent plane at -< a, F(a) > to the surface S that is
(the graph of) F. The equation of this plane is rj — F(a) = dFa(^ — a), and
it is accordingly (the graph of) the affine function (?(Q = dF^i^ — a) + F(a).
Now we know that dFa is the unique T in Hom(F, W) such that AFa{0 =
T(^) + ©(f), and if we set f = f — a, it is easy to see that this is the same as
saying that G is the unique affine map from V to W such that
FU) - G{0 = Kf - «).
That is, M is the unique plane over V that "fits" the surface S around -< a, F(a) >
in the sense of ©-approximation.
However, there is one further geometric fact that greatly strengthens our
feeling that this ixally is the tangent plane.
Theorem 10.2. The plane with equation rj — F(a) = dFa{^ — a) is
exactly the union of all the straight lines through <a, F(a)> in V X W
that are tangent to smooth curves on the surface S = graph F passing
through this point. In other words, the vectors in the subspace dFa of
V X W are exactly the tangent vectors to curves lying in S and passing
through -< a, F(a) >.
Proof. This is nearly trivial. If -< f, ^ > G dFa, then the arc
7(0 = <a + t^,Fia + tO>
in S lying over the line t\-^ a + t^ in V has -< f, dFa(0 ^ = < ^y V^ as its
tangent vector at -<a, F(a) >, by Lemma 8.1 and Theorem 8.2.
Conversely, if ^ i-^ <\(t),F(\(t))> is any smooth arc in S passing through a,
with a = X(^o)> then its tangent vector at <a, F(a) > is
<X\to),dFa{yito))>,
a vector in (the graph of) dFa. □

:5.io
ELEMENTARY APPLICATIONS
163
As an example of the general tangent plane discussed above, let F =
<fij2> be the map from IR^ to U^ defined by/i(x) = (xj - a:^)/2,/2(x) =
.ria:2. The graph of F is a surface over U in
X U . According to our
above discussion, the tangent plane at -< a, F(a) > has the equation y =
dFa(x — a) + F(a). At a = -< 1, 2> the Jacobian matrix of clF^ is
Xi —Xo
_X2 X]
2] = [1 ~2]
lj<l,2> [2 Ij
jirid F(a) = < —f, 2 >. The equation of the tangent plane il/ at -< 1, 2 > is thus
-2I
<yi,y2>
U -2
L2 1
<Xi
1,:^2
2> + <-f,2>.
('omputing the matrix product, we have the scalar equations
y^ = xi - 2x2 + (-1 +4 -f) = xi- 2x2 + f,
2/2 = 2xx + 0:2 + (-2 -2 +2) = 2a:i +0:2-2.
Note that these two equations present the affine space M as the intersection
of the hyperplane in IR^ consisting of all <X\, X2, Vi, 2/2^ such that
0:1 — 20:2 — Vi = —f,
with the hyperplane having the equation
2a:i + 0:2 — 2/2 = 2.
IXERCISES
10.1 Find the maximum value oi f(x, y, z) = x -j- y -{- z on the ellipsoid
^2 _f_ 2y^ + 3^2 = 1.
10.2 Find the maximum value of the linear functional f(x) = ^iCiXi on the unit
^l)licreXli X? = 1.
10.3 Find the (minimum) distance between the two lines
t8^
and
y = s<l,l,l>+ <1,0,-1>
10.4 Show that there is a uniquely determined pair of closest points on the two lines
\ ^ ta-{-l and y = sb + m in IR '^ unless b = A:a for some k. We assume that
u ^ 0 F^ b„ Remember that if b is not of the form ka, then |(a, b)| < ||a||2 ||b||2,
recording to the Schwarz inequality.
10.5 Show that the origin is the only critical point of f(x, y, z) = xy -{- yz-^ zx.
I'iiid a line through the origin along which 0 is a maximum point for/, and find another
liiio along which 0 is a minimum point.

164 THE DIFFERENTIAL CALCULUS 3.11
10.6 In the problem of minimizing the area of a rectangular parallelepiped of given
volume V worked out in the text, it was assumed that
A
<-+^^)
has an absolute minimum at an interior point of the first quadrant. Prove this. Show
first that .1 —> oo if '<^x,y>- approaches the boundary in any way:
x—>0, x—>oo^ ^_>0, or 2/—>^.
10.7 Let /^: IR2 -> [R2 ^g ^^e mapping defined by
yi = sin (xi + X2), 2/2 = cos (xi — X2).
Find the equation of the tangent plane in U^ to the graph of F over the point a =
<7r/4,7r/4>.
10.8 Define F: IR^ _> [R2 ^y
3 3
E2 V^ 3
1 1
Find the equation of the tangent plane to the graph of F in U^ over a = <1,2, —1>.
10.9 Let co(f, rj) be a bounded bilinear mapping from a product normed linear space
F X ir to a normed linear space X. Show that the equation of the tangent plane to the
graph iS of CO in F X ir X X at the point <a,p,y>^Sis
r = co(?,/?) + co(a,77) + co(a,/?).
10.10 Let F be a bounded linear functional on the normed linear space V. Show that
the equation of the tangent plane to the graph of F^ in F X IR over the point a can be
written in the form ij = F^(a)(3F(0 — 2F(a)).
10.11 Show that if the general equation for a tangent plane given in the text is applied
to a mapping P' in Hom(F, W), then it reduces to the equation for F itself [rj = F(^)],
no matter where the point of tangency. (Naturally!)
10.12 Continuing Exercise 9.1, show that the tangent space to the range of F in W
at F(0) is the projection on IF of the tangent space to the graph of F in (R^ X IF at the
point <0, F(0) >. Now define the tangent plane to range F in IF at F(0), and show
that it is similarly the projection of the tangent plane to the graph of F.
10.13 Let /^': F —> IF be differentiable at a. Show that the range of dFa is the
projection on IF of the tangent space to the graph of /^ in F X IF at the point <«, F{a) >.
11. THE IMPLICIT-FUNCTION THEOREM
The formula for the Jacobian of a composite map that we obtained in Section 9
is reminiscent of the chain rule for the differential of a composite map that we
derived earlier (Section 8). The Jacobian formula involves numbers (partial
derivatives) that we multiply and add; the differential chain rule involves linear
maps (partial differentials) that we compose and add. (The similarity becomes
a full formal analogy if we use block decompositions.) Roughly speaking, the

3.11 THE IMPLICIT-FUNCTION THEOREM 165
whole differential calculus goes this way. In the one-variable calculus a
differential is a Hnear map from the one-dimensional space U to itself, and is therefore
multiplication by a number, the derivative. In the many-variable calculus when
we decompose with respect to one-dimensional subspaces, we get blocks of such
numbers, i.e., Jacobian matrices. When we generalize the whole theory to vector
spaces that are not one-dimensional, we get essentially the same formulas but
with numbers replaced by Hnear maps (differentials) and multipHcation by
composition.
Thus the derivative of an inverse function is the reciprocal of the derivative
of the function: if ^ = f~^ and h = f{a), then g\h) = l/j'{a). The differential
of an inverse map is the composition inverse of the differential of the map: if
G= F-' and F{a) = /?, then clG^ = {dFa)~'.
If the equation g{x, y) = 0 defines y implicitly as a function of x, y = f{x),
we learn to compute/'(a) in the elementary calculus by differentiating
g(x,f{x)) ^0,
and we get
where h = f(a). Hence
f'ia) =
dg/dx
dgjdy
We shall see below that if G{^, -q) = 0 defines 77 as a function of ^, 77 = F{^),
and if /? = ^(«), then we calculate the differential clFa by differentiating the
identity G{^, F{^)) = 0, and we get a formula formally identical to the above.
Finally, in exactly the same way, the so-called auxiliary variable method of
solving max-min problems in the elementary calculus has the same formal
structure as our later solution of a "constrained " maximum problem by Lagrange
multipliers.
In this section we shall consider the existence and differentiability of
functions implicitly defined. Suppose that we are given a (vector-valued) function
G{^, rj) of two vector variables, and we want to know whether setting G equal
to 0 defines 77 as a function of ?, that is, whether there exists a unique function F
such that G(^, F{^)) is identically zero. Supposing that such an "implicitly
defined" function F exists and that everything is differentiable, we can try
to compute the differential of F at a by differentiating the equation G(^, F{^)) —
0, or G o <I,F> = 0. We get dG}<cc,3> ° dl^ + dG\cc,^> ° dF^ = 0, where we
have set p = F{a). If dG^ is invertible, we can solve for dF^, getting
clFcc = —(dG^a,p>)~ ° dG^aS>'
Note that this has the same form as the corresponding expression from the
elementary calculus that we reviewed arbove. If F is uniquely determined, then
so is cWa, and the above calculation therefore strongly suggests that we are

166 THE DIFFERENTIAL CALCULUS 3.11
going to need the existence of (dG%a,p>)~^ as a necessary condition for the
existence of a uniquely defined implicit function around the point < a, jS >.
Since ^ is F(a), we also need (?(a, ^) = 0. These considerations will lead us to
the right theorem, but we shall have to postpone part of its proof to the next
chapter. What we can prove here is that if there is an implicitly defined function,
then it must be differentiable.
Theorem 11.1. Let F, W, and X be normed linear spaces, and let (? be a
mapping from an open subset AxBofVxWtoX. Suppose that F is a
continuous mapping from A to B implicitly defined by the equation (?(f, rj) = 0,
that is, satisfying G{^y F(^)) = 0 on A. Finally, suppose that G is
differentiable at <a^ I3>, where 13 — F(a)^ and that dG%a,fi> is invertible. Then
F is differentiable at a and dFa = — (cl(?<a,/3>)""^ ° dG\a,^>'
Proof. Set 77 = AF«(a, so that G{a + ^, 13 + v) = G(a + f, F(a + ©) = 0.
Then
Applying T~^ to this equation, where T = d(?<a,/3>, and solving for 77, we get
This equation is of the form rj = 0(Q + ©(< f, ^>), and since rj = AF«(f) is
an infinitesimal ^(0, by the continuity of F at a. Lemmas 5.1 and 5.2 imply
first that 7} = 0(0 and then that <^,v> = 0(0- Thus e{e(<l r}>)) =
0((00(f))) = ©(?), and we have
^FM) = V = SiO + o(Q,
where S = —(dG%a,p>)~^ ° dG\a,^>-, an element of Hom(F, W). Therefore, F
is differentiable at a and dF^ has the asserted value. D
We shall show in the next chapter, as an application of the fixed-point
theorem, that if F, TF, and X are finite-dimensional, and if G is a continuously
differentiable mapping from an open subset ^IX^ofFxTFtoX such that
at the point <a, fi> we have both G(a, $) = 0 and dG%a,p> invertible, then
there is a uniquely determined continuous mapping F from a neighborhood M of
a to 5 such that F(a) = fi and G(f, F{Q) = 0 on M. The same theorem is true
for the more general class of complete normed linear spaces which we shall study
in the next chapter. For these spaces it is also true that if T~^ exists, then so
does S~^ for all S sufficiently close to T, and the mapping S 1-^ S~^ is
continuous. Therefore dG\y^^v> is invertible for all </x, ^> sufficiently close to
<a, fi>, and the above theorem then implies that F is differentiable on a
neighborhood of a. Moreover, only continuous mappings are involved in the
formula given by the theorem for dF: yL\-^ dFfj,, and it follows that F is in fact
continuously differentiable near a. These conclusions constitute the implicit-
function theorem, which we now restate.

1,11 THE IMPLICIT-FUNCTION THEOREM 167
Theorem 11.2. Let F, TF, and X be finite-dimensional (or, more generally,
complete) normed linear spaces, \et A X B be an open subset of F X TF,
and let G: A X B -^ X he continuously differentiable. Suppose that at the
point <a, I3> in A X B we have both (?(a, 13) = 0 and dG%a,fi> invertible.
Then there is a ball M about a and a uniquely defined continuously
differentiable mapping F from MtoB such that F(a) = fi and G(f, F(f)) = 0 on M.
The so-called inverse-mapping theorem is a special case of the implicit-
fiiitction theorem.
Theorem 11.3. Let H he 2, continuously differentiable mapping from an
open subset E of a finite-dimensional (or complete) normed linear space IF
to a normed linear space F, and suppose that its differential is invertible at
a point fi. Then H itself is invertible near fi. That is, there is a ball M about
a = H(I3) and a uniquely determined continuously differentiable function F
from M to B such that F(a) = p and H{F(^)) = f on M.
^'f^Hff. Set G(^, 7}) = f — H(r}). Then G is continuously differentiable from
I X ^ to F and dG%a,p> = —dH^ is invertible. The implicit-function theorem
lif^i gives us a ball M about a and a uniquely determined continuously differ-
nhable mapping F from M to B such that F(a) = ^ and 0 = G(f, F(^)) =
n{F(0) onM. D
The inverse-mapping theorem is often given a slightly different formulation
hieh we state as a corollary.
Corollary. Under the hypotheses of the above theorem there exists an open
neighborhood U of 13 such that H is injective on U, N = H[U] is open in
V, and H~^ is continuously differentiable on N. (See Fig. 3.11.)
Fig. 3.11
''>/. The proof of the corollary is left as an exercise.
In practice we often have to apply the Cartesian formulations of these
lliiiorems. The student should certainly be able to write these down,, but we
iliii!! state them anyway; starting with the simpler inverse-mapping theorem.
Theorem 11.4. Suppose that we are given n continuously differentiable
real-valued functions (?z(2/i, . . . , yn), i = 1, . . . , n, of n real variables
defined on a neighborhood E of a point b in IR^ and suppose that the Jacobian
determinant
^((?l, . . . , Gn) /j^N
^(2/1, . . . , 2/n)

168
THE DIFFERENTIAL CALCULUS
3.11
is not zero. Then there is a ball M about a = G^(b) in U^ and a uniquely
determined n-tuple F = <Fi, . . . , Fn^ oi continuously differentiable real-
valued functions defined on M such that F(a) = b and G(F(x)) = x on 71/
for i = 1, . . . , n. That is, Gi{Fi{xi, . . . , Xn), . . . , Fn{xi, . . . , Xn)) = /,
for all X in M and ior i = 1, . . . , n.
For example, if x
we have
<y\ + yl, y\ + yl>, then at the point b = <1, 2>
^(yi, 2/2)
det
det
Syl 32/1
[22/1, 22/2J
<1,2>
l2,
12
4
12 9^ 0,
and we therefore know without trying to solve explicitly that there is a unique
solution for y in terms of x near x = < 1^ + 2^, 1^ + 22> = <9, 5>. The
reader would find it virtually impossible to solve for y, since he would quickly
discover that he had to solve a polynomial equation of degree 6. This clearly
shows the power of the theorem: we are guaranteed the existence of a mapping
which may be very difficult if not impossible to find explicitly. (However, in
the next chapter we shall discover an iterative procedure for approximating tlK*
inverse mapping as closely as we want.)
Everything we have said here applies all the more to the implicit-function
theorem, which we now state in Cartesian form.
Theorem 11.5. Suppose that we are given m continuously differentiable
real-valued functions Gi(x, y) = Gi(xi, . . . , Xn, yi, - - - , ym) of n + m real
variables defined on an open subset A X B oi IR^+^ and an (n + m)-tuple
<a, b> = <ai, . . . , an, &i, . . . , ??m>" such that Gi(a, b) = 0 for i ^
1, . . . , m, and such that the Jacobian determinant
d(Gi, . . . , Gm)
^(2/1,'" ,ym)
(a,b)
is not zero. Then there is a ball M about a in U^ and a uniquely determin(Ml
m-tuple F= <Fi, . . . , Fm^" of continuously differentiable real-valued
functions Fj(x) = Fj(xi, . . . , Xn) defined on M such that b = F(a) ami
Gi(x, F(x)) = 0 on M for z = 1, . . . , m. That is, hi = Fi(ai, . . . , an) for
i = 1, . . . , m,andft(a:i, . . . , Xn;Fi(xi, . . . , Xn), . . . ,Fm(xi, . . . , Xn))
0 for all X in M and ior i = 1, . . . , m.
For example, the equations
3 I 3
Xi + X2
x\
xl
y\
yl
0,
2/2 = 0
can be solved uniquely for y in terms of x near -<x, y> = <1, 1, 1, —1>,

3.11 THE IMPLICIT-FUNCTION THEOREM 169
because they hold at that point and because
eiG^^G^ _ d^^
— 22/1, —22/2I _ ^^^, ^,2 .. ..2^
has the value 12 there. Of course, we mean only that the solution functions
diVh 2/2)
i 12 ther€
exist, not that we can explicitly produce them.
3,?; -SIJ = '^'^'' - '^''^
EXERCISES
11.1 Show that <x,y> i-^ <e''-i-e^, e^'-i-e~y> is locally invertible about any
point <a, 6>, and compute the Jacobian matrix of the inverse map.
11.2 Show that <w, t;> 1-^ <e" + e'', e" — e''> is locally invertible about any
point <a,b> in IR^, by computing the Jacobian matrix. In this case the whole
mapping is invertible, with an easily computed inverse. Make this calculation, compute
the Jacobian matrix of the inverse map, and verify that the two matrices are inverses
at the appropriate points.
11.3 Show that the mapping <x,y,z> 1-^ <sin x, cosy, e^> from U^ to U^ is
locally invertible about < 0,7r/2, 0 >. Show that
•<x,y, z> i-^ <sm (x -{- y -{- z), cos (x — y -\~ z), e^^+^~^^>
is locally invertible about <7r/4, —7r/4, 0>.
11.4 Express the second map of the above exercise as the composition of two maps,
and obtain your answer a second way.
11.5 Let F: <x,y> i-^ <u,v> be the mapping from U^ to U^ defined by w =
3.2 _|_ y2^ ^ _ 2xy. Compute an inverse G of F, being careful to give the domain and
range of G. How many inverse mappings are there? Compute the Jacobian matrices
of /^ at < 1, 2> and of G^ at <5, 4>, and show by multiplying them that they are
inverse.
11.6 Consider now the mapping F: <x,y> 1-^ <x^,y^>. Show that dF^0,0^ is
singular and yet that the mapping has an inverse G. What conclusion do we draw
about the differentiability of G at the origin?
11.7 Define /^: IR^ _^ (1^2 y^y. ^x,y> ^-^ <e'' cosy, e"" smy>. Prove that F is
locally invertible about every point.
11.8 Define /^: IR^ _^ (R3 ^y ^ ^ y ^here
yi = ^1 + ^2 + (^3 — 1)^, y2 = X1 + X2+ (x3 — 3x3), 2/3 = ^1 + ^2 + ^3.
Prove that x 1-^ y = F(x) is locally invertible about x = <0, 0, 1 >.
11.9 For a function/: IR -^ IR the proof of local invertibility around a point a where
dfa is nonsingular is much simpler than the general case. Show first that the Jacobian
matrix of / at a is the number/'(a). We are therefore assuming that/'(a:) is continuous
in a neighborhood of a and that/'(a) 9^ 0. Prove that then/ is strictly increasing (or
decreasing) in an interval about a. Now finish the theorem. (See Exercise 1.12.)

170 THE DIFFERENTIAL CALCULUS 3.11
11.10 Show that the equations
i^ + x^ + y^ + z^ = 0, t+x^ + y^+z^ = 4, l + x+y+z = 0
have differentiable solutions x(t), y(t), z(t) around <t, x,y, z> = <0, —1, 1,0>.
11.11 Show that the equations
X I 2y , Su I 4v . X I y , u I V .
e -\- e -\- e -\- e =^4, e-\-e-\-e-\-e=4^
can be uniquely solved for u and v in terms of x and y around the point < 0, 0, 0, 0 >.
11.12 Let S be the graph of the equation
xz -\- sin (xy) + cos {xz) = 1
in IR^. Determine whether in the neighborhood of (0, 1, 1) iS is the graph of a
differentiable function in any of the following forms:
^ = /(^, y), ^ = giy, 2), y = H^, 2)-
11.13 Given functions/and g from R^ to U such that/(a, b, c) =0 and g(a, b, c) = 0,
write down the condition on the partial derivatives of / and g that guarantees the
existence of a unique pair of differentiable functions y = h{x) and z = k(x) satisfying;
h(a) = 6, k(a) = c,
and
fix, y, z) = f{x, Hx), k{x)) = 0,
g{x,y,z) = gi^, H^), k{x)) = O around <a,b,c>.
11.14 Let G(f, 7], f) be a continuously differentiable mapping from V = XI? Vi to 11
such that dGa'. Vs —> W is invertible and G{oi) = G(ai, a2, as) = 0. Prove that there
exists a uniquely determined function f = F(^,7]) defined around <ai,a2> in
Fi X V2 such that G(f, rj, F{^, v)) = 0 and F(ai, a2) = as- Also show that
where f = F(^, rj).
11.15 Let F{^j rj) be a continuously differentiable function from F X W to X, and
suppose that dF\a,^> is invertible. Setting 7 = /^(a, j3), show that there is a product
neighborhood L X ilf X A^ of <y,a,fi> in X X F X W and a unique continuously
differentiable mapping G:LX M -^ N such that on LXM, F{^,G{^,^)) = f.
11.16 Suppose that the equation g(x, y, z) =0 can be solved for z in terms of x and //,
This means that there is a function/(x, y) such that g(x, y,f(x, y)) =0. Suppose also
that everything is differentiable and compute dz/dx.
11.17 Suppose that the equations
g(x, y,z) = 0 and h(x, y,z) =0
can be solved for y and z as functions of x. Compute dy/dx.
11.18 Suppose that g(x, y, u,v) =0 and h(x, y, u, v) =0 can be solved for u and v
as functions of x and y. Compute du/dx.
11.19 Compute dz/dx where x^ -j- y^ -j- z^ = 0 and x^ + 2/^ -f 2^ = 1.
11.20 If ^^ + x^ + 2/^ + 2^ =0 and t^ + x^ + y^ + z^ = 1, then dz/dx is ambiguou.s,
We are obviously goinej to think of two of the variables as functions of the other two.

3.11 THE IMPLICIT-FUNCTION THEOREM 171
Also z is going to be dependent and x independent. But is ^ or ?/ going to be the other
independent variable? Compute dz/dx under each of these assumptions.
11.21 We are given four "physical variables" p, e/, t^ and ip such that each of them is a
function of any two of the other three. Show that di/dip has two quite different
meanings, and make explicit what the relationship between them is by labeling the various
functions that are relevant and applying the implicit differentiation process.
11.22 Again the "one-dimensional" case is substantially simpler. Let G be a
continuously differentiable mapping from IR^ to IR such that G(a, 6) = 0 and
(dG/dy){a,h) = G2(a, 6) > 0.
Show that there are positive numbers e and b such that for each c in (a — 5, a + 5)
the function g{y) = G{Cj y) is strictly increasing on [6 — e, 6 + e] and G(c, 6 — e) <
0 < G{Cj h-\~ i). Conclude from the intermediate-value theorem (Exercise 1.13) that
there exists a unique function F: (a — 8, a -\~ 8) —^ (b — e, b -{- e) such that
G{x,F{x)) = 0.
11.23 By applying the same argument used in the above exercise a second time, prove
that F is continuous.
11.24 In the inverse-function theorem show that d/^a = {dH^)~^. That is, the
differential of the inverse of H is the inverse of the differential of H. Show this
a) by applying the implicit-function theorem;
b) by a direct calculation from the identity i/(/^(f)) = f.
11.25 Again in the context of the inverse-mapping theorem, show that there is a
neighborhood M oi fi m A such that F{H(ri)) = 77 on M. (Don't work at this. Just
apply the theorem again.)
11.26 We continue in the context of the inverse-mapping theorem. Assume the result
(from the next chapter) that if dH'^^ exists, then so does dHJ^, for f sufficiently close
to p. Show that there is an open neighborhood U oi ^inB such that H is injective on U,
H[U] is an open set A^ in V, and H~^ is continuously differentiable on A^.
11.27 Use Exercise 3.21 to give a direct proof of the existence of a Lipschitz
continuous local inverse in the context of the inverse-mapping theorem. [Hint: Apply
Theorem 7.4.]
11.28 A direct proof of the differentiability of an inverse function is simpler than the
implicit-function theorem proof. Work out such a proof, modeling your arguments in a
general way upon those in Theorem 11.1.
11.29 Prove that the implicit-function theorem can be deduced from the inverse-
function theorem as follows. Set
H(^,V) = <^,G{^,v)>,
and show that dH^a,^> has the block diagram
I
dG^
0
dG^
Then show that dH^a,^> ^ exists from the block diagram results of Chapter 1. Apply
the inverse-mapping theorem.

172 THE DIFFERENTIAL CALCULUS 3.12
12. SUBMANIFOLDS AND LAGRANGE MULTIPLIERS
If V and W are finite-dimensional spaces, with dimensions n and m, respectively,
and if F is a continuous mapping from an open subset ^4 of F to W, then (the
graph of) F is a subset ofVxW which we visualize as a kind of "n-dimensional
surface" S spread out over A. (See Section 10.) We shall call F an n-dimensional
patch in V X W. More generally, if X is any (n + m)-dimensional vector space,
we shall call a subset S an n-dimensional patch if there is an isomorphism (p
from X to a product space V X W such that V is n-dimensional and (p[S] is a
patch m V X W. That is, S becomes a patch in the above sense when X is
considered to be F X W. This means that if tti is the projection of X = V X W
onto F, then ttJaS] is an open subset A of F, and the restriction tti \ S is one-to-
one and has a continuous inverse. If 7r2 is the projection on W, then F =
7r2 o (tti f AS)~\is the map from A to W whose graph in F X IF is aS (when
V XW is identified with X).
Now there are important surfaces that aren't such "patch" surfaces.
Consider, for instance, the surface of the unit ball in IR^, S = {x : ^f o:^ = 1}. aS is
obviously a two-dimensional surface in IR^ which cannot be expressed as a graph,
no matter how we try to express IR^ as a direct sum. However, it should be
equally clear that S is the union of overlapping surface patches. If a is any point
on aS, then any sufficiently small neighborhood A/' of a in U^ will intersect aS in a
patch; we take V as the subspace parallel to the tangent plane at a and W as
the perpendicular line through 0. Moreover, this property of aS is a completely
adequate definition of what we mean by a submanifold.
A subset aS of an (n + m)-dimensional vector space X is an n-dimensional
submanifold of X if each a on aS has a neighborhood N in X whose intersection
with aS is an n-dimensional patch.
We say that aS is smooth if all these patches Sa are smooth, that is, if the
function F: A -^ W whose graph in F X IF is the patch aS^ (when X is viewed
asVxW) is continuously differentiable for every such patch aS^.
The sphere we considered above is a two-dimensional smooth submanifold
of IR^
Submanifolds are frequently presented as zero sets of mappings. For
example, our sphere above is the zero set of the mapping G from U^ to U defined
by G(x) = Yll xf — I. It is obviously important to have a condition
guaranteeing that such a null set is a submanifold.
Theorem 12.1. Let (? be a continuously differentiable mapping from an open
subset U of an (n + m)-dimensional vector space X to an m-dimensional
vector space Y such that dOa is surjective for every a in the zero set aS of G.
Then aS is an n-dimensional submanifold of X.
Proof. Choose any point 7 of aS. Since dGy is surjective from the (n + m)-
dimensional vector space X to the m-dimensional vector space F, we know that
the null space V of dGy has dimension n (Theorem 2.4, Chapter 2). Let W be any

3.12 SUBMANIFOLDS AND LAGRANGE MULTIPLIERS 173
complement of F, and think of X as F X TF, so that G now becomes a function of
two vector variables and 7 is a point <a, fi> such that (?(a, fi) = 0. The
restriction of dG^a,?> to W is an isomorphism from W to F; that is, {dG\a,^>)~^
exists. Therefore, by the implicit-function theorem, there is a product
neighborhood S^{a) X Sr{fi) of <a, fi> m X whose intersection with S is the graph
of a function on S^{a). This proves our theorem. D
If aS is a smooth submanifold, then the function F whose graph is the patch
of S around 7 (when X is viewed suitably as F X W) is continuously differentia-
ble, and therefore S has a uniquely determined n-dimensional tangent plane M
at 7 that fits S most closely around 7 in the sense of our e-approximations.
If 7 = 0, this tangent plane is an n-dimensional subspace, and in general it is
the translate through 7 of a subspace N. We call N the tangent space of aS at 7;
its elements are exactly the vectors in X tangent to parametrized arcs drawn
in S through 7. What we are going to do later is to describe an n-dimensional
manifold S independently of any imbedding of aS in a vector space. The tangent
space to aS at a point 7 will still be an invaluable notion, but we are not going to
be able to visualize it by an actual tangent plane in a space X carrying aS.
Instead, we will have to construct the vector space tangent to aS at 7 some-
liow.
The clue is provided by Theorem 10.2, which tells us that if aS is imbedded
as a submanifold in a vector space X, then each vector tangent to aS at 7 can be
presented as the unique tangent vector at 7 to some smooth curve lying in aS.
This mapping from the set of smooth curves in aS through 7 to the tangent space
at 7 is not injective; clearly, different curves can be tangent to each other at 7
and so have the same tangent vector there. Therefore, the object in aS that
corresponds to a tangent vector at 7 is an equivalence class of smooth curves
through 7, and this will in fact be our definition of a tangent vector for a general
manifold.
The notion of a submanifold allows us to consider in an elegant way a
classical "constrained" maximum problem. We are given an open subset U
of a finite-dimensional vector space X, a differentiable real-valued function F
(hifined on C/, and a submanifold aS lying in U. We shall suppose that the
submanifold aS is the zero set of a continuously differentiable mapping G from U
to a vector space Y such that dGy is surjective for each 7 on aS. We wish to
consider the problem of maximizing (or minimizing) F{y) when 7 is
"constrained" to lie on aS. We cannot expect to find such a maximum point 7o by
setting dFy = 0 and solving for 7, because 7o will not be a critical point for F.
('onsider, for example, the function ^(x) = Y^l xf — 1 from IR^ to IR and F(x) =
X2. Here the "surface" defined by ^ = 0 is the unit sphere Y^i xf = 1, and on
this sphere F has its maximum value 1 at <0, 1, 0>. But F is linear, and so
,ll^\ = F can never be the zero transformation. The device known as Lagrange
niultiphers shows that we can nevertheless find such constrained critical points
l)y solving dLy = 0 for a suitable function L.

174 THE DIFFERENTIAL CALCULUS 3.12
Theorem 12.2. Suppose that F has a maximum value on S at the point 7.
Then there is a functional I in F* such that 7 is a critical point of the
function F - (lo Gf).
Proof. By the implicit-function theorem we can express X as V X W in such a
way that the neighborhood of S around 7 is the graph of a mapping H from an
open set ^4 in F to W. Thus, expressing F and G as functions on F X TF, wo
have (?(f, ^) = 0 near 7 = <a, i3> if and only if 77 = H{^), and the restriction
of F(^j 7]) to this zero surface is thus the function K: A -^ U defined by K(^) =
F(^j H(^)). By assumption a is a critical point for this function. Thus
0= dKa = dF\^^^^ + dF\cc,^> ° dHc,.
Also from the identity (?(f, H{^)) = 0, we get
0 = dG^a,fi> + dG%a,^> ° dHa.
Since dG%a,0> is invertible, we can solve the second equation for dHa and
substitute in the first, thus getting, dropping the subscripts for simplicity,
dF^ - dF^ o (dG^)-^ o dG^ = 0.
Let / G F* be the functional dF^ o (dG^)-\ Then we have dF^ = I o dG^ and,
by definition, dF^ = I o dG^. Composing the first equation (on the right) with
TTii F X W -^ V and the second with 7r2, and adding, we get dF^a,^> =
I o dG^a,^>^ That is, d{F - I o G)y = 0. D
Nothing we have said so far explains the phrase "Lagrange multipliers".
This comes out of the Cartesian expression of the theorem, where we have U
an open subset of a Cartesian space IR^, F = IR^, G = <^^ . . . , ^^>, and I in
F* of the form I, : l(y) = ET CiVi. Then F - I o G = F - Zl^ ag^ and
d(F — I o G)^ = 0 becomes
These n equations together with the m equations G = <^\...,^^> =0 give
m + n equations in the m + n unknowns 0:1, ... , Xm ci, . . . , Cm-
Our original trivial example will show how this works out in practice. Wo
want to maximize F(x) = x^ from IR^ to IR subject to the constraint Z\ xf = L
Here ^(x) = 2^f xf — 1 is also from IR^ to IR, and our method tells us to look
for a critical point oi F — eg subject to ^ = 0. Our system of equations is
0 - 2cxi = 0,
1 — 2cx2 = 0,
0 - 2cxs = 0,

3.13 FUNCTIONAL DEPENDENCE 175
The first says that c = 0 or a:i = 0, and the second impHes that c cannot be 0.
Therefore, a:i = 0:3 = 0, and the fourth equation then shows that X2 = ±1.
Another example is our problem of minimizing the surface area A =
2(0:2/ + yz + zx) of a rectangular parallelepiped, subject to the constraint of a
constant volume, xyz = V. The theorem says that the minimum point will be a
critical point oi A — XF for some X, and, setting the differential of this function
equal to zero, we get the equations
2(2/ + z) - Xyz = 0,
2(x + z) — Xxz = 0,
2(x + y) - Xxy = 0,
together with the constraint
xyz = V.
The first three equations imply that x = y = z; the last then gives V^'^ at the
common value.
*13. FUNCTIONAL DEPENDENCE
The question, roughly, is this: If we are given a collection of continuous functions,
all defined on some open set A, how can we tell whether or not some of them are
functions of the rest? For example, if we are given three real-valued continuous
functions /i, /2, and /a, how can we tell whether or not some one of them is a
function of the other two, say/a is a function of/i and/2, which means that there
is a function of two variables g(xy y) such that/a (0 = ^(/i(0>/2(0) for all t in
the common domain ^4 ? If this happens, we say that /a is functionally dependent
on/i and /2. This is very nearly the same as asking when it will be the case that
the range S of the mapping F:t\-^ ^/i(0>/2(0>/3(0^ ^^ ^ two-dimensional
submanifold of U^. However, there are differences in these questions that are
worth noting. If/3 is functionally dependent on /i and /2, then the range of F
certainly lies on a two-dimensional submanifold of IR^, namely, the graph of g.
But this is no guarantee that it itself forms a two-dimensional submanifold.
For example, both /2 and f^ might be functionally dependent on /i, /2 = ^ ° /i,
and /a = /i o /i, in which case the range of F lies on the curve < s, g{s), h(s) > in
R^, which is a one-dimensional submanifold. In the opposite direction, the range
of F can be a two-dimensional submanifold M without fs being functionally
dependent on /2 and /i. All we can conclude in this case is that locally one of the
functions {/J f is a function of the other two, since locally M is a surface patch,
in the language of the last section. But if we move a little bit away on the
curving surface M to the neighborhood of another point, we may have to solve
for a different one of the functions. Nevertheless, if M = range F is a subset of
a two-dimensional manifold, it is reasonable to say that the functions {fi} f are
functionally dependent, and we are led to examine this more natural notion.

176 THE DIFFERENTIAL CALCULUS 3A',\
If we assume that F = </i, /2, fs > is continuously differentiable and that
the rank of dFa is 3 at some point a in A, then the implicit-function theorem
implies that F[A] includes a whole ball in U^ about the point F(a). Thus a
necessary condition for M = range F to lie on a two-dimensional submanifold in
U^ is that the rank of dFa be everywhere less than 3. We shall see, in fact, that
if the rank of dFa is 2 for all a, then M = range F is essentially a two-dimensional
manifold. (There is still a tiny difficulty that we shall explain later.) Our tools
are going to be the implicit-function theorem and the following theorem, which
could well have come much earlier, that the rank of T is a "lower semi
continuous" function of T.
Theorem 13.1. Let V and W be finite-dimensional vector spaces, normed
in some way. Then for any T in Hom(F, W) there is an e such that
\\S - T\\ < € => rank S > rank T.
Proof. Let T have null space N and range R, and let X be any complement of
N in V. Then the restriction of T to X is an isomorphism to R, and hence is
bounded below by some positive m. (Its inverse from i^ to X is bounded by
some 6, by Theorem 4.2, and we set m = 1/6.) Then if ||aS — I'll < m/2, it
follows that S is bounded below on X by m/2, for the inequalities
\\T(a)\\ > m\\a\\ and {{(S - T)(a)\\ < (m/2)||a||
together imply that ||AS(a)ll ^ (W2)ll«ll. In particular, S is infective on Z, and
so rank S = d(range S) > d{X) = d{R) = rank T. D
We can now prove the general local theorem.
Theorem 13.2. Let V and W be finite-dimensional spaces, let r be an integer
less than the dimension of IF, and let F be a continuously differentiable map
from an open subset A CV to W such that the rank of dFy = r for all 7
in A. Then each point 7 in ^4 has a neighborhood U such that F[U] is an
r-dimensional patch submanifold of W.
Proof. For a fixed 7 in ^4 let Vi and Y be the null space and range of dFy^ let
V2 be a complement of Vi in V, and view F as Fi X ¥2- Then F becomes
a function F(f, tj) of two variables, and if 7 = <a, I3> ^ then dF%a,^> is an
isomorphism from V2 to Y. At this point we can already choose the
decomposition W = Wi © W2 with respect to which F[A] is going to be a graph
(locally). We simply choose any direct sum decomposition W = Wi @ W2
such that W2 is a complement of F = range dF^a,^>' Thus TFi might be F,
but it doesn^t have to be. Let P be the projection of W onto TFi, along TF2.
Since F is a complement of the null space of P, we know that P f F is an
isomorphism from F to TFi. In particular, TFi is ?^-dimensional, and
rank P o dF^a,^> = r.

3.13 FUNCTIONAL DEPENDENCE 177
Moreover, and this is crucial, P is an isomorphism from the range of
dF^^^ri> to Wi for all < ^, v^ sufficiently close to <a, ^> . For the above rank
theorem implies that rank P o dF^^^r]> ^ rank P o dF^a,fi> = r on some
neighborhood of <a, i3>. On the other hand, the range of P o dF^^^r]> is
included in the range of P, which is TTi, and so rank P o dF^^^ri> ^ r. Thus
rank P o dF^^.rj> = r for < f, ^> near <a, i3>, and since rank dF^^^r]> = r
by hypothesis, we see that P is an isomorphism on the range of any such dF^^^ri>'
Now define H:Wi X A -^ ^i as the mapping
<t,^,V> ^PoFa,v) - f.
If jLt = P o P(a, j3), then dH%^^a,p> = P o dF\a,^>i which is an
isomorphism from V2 to Wi. Therefore, by the implicit-function theorem there exists
a neighborhood L X M X N of <ijlj a, ^> and a uniquely determined
continuously differentiable mapping G from L X M to N such that
^(f, f, G(f, ?)) = 0
on L X M. That is,
f=PoP(f,G(f, 0)
on L X M.
The remainder of our argument consists in showing that P(f, G(f, f)) is a
function of f alone. We start by differentiating the above equation with respect
to f, getting
0 = P o (dF^ + dF^ o dG^) = P o cZP o </, dG^>.
As noted above, P is an isomorphism on the range of dF^^^r]> for all < f, ^>
sufficiently close to <ay I3>, and if we suppose that L X M is also taken small
enough so that this holds, then the above equation implies that
cZP<^,,> o <I,dG^> = 0
for all < f, f > G L X M. But this is just the statement that the partial
differential with respect to f of P(f, G(f, f)) is identically 0, and hence that
is a continuously differentiable function K of f alone:
Since v = <?(f, f) and f = P o P(f, ^y), we thus have P(f, 77) = K(P o p(f, ^y)),
or
F = Ko p o F,
and this holds on the open set U consisting of those points < ^, v^ in M X A'
such that P o P(f, 77) G L. If we think of IF as TFi X TF2, then F and K
are ordered pairs of functions, F= <F^,F^> and K= <l^k>^P is the
mapping <^,v> i-^ f, and the second component of the above equation is
F^ = ko F\

178
THE DIFFERENTIAL CALCULUS
3.13
Since F^[U] = P o F[U] = L, the above equation says that F[U] is the graph of
the mapping k from L to TF2. Moreover, L is an open subset of the r-dimensional
vector space Wij and therefore F[U] is an r-dimensional patch manifold in
W^WiXW2.U
The above theorem includes the answer to our original question about
functional dependence.
Corollary. Let F = {/*}7 be an m-tuple of continuously differentiable
real-valued functions defined on an open subset ^ of a normed linear space
F, and suppose that the rank of dFa has the constant value r on J., where r
is less than m. Then any point T in J. has a neighborhood 17 over which
m — r oi the functions are functionally dependent on the remaining r.
Proof. By hypothesis the range Y of dFy = <df\,, . . , df^> is an
r-dimensional subspace of W^, We can therefore find a basis for a complementary sub-
space W2 by choosing m — r of the standard basis elements {5*}, and we may
as well renumber the fimctions f so that these are B^'^^, . . . , 5^. Then the
projection P of IR"^ onto IR^ = L(5\ . . . , 5"*) is an isomorphism from Y to W
(since F is a complement of its null space), and by the theorem there is a
neighborhood f7 of 7 over which (I —- P) o F is Si function k of P o F. But this says
exactly that <^+^ . . . ,/^> = fc o <f\ . . . ,f>. That is, k is an (m — r)-
tuple-valued function, k = <fc'+S . . . , fc^>, and f = ¥ o </i, . . . ,f> for
J = r + 1, . . . , m. Q
Fig. 3.12
We mentioned earlier in the section that there was a difficulty in concluding
that if F is a continuously differentiable map from an open subset ^ of F to TF
whose differential has constant rank r less than rf(TF), then S = range F is an
r-dimensional submanifold of S. The flaw can be described as follows. The
definition of a submanifold S of X required that each point of S have a
neighborhood in X whose intersection with aS is a patch. In the case before us, what we

3.14 UNIFORM CONTINUITY AND FUNCTION-VALUED MAPPINGS 179
can conclude is that if jS is a point of aS, then 13 = F(a) for some a in ^, and a
has a neighborhood U whose image under F is a patch. But this image may not
be a full neighborhood of 13 in aS, because S may curve back on itself in such a
way as to intrude into every neighborhood of 13. Consider, for example, the one-
dimensional r imbedded in IR^ suggested by following Fig. 3.12. The curve
begins in the a:2;-plane along the 2;-axis, curves over, and when it comes to the
zy-plane it starts spiraling in to the origin in the xy-plsiue (the point of change
over from the a:2;-plane to the xy-plsine is a singularity that we could smooth out).
The origin is not a point having a neighborhood in IR^ whose intersection with r
is a one-patch, but the full curve is the image of (—-1, 1) under a continuously
differentiable injection.
We would consider F to be a one-dimensional manifold without any difficulty,
but something has gone wrong with its imbedding in IR^, so it is not a
one-dimensional si^bmanifold of IR^.
*14. UNIFORM CONTINUITY AND FUNCTION-VALUED MAPPINGS
In the next chapter we shall see that a continuous function F whose domain is a
bounded closed subset of a finite-dimensional vector space V is necessarily
uniformly continuous. This means that given €, there is a 5 such that
u--v\\<B ^ mo-Fm <e
for all vectors J and tj in the domain of F.
The point is that B depends only on e and not, as in ordinary continuity,
cm the "anchor" point at which continuity is being asserted. This is a very
Important property. In this section we shall see that it underlies a class of
theorems in which a point map is escalated to a function-valued map, and prop-
irties of the point map imply corresponding properties of the function-valued
map. Such theorems have powerful applications, as we shall see in Section 15
find in Section 1 of Chapter 6. An application that we shall get immediately here
Is the theorem on differentiation under the integral sign. However, it is only
Theorem 14.3 that will be used later in the book.
Suppose first that F(?, tj) is a bounded continuous function from a product
i^pen set M X N to a normed linear space X. Holding rj fixed, we have a function
JvW = ^(?> v) which is a bounded continuous function on M, that is, an
r li^ment of the normed linear space Y = (Be(M, X) of all bounded continuous
inaps from M toX. This function is also indicated F(•, ry), so that f^ = F{-, tj).
We are supposing that the uniform norm is being used on Y:
i|/,ii = lub {\\f,m : ? e M} = lub {\\Fil ,)11 -.^gM}.
Theorem 14,1, In the above context, if F is uniformly continuous, then the
mapping r? i-^ /,, (or r; t-^ F(-, tj)) is continuous, in fact, uniformly continuous,
from N to Y,

180 THE DIFFERENTIAL CALCULUS 3.14
Proof, Given €, choose b so that
\\<^.ri> - <ix,v>\\ < b =^ \\F{^, V) - Fill, v)\\ < €.
Taking /jl = ^ and rewriting the right-hand side, we have
h-v\\ < b => \\f,a) -f,m <€
for all ^ Thus
11^ - ^11 < 5 =^ 11/^ -/Jl^ < €. D
We have proved that if a function of two variables is uniformly continuous,
then the mappings obtained from it by the general duality principle are
continuous. This phenomenon lies behind many well-known facts. For example:
Corollary. If F(x, y) is a uniformly continuous real-valued function on the
unit square [0, 1] X [0, 1] in U^, then Jq F(x, y) dx is a continuous function
of 2/.
Proof, The mapping y \-^ Jq^ F{xy y) dx is the composition of the bounded
linear mapping f ^-^ Jq f from e([0, 1] to U with the continuous mapping
y -^ F(' J y) from [0, 1] to ©[0, 1], and is continuous as the composition of
continuous mappings. D
We consider next the differentiability of the above duality-induced mapping.
Theorem 14.2. If F is a bounded continuous mapping from an open product
set M X A^ of a normed linear space F X TT to a normed linear space Z,
and if dF%a,^> exists and is a bounded uniformly continuous function of
<a, fi> on M X Nj then (p: tj -^ F(', rj) is a differentiable mapping from
NtoY= (Be(M, Z), and [d<p^(vm) = dF%^,^^(r}),
Proof, Given €, we choose b by the uniform continuity of dF^, so that
WfjL — ^11 < 5 =^ \\dF%^^^^ — cZF<^,,>|| < €
for all f G M, The corollary to Theorem 7.4 then implies that
Ihll < b => ||An^^>W - dF%^,^^(v)\\ < e\\v\\
for all f G M, all 13 G N, and all rj such that the line segment from 13 to 13 + rj
is in N, We fix ^ and rewrite the right-hand side of the above inequality. This
is the heart of the proof. First
^F%^Mv) = Fa, 0 + v) -Fa,p)
= [f^+.-f^m = wiP + v) - <pma) = i^Mvm)'
Next we can check that if ||(i/^<;,,;,>|| < h for </x, v> G M X N, then the
mapping T defined by the formula [T'(^)](?) = dF\^^^y{ri) is an element of
Hom(TF, Y) of norm at most h. We leave the detailed verification of this as an

3.14 UNIFORM CONTINUITY AND FUNCTION-VALUED MAPPINGS 181
exercise for the reader. The last displayed inequality now takes the form
Ihll < 8 ^ ||[A^^(,) - Tivmn < e\\v\\,
and hence
II.7II < 8 ^ \\A<p^(r}) - T(v)\U < €h||.
This says exactly that the mapping cp is differentiable at ^ and d(p^ = T. U
The mapping (p is in fact continuously differentiable, as can be seen by
arguing a little further in the above manner. The situation is very close to being
an application of Theorem 14.1.
The classical theorem on differentiability under the integral sign is a corollary
of the above theorem. We give a simple case. Note that if 77 is a real variable y,
then the above formula for dcp can be rewritten in terms of arc derivatives:
Corollary. If F(Xf y) is a continuous real-valued function on the unit
square [0, 1] X [0, 1], and if dF/dy exists and is a uniformly continuous
function on the square, then Jq^ F{x, y) dx is a differentiable function of y
and its derivative is ^q (dF/dy) (a:, y) dx.
Proof, The mapping T: y ^-^ jo F{x^ y) dx is the composition of the bounded
linear mapping/ 1-^ Jq f{x) dx from e([0, 1]) to R with the differentiable mapping
ip:y \-^ F{' ^ y) from [0, 1] to e([0, 1]), and is therefore differentiable by the
composite-function rule. Then Theorem 7.2 and the fact that the differential of
a bounded linear map is itself give
T'{y) = f^ [>p'(y)Kx) dx= f^^ (x, y) dx. D
We come now to the situation of most importance to us, where a point to
point map generates a function-to-function map by composition. Let A be an
open set in a normed linear space F, let S be an arbitrary set, and let Gi be the
set of bounded maps / from Sio A. Then a is a subset of the normed linear space
(B(aS, V) of all bounded functions from S ioV under the uniform norm. A
function / G Ct will be an interior point of Ct if and only if the distance from the range
of/ to the boundary of Ct is a positive number 5, for this is clearly equivalent to
saying that a includes a ball in (B(aS, V) about the point /. Now let g be any
bounded mapping from ^4 to a normed linear space TT, and let G\ a -^ (B(aS, W)
be composition by g. That is, h = G(f) if and only if / G Ct and h = g o f. We
can consider both the continuity and differentiability of G, but we shall only
work out the differentiability theorem.
Theorem 14.3. Let the function g: A -^ W he differentiable at each point
a in A, and let dga be a bounded uniformly continuous function of a. Then
the mapping G: d -^ (B(aS, W) defined by (?(/) = ^ 0/ is differentiable at

182 THE DIFFERENTIAL CALCULUS 3.15
any interior point / in a and dG/: (B(aS, V) -^ (B(aS, W) is defined by
[dGf{h)]{8) = dgf,s,(h(s))
for all s gS.
Proof. Given e, choose 8 by the uniform continuity of dg so that
\\a - ^\\ < 8 =^ \\dga - dg^W < e,
and then apply the corollary to Theorem 7.4 once more to conclude that
II^11 < 5 => \\Agaa)-dgM)\\ <eUl
provided the line segment from a to a + f is in A. Now choose any fixed interior
point / in a, and choose 8' < 5 so that ^5'(/) C GL^ Then for any h in (B(aS, F),
\\h\U < 5' => ||Aj//(.)(/i(s)) - dgf^sMs))\\ < 4h(s)\\
for all seS. Define a map T: (&{S, V) •-* (&{S, W) by [T{h)](s) = dgf^s)(h(s)).
Then the above displayed inequality can be rewritten as
\\h\U < S' =^ \\AGf{h) - T(h)\U < €\\h\\^.
That is, AGy = T" + o. We will therefore be done when we have shown that
T G Hom((B(^, V), (&{S, W)).
First, we have
(r(/ii + h2))(s) = dgf^,,(ihi + h2Ks)) = di7/(.)(/ii(s) + h^is))
= %/(.)(/ii(s)) + %/(»)(/i2(s)) = iT{h^))(s) + (T{h2))(s).
Thus T(hi + /12) = ^(^1) + TQ12), and homogeneity follows similarly. Second,
if ?) is a bound to \\dg^\\ on A, then \\T{h)\U = lub {\\{T{h)){s)\\ : s G S} <
lub {||<i^/(s)|| • ||^(s)|| : s G aS} < 6||/i||oo. Therefore, \\T\\ < 6, and we are
finished. D
In the above situation, if g is from A X C/ to TF, so that G{f) is the function
h given by h{t) = g{fit), t), then nothing is changed except that the theorem is
about dg^ instead of dg. If, in addition, F is a product space Vi X F2, so that/
is of the form </i,/2> and [G(f)](t) = g{ji{t)^ f2{t), 0> ^^^^ ^^^ ml^^ about
partial differentials give us the formula
[dGfOiM) = dg},t,{h,{t))+dgj,t,{h2{t)).
*15. THE CALCULUS OF VARIATIONS
The problems of the calculus of variations are simply critical-point problems of a
certain type with a characteristic twist in the way the condition dF^ = 0 is used.
We shall illustrate the subject by proving one of its standard theorems.
Since we want to solve a constrained maximum problem in which the domain
is an infinite-dimensional vector space, a systematic discussion would start off

3.15 THE CALCULUS OF VARIATIONS 183
with a more general form of the Lagrange multiplier theorem. However, for our
purpose it is sufficient to note that if aS is a closed plane M + a, then the
restriction of F to aS is equivalent to a new function on the vector space M, and its
differential at ^ = rj + a in S is clearly just the restriction of dF^ to M. The
requirement that jS be a critical point for the constrained function is therefore
simply the requirement that dF^ vanish on M.
Let F be a uniformly continuous differentiable real-valued function of three
variables defined on (an open subset of) W X W X U, where TF is a normed
linear space. Given a closed interval [a, h] C ff^, let V be the normed linear space
e\[a, 6], W) of smooth arcs /: [a, h] -^ W, with ||/|| taken as ||/||oc + ||/'||oc.
The problem is to maximize the (nonlinear) functional G(f) = fa F(f(t),f'(t),t) dt,
subject to the restraints /(a) = a and /(6) = fi. That is, we consider only
smooth arcs in W with fixed endpoints a and fi, and we want to find that arc
from a to fi which maximizes (or minimizes) the integral. Now we can show
that (? is a continuously differentiable function from (an open subset of) V to R.
The easiest way to do this is to let X be the space 6([a, 6], W) of continuous arcs
under the uniform norm, and to consider first the more general functional K
from Z X X to IR defined by K{f, g) = ja F(f(t), g(t), t) dt. By Theorem 14.3
the integrand map </, g> ^-^ F(f('), g('), •) is differentiable from Z X X to
e([a, h]) and its differential at <f, g> evaluated at <h, k> is the function
dFknt)Mt),t>(hit)) + dF\f^t)M),t>Kt).
Since/1-^ /« f{t) is a bounded linear functional on 6, it is differentiable and equal
to its differential. The composite-function rule therefore implies that K is
differentiable and that
dK<f^g>{h, k) = [^ [dF\h(t)) + dF^(k(t))] dt,
Ja
where the partial differentials in the integrand are at the point <f(t)y git),t>.
Now the pairs <f, g> such that/' exists and equals g form a closed subspace of
X X X which is isomorphic to V. It is obvious that they form a subspace, but
to see that it is closed requires the theory of the integral for parametrized arcs
from Chapter 4, for it depends on the representation f(t) = f(a) + Ji/'(s) ds
and the consequent norm inequality \\f{t) — f(a)\\ < (t — a)||/'||oo. Assuming
this, we see that our original functional G is just the restriction of K to this sub-
space (isomorphic to) V, and hence is differentiable with
dGf(h) = f^dF\h(t)) +dF^{h'{t)) dt.
Ja
This differential dGj is called the first variation of G about /.
The fixed endpoints a and fi for the arc / determine in turn a closed plane P
in F, for the evaluation maps (coordinate projections) TTx'.f^-^ f(x) are bounded
and P is the intersection of the hyperplanes Ta = a and tt?, = jS. Since P is a
translate of the subspace M = {f gV: f(a) = f(h) = 0}, our constrained

184 THE DIFFERENTIAL CALCULUS 3.15
maximum equation is
dGf{h) = [^ ldF\h{t)) + dF^{h'{t))] dt = 0
J a
for all h in M.
We come now to the special trick of the calculus of variations, called the
lemma of Du Bois-Reymond.
Suppose for simplicity that W = R. Then F is a function F{x, 2/, t) of three
real variables, the partial differentials are equivalent to ordinary partial
derivatives, and our critical-point equation is
'^^^^^=L($-'+%-^)='-
If we integrate the first term in the integral by parts and remember that h{a)
h{b) = 0, we see that the equation becomes
f /c)W f c)F\
where g = h\ Since h is an arbitrary continuously differentiable function except
for the constraints h(a) = hQ)) = 0, we see that g is an arbitrary continuous
function except for the constraint ja g{t) dt = 0. That is, dF/dy — jdF/dx is
orthogonal to the null space N of the linear functional g 1-^ ja g{t) dt. Since the
one-dimensional space N-^ is clearly the set of constant functions,
rb rb
g = 0 ^ Cg = 0.
J a J a
our condition becomes
1^ im,f'{t), 0 = j^ ^ im,f'(s), s) ds+c.
This equation implies, in particular, that the left member is differentiable. This
is not immediately apparent, since /' is only assumed to be continuous.
Differentiating, we conclude finally that / is a critical point of the mapping G if and
only if it is a solution of the differential equation
which is called the Euler equation of the variational problem. It is an ordinary
differential equation for the unknown function /; when the indicated derivative
is computed, it takes the form
d^F d'F d'F dF ^
dy^^ ~^dydx^ ~^dydt dx
If W is not IR, we get exactly the same result from the general form of the
integration by parts formula (using Theorem 6.3) and a more sophisticated

3.15 THE CALCULUS OF VARIATIONS 185
version of the above argument. (See Exercise 10.14 and 10.15 of Chapter 4.)
That is, the smooth arc / with fixed endpoints a and jS is a critical point of the
mapping g \-^ j^ F(g{t)^ (/{t)^ t) dt if and only if it satisfies the Euler differential
equation
■j^dF<f(^t)j'{t),t> = dF<fi^t)jf{t),t>'
This is now a vector-valued equation, with values in TT*. If W is
finite-dimensional, with dimension n, then a choice of basis makes TT* into IR'*, and this
vector equation is equivalent to n scalar equations
where F is now a function of 2n + 1 real variables,
F{x, y, t) = F{xi, . . . , a:n, 2/1, . . . , 2/n, t).
Finally, let us see what happens to the simpler variational problem {W = R)
when the endpoints of/are not fixed. Now the critical-point equation is dGfQi) =
0 for all h in F, and when we integrate by parts it becomes
-,6 rh
dy
Ja J a \
for all h in V. We can reason essentially as above, but a little more closely, to
conclude that a function / is a critical point if and only if it satisfies the Euler
equation
dt \dy/ dx
and also the endpoint conditions
= 0.
dF^
dy
dl
dy
t=b
This has been only a quick look at the variational calculus, and the interested
reader can pursue it further in treatises devoted to the subject. There are many
more questions of the general type we have considered. For example, we may
want neither fixed nor completely free endpoints but freedom subject to
constraints. We shall take this up in Chapter 13 in the special case of the
variational equations of mechanics. Or again, / may be a function of two or more
variables and the integral may be a multiple integral. In this case the Euler
equation may become a system of partial differential equations in the unknown/.
Finally, there is the question of sufficient conditions for the critical function to
give-a maximum or minimum value to the integral. This will naturally involve a
study of the second differential of the functional (?, or its second variation, as it is
known in this subject.

186 THE DIFFERENTIAL CALCULUS 3.1(')
*16. THE SECOND DIFFERENTIAL AND
THE CLASSIFICATION OF CRITICAL POINTS
Suppose that V and W are normed linear spaces, that A is an open subset of \\
and that F: A -^ W is sl continuously differentiable mapping. The first
differential of F is the continuous mapping clF: 7 i—> dFy from A to Hom(F, W). We
now want to study the differentiability of this mapping at the point a. Prc^-
sumably, we know what it means to say that dF is differentiable at a. By
definition d(dF)a is a bounded linear transformation T from V to Hom(F, W)
such that A(dF)a{v) - T(v) = o(v). That is, dF«+^ - dFa - T(r}) is an
element of Hom(F, W) of norm less than e||77|| for r} sufficiently small. We set
d^Fa = d{dF)a and repeat: c/^F^ = d^F^i-) is a linear map from V to Hom(F, W),
d^Fa(v) = d^Fa{v)(') is an element of Hom(F, IF), and d^Fa(v)(^) is a vector
in W. Also, we know that d^Fa is equivalent to a bounded bilinear map
coiFX F->IF,
where co(77, ?) = d^FMiO-
The vector d^Fa(v)(0 clearly ought to be some kind of second derivative of
F Sit a, and the reader might even conjecture that it is the mixed derivative in
the directions ^ and rj.
Theorem 16.1. If F: A —> W is continuously differentiable, and if the
second differential d^Fa exists, then for each fixed /x G F the function
D^F\ 7 K-> D^F{7) from A to IF is differentiable at a and D,(D^F)(a) =
Proof. We use the evaIuation-at-/>t map ev^: Hom(F, IF) —> IF defined for a
fixed jLt in F by ev^(T) == T(ijl). It is a bounded linear mapping. Then
(D,F)ia) = dFM - ev,(6/F«) = (ev, o dF){a),
so that the function D^F is the composition ev^ o dF. It is differentiable at a
because d(dF)a exists and ev^ is linear. Thus (D^(D^F))(a) = d{D^F)a{v) =
d(ev, o dFUv) = (ev, o d{dF)^){v) = eY,[{d^F,){v)] = {d^F^{v)){ix). D
The reader must remember in going through the above argument that Dy,F
is the function (D^F)('), and he might prefer to use this notation, as follows
D,{(D,F){-))\^ = d{{D,F)('))^{v) = diev, o dF,.,Uv)
= [ev, o d{dF,.X]{^) = eY,{d'Fa(p)).
If the domain space V is the Cartesian space W^, then the differentiability of
(D8jF)(') = (dF/dxj)(') at a implies the existence of the second partial deriva
tives (d^F/dxi dxj)(a) by Theorem 9.2, and with b and c fixed, we then have
dxj dxi

3.16 SECOND differential; classification of critical points 187
Thus,
Corollary 1. If F = W^ in the above theorem, then the existence of d^F^
implies the existence of all the second partial derivatives (d^F/dxidxj)(a)
and
Moreover, from the above considerations and Theorem 9.3 we can also
conclude that:
Theorem 16.2. If F = IR^, and if all the second partial derivatives
(d^F/dxi dxj)(a) exist and are continuous on the open set A, then the
second differential d^F^^ exists on A and is continuous.
Proof. We have directly from Theorem 9.3 that each first partial derivative
(dF/dxj)(') is differentiable. But dF/dxj = evg^" o dF, and the corollary is then a
consequence of the following general principle. D
Lemma. If {Si}\ is a finite collection of linear maps on a vector space W
such that S = <Sij . . . ^ Sk> is invertible, then a mapping F: A -^ W is
differentiable at a if and only if Si o F is differentiable at a for all i.
Proof. For then S o F and F = S~^ o S o F 2ire differentiable, by Theorems 8.1
and 6.2. D
These considerations clearly extend to any number of differentiations. Thus,
if d'^Ff^.)'. y —> d'^Fy is differentiable at a, then for fixed b and c the evaluation
d^F(.)(b, c) is differentiable at a and the formula
dxjdxi
{D^{D^F)){.) = d%.,(b, a) = E b,c,^^{-)
shows (for special choices of b and c) that all the second partials (d^F/dxj dxi)(')
are differentiable at a, with
(Da/)c-DbF)(a) = Da({D,D^F)(-))l = Z^ b.cA^^J^J^^^ (a).
Conversely, if all the third partials exist and are continuous on A, then the
second partials are differentiable on A by Theorem 9.3, and then d^F(^.) is
differentiable by the lemma, since (d^F/dxi dxj)(') = ev^5\d^'y, o d^F(^.y
As the reader will remember, it is crucially important in working with
liigher-order derivatives that d^F/dxi dxj = d^F/dxj dxi, and we very much
need the same theorem here.
Theorem 16.3. The second differential is a symmetric function of its two
arguments: {d^F^{n)){i) = id^F^{0){r}).

188 THE DIFFERENTIAL CALCULUS 3.16
Proof. By the definition of d(dF)a, given €, there is a 5 such that
MdFUv) - d(dFUv)\\ < eM
whenever ||77|| < 8. Of course, A{dF)a('n) = dFaJ^t) ~ ^^a- If we write down
the same inequality with rj replaced by 77 + f, then the difference of the
transformations in the left members of the two inequalities is
(dF^+, - dF,^,^^) - d^F^(-0,
and the triangle inequality therefore implies that
\\(dF„+, - rfF«+,+r) - d^F^{-^)\\ < e(h\\ + h + fll) < 2e(|h|| + ||f||),
provided that both rj and ^ + T have norms at most 8. We shall take ||f || < 5/3
and 117711 < 25/3. If we hold f fixed, and if we set T = d^F^{-^) and G(f) =
^(f) - ^(f + f), then this inequality becomes pG^^.^ - T\\ < 2€{\\'n\\ + ||f ||),
and since it holds whenever ||77|| < 25/3, we can apply the corollary to Theorem
7.4 and conclude that IIAG«4.^(f) - T{^)\\ < 2€{\\'n\\ + ||f||)||f||, provided that 77
and 77 + f have norms at most 25/3. This inequality therefore holds if 77, f, and f
all have norms at most 5/3. If we now set ^ = —77, we have
and AG«4.^(f) = F(a + 77 + f) - F{a + 77) - F{a + f) + F{a). This function
of 77 and f is called the second difference of F at a, and is designated A^Faivy ?)•
Note that it is symmetric in f and 77. Our final inequality can now be rewritten as
\\A'F„(r,,^)- {d'FM)m <46||„||||?||.
Reversing 77 and f, and using the symmetry of A^F^, we see that
\\(d^FM)iO - {d'Faa))m < 86h|| ll^ll,
provided 77 and f have norms at most 5/3. But now it follows by the usual
homogeneity argument that this inequality holds for all 77 and J. Finally, since
€ is arbitrary, the left-hand side is zero. D
The reader will remember from the elementary calculus that a critical
point a for a function / [/'(a) = 0] is a relative extremum point if the second
derivative/"(a) exists and is not zero. In fact, if/"(^) < 0> then/has a relative
maximum at a, because/"(^) < 0 implies that/' is decreasing in a neighborhood
of a and the graph of / is therefore concave down in a neighborhood of a.
Similarly, / has a relative minimum at a if/'(a) = 0 and /"(a) > 0. If/"(a) = 0,
nothing can be concluded.
If / is a real-valued function defined on an open set ^4 in a finite-dimensional
vector space F, if a G ^4 is a critical point of /, and if d^fa exists and is a non-

3.16
SECOND differential; classification of critical points
189
singular element of Hom(F, F*), then we can draw similar conclusions about the
behavior of / near a, only now there is a richer variety of possibilities. The
reader is probably already familiar with what happens for a function / from U^
to U. Then a may be a relative maximimi point (a "cap" point on the graph
of/), a relative minimimi point, or a saddle point as shown in Fig. 3.13 for the
graph of the translated function A/a. However, it must be realized that new
axes may have to be chosen for the orientation of the saddle to the axes to look as
shown. Replacing / by A/a amounts to supposing that 0 is the critical point and
that /(O) = 0. Note that if 0 is a saddle point, then there are two
complementary subspaces, the coordinate axes in the Fig. 3.13, such that 0 is a relative
maximum for / when / is restricted to one of them, and a relative minimima point
for the restriction of / to the other.
Fig. 3.13
We shall now investigate the general case and find that it is just like the
two-dimensional case except that when there is a saddle point the subspace on
which the critical point is a maximum point may have any dimension from 1
to n — 1 [where d(V) = n]. Moreover, this dimension is exactly the nimiber of
—Ts in the standard orthonormal basis representation of the quadratic form
Our hypotheses, then, are that / is a continuously diff erentiable real-valued
function on an open subset of a finite-dimensional normed linear space F, that
a G A is a critical point for f (dfa ~ 0), and that the mapping d^fa'. V —^V*
exists and is nonsingular. This last hypothesis is equivalent to assimiing that the
bilinear form co(J, rj) ~ d^fai^y v) has a nonsingular matrix with respect to any
basis for V. We now use Theorem 7.1 of Chapter 2 to choose an w-orthonormal
basis {a*} 1. Remember that this means that co(ai, aj) ~ 0 if i 9^ j\ co(at, ai) ~ 1
ioT i = 1, . . . , p, and co(a/, ai) = —1 for i == p + 1, . . . , n. There cannot be
any 0 values for o){ai, ai) because the matrix Uj = o){ai, aj) is nonsingular: if
o){ai, ai) = 0, then the whole ith colimin is zero, the colimin space has
dimension < n — 1, and the matrix is singular.
We can use the basis isomorphism <p to replace F by IR^ (i.e., replace / by
/ o (p)^ and we can therefore suppose that V = U^ and that the standard basis is

190 THE DIFFERENTIAL CALCULUS 3.16
w-orthonormal, with co(x, y) = ll? Xiyi — YZ+i ^^Vi- Since
co(5^ 50 = <fm\ S') = D,'D,'fi^) = ^^ (a),
our hypothesis of co-orthogonality is that {d^f/dxi dxj){a) = 0 for i 9^ j\
d^f/dxf = 1 for 1=1,...,/), and d^f/dxf = -1 for z = p + 1, . . . , n.
Since p can have any value from 0 to n, there are n + 1 possibilities. We show
first that if p = n, then a is a relative minimum of/. In this case the quadratic
form q is said to be positive definite^ since q(x) = w(x, x) is positive for every
nonzero x. We also say that the bilinear form co(x, y) = <i^/a(x, y) is positive
definite^ and, in the language of Chapter 5, that co is a scalar product.
Theorem 16.4. Let / be a continuously differentiable real-valued function
defined on an open subset A of IR^, and let a G ^4 be a critical point of / at
which d^f exists and is positive definite. Then / has a relative minimum at a.
Proof. We suppose, as above, that the standard basis {5*} 1 is co-orthonormal.
By the definition of cZ^/a, given €, there is a 5 such that
|(cZA+,(x) - dL(x)) - d'f^ix, y)| < €||x|| ||y||
whenever ||y|| < 8. Now d/a = 0, since a is a critical point of/, and d^f^(xy y) =
Hi Xiijij by the assumption that {5*] 1 is co-orthonormal. Therefore, if we use
the two-norm on U and set y = ^x, we have
\dUtM - t\\x\\'\ < et\\xf.
Also, if h(t) = /(a + ^x), then h\s) = d/a+sxCx), and this inequality therefore
says that (1 — €)^||a:|p < h\t) < (1 + €)^||a:||^. Integrating, and remembering
that h(l) ~ h(0) = /(a + x) - /(a) = A/a(x), we have
(-i) nr ^.,.(., _< {ip)
Ixf
whenever ||a:|| < 8. This shows not only that a is a relative minimum point
but also that A/a lies between two very close paraboloids when x is sufficiently
small. D
The above argument will work just as well in general. If
g(x) = Z) ^? — ^ Xi
,.2
1 P+i
is the quadratic form of the second differential and ||x||2 = Hi xf, then
replacing ||x||^ inside the absolute values in the above inequalities by g(x), we conclude
that

3.17 THE TAYLOR FORMULA 191
or
J (i: (1 - e)xf -1: (1 + e)xf) < AUix)
\ 1 p+i /
\ 1 p+i /
This shows that A/a lies between two very close quadratic surfaces of the
same type when ||x|| < 8. Ifl<p<n — 1 and a = 0, then/has a relative
minimum on the subspace Vi = L({5*}i) and a relative maximum on the
complementary space V2 = L{{8^}p^i).
According to our remarks at the end of Section 2.7, we can read off the type
of a critical point for a function of two variables without orthonormalizing by
looking at the determinant of the matrix of the (assumed nonsingular) form d^f^.
This determinant is
hit
If it is positive, then a is either a relative minimum or a relative maximum. We
can tell which by following / along a single line, say the a;i-axis. Thus, if
d^f/dxl < ^j then a is a relative maximum point. On the other hand, if the
above expression is negative, then a is a saddle point.
It is important for the calculus of variations that Theorem 16.4 remains
true when the domain space is replaced by a space of the general type that we
shall study in the next chapter, called a Banach space. The hypotheses now are
that a is a critical point of/, that q{0 = dfa{^, f) is positive definite, and that
the scalar product norm q^'^ (see Chapter 5) is equivalent to the given norm on
V. The proof remains virtually unchanged.
♦17. HIGHER ORDER DIFFERENTIALS. THE TAYLOR FORMULA
We have seen that if V and W are normed linear spaces, and if F is a differ-
entiable mapping from an open set A in 7 to TF, then its differential dF = dF(^.)
is a mapping from A to Hom(7, W). If this mapping is differentiable on A, then
its differential d{dF) = d{dF)(^.) is a mapping from A to Hom(7, Hom(7, W)).
We remember that an element of Hom(F, Hom(7, W)) is equivalent by duality
to a bilinear mapping from 7 X 7 to T7, and if we designate the space of all such
bilinear mappings by Hom^(7, T7), then d{dF) can be considered to be from A to
Hom^(7, W), We write d{dF) = d'^F, and call this mapping the second
differential of F. In Section 16 we saw that d^Fai^j v) = D^{Dr,F){a)j and that if
7 = R^, then
rfX(b, c) = Z)bZ)eF(a) = 2 6,cy^0^ (a).
The differentials of higher order are defined in the same way. If
d^F:A-^ Hom2(F, W)

192 THE DIFFERENTIAL CALCULUS 3.17
is differentiable on A, then its differential, d{d^F) = d^F, is from A to
Hom(7, Hom^(7, W)) = Hom^(F, W), the space of all trilinear mappings from
V^ =V XV XV ioW, Continuing inductively, we arrive at the notion of the 7?th
differential of /^ on A as a mapping from A to Hom(F, Hom^~"^(F, W)) =
Hom'^(F, W)j the space of all n-linear mappings from V^ to W. The theorem
that d^Fa is a symmetric element of Hom^(F, W) extends inductively to show
that d^Fa is a symmetric element of Hom^(F, W). We shall omit this proof.
Our theorem on the evaluation of the second differential by mixed directional
derivatives also generalizes by induction to give
Z)^,,...,Z)^/(a) = d^/^«(fi,...,U,
for starting from the left-hand term, we have
Dj,(Z)f„ . . . , i)j/)(-)U = d(D^„ ..., Dj„F(-))a(?i)
= <^(ev<{, j„> o d'-iF(.))„(^i)
= [ev<j, j„> o d(d"-iF(.))a](^i)
= ey<i, („>(d^FMi))
= (d"F„(^i))(^2, ...,^n) = d'^FMu ■.., U-
If F = R^, then our conclusions about partial derivatives extend inductively
in the same way to show that F has continuous differentials on A up through
order m if and only if all the mth-order partial derivatives d^F/dXi^j . . . , dxi^
exist and are continuous on A, with
d-F^{c\ . . . , O = Z ci, ...,cZ (a).
We now consider the behavior of F along the line t ^-^ a -{- t-q, where, of
course, a and 77 are fixed. If \{t) = F{a-[- tri), then we can prove by induction
that
d'\/dt' = {Dr,yF{a + trj).
We know this to be true for j = 1 by Theorem 7.2, and assuming it for j = m,
we have, by the same theorem,
^^ = \^) (0 = d{D-F)^^t,{ri) = D,{D-F){a + tri) = D--^'F{a + tv).
Now suppose that Fis real-valued {W = R). We then have Taylor's formula:
\{t) = x(0) + a'(0) + • ■ ■ + ^ x""'(0) + (J^^), x^-'+^'Cto)
for some k between 0 and 1. Taking t = 1 and substituting from above, we have
F(a + v) = F(«) + D,F{a) + ■ ■ ■ + ^_ Dy{a) + ^^|^^, D'^+'Fia + fc,),

3.17 THE TAYLOR FORMULA 193
which is the general Taylor formula in the normed linear space context. In
terms of differentials, it is
J_
ml
F{a +v) = F{a) + dFM + • • • + ^l ^"^«(^^ • • - ^)
~^ (m + 1)! ^'^ Fa+kv(Vj . . . , ^)-
If V = Wy then DyG = J^\ yi dG/dxij and so the general term in the
Taylor expansion is
m! VY ^' axi/ ' ^"^ - m! i.,..t?„=i ^'^ ' ' ' ^'- dxi, ■ • • dxi„
li m = n = 2j and if we use the notation x= -<^Xj y>, s = <s, t> j then
The above description is logically simple, but it is inefficient in that it repeats
identical terms such as 2/i2/2(^^^/^^i ^^2) and 2/22/1 (^^^/^^2 ^^i)- We conclude
by describing for the interested reader the modern "multi-index" notation for this
very complicated situation.
Remember that we are looking at the mth term of the Taylor formula for Fj
and that F has n variables.
For any n-tuple k = </ci, . . . , /cn> of nonnegative integers, we define |k|
as 2^1 ki, and for x G W, we set x*" = 0:1^^X2^^ • • • Xn^". Also we set F^ =
Fk^k2"'Ki 01* better, if DjF = dF/dxj, we set
D'^F = Di^'D2^' • • • Dn^-F = F^.
Finally, we set k! = /ci!/c2! • • • /cn!, and if p > |k|, we set (i) = p!/k!(p — |k|)!.
Then the mth term of the Taylor expansion of F is
a,£(k)°'^<""^
which is surely a notational triumph.
The general Taylor formula is too cumbersome to be of much use in practice;
it is principally of theoretical value. The Taylor expansions that we actually
compute are generally found by other means, such as substitution of a
polynomial (or power series) in a power series. For example.
3! ' 5!
= x +
y 3! 2 "^ \5! 2 / "^ V 4! 3!/
A mapping from A to W which has continuous differentials of all orders
through k is said to be of class C* on 4, and the collection of all such mappings

194 THE DIFFERENTIAL CALCULUS 3.17
is designated C^(A, W) or e^(A, W). It is clear that (7^(A, W) is a vector space
(induction). Moreover, it can also be shown by induction that a composition
of C^-maps is itself of class C^. This depends on recognizing the general form of
the mth differential of a composition F o (? as being a finite sum, each term of
which is a composition of functions chosen from F, dF,. . . , d^F, G, dGj . . . , d^G.
Functions of many variables are involved in these calculations, and it is
simplest to treat each as a function of a single n-tuplet variable and to apply the
obvious corollary of Theorem 8.1 that if G\ . . . , G"^ are of class C^, then so is
G= < (?\ . . . , (?^>, with d^G = <d^G\ . . . , d^(?^>. As a special case of
composition, we can conclude that a product of C^-maps is of class C^.
We shall see in the next chapter that <p: T ^-^ T~^ is a differentiable map on
the open set of invertible elements in Hom V (if F is a Banach space) and that
d<pT{H) = - T-^HT-\ Since <S, H, T> ^S-^HT-Hhen has continuous
partial differentials, we can continue, and another induction shows that (p is of class
C^ for every k and that d'^ipriHij . . . , Hm) is a finite sum of finite products of
T~^, Hij . . . , H^. It then follows that a function F defined implicitly by a C^-
function G is also of class C^, for its differential, as computed in the implicit-
function theorem, is then a composition of maps of class C^~^.
A mapping F which is of class C^ for all k is said to be of class (7°°, and it
follows from our remarks above that the family of C*-maps is closed under all
the operations that we have met in the calculus. If the domain of F is an open
set in W, then F G e°°(A, W) if and only if all the partial derivatives of F exist
and are continuous on A.

CHAPTER 4
COMPACTNESS AND COMPLETENESS
In this chapter we shall investigate two properties of subsets of a normed linear
space V which are concerned with the fact that in a certain sense all the points
which ought to be there really are there. These notions are largely independent
of the algebraic structure of V, and we shall therefore study them in their own
most natural setting, that of metric spaces. The stronger of these two properties,
compactness, helps to explain why the theory of finite-dimensional spaces is so
simple and satisfactory. The weaker property, completeness, is shared by
important infinite-dimensional normed linear spaces, and allows us to treat
these spaces in almost as satisfactory a way.
It is these properties that save the calculus from being largely a formal
theory. They allow us to define crucial elements by limiting processes, and are
responsible, for example, for an infinite series having a sum, a continuous real-
valued function assuming a maximum value, and a definite integral existing.
For the real number system itself, the compactness property is equivalent to the
least upper bound property, which has already been an absolutely essential tool
in our construction of the differential calculus in Chapter 3.
In Sections 8 through 10 we shall apply completeness to the calculus. The
first of these sections is devoted to the existence and differentiability of functions
defined by power series, and since we want to include power series in an operator
Tj we shall take the occasion to introduce and exploit the notion of a Banach
algebra. Next we shall prove the contraction mapping fixed-point theorem, which
is the missing ingredient in our unfinished proof of the implicit-function theorem
in Chapter 3 and which will be the basis for the fundamental existence and
uniqueness theorem for ordinary differential equations in Chapter 6. In Section
10 we shall prove a simple extension theorem for linear mappings into a complete
normed linear space and apply it to construct the Riemann integral of a param-
atrized arc.
1. METRIC SPACES; OPEN AND CLOSED SETS
In the preceding chapter we occasionally treated questions of convergence and
continuity in situations where the domain was an arbitrary subset A of a normed
linear space V. In such discussions the algebraic structure of V fades into the
background, and the vector operations of V are used only to produce the combi-
195

196 COMPACTNESS AND COMPLETENESS 4.1
nation ||a — ^\\j which is interpreted as the distance from a to ^. If we distill
out of these contexts what is essential to the convergence and continuity
arguments, we find that we need a space A and a function p: A X A -^ Uj p{xj y)
being called the distance from x to y, such that
1) P(^j y) > 0 if X 7^ ?/; and p{x, x) = 0;
2) p{x, y) = p{y, x) for all x,yGA;
3) p(x, z) < p{x, y) + p{y, z) for all x,y,z^ A.
Any set A together with such a function p from A X A to R is called a metric
space] the function p is the metric. It is obvious that a normed linear space is a
metric space under the norm metric p(a, p) — \\a — I3\\ and that any subset B
of a metric space A is itself a metric space under p \ B X B. If we start with a
nice intuitive space, like W^ under one of its standard norms, and choose a weird
subset Bj it will be clear that a metric space can be a very odd object, and may
fail to have almost any property one can think of.
Metric spaces very often arise in practice as subsets of normed linear
spaces with the norm metric, but they come from other sources too. Even in the
normed linear space context, metrics other than the norm metric are used.
For example, aS might be a two-dimensional spherical surface in R^, say S =
{x : X!i ^i = I}; and p(x, y) might be the great circle distance from x to y. Or,
more generally, aS might be any smooth two-dimensional surface in R^, and
p(x, y) might be the length of the shortest curve connecting x to y in aS.
In this chapter we shall adopt the metric space context for our arguments
wherever it is appropriate, so that the student may become familiar with this
more general but very intuitive notion. We begin by reproducing the basic
definitions in the language of metrics. Because the scalar-vector dichotomy is
not a factor in this context, we shall drop our convention that points be
represented by Greek or boldface roman letters and shall use whatever letters we wish.
Definition. If X and Y are metric spaces, then f: X -^ Y is continuous at
a G X if for every e there is a 5 such that
p{x,a) < 8 =^ p(f{x),f(a)) < e.
Here we have used the same symbol 'p^ for metrics on different spaces, just
as earlier we made ambiguous use of the norm symbol.
Definition. The (open) ball of radius r about p, Br{p)j is simply the set of
points whose distance from p is less that r:
Br(p) = {x:p{x,p) < r}.
Definition. A subset A C X is open if every point p in A is the center of
some ball included in A, that is, if
(VpGA)(3r>«)(B.(p)c4).

4.1 METRIC spaces; open and closed sets 197
Lemma 1.1. Every ball is open; in fact, if g G Br{p) and 8 = r — p{pj q),
then B8{q)cBr(p).
Proof. This amounts to the triangle inequality. For, if x G ^5(5), then p{x, q) <
d and p(x, p) < p{xj q) + p{qj p) < d + p{pj q) = r, so that x G Br{p).
Thus Bsiq) C Br{p). D
Lemma 1.2. If p is held fixed, then p(p, x) is a continuous function of x.
Proof. A symbol-by-symbol paraphrase of Lemma 3.1 of Chapter 3 shows that
\p(Py^) ~ P(P)y)\ ^ P(^)y)y so that p(p,x) is actually a Lipschitz function
with constant 1. D
Theorem 1.1. The family 3 of all open subsets of a metric space aS has the
following properties:
1) The union of any collection of open sets is open; that is, if {Ai: i G 1} C
3, then \Jisi Ai G 3.
2) The intersection of two open sets is open; that is, if A,B ^ 3, then
A n ^ G 3.
3) 0, 7 G 3.
Proof. These properties follow immediately from the definition. Thus any
point pm\JiAi lies in some Ay, and therefore, since Aj is open, some ball about p
is a subset of Aj and hence of the larger set \Ji Ai. D
Corollary. A set is open if and only if it is a union of open balls.
Proof. This follows from the definition of open set, the lemma above, and
property (1) of the theorem. D
The union of all the open subsets of an arbitrary set A is an open subset of
A, by (1), and therefore is the largest open subset of A. It is called the interior
of A and is designated A^^^. Clearly, p is in A^^^ if and only if some ball about p
is a subset of A, and it is helpful to visualize A^^^ as the union of all the balls
that are in A.
Definition. A set A is dosed if A' is open.
The theorem above and De Morgan's law (Section 0.11) then yield the
following complementary set of properties for closed sets.
Theorem 1.2
1) The intersection of any family of closed sets is closed.
2) The union of two closed sets is closed.
3) 0 and V are closed.
Proof. Suppose, for example, that {Bi: i g /} is a family of closed sets. Then
the complement B'i is open for each z, so that \ji Bl is open by the above theorem.

198 COMPACTNESS AND COMPLETENESS 4.1
Also, \JiBl = (CliBiY by De Morgan's law (see Section 0.11). Thus OiBi is
the complement of an open set and is closed. D
Continuing our "complementary" development, we define the closurej A, of
an arbitrary set A as the intersection of all closed sets including A, and we have
from (1) above that A is the smallest closed set including A. De Morgan's law
implies the important identity
(I)' = {Ay^'K
For /^ is a closed superset of A if and only if its complement U = 7^' is an open
subset of A'. By De Morgan's law the complement of the intersection of all such
sets F is the union of all such sets U. That is, the complement of Z is (A')'"*.
This identity yields a direct characterization of closure:
Lemma 1.3. A point p is in A if and only if every ball about p intersects A.
Proof. A point p is not in A if and only if p is in the interior of A', that is, if and
only if some ball about p does not intersect A. Negating the extreme members
of this equivalence gives the lemma. D
Definition. The boundary, dA, of an arbitrary set A is the difference
between its closure and its interior. Thus
dA = A - A
int
Since A — ^ = A fi ^', we have the symmetric characterization dA =
A n (A^). Therefore, dA = a(A'); also,
p G dA if and only if every ball about p intersects both A and A'.
Example. A ball Br(a) is an open set. In a normed linear space the closure of
Br(a) is the closed ball about a of radius r, {f : p(f, a) < r}. This is easily seen
from Lemma 1.3. The boundary dBr(a) is then the spherical surface of radius r
about a, {f : p(f, a) = r}. If some but not all of the points of this surface are
added to the open ball, we obtain a set that is neither open nor closed. The
student should expect that a random set he may encounter will be neither open
nor closed.
Continuous functions furnish an important source of open and closed sets
by the following lemma.
Lemma 1.4. If X and Y are metric spaces, and if / is a continuous mapping
from X to F, then f~^[A] is open in X whenever A is open in F.
Proof. If p Ef~^[A], then f(p) G A, and, since A is open, some ball ^e(/(p))
is a subset of A. But the continuity of / at p says exactly that there is a 5 such
that flB8{p)]CB,(f{p)). In particular, flB^ip)] C A and B^ip) Cf'^lA].
Thus for each p in/~^[A] there is a ball about p included in/~'^[A], and this set
is therefore open. D

4.1 METRIC spaces; open and closed sets 199
Since/"^[A'] = (/"^[A])', we also have the following corollary.
Corollary. If/: X -^ Y is continuous, then/~'^[C] is closed in X whenever C
is closed in Y.
The converses of both of these results hold as well. As an example of the use
of this lemma, consider for a fixed a G X the continuous function f: X -^ U
defined by/(f) = p(f, a). The sets (—r, r), [0, r], and {r} are respectively open,
closed, and closed subsets of U. Therefore, their inverse images under/—the ball
Br{a), the closed ball {f : p(f, a) < r}, and the spherical surface {f : p(f, a) = r}
—'are respectively open, closed, and closed in X. In particular, the triangle
inequality argument demonstrating directly that Br{a) is open is now seen to be
unnecessary by virtue of the triangle inequality argument that demonstrates the
continuity of the distance function (Lemma 1.2).
It is not true that continuous functions take closed sets into closed sets in the
forward direction. For example, if /: R -^ R is the arc tangent function, then
f[U] = range / = (—7r/2,7r/2), which is not a closed subset of U. The reader
may feel that this example cheats and that we should only expect the /-image of a
closed set to be a closed subset of the metric space that is the range of /. He
might then consider f{x) = 2x/(l + x^) from U to its range [—1, 1]. The set
of positive integers /"*" is a closed subset of K, but/[/"*"] = {2r?/(l + ^^)}T is not
closed in [—1, 1], since 0 is clearly in its closure.
The distance between two nonempty sets A and Bj p{Aj B), is defined as
gib {p{aj h) : a G A and h G B}. If A and B intersect, the distance is zero.
If A and B are disjoint, the distance may still be zero. For example, the interior
and exterior of a circle in the plane are disjoint open sets whose distance apart is
zero. The x-axis and (the graph of) the function f{x) = 1/x are disjoint closed
sets whose distance apart is zero. As we have remarked earlier, a set A is closed
if and only if every point not in A is a positive distance from A. More generally,
for any set A a point 7? is in A if and only if p(p. A) = 0.
We list below some simple properties of the distance between subsets of a
normed linear space.
1) Distance is unchanged by a translation: p{Aj B) = p{A + 7, B + 7)
(becaiuse ||(a + 7) - (^ + 7)|| = ||a - ^||).
2) p{kA, kB) = \k\p(A, B) (because \\ka - kp\\ = \k\ \\a - ^||).
3) If iV is a subspace, then the distance from ^ to iV is unchanged when we
translate B through a vector in N: p{Nj B) = p{N, B -\- ri) \i -q g N
(because N — -q = N).
4) UT G Hom(7, W), then p{T[Al T[B]) < \\T\\p{A, B) (because
\\na)-Tm < wn-wa-m-
Lemma 1.5. If iV is a proper closed subspace and 0 < ^ < 1, there exists an
a such that ||a|| = 1 and p(a, N) > I — e.

200 COMPACTNESS AND COMPLETENESS 4.1
Proof. Choose any fi i N. Then p{^, N) > 0 (because N is closed), and there
exists an 77 G AT such that
11/3 - ,11 < p(/3, N)/{1 - 6)
[by the definition of p{0, N)]. Set « = (/3 - r,)/\\0 - r,\\. Then ||a|| = 1 and
p{a, N) = p{fi - V, N)/\\0 - 7,11
= pifi, N)/\\p - ,11 > p(/3, N){1 - e)/piP, N) = l-e,
by (2), (3), and the definition of 77. D
The reader may feel that we ought to be able to improve this lemma. Surely,
all we have to do is choose the point in N which is closest to jS, and so obtain
lli^ — ^11 = p(Pj N)j giving finally a vector a such that ||a|| = 1 and p(a, N) = 1.
However, this is a matter on which our intuition lets us down: if N is infinite-
dimensional, there may not be a closest point 77! For example, as we shall see
later in the exercises of Chapter 5, if V is the space ©([ — 1, 1]) under the two-
norm 11/11 = (Jii/^)^^^, and if iV is the set of functions (/in 7 such that Jo^ ^ = 0,
then iV is a closed subspace for which we cannot find such a "best" a. But if N is
finite-dimensional, we can always find such a point, and if F is a Hilbert space,
(see Chapter 5) we can also.
EXERCISES
1.1 Write out the proof of Lemma 1.2.
1.2 Prove (2) and (3) of Theorem 1.1.
1.3 Prove (2) of Theorem 1.2.
1.4 It is not true that the intersection of a sequence of open sets is necessarily open.
Find a counterexample in R.
1.5 Prove the corollary of Lemma 1.4.
1.6 Prove that p G A ii and only if p(p. A) = 0.
1.7 Let X and Y be metric spaces, and let/: X —> F have the property that/~^[5]
is open in X whenever B is open in Y. Prove that / is continuous.
1.8 Show that p{x, A) = p{x, A).
1.9 Show that p{Xj A) is a continuous function of x. (In fact, it is Lipschitz
continuous.)
1.10 Invent metric spaces S (by choosing subsets of U^) having the following
properties :
1) S has n points.
2) S is infinite and pix^y) > 1 if a: 9^ y.
3) S has a ball Bi{a) such that the closed ball {x: p(x, a) < 1} is not the same as
the closure oi Bi(a).

4.2 TOPOLOGY 201
1.11 Prove that in a normed linear space a closed ball is the closure of the
corresponding open ball.
1.12 Show that if /: X —> Y and g^: F —> Z are continuous (where X, Y, and Z are
metric spaces), then so is g' © /.
1.13 Let X and Y be metric spaces. Define the notion of a product metric on Z =
X X Y. Define a 1-metric pi and a uniform metric Poo on Z (showing that they are
metrics) in analogy with the 1-norm and uniform norm on a product of normed linear
spaces, and show that each is a product metric according to your definition above.
1.14 Do the same for a 2-metric p2 on Z = X X F.
1.15 Let X and Y be metric spaces, and let F be a normed linear space. Let/: X —> R
and g: F —> F be continuous maps. Prove that
<x,y>^f{x)g{y)
is a continuous map from XX F to F.
*2. TOPOLOGY
If X is an arbitrary set and 3 is any family of subsets of X satisfying properties
(1) through (3) in Theorem 1.1, then 3 is called a topology on X. Theorem 1.1
thus asserts that the open subsets of a metric space X form a topology on X.
The subsequent definitions of interior, closed set, and closure were purely
topological in the sense that they depended only on the topology 3, as were
Theorem 1.2 and the identity (A)' = (A'y^^. The study of the consequences of
the existence of a topology is called general topology.
On the other hand, the definitions of balls and continuity given earlier were
metric definitions, and therefore part of metric space theory. In metric spaces,
then, we have not only the topology, but also our e-definitions of continuity and
balls and the spherical characterizations of closure and interior.
The reader may be surprised to be told now that although continuity and
convergence were defined metrically, they also have purely topological
characterizations and are therefore topological ideas. This is easy to see if one keeps
in mind that in a metric space an open set is nothing but a union of balls. We
have:
/ is continuous at p if and only if for every open set A containing f{p) there
exists an open set B containing p such that/[^] C A.
This local condition involving behavior around a single point p is more
fluently rendered in terms of the notion of neighborhood. A set A is a
neighborhood of a point p if p G A^^^. Then we have:
/ is continuous at p if and only if/or every neighborhood N of/(p), f^lN]
is a neighborhood of p.
f
Finally there is an elegant topological characterization of global continuity.
Suppose that S\ and S2 are topological spaces. Then /: iSi —> S2 is continuous

202 COMPACTNESS AND COMPLETENESS 4:.'.)
(everywhere) if and only if/~^[^] is open whenever A is open. Also, / is
continuous if and only if/~^[^] is closed whenever B is closed. These conditions
are not surprising in view of Lemma 1.4.
3. SEQUENTIAL CONVERGENCE
In addition to shifting to the more general point of view of metric space theory,
we also want to add to our kit of tools the notion of sequential convergence,
which the reader will probably remember from his previous encounter with the
calculus. One of the principal reasons why metric space theory is simpler and
more intuitive than general topology is that nearly all metric arguments can be
presented in terms of sequential convergence, and in this chapter we shall
partially make up for our previous neglect of this tool by using it constantly and
in preference to other alternatives.
Definition. We say that the infinite sequence {Xn} converges to the point a
if for every e there is an N such that
n > N ==> p{xn, a) < e.
We also say that Xn approaches a as n approaches (or tends to) infinity, and
we call a the limit of the sequence. In symbols we write x^-^aasn-^oo,or
lim„_,oo Xn = a. Formally, this definition is practically identical with our earlier
definition of function convergence, and where there are parallel theorems the
arguments that we use in one situation will generally hold almost verbatim in
the other. Thus the proof of Lemma 1.1 of Chapter 3 can be alternated slightly
to give the following result.
Lemma 3.1. If {fj and {rji} are two sequences in a normed linear space V^
then
^i -^ a and rji-^ P =^ ^i + 77i "^ « + i^-
The main difference is that we now choose N as max {iVi, N2} instead of
choosing b as min {5i, 52}. Similarly:
Lemma 3.2. If f^ -^ amV and Xi -^ a in R, then Xi^i -^ aa.
As before, the definition begins with three quantifiers, (Ve)(3iV)(Vn). A
somewhat more idiomatic form can be obtained by rephrasing the definition in
terms of balls and the notion of "almost all n". We say that P{n) is true for
almost all n if P{n) is true for all but a finite number of integers n, or equivalently,
if (3iV)(Vn>^)P(n). Then we see that
lim Xn = aii and only if every ball about a contains almost all the x„.
The following sequential characterization provides probably the most
intuitive way of viewing the notion of closure and closed sets.

4.3 SEQUENTIAL CONVERGENCE 203
Theorem 3.1. A point x is in the closure A of a set A if and only if there is a
sequence {xn} in A converging to x.
Therefore, a set A is closed if and only if every convergent sequence lying
in A has its limit in A.
Proof. If {xn} C A and Xn -^ Xj then every ball about x contains almost every
Xn, and so, in particular, intersects A. Thus x G Ahy Lemma 1.3. Conversely,
if X G Aj then every ball about x intersects A, and we can construct a sequence in
A that converges to x by choosing Xn as any point in Bunix) 0 A. Since A is
closed if and only if A = Aj the second statement of the theorem follows from
the first. D
There is also a sequential characterization of continuity which helps greatly
in using the notion of continuity in a flexible way. Let X and Y be metric spaces,
and let / be any function from X to Y.
Theorem 3.2. The function / is continuous at a if and only if, for any
sequence {xn} in X, if Xn -^ a, then f{xn) -^ f{a).
Proof. Suppose first that / is continuous at a, and let {Xn} be any sequence
converging to a. Then, given any e, there is a 5 such that
p{x , a)< 8 => p{f[x) , /(a)) < e,
by the continuity of / at a, and for this b there is an N such that
n > N => p(Xn , a) < 5,
because Xn -^ a. Combining these implications, we see that given e we have
found N so that n > N => pUM , /(«)) < e. That is, f{xn) -^f{a).
Now suppose that / is not continuous at a. In considering such a negation
it is important that implicit universal quantifiers be made explicit. Thus,
formally, we are assuming that '^(Ve)(35)(Vx)(p(x , a) < 5 =» p{f{x), /(a)) < e),
that is, that (3e)(V5)(3x)(p(x , a) < 8 & p(f{x) , /(a)) > e). Such symbolization
will not be necessary after the reader has had some practice in computing logical
negations; the experienced thinker will intuit the correct negation without a
formal calculation. In any event, we now have a fixed e, and for each d of the
form 8 = 1/n we can let Xn be a corresponding x. We then havep(x„ , a) < 1/n
and p(f{xn) , /(a)) > e for all n. The first inequality shows that Xn -^ a; the
second shows that f(xn) -h/(a). Thus, if / is not continuous at a, then the
sequential condition is 7wt satisfied. D
The above type of argument is used very frequently and almost amounts to
an automatic proof procedure in the relevant situations. We want to prove, say,
that {Vx)(3y)(Vz)P(x, ?/, z). Arguing by contradiction, we suppose this false, so
that {3x)(Vy)(3z)^P(Xj y^ z). Then, instead of trying to use all numbers y, we
let y run through some sequence converging to zero, such as [1/^?), and we choose

204 COMPACTNESS AND COMPLETENESS 4.3
one corresponding 2;, Zn, for each such y. We end up with r^P{x, 1/n, Zn) for the
given X and all n, and we finish by arguing sequentially.
The reader will remember that two norms p and g on a vector space V are
equivalent if and only if the identity map f j-^ f is continuous from <V,p> to
<Vjq> and also from <Vj q> to <Vjp>. By virtue of the above theorem
we now see that:
Theorem 3.3. The norms p and q are equivalent if and only if they yield
exactly the same collection of convergent sequences.
Earlier we argued that a norm on a product V XW oi two normed linear
spaces should be equivalent to ||<a:, ^^\\i = \W\\ + llfll- Now with respect to
this sum norm it is clear that a sequence <«„, fn>^ in F X TF converges to
< a, f > if and only if «„ -^ a in 7 and f„ -^ f in W. We now see (again by
Theorem 3.2) that:
Theorem 3.4. A product norm on F X TF is any norm with the property
that <anj ^n> -^ <0Lj ^>' in V XW \i and only if «„ -^ a in V and
f n -^ f in W.
EXERCISES
3.1 Prove that a convergent sequence in a metric space has a unique limit. That is,
show that \i Xn-^ a and Xn —> &, then a — h,
3.2 Show that Xn—^xm the metric space X if and only if p{Xnj x) —> 0 in R.
3.3 Prove that if o^n —> a in R and a^n ^ 0 for all n, then a > 0.
3.4 Prove that if o^n —> 0 in R and \yn\ ^ Xn for all n, then 2/n —^ 0.
3.5 Give detailed e, A^-proofs of Lemmas 3.1 and 3.2.
3.6 By applying Theorem 3.2, prove that if X is a metric space, F is a normed linear
space, and F and G are continuous maps from X to F, then iP + G is continuous.
State and prove the similar theorem for a product FG.
3.7 Prove that continuity is preserved under composition by applying Theorem 3.2.
3.8 Show that (the range of) a sequence of points in a metric space is in general not
a closed set. Show that it may be a closed set.
3.9 The fact that in a normed linear space the closure of an open ball includes the
corresponding closed ball is practically trivial on the basis of Lemma 3.2 and Theorem
3.1. Show that this is so. ^
3.10 Show directly that if the maximum norm \\<a, f >|| = max {||a:||, ||f||} is used
on F = Fi X F2, then it is true that
<oin, fn> —^ <oi, f > in F
if and only if
an—^oc in Vi and Jn —^ J in F2.

4.4 SEQUENTIAL COMPACTNESS 205
3.11 Show that if || || is any increasing norm on R^ (see the remark after Theorem 4.3
of Chapter 3), then
p«xi,yi>,<X2yy2>) = ||<p(^i,^2),p(2/i,2/2)>||
is a metric on the product X X F of two metric spaces X and Y.
3.12 In the above exercise show that <Xn,yn^ —> <x,y> in X X F if and only if
a^n —^ ^ in X and 2/n —^ 2/ in Y. This property would be our minimal requirement for a
product metric,
3.13 Defining a product metric as above, use Theorem 3.2 to show that
<f,g>:S-^XXY
is continuous if and only if/: S —^ X and g: S -^ Y are both continuous.
3.14 Let X, Y, and Z be metric spaces, and let /: X X F —> Z be a mapping such
that f{x, y) is continuous in the variables separately. Suppose also that the continuity
in X is uniform over y. That is, suppose that given e and xqj there is a 5 such that
p{x,xo) < 8 => p(f{x,y),f{xo,y)) < e
for every value of y. Show that then / is continuous on X X Y.
3.15 Define the function / on the closed unit square [0, 1] X [0, 1] by
/(O, 0) = 0, fix, y) = T-^^ if <x, y> ^ <0, 0>.
Then/is continuous as a function of x for each fixed value of y, and conversely. Show,
however, that / is not continuous at the origin. That is, find a sequence < Xn, yn >"
converging to < 0, 0 > in the plane such that f{Xn, yn) does not converge to 0. This
example shows that continuity of a function of two variables is a stronger property
than continuity in each variable separately.
4. SEQUENTIAL COMPACTNESS
The reader is probably familiar with the idea of a subsequence. A subsequence of
a sequence {xn\ is a new sequence {ym} that is formed by selecting an infinite
number, but generally not all, of the terms x^ and counting them off in the
order of the selected indices. Thus, if ni is the first selected n, 712 the next, and
so on, and if we set ym = Xn^, then we obtain the subsequence
1^1? y2i • • • J ymi • • •! = xp^nii ^n2) • • • ; *^w„j • • •/ ^^ xViniin = l*^n„/m«
Strictly speaking, this counting off of the selected set of indices nis a sequence
m y-^ Um from Z"*" to Z"*" which preserves order: nm-\-i > nm for all m. And the
subsequence m >-^ Xn^ is the composition of the sequence nv-^ Xn and the
selector sequence.
In order to avoid subscripts on subscripts, we may use the notation n{m)
instead of n^. In either case we are being conventionally sloppy: we are using
the same symbol 'n' as an integer-valued variable, when we write Xnj and as the
selector function, when we write n{m) or n^. This is one of the standard nota-

206 COMPAGTJS^ESS AND COMPLETENESS 4.4
tional ambiguities which we tolerate in elementary calculus, because the cure is
considered worse than the disease. We could say: let / be a sequence, i.e., a
function from Z"*" to U. Then a subsequence of / is a composition / © gr, where g
is a mapping from /"*" to /"*" such that g(m + 1) > g{m) for all m.
If you have grasped the idea of subsequence, you should be able to see that
any infinite sequence of O's and I's, say {0, 1, 0, 0, 1, 0, 0, 0, 1, . . .}, can be
obtained as a subsequence of {0, 1, 0, 1, 0, 1, . . . , [1 + ( —1)'*]/2, . . .}.
If Xn -^ a, then it should be clear that every subsequence also converges to a.
We leave the details as an exercise. On the other hand, if the sequence {xn}
does not converge to a, then there is an e such that for every N there is some
larger n at which p(Xn, a) > e. Now we can choose such an n for every A^,
taking care that n^r+i > n^v, and thus choose a subsequence all of whose terms
are at a distance at least e from a. Then this sequence has no subsequence
converging to a. Thus, if {xn} does not converge to a, then it has a subsequence
no (sub)subsequence of which converges to a. Therefore,
Lemma 4.1. If the sequence {xn} and the point a are such that every
subsequence of {xn} has itself a subsequence that converges to a, then
Xn -^ a.
This is a wild and unlikely sounding lemma, but we shall use it to prove a
most important theorem (Theorem 4.2).
Definition. A subset ^ of a metric space is sequentially compact if every
sequence in A has a subsequence that converges to a point of A.
Here, so to speak, we create convergence out of nothing. One would expect
a compact set to have very powerful properties, and perhaps suspect that there
aren't many such sets. We shall soon see, however, that every boimded closed
subset of R'* is compact, and it is in the theory of finite-dimensional spaces that
we most frequently use this notion. Sequential compactness in
infinite-dimensional spaces is a much rarer phenomenon, but when it does occur it is very
important, as we shall see in our brief look at Sturm-Liouville theory in Chapter 6.
We begin with a few simple but important general results.
Lemma 4.2. If ^ is a sequentially compact subset of a metric space 5,
then A is closed and bounded.
Proof. Suppose that {xn} C A and that Xn -^ h. By the compactness of A
there exists a subsequence {xn(i)}i that converges to a point a G A. But a
subsequence of a convergent sequence converges to the same limit. Therefore,
a = h and 6 G A. Thus A is closed.
Boundedness here will mean lying in some ball about a given point b. If A
is not bounded, for each n there exists a point Xn ^ A such that p{xnj b) > n.
By compactness a subsequence {xn(t)}t converges to a point a G A, and
p{xn(t), b) -^ p(a, b).
This clearly contradicts p{xn{i), b) > n(i) > i. D

4.4 SEQUENTIAL COMPACTNESS 207
Continuous functions carry compact sets into compact sets. The proof of
the following result is left as an exercise.
Theorem 4.1. If/is continuous and A is a sequentially compact subset of its
domain, then/[A] is sequentially compact.
A nonempty compact set A C B^ contains maximum and minimum elements.
This is because lub A is the limit of a sequence in A, and hence belongs to A
itself, since A is closed. Combining this fact with the above theorem, we obtain
the following well-known corollary.
Corollary. If/is a continuous real-valued function and dom (/) is nonempty
and sequentially compact, then / is bounded and assumes maximum and
minimum values.
The following very useful result is related to the above theorem.
Theorem 4.2. If / is continuous and bijective and dom (/) is sequentially
compact, then/~^ is continuous.
Proof, We have to show that if Vn -^ V in the range of /, and if Xn = f^iVn)
and X = f~^{y), then Xn -^ x. It is sufficient to show that every subsequence
{^n(t)}t has itself a subsequence converging to x (by Lemma 4.1). But, since
dom (/) is compact, there is a subsequence {xn{iU))}j converging to some z, and
the continuity of / implies that f{z) = lim^-^^ fixnuu))) = limy^^ ynHU)) = V-
Therefore, z = S~^{y) = x, which is what we had to prove. Thus f^^ is
continuous. D
We now take up the problem of showing that bounded closed sets in R"^ are
compact. We first prove it for U itself and then give an inductive argument
for R^.
A sequence {xn} C K is said to be increasing if Xn < Xn+i for all n. It is
strictly increasing if Xn < Xn+i for all n. The notions of a decreasing sequence
and a strictly decreasing sequence are obvious. A sequence which is either
increasing or decreasing is said to be monotone. The relevance of these notions here lies
in the following two lemmas.
Lemma 4.3. A bounded monotone sequence in U is convergent.
Proof, Suppose that {xn} is increasing and bounded above. Let I be the least
upper bound of its range. That is, Xn < I for all n, but for every e, Z — € is not an
upper bound, and so I — e < xn for some N. Then
n > N =^ I — e < Xn < Xn < ly
and so \xn — /[ < e. That is, o^n -^ ^ as n -^ oo. D
Lemma 4.4. Any sequence in R has a monotone subsequence.
Proof, Call Xn a peak term if it is greater than or equal to all later terms. If
there are infinitely many peak terms, then they obviously form a decreasing

208 COMPACTNESS AND COMPLETENESS 4.4
subsequence. On the other hand, if there are only finitely many peak terms, then
there is a last one Xno (or none at all), and then every later term is strictly less
than some other still later term. We choose any rii greater than no, and then we
can choose n2 > ni so that Xm < Xn^i etc. Therefore, in this case we can choose
a strictly increasing subsequence. We have thus shown that any sequence {xn}
in R has either a decreasing subsequence or a strictly increasing subsequence. D
Putting these two lemmas together, we have:
Theorem 4.3. Every bounded sequence in R has a convergent subsequence.
Now we can generalize to R'* by induction.
Theorem 4.4. Every bounded sequence in R'* has a convergent subsequence
(using any product norm, say || ||i).
Proof. The above theorem is the case n = 1. Suppose then that the theorem is
true for n — 1, and let {x'"},^ be a bounded sequence in W^, Thinking of W^ as
W""^ X K, we have x^ = <y'^, ^m>, and {y'^]m is bounded in W"^, because if
X = <y, 2;>, then ||x||i = ||y||i + \z\ > ||y||i. Therefore, there is a subsequence
{y^^*^}t converging to some y in [R^"~\ by the inductive hypothesis. Since
{znii)} is bounded in U, it has a subsequence {zn(i(p))}p converging to some x in
U. Of course, the corresponding subsubsequence {y'*^*^^^^}^ still converges to y
in U'''^\ and then {x''^^^^^^}^ converges to x = <y, ^> in R'' = R'*-^ X U,
since its two component sequences now converge to y and Zj respectively. We
have thus found a convergent subsequence of {x'*}. D
Theorem 4.5. If ^ is a bounded closed subset of U^j then A is sequentially
compact (in any product norm).
Proof, If {xn} C A, then there is a subsequence {xn(i)}i converging to some x
in U^, by Theorem 4.4, and a; is in A, since A is closed. Thus A is compact. D
We can now fill in one of the minor gaps in the last chapter.
Theorem 4.6. All norms on R'* are equivalent.
Proof, It is sufficient to prove that an arbitrary norm || || is equivalent to || || i.
Setting a = max {||5*||}^, we have
Jlxisi <J^\xi\ \\8'\\ < a||x||i,
1 " 1
so one of our inequalities is trivial. We also have | ||a;|| — \\y\\ \ < \\x — y\\ <
a\\x — 2/111, so ||a;|| is a continuous function on U^ with respect to the one-norm.
Now the unit one-sphere S = {x : ||x||i = 1} is closed and bounded and so
compact (in the one-norm). The restriction of the continuous function ||a;|| to
this compact set S has a minimum value m, and m cannot be zero because S
does not contain the zero vector. We thus have ||a;|| > m||a;||i on Sj and so
||a;|| > m||a;||i on R^, by homogeneity. Altogether we have found positive
constants a and m such that m|| ||i < || || < a\\ ||i. D

4.4 SEQUENTIAL COMPACTNESS 209
Composing with a coordinate isomorphism, we see that all norms on any
finite-dimensional vector space are equivalent.
Corollary. If M is a finite-dimensional subspace of the normed linear space
Vj then M is a closed subspace of V.
Proof. Suppose that {fn} CM and ^n —^ a gV. We have to show that a is in
M. Now {fn} is a bounded subset of Mj and its closure in M is therefore
sequentially compact, by the theorem. Therefore, some subsequence converges to
a point i9 in M as well as to a, and so a = fi G M. U
EXERCISES
4.1 Prove by induction that if /: Z"^ ^ Z+ is such that /(n + 1) > /(n) for all n,
then f{n) > n for all n.
4.2 Prove carefully that if Xn ^ a as n ^ oo, then Xnim) ^ a as m ^ oo for any
subsequence. The above exercise is useful in this proof.
4.3 Prove that if {xn} is an increasing sequence in U (xn+i ^ Xn for all n), and if
{xn} has a convergent subsequence, then {xn} converges.
4.4 Give a more detailed version of the argument that if the sequence {xn} does not
converge to a, then there is an e and a subsequence {xn{m)}m such that p{Xnim), o) ^ ^
for all m.
4.5 Find a sequence in R having no convergent subsequence.
4.6 Find a nonconvergent sequence in R such that the set of limit points of
convergent subsequence consists exactly of the number 1.
4.7 Show that there is a sequence {xn) in [0, 1] such that for any y G [0, 1] there is a
subsequence Xn^ converging to y.
4.8 Show that the set of limits of convergent subsequences of a sequence {xn} in a
metric space X is a closed subset of X.
4.9 Prove Theorem 4.1.
4.10 Prove that the Cartesian product of two sequentially compact metric spaces is
sequentially compact. (The proof is essentially in the text.)
4.11 A metric space is boundedly compact if every closed bounded set is sequentially
compact. Prove that the Cartesian product of two boundedly compact metric spaces is
boundedly compact (using, say, the maximum metric on the product space).
4.12 Prove that the sum A -{- B oi two sequentially compact subsets of a normed
linear space is sequentially compact.
4.13 Prove that the sum A -i- B oi sl closed set and a compact set is closed.
4.14 Show by an example in U that the sum of two closed sets nee^ not be closed.
4.15 Let {Cn} be a decreasing sequence (Cn+i CCn for all n) of nonempty closed
subsets of a sequentially compact metric space S. Prove that Oi (^n is nonempty.
4.16 Give an example of a decreasing sequence {Cn} of nonempty closed subsets of
a metric space such that Cif Cn — 0.

210 COMPACTNESS AND COMPLETENESS 4.5
4.17 Suppose the metric space S has the property that every decreasing sequence
{Cn] of nonempty closed subsets of S has nonempty intersection. Prove that then *S
must be sequentially compact. [Hint: Given any sequence {xi} C S, let Cn be the
closure of {xi: i > n].]
4.18 Let .1 be a sequentially compact subset of a nls F, and let B be obtained from .1
by drawing all line segments from points of .1 to the origin (that is,
B = {fa : a G .4 and < G [0, 1]}).
Prove that B is compact.
4.19 Show by applying a compactness argument to Lemma L5 that if A^ is a proper
closed subspace of a finite-dimensional vector space F, then there exists a in F such
that ||a|| = p(a, N) = L
5. COMPACTNESS AND UNIFORMITY
The word * uniform' is frequently used as a qualifying adjective in mathematics.
Roughly speaking, it concerns a "point" property P{y) which may or may not
hold at each point ?/ in a domain A and whose definition involves an existential
quantifier. A typical form for P{y) is (yc){M)Q{y, c, d). Thus, if P{y) is V is
continuous at y\ then P{y) has the form (ye){^b)Q{y, e, b). The property holds
on A if it holds for all y \n A, that is, if
(yy^^)[(yc){M)Q{y, c, d)].
Here d will, in general, depend both on y and c; if either ?/ or c is changed, the
corresponding d may have to be changed. Thus b in the definition of continuity
depends both on e and on the point y at which continuity is being asserted. The
property is said to hold uniformly on A, or uniformly in y, if a value d can be
found that is independent of y (but still dependent on c). Thus the property holds
uniformly in y if
{Vc)(M){Vy^^)Q{y,c,d);
the uniformity of the property is expressed in the reversal of the order of the
quantifiers (^y^^) and (3d). Thus/is uniformly continuous on A if
(Ve)(35)(V2/, 2^^)[p(2/, 2) < h=^p{M,f{z)) < e].
Now b is independent of the point at which continuity is being asserted, but still
dependent on e, of course.
We saw in Section 14 of the last chapter how much more powerful the point
condition of continuity becomes when it holds uniformly. In the remainder of
this section we shall discuss some other imiform notions, and shall see that the
uniform property is often implied by the point property if the domain over which
it holds is sequentially compact.
The formal statement forms we have examined above show clearly the
distinction between uniformity and nonuniformity. However, in writing an
argument, we would generally follow our more idiomatic practice of dropping out

4.5 COMPACTNESS AND UNIFORMITY 211
the inside universal quantifier. For example, a sequence of functions [fn] C W^
converges pointwise to/: A —> IF if it converges to/at every point p in A, that
is, if for every point p in A and for every e there is an N such that
n>N =^ pifn(p),Kp)) <e.
The sequence converges uniformly on A if an AT exists that is independent of p,
that is, if for every e there is an N such that
n > N => pifn(p)J(p)) < e for every p in A.
When p(f, rj) = II f — 77I!, saying that p(/n(p), f(p)) < e for all p is the same as
saying that \\fn — f\\oo < e. Thus fn -> /uniformly if and only if ||/^ — /||^ -> 0;
this is why the norm ||/||oo is called the uniform norm.
Pointwise convergence does not imply uniform convergence. Thus/n(x) =
x^ on A = (0, 1) converges pointwise to the zero function but does not converge
uniformly.
Nor does continuity on A imply uniform continuity. The function f{x) =
1/x is continuous on (0, 1) but is not uniformly continuous. The function
sin (l/x) is continuous and bounded on (0, 1) but is not uniformly continuous.
Compactness changes the latter situation, however.
Theorem 5.1. If/is continuous on A and A is compact, then/is uniformly
continuous on A.
Proof. This is one of our "automatic" negation proofs. Uniform continuity
(UC) is the property
(V6>^)(35>«)(Vx,i/^^)[p(x,i/) < 8=>p(f{x)J{y)) < e].
Therefore, -UC ^ (3e)(V5)(3x, y)[p(x, y) < b and p(/(x), /(?/)) > e]. Take
5 = l/rij with corresponding Xn and y^ Thus, for all n, p{xnj yn) < V^ and
p(f{^n),f{yn)) ^ e, where e is a fixed positive number. Now {xn} has a
convergent subsequence, say XnH) —> x, by the compactness of A. Since
we also have yn{i) —> x. By the continuity of / at x,
p(KXn(i))J{yMi))) < p(f{Xnii))J(x)) + p(f{x), f{ynii))) -> 0,
which contradicts p{f(xnii)),f(ynii))) ^ c. This completes the proof by
negation. D
The compactness of A does not, however, automatically convert the point-
wise convergence of a sequence of functions on A into uniform convergence. The
"piecewise linear" functions /n- [0, 1] —> [0, 1] defined by the graph shown in
Fig. 4.1 converge pointwise to zero on the compact domain [0, 1], but the
convergence is not uniform. (However, see Exercise 5.4.)

212 COMPACTNESS AND COMPLETENESS
1| JJ 1 1
4.5
Fig. 4.2
We pointed out earlier that the distance between a pair of disjoint closed
sets may be zero. However, if one of the closed sets is compact, then the distance
must be positive.
Theorem 5.2. If A and C are disjoint nonempty closed sets, one of which is
compact, then p(A,C) > 0.
Proof. The proof is by automatic contradiction, and is left to the reader.
This result is again a uniformity condition. Saying that a set A is disjoint
from a closed set C is saying that (Va:^^)(3r>^)(^,(x) nC = 0). Saying that
p{A, C) > 0 is saying that (3r>^)(Vx^^) . . .
As a last consequence of sequential compactness, we shall establish a very
powerful property which is taken as the definition of compactness in general
topology. First, however, we need some preparatory work. If A is a subset of a
metric space 5, the r-neighhorhood of A, Br[A], is simply the union of all the balls
of radius r about points of A:
Br[A] = \J{Br{a) :aGA} = {x: (3a^^)(p(x, a) < r)}.
A subset A C >S is r-dense in 5 if 5 C ^r[-4], that is, if each point of S is closer
than r to some point of A.
A subset A of a metric space S is dense m S \i A = S. This is the same as
saying that for every point p in aS there are points of A arbitrarily close to p.
The set Q of all rational numbers is a dense subset of the real number system R,
because any irrational real number x can be arbitrarily closely approximated by
rational numbers. Since we do arithmetic in decimal notation, it is customary to
use decimal approximations, and if 0 < x < 1 and the decimal expansion of
X \Q X = 2Z? an/10'', where each an is an integer and 0 < an < 10, then
YJi ^n/lO^ is a rational number differing from x by less than 10~^. Note that A
is a dense subset of B if and only if A is r-dense in B for every positive r.
A set B is said to be totally hounded if for every positive r there is a finite set
which is r-dense in B. Thus for every positive r the set B can be covered by a
finite number of balls of radius r. For example, the n — 1 numbers {z/n}^~^ are
(l/n)-dense in the open interval (0, 1) for each n, and so (0, 1) is totally bounded.

4.5 COMPACTNESS AND UNIFORMITY 213
Total boundedness is a much stronger property than boundedness, as the
following lemma shows.
Lemma 5.1. If the normed linear space V is infinite-dimensional, then its
closed unit ball Bi = {f : ||f|| < 1} cannot be covered by a finite number
of balls of radius §.
Proof. Since V is not finite-dimensional, we can choose a sequence {a-n} such
that an+i is not in the linear span Mn of {ai, ... , an}j for each n. Since Mn is
closed in 7, by the corollary of Theorem 4.6, we can apply Lemma L5 to find
a vector f^ in Mn such that ||fn|| = 1 and p{^nj ^n-i) > § for all n > 1.
We take f i = a:i/||ai||, and we have a sequence {fn} C Bi such that
I sm sn|| ^
if m 7^ n. Then no ball of radius J can contain more than one f i, proving the
lemma. D
For a concrete example, let V be e([0, 1]), and let fn be the "peak" function
sketched in Fig. 4.2, where the three points on the base are l/(2n + 2), l/(2n+ 1),
and l/2n. Then/n+i is "disjoint" from fn (that is, fn+ifn = 0), and we have
||/n||oo = 1 for all n and ||/n — /mllw = I \i n 9^ m. Thus no ball of radius J can
contain more than one of the functions/n, and accordingly the closed unit ball in
V cannot be covered by a finite number of balls of radius ^.
Lemma 5.2. Every sequentially compact set A is totally bounded.
Proof. If A is not totally bounded, then there exists an r such that no finite
subset F is r-dense in A. We can then define a sequence {pn} inductively by
taking pi as any point of A, p2 as any point of A not in Br{pi)j and pn as any
point of A not in Br[\J\~^ pi] = \JT~^ Br{pi). Then {pn} is a sequence in A
such that p(pij pj) > r for all i 9^ j. But this sequence can have no convergent
subsequence. Thus, if A is not totally bounded, then A is not sequentially
compact, proving the lemma. D
Corollary. A normed linear space V is finite-dimensional if and only if its
closed unit ball is sequentially compact.
Proof. This follows from Theorem 4.4 in one direction and from the above two
lemmas in the other direction. D
Lemma 5.3. Suppose that A is sequentially compact and that {Ei: i G 1}
is an open covering of A (that is, {Ei} is a family of open sets and A c \JiEi).
Then there exists an r > 0 with the property that for every point pin A the
ball Br{p) is included in some Ej.
Proof. Otherwise, for every r there is a point pin A such that Br{p) is not a
subset of any ^y. Taker= 1/n, with corresponding sequence {pn}. Thus ^i/n(pn)
is not a subset of any Ej. Since A is sequentially compact, {pn} has a convergent
subsequence, Pn{m) —> p as m —> 00. Since {E^} covers A, some Ej contains p,

214 COMPACTNESS AND COMPLETENESS 4.5
and then B^{p) c Ej for some e > 0, since Ej is open. Taking m large enough so
that 1/m < e/2 and also p{pn(m)j v) < ^Z^, we have
Bilnim){Pn{m)) C B^(p) C Ej,
contradicting the fact that Bun(pn) is not a subset of any Ei. The lemma has
thus been proved. D
Theorem 5.3. If ^ is an open covering of a sequentially compact set A,
then some finite subfamily of ^ covers A.
Proof. By the lemma immediately above there exists an r > 0 such that for
every pm A the ball Br{p) lies entirely in some set of 9^, and by the first lemma
there exist Pij. - - ,Pn iii A such that -A C Ui Br{pi). Taking corresponding sets
Ei in ^ such that Br{pi) C Ei for i = 1, . . . , n, we clearly have A C U i ^i- D
In general topology, a set A such that every open covering of A includes a
finite covering is said to be compact or to have the Heine-Borel property. The
above theorem says that in a metric space every sequentially compact set is
compact. We shall see below that the reverse implication also holds, so that the
two notions are in fact equivalent on a metric space.
Theorem 5.4. If A is a compact metric space, then A is sequentially
compact.
Proof. Let {xn} be any sequence in A, and let ^ be the collection of open balls B
such that B contains only finitely many Xi. If ^ were to cover A, then by
compactness A would be the union of finitely many balls in 9^, and this would clearly
imply that the whole of A contains only finitely many Xi, contradicting the fact
that {xi} is an infinite sequence. Therefore, ^ does not cover A, and so there is a
point X in A such that every ball about x contains infinitely many of the Xi.
More precisely, every ball about x contains Xi for infinitely many indices i. It can
now be safely left to the reader to see that a subsequence of {xn} converges to x. U
EXERCISES
5.1 Show that/„(a:) = x"" does not converge uniformly on (0, 1).
5.2 Show that/(a:) = 1/x is not uniformly continuous on (0, 1).
5.3 Define the notion of a function K: X X F —> F being uniformly Lipschitz in its
second variable over its first variable.
5.4 Let *S be a sequentially compact metric space, and let {/„] be a sequence of
continuous real-valued functions on S that decreases pointwise to zero (that is, {fnip)]
is a decreasing sequence in U and/n(p) —> 0 as n —> «> for each pin S), Prove that the
convergence is uniform. (Try to apply Exercise 4.15.)
5.5 Restate the corollaries of Theorems 15.1 and 15.2 of Chapter 3, employing the
weaker hypotheses that suffice by virtue of Theorem 5.1 of the present section.

4.6 EQUICONTINUITY 215
5.6 Prove Theorem 5.2.
5.7 Prove that if .1 is an r-dense subset of a set X in a normed Hnear space F, and
if B is an s-dense subset of a set F C V, then A -]- B is (r -\- s)-dense in X + F.
Conclude that the sum of two totally bounded subsets of V is totally bounded.
5.8 Suppose that the n points [pi] ^ are r-dense in a metric space X. Let .1 be any
subset of X. Show that A has a subset of at most n points that is 2r-dense in .1.
Conclude that any subset of a totally bounded metric space is itself totally bounded.
5.9 Prove that the Cartesian product of two totally bounded metric spaces is totally
bounded.
5.10 Show that if a metric space X has a dense subset A that is totally bounded, then
X is totally bounded.
5.11 Show that if two continuous mappings/and g from a metric space X to a metric
space Y are equal on a dense subset of X, then they are equal everywhere.
5.12 Write out in explicit quantified form involving the existence of balls the
statement that the interiors of the sets {Ai} cover the metric space A. Then show that the
conclusion of Lemma 5.3 is another uniformity assertion.
5.13 Reprove the theorem that a continuous function on a compact domain is
bounded on the basis of Theorem 5.3.
5.14 Reprove the theorem that a continuous function on a compact domain is
uniformly continuous from Theorem 5.3.
6. EQUICONTINUITY
The application of sequential compactness that we shall make in an infinite-
dimensional context revolves around the notion of an equicontinuous family of
functions. If A and B are metric spaces, then a subset ^ C B^ is said to be
equicontinuous at po in A if all the functions of ^ are continuous at po and if
given e, there is a 5 which works for them all, i.e., such that
piPyPo) < 5 =» p{f{p)J{Po)) < e for every /in 9^.
The family ^ is uniformly equicontinuous if 8 is also independent of poj and so is
dependent only on e. Our quantifier string is thus (Ve)(35)(Vp, g^'^)(V/^^).
For example, given m > 0, let 9^ be a collection of functions / from (0, 1) to
(0, 1) such that/' exists and |/'| < m on (0, 1). Then \f{x) — f{y)\ < m\x — y|,
by the ordinary mean-value theorem. Therefore, given any e, we can take
d = e/m and have
[x-y\ < d =^ \f{x) -f{y)\ < e
for all Xj y G {Oy 1) and all / G 9^. The collection ^ is thus uniformly
equicontinuous.
Theorem 6.1. If A and B are totally bounded metric spaces, and if 9^ is a
uniformly equicontinuous subfamily of B^, then ^ is totally bounded in the
uniform metric.

216 COMPACTNESS AND COMPLETENESS 4.7
Proof. Given e > 0, choose b so that for all/in ^ and all pi, p2 in A, p(pi, ^2) <
b =» p(/(pi), f{p2)) < c/4. Let D be a finite subset of A which is 5-dense in Aj
and let ^ be a finite subset of B which is (e/4)-dense in B. Let G be the set E^ of
all functions on D into E. G is of course finite; in fact, #G = n^, where m = #D
and n = ^E. Finally, for each g G Glet ^g be the set of all functions f G^ such
that
p(/(p), gip)) < e/4 for every p G D,
We claim that the collections ^g cover ^ and that each ^g has diameter at most €.
We will then obtain a finite e-dense subset of ^ by choosing one function from
each nonempty 9^^, and the theorem will be proved.
To show that every / G 9^ is in some ^g, we simply construct a suitable g.
For each p in D there exists a g in ^ whose distance from f{p) is less than e/4.
If we choose one such qinE for each p in D, we have a function g inG such that
The final thing we have to show is that li f,h G 9^^, then p(/, h) < e. Since
P{hy g) < e/4 on D and p(/, g) < e/4 on D, it follows that
p{f{p)y Hp)) < e/2 for every p G D.
Then for any p' G A we have only to choose p G D such that p(p', p) < 5, and
we have
p(/(p'), Hp')) < p{f{p')J{p)) + p{f{p), h{p)) +p{h{p), Hp'))
< e/4 + e/2 + e/4 = e. D
The above proof is a good example of a mathematical argument that is
completely elementary but hard. When referring to mathematical reasoning, the
words 'sophisticated' and 'difficult' are by no means equivalent.
7. COMPLETENESS
If x„ -^ a as n -^ 00, then the terms Xn obviously get close to each other as n
gets large. On the other hand, if {Xn} is a sequence whose terms get arbitrarily
close to each other as n -^ 00, then {x„} clearly ought to converge to a limit.
It may not, however; the desired limit point may be missing from the sp^ce.
If a metric space S is such that every sequence which ought to converge actually
does converge, then we say that S is complete. We now make this notion precise.
Definition. {x„} is a Cauchy sequence if for every e there is an N such that
m > N and n > N => p{Xmj Xn) < e.
Lemma 7.1. If {Xn} is convergent, then {xn} is Cauchy.
Proof. Given e, we choose N such that n > N =^ p{Xnj o) < e/2, where a is
the limit of the sequence. Then if m and n are both greater than AT, we have
p(xm, Xn) < p(xm, o) + p{a, Xn) < e/2 + e/2 = e. D

4.7 COMPLETENESS 217
Lemma 7.2. If {xn} is Cauchy, and if a subsequence is convergent, then
{xn} itself converges.
Proof. Suppose that Xn(i) -^ a as i -^ oo. Given e, we take N so that mjn> N=^
p{Xn, Xm) < e. Because Xn^i) -^ cl as i-^ oo, we can choose an i such that
n(i) > N Sindp{Xn(i)ya) < e. Thus if m > AT, we have
p{Xmj a) < p{x i)) + p{xn(i)i a) < 2e,
and so x^ -^ a. D
Actually, of course, ii rrij n > N => p{Xm, Xn) < e, and if Xn -^ a, then for
any m > AT' it is true that p(x^, a) < e. Why?
Lemma 7.3. If A and B are metric spaces, and if T is a Lipschitz mapping
from A to B, then T carries Cauchy sequences in A into Cauchy sequences in
B, This is true in particular if A and B are normed linear spaces and T is an
element of Hom(A, B).
Proof. Let {x„} be a Cauchy sequence in A, and set yn = T{Xn). Given e,
choose N so that m, n > N =^ p{Xmj Xn) < ^/Cj when C is a Lipschitz constant
iorF, Then
m,n> N =» p(?/^, Vn) = p{T{xm), T{Xn)) < Cp{xm, Xn) < Ce/C = e. D
This lemma has a substantial generalization, as follows.
Theorem 7.1. If A and B are metric spaces, {x„} is Cauchy in A, and
F: A -^ B is uniformly continuous, then {F{xn)} is Cauchy in B,
Proof. The proof is left as an exercise.
The student should try to acquire a good intuitive feel for the truth of these
lemmas, after which the technical proofs become more or less obvious.
Definition. A metric space A is complete if every Cauchy sequence in A
converges to a limit in A. A complete normed linear space is called a Banach
space.
We are now going to list some important examples of Banach spaces. In
each case a proof is necessary, so the list becomes a collection of theorems.
Theorem 7.2. U is complete.
Proof. Let {xn} be Cauchy in U. Then {x„} is bounded (why?) and so, by
Theorem 4.3, has a convergent subsequence. Lemma 7.2 then implies that
{xn} is convergent. D
Theorem 7.3. If A is a complete metric space, and if / is a continuous
bijective mapping from A to a metric space B such that f~^ is Lipschitz
continuous, then B is complete. In particular, if F is a Banach space, and if
T in Hom(F, W) is invertible, then TF is a Banach space.

218 COMPACTNESS AND COMPLETENESS 4.7
Proof, Suppose that {yn} is a Cauchy sequence in Bj and set Xi = f~^{yi) for
all i. Then {xi} is Cauchy in Aj by Lemma 7.3, and so converges to some x'm A,
since A is complete. But then yn = f(xn) -^ f(x), because / is continuous.
Thus every Cauchy sequence in B is convergent and B is complete. D
The Banach space assertion is a special case, because the invertibility of T
means that T~^ exists in Hom(IF, V) and hence is a Lipschitz mapping.
Corollary. If p and q are equivalent norms on V and <V, p> is complete,
then so is <V, q>.
Theorem 7.4. If Vi and V2 are Banach spaces, then so is Vi X ¥2-
Proof. If {< ^n, Vn'>^} is Cauchy, then so are each of {fn} and {rjn} (by
Lemma 7.3, since the projections Ti are bounded). Then fn -^ « and Vn —^ f^
for some a G 7i and /3 G V2. Thus < ^n, Vn> -^ <ol, P> in Vi X ¥2- (See
Theorem 3.4.) D
Corollary 1. If {Vi] \ are Banach spaces, then so is 11?= 1 Vi-
Corollary 2. Every finite-dimensional vector space is a Banach space
(in any norm).
Proof. R^ is complete (in the one-norm, say) by Theorem 7.2 and Corollary 1
above. We then impose a one-norm on V by choosing a basis, and apply the
corollary of Theorem 7.3 to pass to any other norm. D
Theorem 7.5. Let TF be a Banach space, let A be any set, and let (B(A, W)
be the vector space of all bounded functions from A toW with the uniform
norm \\f\\^ = lub {||/(a)|| : a G A}. Then (B(A, W) is a Banach space.
Proof. Let [fn] be Cauchy, and choose any a G A. Since ||/n(a) — fm{ci)\\ ^
Wfn — /m||x, it follows that {/n(a)} is Cauchy in W and so convergent. Define
g: A —> W hy g(a) = lim/^(a) for each a G A. We have to show that g is
bounded and that fn -^ {/.
Given e, we choose N so that m, 71 > N =^ \\fm — /n|U < e. Then
WfUa) - g{a)\\ = lim ||/^(a) - fn{a)\\ < e.
n—>oo
Thus, lim > Nj then ||/^(a) — g{a)\\ < e for all a ^ A, and hence ||/,,i — (;||oo <
e. This implies both that fm — g ^ (B(A, IF), and so
g = fm-{fm- g) e (B(A, TF),
and that fm-^g in the uniform norm. D
Theorem 7.6. If F is a normed linear space and IF is a Banach space, then
Hom(F, IF) is a Banach space.
The method of proof is identical to that of the preceding theorem, and we
leave it as an exercise. Boundedness here has a different meaning, but it is used

4.7
COMPLETENESS
219
in essentially the same way. One additional fact has to be established, namely,
that the limit map (corresponding to g in the above theorem) is linear.
Theorem 7.7. A closed subset of a complete metric space is complete. A
complete subset of any metric space is closed.
Proof. The proof is left to the reader.
It follows from Theorem 7.7 that a complete metric space A is absolutely
closed, in the sense that no matter how we extend A to a larger metric space
Bj A is always a closed subset of B. Actually, this property is equivalent to
completeness, for if A is not complete, then a very important construction of
metric space theory shows that A can be completed. That is, we can construct
a complete metric space B which includes A. Now, if A is not complete, then
the closure of A in B, being complete, is different from A, and A is not absolutely
closed.
See Exercise 7.21 through 7.23 for a construction of the completion of a
metric space. The completion of a normed linear space is of course a Banach
space.
Theorem 7.8. In the context of Theorem 7.5, let A be a metric space, let
e(A, W) be the space of continuous functions from A to W, and set
(Be(A, w) = (B(A, w) n e(A, w).
Then (BC is a closed subspace of (B.
}<e/3
Fig. 4.3
Proof, We suppose that {/n} C (BC and that \\fn — oWoo -^ 0, where gf E (B.
We have to show that g is continuous. This is an application of a much used
"up, over, and down" argument, which can be schematically indicated as in
Fig. 4.3.
Given e, we first choose any n such that \\fn — (/|U < e/3. Consider now
any a G A, Since/„ is continuous at a, there exists a 8 such that
p(x,a) < 8 => \\fn(x) -fn(a)\\ < e/3.

220 COMPACTNESS AND COMPLETENESS 4.7
Then
p{x,a) < 8 => \\g{x) - g{a)\\ < \\g{x) - fn{x)\\ + \\fn{x) - fn{a)\\
+ Wfnia) - g{a)\\ < e/3 + e/3 + e/3 = c.
Thus g is continuous at a for every a G Aj and so ^ G (BC. D
This important classical result is traditionally stated as follows: The limit of
a uniformly convergent sequence of continuous functions is continuous.
Remark. The proof was slightly more general. We actually showed that if
fn-^f uniformly, and if each fn is continuous at a, then / is continuous at a.
Corollary. (Be(A, W) is a Banach space.
Theorem 7.9. If A is a sequentially compact metric space, then A is
complete.
Proof. A Cauchy sequence in A has a subsequence converging to a limit in A,
and therefore, by Lemma 7.2, itself converges to that limit. Thus A is complete. D
In Section 5 we proved that a compact set is also totally bounded. It can be
shown, conversely, that a complete, totally bounded set A is sequentially
compact, so that these two properties together are equivalent to compactness.
The crucial fact is that if A is totally bounded, then every sequence in A
has a Cauchy subsequence. If A is also complete, this Cauchy subsequence will
converge to a point of A. Thus the fact that total boundedness and
completeness together are equivalent to compactness follows directly from the next
lemma.
Lemma 7.4. If A is totally bounded, then every sequence in A has a Cauchy
subsequence.
Proof. Let {pm} be any sequence in A. Since A can be covered by a finite
number of balls of radius 1, at least one ball in such a covering contains infinitely
many of the points {pm}- More precisely, there exists an infinite set Mi C ^"^
such that the set {pm : m G Mi} lies in a single ball of radius I. Suppose that
Ml, . . . , Mn C /"*" have been defined so that Mi^i C Mifori = 1, . . . , n — 1,
Mn is infinite, and {pm - r^ ^ Mi} is a subset of a ball of radius l/iiori= 1,..., n.
Since A can be covered by a finite family of balls of radius l/(n + 1), at least
one covering ball contains infinitely many points of the set {pm ' 'm G Mn}. More
precisely, there exists an infinite set M^+i C Mn such that {pm • rn G M^n+i}
is a subset of a ball of radius l/(n + 1). We thus define an infinite sequence
{Mn} of subsets of /"*" having the above properties.
Now choose mi G Mij m2 G M2 so that m2 > mi, and, in general, mnj^i G
MnJfi SO that mnjfi > mn- Then the subsequence {pm^n is Cauchy. For
given e, we can choose n so that 1/n < e/2. Then i, j ":> n => mij mj g Mn =>
p{pmi, Pnij) < 2(l/n) < e. This proves the lemma, and our theorem is a
corollary. D

4.7 COMPLETENESS 221
Theorem 7.10. A metric space aS is sequentially compact if and only if aS is
totally bounded and complete.
The next three sections will be devoted to applications of completeness to
the calculus, but before embarking on these vital matters we should say a few
words about infinite series. As in the ordinary calculus, if {fn} is a sequence in a
normed linear space F, we say that the series ^^i converges and has the sum a,
and write J^°^ ^i = a, if the sequence of partial sums converges to a. This means
that (Jn -^ « as n -^ oo, where (Tn is the finite sum ^^ ^i for each n. We say that
^^i converges absolutely if the series of norms II||fi|| converges in U. This is
abuse of language unless it is true that every absolutely convergent series
converges, and the importance of the notion stems from the following theorem.
Theorem 7.11. If F is a Banach space, then every absolutely convergent
series in V is convergent.
Proof, Let ^^i be absolutely convergent. This means that 2211 fill converges in
Uj i.e., that the sequence {sn} converges in R, where Sn = H^ ||fi||. If m < n,
then
Wn ~ (Tm =
m + 1
m + 1
Since {si} is Cauchy in R, this inequality shows that {(Ti} is Cauchy in V and
therefore, because V is complete, that {(Tn} is convergent in V. That is, X^ ^i is
convergent in V. D
The reader will be asked to show in an exercise that, conversely, if a normed
linear space V is such that every absolutely convergent series converges, then V
is complete. This property therefore characterizes Banach spaces.
We shall make frequent use of the above theorem. For the moment we
note just one corollary, the classical Weierstrass comparison test.
Corollary. If {fn} is a sequence of bounded real-valued (or TF-valued, for
some Banach space W) functions on a common domain A, and if there is a
sequence {Mn} of positive constants such that X ^n is convergent and
11/nil 00 < Mn for each n, then X^/n is uniformly convergent.
Proof, The hypotheses imply that 2Z||/n||oo converges, and so J2 fn converges in
the Banach space (B(4, W) by the theorem. But convergence in (B(A, W) is
imiform convergence. D
EXERCISES
7.1 Prove that a Cauchy sequence in a metric space is a bounded set.
7.2 Let F be a normed linear space. Prove that the sum of two Cauchy sequences
in V is Cauchy.
7.3 Show also that if {f n} is Cauchy in V and {an} is Cauchy in U, then {an^n} is
Cauchy in V,

222 COMPACTNESS AND COMPLETENESS 4.7
7.4 Prove that if {f„} is a Cauchy sequence in a normed linear space V, then
[\\^n\\] is a Cauchy sequence in U.
7.5 Prove that if [xn] and f^/n) are two Cauchy sequences in a metric space S,
then [p(xn, Vn)] is a Cauchy sequence in U.
7.6 Prove the statement made after the proof of Lemma 7.2.
7.7 The rational number system is an incomplete metric space. Prove this by
exhibiting a Cauchy sequence of rational numbers that does not converge to a rational
number.
7.8 Prove Theorem 7.1.
7.9 Deduce a strengthened form of Theorem 7.3 from Theorem 7.1.
7.10 Write out a careful proof of Theorem 7.6, modeled on the proof of Theorem 7.5.
7.11 Prove Theorem 7.7.
7.12 Let the metric space .Y have a dense subset Y such that every Cauchy sequence
in Y is convergent in X. Prove that X is complete.
7.13 Show that the set W of all Cauchy sequences in a normed linear space V is
itself a vector space and that a seminorm p can be defined on W by p({^n]) = Hm ||fn||.
(Put this together from the material in the text and the preceding problems.)
7.14 Continuing the above exercise, for each ^ E: V, let ^^ be the constant sequence
all of whose terms are f. Show that ^: f»—> f*" is an isometric linear injection of V into IT
and that dlV] is dense in W in terms of the seminorm from the above exercise.
7.15 Prove next that every Cauchy sequence in d[V] is convergent in W. Put
Exercises 4.18 of Chapter 3 and 7.12 through 7.14 of this chapter together to conclude that
if N is the set of null Cauchy sequences in W, then W/N is a Banach space, and that
J 1-^ ^^ is an isometric linear injection from V to a dense subspace of W/N'. This
constitutes the standard completion of the normed linear space V.
7.16 We shall now sketch a nonstandard way of forming the completion of a metric
space S. Choose some point po in S, and let V be the set of real-valued functions on S
such that/(po) = 0 and/is a Lipschitz function. For/in V define ||/|| as the smallest
Lipschitz constant for /. That is,
11/11 =lub{|/(p)-/(5)|/p(p,g)}.
Prove that T^ is a normed linear space under this norm. (F actually is complete, but
we do not need this fact.)
7.17 Continuing the above exercise, we know that the dual space F* of all bounded
linear functionals on V is complete by Theorem 7.6. We now want to show that S can
be isometrically imbedded in F*; then the closure of aS as a subset of F will be the
desired completion of S. For each p G S, let dp'. V -^ Uhe "evaluation at p". That is,
Opif) = fip)- Show that dp G F* and that \\dp — d^W < p{p, q).
7.18 In order to conclude that the mapping 6: p^-^ ^^ is an isometry (i.e., is distance-
preserving), we have to prove the opposite inequality \\dp — ^^H > p(p, q). To do this,
choose p and consider the special function/(a:) = p{p, x) — p{p, po). Show that/ is
in F and that ||/|| = 1 (from an early lemma in the chapter). Now apply the definition
of \\6p — 6q\\ and conclude that 6 is an isometric injection of S into F*. Then d[S] is
our constructed completion.

4.8 A FIRST LOOK AT BANACH ALGEBRAS 223
7.19 Prove that if a normed linear space V has the property that every absolutely
convergent series converges, then V is complete. (Let {an] be a Cauchy sequence.
Show that there is a subsequence {a^J » such that if ft = an^+i — ar^, then \\^.i\\ < 2~'\
Conclude that the subsequence converges and finish up.)
7.20 The above exercise gives a very useful criterion for V to be complete. Use it to
prove that if F is a ]3anach space and iV is a closed subspace, then Y/N is a Banach
space (see Exercise 4.14 of Chapter 3 for the norm on YIN).
7.21 Prove that the sum of a uniformly convergent series of infinitesimals (all on the
same domain) is an infinitesimal.
8. A FIRST LOOK AT BANACH ALGEBRAS
When we were considering the implicit-function theorem and the inverse-function
theorem in the last chapter, we saw how useful it is to know that if a
transformation T has an inverse T"^^ then so does aS whenever ||aS — r|| is small enough,
and that the mapping T h^ T~^ is continuous on the open set of all invertible
elements. When the spaces in question are finite-dimensional, these facts can
be made to follow from the continuity of the determinant function T h^ A(r)
from Hom 7 to R. It is also possible to produce them by arguing directly in
terms of upper and lower bounds for T and its close approximations aS. But the
most natural, most elegant,, and—in the case of Banach spaces—easiest way to
prove these things is to show that if F is a Banach space and T in Hom Y has
norm less than one, then the sum of the geometric series l^o ^^ is the inverse of
I — T, just as in the elementary calculus. But in making this argument, the
fact th^t r is a linear transformation has little importance, and we shall digress
for a moment to explore this situation.
Lit us summarize the norm and algebraic properties of Hom Y when 7 is a
Banach space. First of all, we know that Hom Y is also a Banach space. Second,
it is an algebra. That is, it possesses an associative multiplication operation
(composition) that relates to the linear operations according to the following
laws:
{S,+S2)T=SiT + S2T,
c{ST)= {cS)T=S(cT).
Finally, multiplication is related to the norm by
ll^rll < ll^ll llrll and 11711 = 1.
This list of properties constitutes exactly the axioms for a Banach algebra.
Just as we can see certain properties of functions most clearly by forgetting
that they are functions and considering them only as elements of a vector space,
now it turns out that we can treat certain properties of transformations in
Hom(F) most simply by forgetting the complicated nature of a linear
transformation and considering it merely as an element of an abstract Banach algebra A.

224 COMPACTNESS AND COMPLETENESS 4.8
The most important simple thing we can do in a Banach algebra that we
couldn't do in a Banach space is to consider power series. The following theorem
shows that the geometric series, in particular, plays the same central role here
that it plays in elementary calculus. Since we are not thinking of the elements of
A as transformations, we shall designate them by lower-case letters; e is the
identity of A,
Theorem 8.1. If A is a Banach algebra, and if x in A has norm less than one,
then (e — x) is invertible and its inverse is the sum of the geometric series
in x:
(e - x)-' = Z
X
0
Also, \\e — (e — x) ^\\ < r/(l — r), where r = ||x||.
Proof. Since ||x^|| < ||x||^ = r^, the series Yl ^^ is absolutely convergent when
||.t|| < 1 by comparison with the ordinary geometric series Yl ^'^- It is therefore
convergent, and if ?/ = 2Zo ^^j then
(e — x)y = lim {e — x)^ x'^ = lim (e — x^'^^) =
6,
0
since ||x||^+^ < r^+^ -> 0. That is, y = (e — x)~\ Finally,
\\e — (e — xY
<J2r''= r/(l - r). D
Theorem 8.2. The set 91 of invertible elements in a Banach algebra A is
open and the mapping x »—> x~^ is continuous from 91 to 91. In fact, if y~^
exists and m = \\y~'^\\j then (y — h)~^ exists whenever \\h\\ < 1/m and
Uy-h)-' -y~'\\ < m'\\h\\/{l - m\\h\\).
Proof, Set X = y~^h. Then (y — h) = y(e — a;), where ||x|| = ||2/~^/«'|| ^
ni\\h\\, and so by the above theorem y — h will be invertible, with (y — h)~^ =
(e — x)~^y~^j provided \\h\\ < 1/m, Then also
\\y-' - (y- h)-'\\ < \\e - (e - x)-'\\ • m,
and this is bounded above by
mr/(l - r) < m^h\\/(l - m\\h\\),
by the last inequality in the above theorem. D
Corollary. If F and W are Banach spaces, then the invertible elements in
Hom(7, W) form an open set, and the map T h^ T~^ is continuous on this
domain.
Proof, Suppose that T^^ exists, and set m = ||r-^||. Then if ||r - aS|| < 1/m,
wehave||/- ^-^aSH < ||r-^|| \\T - S\\ < 1, and so T-^aS = I - {I - T'^S)
is an invertible element of Hom V, Therefore, S = T{T~^S) is invertible and
^_i _ ^y,_i^^_iy,_i rj.^^ continuity of S ^ S'^'^ is left to the reader. D

4.8 A FIRST LOOK AT BANACH ALGEBRAS 225
We saw above that the map x ^-^ (e — x)~^ from the open unit ball ^i(O) in
a Banach algebra A to A is the sum of the geometric power series. We can define
many other mappings by convergent power series, at hardly any greater effort.
Theorem 8.3. Let A be a Banach algebra. Let the sequence {an} C A and
the positive number 8 be such that the sequence {\\an\\ d^} is bounded. Then
J2 0,71^^ converges for x in the ball B^{Q) in A, and if 0 < 5 < 5, then the
series converges uniformly on ^s(O).
Proof. Set r = s/b, and let 6 be a bound for the sequence {||an||5^}. On the
ball Bs{0) we then have ||ana:''|| < ||an||5'' = ||an||5''r'' < hr"^, and the series
therefore converges uniformly on this ball by comparison with the geometric
series h^L t^, since r < L D
The series of most interest to us will have real coefficients. They are included
in the above argument because the product of the vector x and the scalar t is
the algebra product {te)x. In addition to dealing with the above geometric series,
we shall be particularly interested in the exponential function e^ =^ YI'q x'^/n\.
The usual comparison arguments of the elementary calculus show just as easily
here that this series converges for every x in A and uniformly on any ball.
It is natural to consider the differentiability of the maps from A to A defined
by such convergent series, and we state the basic facts below, starting with a
fundamental theorem on the differentiability of a limit of a sequence.
Theorem 8.4. Let {F^} be a sequence of maps from a ball ^ in a normed
linear space 7 to a normed linear space W such that F^ converges pointwise
to a map F onB and such that {clF^} converges for each a and uniformly over
a. Then F is differentiate on B and dF^ = lim dF^ for each /? in B.
Proof, Fix /? and set T = lim dF"^, By the uniform convergence of {dF""},
given e, there is an N such that WdF"^ — dF^\\ < e for all n > iV and for all a
in B. It then follows from the mean-value theorem for differentials that
\\{AFm - AF^(^)) - {dFm - dF^im < 2eU\\
for all n > AT and all f such that /? + ^ G B, Letting n —> oo and regrouping,
we have
ii(AF^(^) - m)) - (AF^(^) - dF^Wn < 2eU\\
for all such f. But, by the definition of dF^ there is a 5 such that
iiAF^(^) - dF^m < m
when llfll < d. Putting these last two inequalities together, we see that
m < 8 => \\AF^U)-Tm <36||f||.
Thus F is differentiable at /3 and dF^ = T. U
The remaining proofs are left as a set of exercises.

226 COMPACTNESS AND COMPLETENESS 4.8
Lemma 8.1. Multiplication on a Banach algebra A is differentiable (from
A X A to A). If we let p be the product function, so that p{Xj y) = xy,
then dp^a,h>{x, y) = ay + xh.
Lemma 8.2. Let A be a commutative Banach algebra, and let p be the
monomial function p(x) = ax^. Then p is everywhere differentiable and
dpy{x) = 7my^~^x,
Lemma 8.3. If {||anlh'^j is a bounded sequence in U, then {n||an||s^} is
bounded for any 0 < s < r, and therefore J^ nanX^~^ converges uniformly
on any ball in A smaller than ^^(0).
Theorem 8.5. If ^ is a commutative Banach algebra and {a^} C -A is such
that nknlb'^] is bounded in R, then F(x) = Y.o (^n^^ is defined and
differentiable on the ball Br(0) in A, and
dFy(x) =: (Z nany''-'^
It is natural to call the element X!^ nany^~^ the derivative of F Sit y and to
designate it F^(y)j although this departs from our rule that derivatives are
vectors obtained as limits of difference quotients. The remarkable aspect of the
above theorem is that for this kind of differentiable mapping from an open
subset of ^ to ^ the linear transformation dFy is multiplication by an element
of ^: dFy(x) = F'{y)'X,
In particular, the exponential function exp {x) = e"" = Y.Z ^^/^J is its own
derivative, since Xl"]^ 7?x^~V^^- ^ Z!o ^'"/^v ^^^ hova this fact (see the
exercises) or from direct algebraic manipulation of the series in question, we can
deduce the law of exponents e^'^^ = e^e^. Remember, though, that this is on a
commutative Banach algebra. The function x ^-^ e^ = 2Zo ^^/^- can be defined
just as easily on any Banach algebra A, but it is not nearly as pleasant when A
is noncommutative. However, one thing that we can always do, and often
thereby save the day, is to restrict the exponential mapping to a commutative
subalgebra of A, say that generated by a single element x. For example, we can
consider the parametrized arc l{t) = e^^ {x fixed) into ayiy Banach algebra A,
and, because its range lies in the commutative subalgebra X generated by x, we
can apply Theorem 7.2 of Chapter 3 to conclude that 7 is differentiable and that
l\t) = d QXY>tx {x) = xe'^.
This can also easily be proved directly from the law of exponents:
Mt{h) = e^'+^^^^ - e'^ = e'-^(e'^^ - 1),
and since it is clear from the series that {e^^ — l)//i —> x as /i —> 0, we have that

4.8 A FIRST LOOK AT BAXACH ALGEBRAS 227
EXERCISES
8.1 Finish the proof of the corollary of Theorem 8.2.
8.2 Let .1 be a J^anach algebra, and let [a„] C ^ and x G A he such that 23 ^i-^"'
converges. Suppose also that x satisfies a polynomial indentity p{x) = ^obiX^ = 0,
where [bi] C ^ and bn 9^ 0. Prove that the element 5ZT otio:' is a polynomial in x of
degree < n — 1. (Let .1/ be the linear span of {^^]o~S and show first that x^ E M
for all I.)
8.3 Let .1 be any l^anach algebra, let a: be a fixed element in J, and let X be the
smallest closed subalgebra of .1 containing x. Prove that A^ is a commutative Banach
algebra. (The set of polynomials p(x) = 5Zo ^i^' is the smallest algebra containing x.
Consider its closure in X.)
8.4 Prove Lemma 8.1. [Hint: <x, y> t-^ xy is a, bounded bilinear map.]
8.5 Prove Lemma 8.2 by making a direct A-estimate from the binomial expansion,
as in the elementary calculus.
8.6 Prove Lemma 8.2 by induction from Lemma 8.1.
8.7 Let .1 be any Banach algebra. Prove that p-.x^-^x^ is differentiable and that
dpa(^) = ^0? -\- axa -\- a^x.
8.8 Prove by induction that if q{x) = x"", then q is differentiable and
n-l
dqa{x) = ^ a^a:a^"-i-^\
1=0
Deduce Lemma 8.2 as a corollary.
8.9 Let. I be any Banach algebra. Prove that r: x^-^ x~^ is everywhere differentiable
on the open set U of invertible elements and that
dVaix) = — a~^xa~^,
[Hint: Examine the proofs of Theorems 8.1 and 8.2.]
8.10 Let .4 be an open subset of a normed linear space V, and let F and G be mappings
from A to a Banach algebra X that are differentiable at a. Prove that the product
mapping FG is differentiable at a and that d(FGa) = F{a) dGa + dFaG{a). Does it
follow that d{F^)a = 2F{a) dFJ
8.11 Continuing the above exercise, show that if X is a commutative Banach algebra,
then d{F'')a = nF^'-^a) dFa>
8.12 Let F: A —> X be a differentiable map from an open set .1 of a normed linear
space to a Banach algebra X, and suppose that the element F(^) is invertible in .Y
for every ^ in A. Prove that the map G: ^^-^ [F{^)]~^ is differentiable and that
dGaiO = —F(a)-^ dFaiOFM' Sho w also that if/''is a parametrized arc (A = /CK),
then G\a) = -F(a)-'^ • F'{a) • F{a)-K
8.13 Prove Lemma 8.3.
8.14 Prove Theorem S.5 by showing that Lemma 8.3 makes Theorem 8.4 applicable.
8.15 Show that in Theorem 8.4 the convergence oi F^ to F needs only to be assumed
at one point, provided we know that the codomain space W is a Banach space.

228 COMPACTNESS AND COMPLETENESS 4.9
8.16 We want to prove the law of exponents for the exponential function on a
commutative Banach algebra. Show first that (exp (—a:))(exp x) = e by applying Exercise
7.13 of Chapter 3, the above Exercise 8.10, and the fact that d expa (x) = (exp a)x.
8.17 Show that if Z is a commutative Banach algebra and F: X ^^ X is a
different iable map such that dFaiO = ?^'X^)j then F(^) = p exp J for some constant p.
[Consider the differential of F{^) exp (—J).]
8.18 Now set F{^) = exp (J + r;) and prove from the above exercise that
exp {^+ v) = exp (J) exp (rj).
You will also need the fact that exp 0 = 1.
8.19 Let 2 be a nilpotent element in a commutative Banach algebra A". That is,
z^ = 0 for some positive integer p. Show by an elementary estimate based on the
binomial expansion that if ||a:|| < 1, then ||a:+2||'^ < knP\\x\\^~p for n > p. The
series of positive terms XI ^^^'^ converges for r < 1 (by the ratio test). Show, therefore,
that the series for log (l — {x -\- z)) and for (l — {x -\- z))~^ converge when ||a:|| < 1.
8.20 Continuing the above exercise, show that F{y) = log (1 — y) is defined and
differentiable on the ball \\y — 2;|| < 1 and that dFa(x) = —(1 — a)~^ • x. Show,
therefore, that exp (log {1 — y)) = 1 — y on this ball, either by applying the inverse
mapping theorem or by applying the composite function rule for differentiating.
Conclude that for every nilpotent element z in X there exists sl u in X such that
exp u = 1 — z.
8.21 Let Zi, . . . , Z„ be Banach algebras. Show that the product Banach space
X = IJi Zi becomes a Banach algebra if the product xy = <x,. .. ,XnXyi,. .. ,yn>
is defined as <xiyi, . . . , Xnyn^ and if the maximum norm is used on X.
8.22 In the above situation the projections iri have now become bounded algebra
homomorphisms. In fact, just as in our original vector definitions on a product space,
our definition of multiplication on X was determined by the requirement that Tri{\y) =
Tri{x)Tri(y) for all i. State and prove an algebra theorem analogous to Theorem 3.4 of
Chapter 1.
8.23 Continuing the above discussion, suppose that the series ^ anx"" converges in X,
with sum y. Show that then ^(an)i(xi)'' converges in Xi to yi for each i, where, of
course, y = <yi, . . . ,yn>. Conclude that e"" = <e^i, . . . , e^»> for any x =
■<xi, . . . , Xn^ in Z.
8.24 Define the sine and cosine functions on a commutative Banach algebra, and
show that sin' = cos, cos' = —sin, sin^ + cos^ = e.
9. THE CONTRACTION MAPPING FIXED-POINT THEOREM
In this section we shall prove the very simple and elegant fixed-point theorem for
contraction mappings, and then shall use it to complete the proof of the implicit-
function theorem. Later, in Chapter 6, it will be the basis of our proof of the
fundamental existence and uniqueness theorem for ordinary differential
equations. The section concludes with a comparison of the iterative procedure of the
fixed-point theorem and that of Newton's method.

4.9 THE CONTRACTION MAPPING FIXED-POINT THEOREM 229
A mapping K from a metric space X to itself is a contraction if it is a Lipschitz
mapping with constant less than 1; that is, if there is a constant C with 0 < C < 1
such that p(K(x), K{y)) < Cp{x, y) for all x, y E: X. A fixed point of K is, of
course, a point x such that K(x) = x.
A contraction K can have at most one fixed point, since if K(x) = x and
K(y) = y, thenp(x, y) = p(K(x), K{y)) < Cp{x, y), and so (1 — C)p{x, y) < 0.
Since C < 1, this implies that p(x, y) = 0 and x = y.
Theorem 9.1. Let X be a nonempty complete metric space, and let
K: X -^ X be a contraction. Then K has a (unique) fixed point.
Proof. Choose any Xq in X, and define the sequence {xn} o inductively by setting
Xi = K(xo)j X2 = K(xi) = K^{xo), and Xn = K(xn-i) = K^(xo). Set 8 =
p(xi, Xq), Thenp(x2, Xi) = p(K(xi), K{xo)) < Cp(xi, Xq) = C8, and, by
induction,
p(Xn + U Xn) = p{K(Xn), K(Xn-l)) < Cp{Xn, Xn-l) < C ' C^'' d = C^8.
It follows that {xn} is Cauchy, for if m > n, then
m —1 m —1
p(Xm, Xn) < Z p{Xi+i, Xi) < ^ C'S < C"5/(l - C),
n n
and C^ -^ 0 as n -^ oo, because C < 1. Since X is complete, {xn] converges to
some a in X, and it then follows that K{a) = lim K{Xn) = lim x'n+i = a, so
that a is a fixed point. D
In practice, we meet mappings K that are contractions only near some
particular point p, and we have to establish that a suitable neighborhood of p
is carried into itself by K. We show below that if K is a contraction on a ball
about p, and if K doesn't move the center of p very far, then the theorem can
be applied.
Corollary 1. Let Z) be a closed ball in a complete metric space X, and let
K: D -^ X be a contraction which moves the center of D a distance at
most (1 — C)r, where r is the radius of D and C is the contraction constant.
Then K has a unique fixed point and it is in D.
Proof, We simply check that the range of K is actually in D. If p is the center
of D and x is any point in D, then
p{K(xl p) < p(K(x), K(p)) + p{K(pl p)
< Cp{x, p) + {l- C)r < Cr + (1 - C)r = r. U
Corollary 2. Let B be an open ball in a complete metric space X, and let
K: ^ -^ X be a contraction which moves the center of ^ a distance less than
(1 — (7)r, where r is the radius of B and C is the contraction constant.
Then K has a unique fixed point.
Proof. Restrict K to any slightly smaller closed ball D concentric with B, and
apply the above corollary. D

230 COMPACTNESS AND COMPLETENESS 4.9
Corollary 3. Let K be a contraction on the complete metric space X, and
suppose that K moves the point x a distance d. Then the distance from x to
the fixed point is at most d/(l — C), where C is the contraction constant.
Proof, Let D be the closed ball about x of radius r = d/(l — C), and apply
Corollary 1 to the restriction of K to D. It implies that the fixed point is in D. D
We now suppose that the contraction K contains a parameter s, so that K
is now a function of two variables K(Sj x). We shall assume that K is a
contraction in X uniformly over s, which means that p(K(Sj x), K(Sj y)) < Cp{x, y)
for all X, y, and s, where 0 < C < L We shall also assume that K is a
continuous function of 5 for each fixed x.
Corollary 4. Let K be a mapping from aS X X to X, where X is a complete
metric space and S is any metric space, and suppose that K(Sj x) is a
contraction in X uniformly over 5 and is continuous in 5 for each x. Then the
fixed point Ps is a continuous function of s.
Proof. Given e, we use the continuity of K in its first variable around the point
<t, pt> to choose 8, so that if p{s, t) < 8, then the distance from K{s, pt) to
K(t, Pt) is at most e. Since K(t, pt) = pt, this simply says that the contraction
with parameter value s moves pt a distance at most e, and so the distance from
Pt to the fixed point Ps is at most e/(l — C) by Corollary 3. That is, p(s, t) <
8 =» p{ps, Pt) < ^/(l — C)j where C is the uniform contraction constant, and
the mapping s ^-^ ps is accordingly continuous at t. D
Combining Corollaries 2 and 4, we have the following theorem.
Theorem 9.2. Let ^ be a ball in a complete metric space X, let S be any
metric space, and let K be a mapping from aS X ^ to X which is a contraction
in its second variable uniformly over its first variable and is continuous in its
first variable for each value of its second variable. Suppose also that K
moves the center of ^ a distance less than (1 — C)r for every 5 in S, where r
is the radius of B and C is the uniform contraction constant. Then for each 5
in S there is a unique p in B such that K{Sj p) = p, and the mapping
s ^-^ p is continuous from aS to ^.
We can now complete the proof of the implicit-function theorem.
Theorem 9.3. Let V, W, and X be Banach spaces, let A X ^ be an open
subset of V X W, and let G: A X ^ —> X be continuous and have a
continuous second partial differential. Suppose that the point <«,/?> in
A X ^ is such that G(a, 0) = 0 and dG\aS> is invertible. Then there are
open balls M and N about a and /?, respectively, such that for each f in M
there is a unique -q in N satisfying G(f, 77) = 0. The function F thus
uniquely defined near <«,/?> by the condition G{^,F{^)) = Qis continuous.
Proof, Set T = dG\aS> and jf^(f, 77) = 77 - T-\G{^, 77)). Then K is a
continuous mapping from A X B to W such that K(a, 0) = P, and K has a con-

4.9 THE CONTRACTION MAPPING FIXED-POINT THEOREM 231
tinuous second partial differential such that dK%a,^> = 0. Because dK%^,j,>^ is
a continuous function of < /i, ^ >, we can choose a product ball M X N about
<aj ^> on which dK\y,^yy, is bounded by J, and we can then decrease the ball M
if necessary so that for /x in M we also have \\K{yij 0) — P\\ < r/2j where r is the
radius of the ball N, The mean-value theorem for differentials implies that K is
a contraction in its second variable with constant ^. The preceding theorem
therefore shows that for each f in M there is a unique rjinN such that K(^, rj) =
7} and the mapping F: ^ ^-^ rj is continuous. Since K{^j 77) == 77 if and only if
G(^, v) = 0, we are done. D
Theorems 8.2 and 9.3 complete the list of ingredients of the implicit-function
theorem. (However, see Exercise 9.8.)
We next show, in the other direction, that if a contraction depending on a
parameter is continuously differentiable, then the fixed point is a continuously
differentiable function of the parameter.
Theorem 9.4. Let V and W be Banach spaces, and let iiC be a differentiable
mapping from an open subset AxBoiVxWtoW which satisfies the
hypotheses of Theorem 9.2. Then the function F from A to ^ uniquely
defined by the equation iiC(f, F(^)) = F(^) is differentiable.
Proof. The inequality ||K(f, 77') - K(f, t?")!! < CWv' - -q'^ is equivalent to
||^^<a.^>|| < C for all <a, p> m Ax B. We now define G by {?(f, 77) =
rj — K(^j 77), and observe that dG^ = I — dK^ and that dG^ is therefore
invertible by Theorem 8.1. Since G(^,F(0) = 0,it follows from Theorem 11.1
of Chapter 3 that F is differentiable and that its differential is obtained by
differentiating the above equation. D
Corollary. If K is continuously differentiable, then so is F.
* We should emphasize that the fixed-point theorem not only has the implicit-
function theorem as a consequence, but the proof of the fixed-point theorem
gives an iterative procedure for actually finding the value of F{^)j once we
know how to compute T~^ (where T = dG%a,p>)' In fact, for a given value of
f in a small enough ball about <a, i9> consider the function G(f, •)• If we
set K(^y rj) = Tj — T~^G(^y rj), then the inductive procedure
Vi+i = K(^, rji)
becomes
(vi+i -Vi) = -T-'Ga,vi). (9.1)
The meaning of this iterative procedure is easily seen by studying the graph of
the situation where V = W = R^. (See Fig. 4.4.) As was proved above, under
suitable hypotheses, the series Slki+i ~ Vi\\ converges geometrically.
It is instructive to compare this procedure with Newton's method of
elementary calculus. There the iterative scheme (9.1) is replaced by
ivi+i - Vi) = S7'G(i, 77,), (9.2)

232 COMPACTNESS AND COMPLETENESS
4.9
G(a,-),
G{x,')
G{x, ^o)>
Fig. 4.4
2/3 2/2 2/1
where Si = dG\^^y^.>^. (See Fig. 4.5.) As we shall see, this procedure (when it
works) converges much more rapidly than (9.1), but it suffers from the
disadvantage that we must be able to compute the inverses of an infinite number of
linear transformations Si,
Fig. 4.5
Let us suppress the f which will be fixed in the argument and consider a map
G defined in some neighborhood of the origin in a Banach space. Suppose that G
has two continuous differentials. For definiteness, we assume that G is defined
in the unit ball, B, and we suppose that for each x ^B the map dGx is invertible
and, in fact,
\\dG7^\\ < K, \\d^Gj\ < K,
Let Xo == 0 and, assuming that Xn has been defined, we set
Xfi^l = Xfi K^n ^\^n))
where Sn = dGx^- We shall show that if ||G(0)|| is sufficiently small (in terms
of K)j then the procedure is well defined (that is, Hxn+iH < 1) and converges
rapidly. In fact, if r is any real number between one and two (for instance
T = f)? we shall show that for some c (which can be made large if ||G(0)|| is
small)
\\Xn-Xn-i\\ <6--". (*)

4.9 THE CONTRACTION MAPPING FIXED-POINT THEOREM 233
Note that if we can establish (*) for large enough c, then ||x„|| < 1 follows.
In fact,
1 1 1 It
which is < 1 if c is large. Let us try to prove (*) by induction. Assuming it true
for Uj we have
ll^n + l — Xn\\ = ||AS;r^G(^n)||
< K{\\G(Xn-l) - c/(?x„_,S-_ii(?(Xn_i)|| + K\\Xn - X,_if}
by Taylor's theorem. Now the first term on the right of the inequality vanishes,
and we have
\\Xn+i - x„\\ < K^Xn - a;„_i|r < Kh-^"\
For the induction to work we must have
or
Since r < 2, this last inequality can be arranged by choosing c sufficiently
large. We must still verify (*) for n == 1. This says that
or
||G(0)||<^- (***)
In summary, for 1 < r < 2 choose c so that K^ < e^'^~'^^^'^ and
^-c(r-l)
< 1.
1 — e-c(r-l) -
Then if (***) holds, the sequence Xn converges exponentially, that is, (*) holds.
If X = limxi, then G{x) = lim G{Xn) = WxnSniXnJfi ~ ^n) = 0. This is
Newton's method.
As a possible choice of c and r, let r == f, and let c be given by K^ = e^^^^,
so that (**) just holds. We may also assume that K > 2^'^, so that e^^'^ > 4^^^
or e^ > 4:j which guarantees that e~^'^ < J, implying that e~^'^/(l — e~^'^) <
1. Then (***) becomes the requirement G(0) < K~^,
We end this section with an example of the fixed-point iterative procedure in
its simplest context, that of the inverse-mapping theorem. We suppose that
^(0) = 0 and that cIHq^ exists, and we want to invert H near zero, i.e., solve
the equation H^ri) -— ^ = 0 for rj in terms of f. Our theory above tells us that
the 7} corresponding to f will be the fixed point of the contraction K(f, 77) =
77 — T~^H(r]) + r"~^(f), where T — dHo, In order to make our example as

Ui = X,
U2 = X -
Us = X -
U4 = X -
-v\
-{y-
~\v~
x^f
(X-
234 COMPACTNESS AND COMPLETENESS 4.1)
simple as possible, we shall take U from R^ to R^ and choose it so that dU^ = I.
Also, in order to avoid indices, we shall use the mongrel notation x = <x, y>,
u = <u,v>.
Consider the mapping x = H(u) defined by x = u + v^j y = u^ + v.
The Jacobian matrix
" 1 2v]
[3^2 1
is clearly the identity at the origin. Moreover, in the expression K(x, u) =
X + u — H(u)j the difference H(u) ~ u is just the function J(u) = < v^j u^ >.
This cancellation of the first-order terms is the practical expression of the fact
that in forming K(f, r;) = r; — T~^G{^, rj), we have acted to make dK^ == 0 at
the "center point" (the origin here). We naturally start the iteration with
uq = 0, and then our fixed-point sequence proceeds
Ui = K(x, Uo) = K{x, 0), . . . , Un = K{x, Wn-l)'
Thus Uo = 0 and u„ = K(x, u„_i) = x — J(u„_i), giving
vi = y,
V2= y — x^,
V3 = y - (x - 2/')^
yy?. v, = y~[x-{y- x'r]\
We are guaranteed that this sequence u^ will converge geometrically provided
the starting point x is close enough to 0, and it seems clear that these two
sequences of polynomials are computing the Taylor series expansions for the
inverse functions u(x, y) and v(Xj y). We shall ask the reader to prove this in an
exercise. The two Taylor series start out
u{x, y) = X — y'^ — 2yx^ -\ ,
v{x, y)= y ~ x^ + Sx^y^ H
EXERCISES
9.1 Let 5 be a compact subset of a normed linear space such that rB G B for all
r G [0, 1]. Suppose that F:B—^B is a Lipschitz mapping with constant 1 (i.e.,
\\F(h — ^'\v)\\ < U — v\\ for all J, r; G B). Prove that F has a fixed point. [Hint:
Consider first G = rF for 0 < r < 1.]
9.2 Give an example to show that the fixed point in the above exercise may not be
unique.
9.3 Let Z be a compact metric space, and let K:X —^ X "reduce each nonzero
distance". That is, p{K{x), K{y)) < p(x, y) ii x 9^ y. Prove that K has a unique
fixed point. (Show that otherwise gib {p{K(x),x)} is positive and achieved as a
minimum. Then get a contradiction.)
9.4 Let i^ be a mapping from S X Z to X, where Z is a complete metric space and S
is any metric space, and suppose that K(s, x) is a contraction in s uniformly over x and

4.9 THE CONTRACTION MAPPING FIXED-POINT THEOREM 235
is Lipschitz continuous in x uniformly over x. Show that the fixed point ps is a Lipschitz
continuous function of s. [Hint: Modify the €,5-beginning of the proof of Corollary
4 of Theorem 9.1.]
9.5 Let D be an open subset of a Banach space F, and let /^: D —> F be such that
/ — /^ is Lipschitz with constant J.
a) Show that if BM) C Z) and /? = K{a), then 5w2(/3) C K[D]. (Apply a corollary
of the fixed-point theorem to a certain simple contraction mapping.) v
b) Conclude that K is injective and has an open range, and that K~^ \^ Lipschitz
with constant 2.
9.6 Deduce an improved version of the result in Exercise 3.20, Chapter 3, from the
result in the above exercise.
9.7 In the context of Theorem 9.3, show that dG\n,v> is invertible if ||c^K<^,„> || < 1.
(Do not be confused by the notation. We merely want to know that S is invertible if
||7_ T-^oS\\ < 1.)
9.8 There is a slight discrepancy between the statements of Theorem 11.2 in Chapter
3 and Theorem 9.3. In the one case we assert the existence of a unique continuous
mapping from a ball M, and in the other case, from the ball M to the ball A^. Show
that the requirement that the range be in N can be dropped by showing that two
continuous solutions must agree on M. (Use the point-by-point uniqueness of
Theorem 9.3.)
9.9 Compute the expression for dFa from the identity G{^, F{^)) = 0 in Theorem
9.4, and show that if K is continuously differentiate, then all the maps involved in the
solution expression are continuous and that a»—> dFa is therefore continuous.
9.10 Going back to the example worked out at the end of Section 9, show by induction
that the polynomials Un — Un—i and Vn — Vn—i contain no terms of degree less than n.
9.11 Continuing the above exercise, show therefore that the power series defined by
taking the terms of degree at most n from Un is convergent in a ball about 0 and that its
sum is the first component u{x, y) of the mapping inverse to H.
9.12 The above conclusions hold generally. Let J = <K,L> be any mapping
from a ball about 0 in R^ to U^ defined by the convergent power series
K{x, 0 = Z cLijx'y', L(x, 2/) = S bij^'y^
in which there are no terms of degree 0 or L With the conventions x = <x,y> and
u = <u,v>, consider the iterative sequence
UO = 0, Un = X — J(Un-l).
Make any necessary assumptions about what happens when one power series is
substituted in another, and show by induction that Un — Un-i contains no terms of
degree less than n, and therefore that the Un define a convergent power series whose
sum is the function u(x, y) = <u{x, y), v(x, y) > inverse to /f in a neighborhood of 0.
[Remember that ^(77) = H^rj) — 77.]
9.13 Let A be a Banach algebra, and let x be an element of A of norm less than 1.
Show that

236 COMPACTNESS AND COMPLETENESS 4.10
This means that if Wn is the partial product YLi (1 + ^^^)j then Wn -^ (e — x)~^.
[Hint: Prove by induction that {e — x)7rn-i = e — x^^.]
This is another example of convergence at an exponential rate, like Newton's
method in the text.
10. THE INTEGRAL OF A PARAMETRIZED ARC
In this section we shall make our final application of completeness. We first
prove a very general extension theorem, and then apply it to the construction of
the Riemann integral as an extension of an elementary integral defined for step
functions.
Theorem 10.1. Let U he Si subspace of a normed linear space 7, and let T
be a bounded linear mapping from f/ to a Banach space W. Then T has a
uniquely determined extension to a bounded linear transformation aS from
the closure U to W, Moreover, ||aS|| = \\T\\.
Proof, Fix aGU and choose {f„] C f/ so that f„ —> a. Then {f„} is Cauchy and
{T{^n)] is Cauchy (by the lemmas of Section 7), so that {T{^n)}
converges to some 0 G W. If {77^} is any other sequence in U converging to a,
then fn - ^n -> 0, T{^n) - T{7in) = T{^n - Vn) "> 0, and so Tirjr,) -> /3 also.
Thus /? is hidependent of the sequence chosen, and, clearly, /? must be the value
S(a) at a of any continuous extension S oi T. If a G U, then /? = lim T{an) =
T{a) by the continuity of T. We thus have aS uniquely defined on U by the
requirement that it be a continuous extension of T.
It remains to be shown that aS is linear and bounded by || r||. For any a, 0gU
we choose {fn}, {Vn] C f/, so that fn —^ « and rjn —> /?. Then x^n + yVn —^
^f + 2/^j so that
S{xa + yp) = lim T{x^n + VVn) = x lim T{^n) + V lim ^(77,) = xS{a) + yS{^).
Thus aS is linear. Finally,
IIaSWII = lim \\T{^n)\\ < \\T\\ lim UnW = \\T\\ • ||aH,
Thus 11^11 is a bound for aS, and, since aS includes T, \\S\\ = \\T\\. D
The above theorem has many applications, but we shall use it only once, to
obtain the Riemann integral ja fit) dt of a continuous function / mapping a
closed interval [a, h] into a Banach space W as an extension of the trivial integral
for step functions. If TF is a normed linear space and /: [a, 6] —> TF is a
continuous function defined on a closed interval [a, 6] C [f^, we might expect to be
able to define Ja f{t) dt as a suitable vector in W and to proceed with the integral
calculus of vector-valued functions of one real variable. We haven't done this
until now because we need the completeness of W to prove that the integral
exists!
At first we Shall integrate only certain elementary functions called step
functions. A finite subset A of [a, h] which contains the two endpoints a and h

4.10 THE INTEGRAL OF A PARAMETRIZED ARC 237
will be called a partition of [a, h]. Thus A is (the range of) some finite sequence
{^i} Oj where a = to < ti < • • • < tn = h, and A subdivides [a, h] into a sequence
of smaller intervals. To be definite, we shall take the open intervals (^i_i, ti),
i == 1, . . . , n, as the intervals of the subdivision. If A and B are partitions and
A C B,we shall say that -B is a refinement of A, Then each interval (5y_i, 5y) of
the ^-subdivision is included in an interval (^i_i, ti) of the A-subdivision; ti^i
is the largest element of A which is less than or equal to 5y_i, and ti is the smallest
greater than or equal to Sj. A step function is simply a map /: [a, 6] —> TF which
is constant on the intervals of some subdivision A = {ti}^. That is, thej^e exists
a sequence of vectors {aj i such that /(J) = aj when f g (^i_i, ^i). The values
of / at the subdividing points may be among these values or they may be different.
For each step function / we define ja f{t) dt as X?=i oii Mi, where/ = ai on
(^i_i, ti) and Ati = ti — ti^i. If/ were real-valued, this would be simply the
sum of the areas of the rectangles making up the region between the graph of /
and the ^-axis. Now / may be described as a step function in terms of many
different subdivisions. For example, if/ is constant on the intervals of A, and
if we obtain B from A by adding one new point s, then / is constant on the
(smaller) intervals of B, We have to be sure that the value of the integral of /
doesn't change when we change the describing subdivision. In the case just
mentioned this is easy to see. The one new point 5 lies in some interval (^i_i, ti),
defined by the partition A. The contribution of this interval to the A-sum is
oii(ti — ti^i), while in the B-sum it splits into ai(ti — s) -\- ai(s — ^i_]). But
this is the same vector. The remaining summands are the same in the two sums,
and the integral is therefore unchanged. In general, suppose that / is a step
function with respect to A and also with respect to C, Set B = A \J C, the
''common refinement" of A and C. We can pass from A to -B in a sequence of
steps at each of which we add one new point. As we have seen, the integral
remains unchanged at each of these steps, and so it is the same for A as for B,
It is similarly the same for C and B, and so for A and C. We have thus shown
that ja f is independent of the subdivision used to define /.
Now fix [a, h] and W, and let 8 be the set of all step functions from [a, h]
to W. Then 8 is a vector space. For, if/and gf in 8 are step functions relative to
partitions A and B, then both functions are constant on the intervals of C =
A \J B, and therefore xf + yg is also. IMoreover, if C = {ti}^, and if on (^i_i, ti)
we have/ = ai and g = fii, so that xf + yg = xai -\- y^i there, then the equation
Y, {xai + y^i) Mi = X {T, ai Mi) + ?/ f Z ^i ^u)
is just ja (xf + yg) = X ja f+y ja 0- The map f ^-^ ja f is thus linear from 8 to
W. Finally,
/
J a
1
< E \WMh < \\f\Ub - a),

238 COMPACTNESS AND COMPLETENESS 4.10
where ||/|U = lub {||/(0|| :tG[a, h]} = max {\\ai\\ : 1 < ^ < n}. That is, if
we use on 8 the uniform norm defined from the norm of TF, then the linear
mapping /»—> /„/is bounded by {h — a). If TF is complete, this transformation
therefore has a unique bounded linear extension to the closure 8 of 8 in
(B([a, h], W) by Theorem 10.1. But we can show that 8 includes the space
e([a, h]y W) of all continuous functions from [a, h] to W, and the integral of a
continuous function is thus uniquely defined.
Lemma 10.1. e([a, h], W) C 8.
Proof, A continuous function / on [a, h] is uniformly continuous (Theorem 5.1).
That is, given e>^, there exists 5>^ such that |s — ^| < 5 =^ ||/(s) — /(0|| < e.
Now take any partition A = {^J q on [a, h] such that A^^ = ti — ti^i < 8 for all
^, and take ai as any value of/on (<i_i, ti). Then ||/(^) — (Xi\\ < e on [^i_i, ti].
Thus, if g is the step function with value ai on (<i_i, ti] and g(a) = ai, then
11/ — ^||oo ^ e. Thus/ is in 8, as desired. D
Our main theorem is a recapitulation.
Theorem 10.2. If TF is a Banach space and V = <B>{[a, h], W) under the
uniform norm, then there exists di, J E. Hom(F, W) uniquely determined by
setting J(/) = lim/o/„, where {/„} is any sequence in 8 converging to /
and Ja/„ is the integral on 8 defined above. Moreover, ||J|| < (6 — a).
If/is elementary from [a, h] to W and c G [a, 6], then of course/is elementary
on each of [a, c] and [c, 6]. If c is added to a subdivision A used in defining/, and
if the sum defining J^ / with respect to B = A KJ {c} is broken into two sums
at c, we clearly have JJ' / = ^a f -\- jc /• This same identity then follows for any
continuous function / on [a, h], since J^ / == lim j^ fn = lim {ja fn + jc fn) =
limj^U + lim//A = j^f+je'f-
The fundamental theorem of the calculus is still with us.
Theorem 10.3. If / G e([a, b], W) and F: [a, 6] -> IF is defined by F(x) =
ja f{t) dt, then F' exists on (a, b) and 7^'(x) = /(x).
Proof. By the continuity of / at Xq, for every e there exists a 8 such that
ll/(:ro)-/(x)|| <e
whenever |x — Xo| < S. But then
||r(/(a:o) -m)dt\\ < e|a;-Xo|,
and since j^^ /(^o) <^^ = /(^o) (^ "~ ^o) by the definition of the integral for an
elementary function, we see that
||/(^o) - (rf(t)dt/(x-Xo))\\ < e.
W \Jxo /It
Since jxof{t) dt = F{x) — F{xo), this is exactly the statement that the
difference quotient for F converges to /(xq), as was to be proved. D

4.10 THE INTEGRAL OF A PARAMETRIZED ARC 239
EXERCISES
10.1 Prove the following analogue of Theorem 10.1. Let A be a subset of a metric
space Bj let C be a complete metric space, and let F: A —> C be uniformly continuous.
Then F extends uniquely to a continuous map from A to C.
10.2 In Exercises 7.16 through 7.18 we have constructed a completion of S, namely,
6[S] in F*. Prove that this completion is unique to within isometry. That is, supposing
that (p is some other isometric imbedding of /S in a complete space X, show that the
identification of the two images oi S by (p o d~^ (from d[S] to (p[S]) extends to an
isometric bijection from 6[S] to (p[S]. [Hint: Apply the above exercise.]
10.3 Suppose that /S is a normed linear space X and that X is a dense subset of a
complete metric space Y. This means, remember, that every point of Y is the limit of a
sequence lying in the subset X. Prove that the vector space structure of X extends in a
unique way to make Y a Banach space. Since we know from Exercise 7.18 that a metric
space can be completed, this shows again that a normed linear space can always be
completed to a Banach space.
10.4 In the elementary calculus, if/is continuous, then
^'mdt==f(xXb-a)
/
Ja
for some x in (a, b). Show that this is not true for vector-valued continuous functions/
by considering the arc /: [0, tt] —> U^ defined by
f{t) = <sin t, cos t>.
10.5 Show that integration commutes with the application of linear transformations.
That is, show that if / is a continuous function from [a, b] to a Banach space W, and if
T G Hom(Tr, X), where X is a Banach space, then
j'T(m)dt^ T^j'mdt^.
[Hint: Make the computation directly for step functions.]
10.6 State and prove the theorem suggested by the following identity:
f' <m, g{t)> dt^ ^ I'm dt, 1^ g{t) dt y.
(Apply the above exercise.)
10.7 Let W be any normed linear space, {aJia finite set of vectors in W, and
{/t}i a corresponding set of real-valued continuous functions on [a, b]. Define the
arc 7 by
n
y{t) = Z/<(0«.-.
Prove that Xf y{t) dt exists and equals

240 COMPACTNESS AND COMPLETENESS 4.11
10.8 Let / be a continuous function from U^ to a Banach space W. Describe how
one might set up a theory of a double integral
//■
f(s, t) ds dt,
IXJ
where / X / is a closed rectangle.
10.9 Prove that if /„ converges uniformly to /, then
Cfn{t)dt~^ Cmdt.
J a J a
This is trivial if you have understood the definition and properties of the integral.
10.10 Suppose that {/„} is a sequence of smooth arcs from [a, h] to a Banach space W
such that Yl\ fnit) is uniformly convergent. Suppose also that J2T fnid) is convergent.
Prove that then ^ fnif) is uniformly convergent, that / = S^/n is smooth, and that
/' = Er/n. (Use the above exercise and the fundamental theorem of the calculus.)
10.11 Prove that even if W is not a Banach space, if the arc /: [a, h] —> W has a
continuous derivative, then X?/' exists and equals/(6) —f{o).
10.12 Let Z be a normed linear space, and set (Z, J) = Z( J) for ^E: X and I G Z*.
Now let / and g be continuously differentiable functions (arcs) from the closed interval
[a, h] to X and Z*, respectively. Prove the integration by parts formula:
(sQ>),Ah)) - {g{a),m) = C im, g'{t)) dt + C {f'{t), g{t)} dt.
J a J a
[Hint: Apply Theorem 8.4 from Chapter 3.]
10.13 State the generalization of the above integration by parts formula that holds
for any bounded bilinear mapping co: 7 X IF —> Z, where Z is a Banach space.
10.14 Let i»—» Z^ be a fixed continuous map from a closed interval [a, h] to the dual TF*
of a Banach space IF. Suppose that for any continuous map g from [a, h] to W
f g(f) dt = 0 => f lt{g{t)) dt = 0.
J a J a
Show that there exists a fixed L E W* such that
1^ lt{g{t))dt = l(J^ g{t)dt^
for all continuous arcs g: [a, b] —> M\ Show that it then follows that U = L for all t.
10.15 Use the above exercise to deduce the general Euler equation of Section 3.15.
11. THE COMPLEX NUMBER SYSTEM
The complex number system C is the third basic number field that must be
studied, after the rational numbers and the real numbers, and the reader surely
has had some contact with it in the past.
Almost everybody views a complex number J as being equivalent to a pair
of real numbers, the ''real and imaginary parts" of J, and the complex number
system C is thus viewed as being Cartesian 2-space K^ with some further struc-

4.11 THE COMPLEX NUMBER SYSTEM 241
ture. In particular, a complex-valued function is simply a certain kind of vector-
valued function, and is equivalent to an ordered pair of real-valued functions,
again its real and imaginary parts.
What distinguishes the complex number system C from its vector substratum
R^ is the presence of an additional operation, complex multiplication. The
vector operations of R^ together with this complex multiplication operation
make C into a commutative algebra. Moreover, it turns out that < 1, 0> is the
unique multiplicative identity in C and that every nonzero complex number ^
has a multiplicative inverse. These additional facts are summarized by saying
that C is a field, and they allow us to use C as a new scalar field in vector space
theory. In fact, the whole development of Chapters 1 and 2 remains valid when
R is replaced everywhere by C. Scalar multiplication is now multiplication by
complex numbers. Thus C^ is the vector space of ordered n-tuples of complex
numbers < ^i, . . . , ^n>^, and the product of an n-tuple by a complex scalar a
is defined by «< ^i, . . . , ^n>^ = <«^i, • • • , «^n>^, where a^i is complex
multiplication.
It is time to come to grips with complex multiplication. As the reader
probably knows, it is given by an odd looking formula that is motivated by thinking
of an element ^ = <Xi, X2>^ as being in the form Xi + ix2j where i^ = — 1,
and then using the ordinary laws of algebra. Then we have
^-n = (xi + ix2){yi + iy^)
= xiyi + ixiy2 + ^^22/i + ^'^^22/2 = (xiyi — X22/2) + ^(^i2/2 + ^22/1),
and thus our definition is
<xuX2> <yi,y2> = <xiyi — X22/2, ^12/2 + ^22/i>.
Of course, it has to be verified that this operation is commutative and satisfies
the laws for an algebra. A straightforward check is possible but dull, and we
shall indicate a neater way in the exercises.
The mapping x ^-^ <x, 0> is an isomorphic injection of the field U into the
field C. It clearly preserves sums, and the reader can check in his mind that it
also preserves products. It is conventional to identify x with its image < x, 0 >,
and so to view R as a subfield of C.
The mysterious i can be identified in C as the pair < 0, 1 >, since then
i^ = <0, 1X0, 1> = < — 1, 0>, which we have identified with —1. With
these identifications we have <x,y> = <x, 0> + <Ojy> = <XyO> +
<0, 1> <y,0> = X + iyj and this is the way we shall write complex numbers
from now on.
The mapping x + iy ^-^ x — iy is a field isomorphism of C with itself.
That is, it preserves both sums and products, as the reader can easily check.
Such a self-isomorphism is called an automorphism. The above automorphism is
called complex conjugation, and the image x — iy oi ^ = x -\- iy is called the
conjugate of f, and is designated f. We shall ask the reader to show in an exercise

242 COMPACTNESS AND COMPLETENESS 4.11
that conjugation is the only automorphism of C (except the identity
automorphism) which leaves the elements of the subfield R fixed.
The Euclidean norm oi ^ = x + iy = <x, y> is called the absolute value
of f, and is designated |f|, so that |f| = \x + zy\ = (x^ + y^^^. This is
reasonable because it then turns out that |fT| = |f| |7|. This can be verified by
squaring and multiplying, but it is much more elegant first to notice the
relationship between absolute value and the conjugation automorphism, namely,
[(X + iy)(x - iy) = x^- {iyY = x''+ y\ Then \^l\'= (n)(n) = (fD(T7) =
|f|^|7|^, and taking square roots gives us our identity. The identity ff = |f|^
also shows us that if f ^ 0, then f/|f|^ is its multiplicative inverse.
Because the real number system IR is a subfield of the complex number
system C, any vector space over C is automatically also a vector space over U:
multiplication b}^ complex scalars includes multiplication by real scalars. And
any complex linear transformation between complex vector spaces is
automatically real linear. The converse, of course, does not hold. For example, a
real linear mapping T from R^ to R^ is not in general complex linear from C to C,
nor does a real linear aS in Hom R^ become a complex linear mapping in Hom C^
when U^ is viewed as C^. We shall study this question in the exercises.
The complex differentiability of a mapping F between complex vector spaces
has the obvious definition AFa = T -\- e, where T is complex linear, and then
F is also real differentiable, in view of the above remarks. But F may be real
differentiable without being complex differentiable. It follows from the
discussion at the end of Section 8 that if {an} C C and {|a^| 5^} is bounded, then the
series J2 (^n^'' converges on the ball ^^(O) in the (real) Banach algebra C, and
F(0 = Lo «nf"" is real differentiable on this ball, with dF^{^) = (E^ na^/3^)r ==
F\p) • f. But multiplication by F\p) is obviously a complex linear operation
on the one-dimensional complex vector space C. Therefore, complex-valued
functions defined by convergent complex power series are automatically
complex differentiable. But we can go even further. In this case, if f ^ 0, we can
divide by f in the defining equation
AF^{0 = F'(p)^^ + o(0
to get the result that
^m _ F'iP) as f -> 0.
That is, F'(j5) is now an honest derivative again, with the complex infinitesimal f
in the denominator of the difference quotient.
The consequences of complex differentiability are incalculable, and we shall
mostly leave them as future pleasures to be experienced in a course on functions
of complex variables. See, however, the problems on the residue calculus at the
end of Chapter 12 and the proof in Chapter 11, Exercise 4.3, of the following
fundamental theorem of algebra.

4.11 THE COMPLEX NUMBER SYSTEM 243
Theorem. Every polynomial with complex coefficients is a product of
linear factors.
A weaker but equivalent statement is that every polynomial has at least one
(complex) root. The crux of the matter is that x^ + 1 cannot be factored over U
(i.e., it has no real root), but over C we have x^ + 1 = (x + i)(x -- z), with the
two roots ± i.
For later use we add a few more words about the complex exponential
function exp f = e^ = 2ZS f""/^'- li i = x + iy, we have e^ = e'"'^'^ = e'^e'^, and
e'' = Zo {iyr/n] =(1- 2/V2! + 2/V4! ) + i{y - yVSl + y'/dl ) =
cos y + i sin y. Thus c"^"^*^ = ^'^(cos y + i sin y). That is, the real and imaginary
parts of the complex-valued function exp (x + iy) are e"^ cos y and e^ sin ?/,
respectively.
EXERCISES
11.1 Prove the associativity of complex multiplication directly from its definition.
11.2 Prove the distributive law,
for complex numbers.
11.3 Show that scalar multiplication by a real number a, a<a:, 2/> = <ax, ay>,\n
C = R^ is consistent with the interpretation of a as the complex number <a, 0 > and
the definition of complex multiplication.
11.4 Let 6 be an automorphism of the complex number field leaving the real numbers
fixed. Prove that d is either the identity or complex conjugation. [Hint: {d(i))^ =
6(i^) ~ d{—l) = —1. Show that the only complex numbers a: + iy whose squares are
— 1 are dzi, and then finish up.]
11.5 If we remember that C is in particular the two-dimensional real vector space R^,
we see that multiplying the elements of C by the complex number a + ib must define
a linear transformation on R^. Show that its matrix is
[b a
11.6 The above exercise suggests that the complex number system may be like the
set .4 of all 2 X 2 real matrices of the form
[: i
Prove that ^ is a subalgebra of the matrix algebra R^^^ (that is, A is closed under
multiplication, addition, and scalar multiplication) and that the mapping
t -3
a + ife
is a bijection from A to C that preserves all algebra operations. We therefore can
conclude that the laws of an algebra automatically hold for C. Why?

244 COMPACTNESS AND COMPLETENESS 4.1 J
11.7 In the above matrix model of the complex number system show that the
absolute value identity |f7| = |f| |7| is a determinant property.
11.8 Let ir be a real vector space, and let V be the real vector space W X W.
Show that there is a ^ in Hom V such that 6^ = —I. (Think of C as being the real
vector space R^ = R X R under multiplication by i.)
11.9 Let F be a real vector space, and let ^ in Hom F satisfy ^^ = —/. Show that V
becomes a complex vector si)ace if ia is defined as 6(a). If the complex vector space T'
is made from the real vector space W as in this and the above exercise, we shall call
F the complexification of W. We shall regard IF itself as being a real subspace of V
(actually W X {0]), and then V = W ® iW.
11.10 Show that the complex vector space C" is the complexification of R". Show
more generally that for any set .1 the complex vector space C^ is the complexification
of the real vector space R^.
11.11 Let F be the complexification of the real vector space IF. Define the operation
of complex conjugation on F. That is, show that there is a real linear mapping <p such
that (p^ = I and (p{ia) = —i(p(a). Show, conversely, that if F is a comi)lex vector
space and v? is a conjugation on F [a real linear mapping (p such that (p^ = I and
(p(ia) = —i(p{a)], then F is (isomorphic to) the complexification of a real linear space
IF. (Apply Theorem 5.5 of Chapter 1 to the identity <p^ — I = 0.)
11.12 Let IF be a real vector space, and let F be its complexification. Show that
every T in Hom IF "extends" to a complex linear S in Hom F which commutes with the
conjugation (p. By S extending T we mean, of course, that S \ (IF X {0}) = T.
Show, conversely, that if S in Hom F commutes with conjugation, then S is the
extension of a T in Hom W.
11.13 In this situation we naturally call S the complexification of T. Show finally
that if S is the complexification of T, then its null space .Y in F is the direct sum
X = N ® iN, where A^ is the null space of T in W. Remember that we are viewing F
as IF e iW.
11.14 On a complex normed linear space F the norm is required to be complex
homogeneous:
llXall = |X| • ||a||
for all complex numbers X. Show that the natural definitions of || ||i, || ||2, and || ||oo
on C" have this property.
11.15 If a real normed linear space W is complexified to F = W 0 ilF, there is no
trivial formula which converts the real norm for W into a complex norm for F. Show
that, nevertheless, any product norm on F (which really is IF X IF) can be used to
generate an equivalent complex norm. [Hint: Given < ^, 77 > G F, consider the set ol"
numbers {\\{x + iy) < f, 77 > || : |x + iy\ = 1}, and try to obtain from this set a single
number that works.]
11.16 Show that every nonzero complex number has a logarithm. That is, show that
\iu -\- iv 9^ 0, then there exists an x + iy such that e*+'^ = t/ + iv. (Write the equation
e'^(cos 2/ + i sin 2/) = u-^ iv, and solve by being slightly clever.)
11.17 The fundamental theorem of algebra and Theorem 5.5 of Chapter 1 imply
that if F is a complex vector space and T in Hom F satisfies p{T) = 0 for a polynomial

4.12 WEAK METHODS 245
p, then there are subspaces [Vi] i of F, complex numbers [X^] i, and integers [jUi] ? such
that V = @i Vi, Vi is T-invariant for each, and (T — \iT)^i = 0 on Vi for each i.
Show that this is so. Show also that if V is finite-dimensional, then every T in Hom V
must satisfy some polynomial equation p(t) = 0. (Consider the linear independence or
dependence of the vector /, T, T^, . . . , T"^, . . . , in the vector space Hom F.)
11.18 Suppose that the polynomial p in the above exercise has real coefficients. Use
the fact that complex conjugation is an automorphism of C to prove that if X is a root
of p, then so is X.
Show that if F is the complexification of a real space W and T is the complexifica-
tion of i? G Hom W, then there exists a real polynomial p such that p{T) = 0.
11.19 Show that if ]F is a finite-dimensional real vector space and R G Hom W is an
isomorphism, then there exists an .1 G Hom W such that R = e^ (that is, log R exists).
This is a hard exercise, but it can be proved from Exercises 8.19 through 8.23, 11.12,
11.17, and 11.18.
*12. WEAK METHODS
Our theorem that all norms are equivalent on a finite-dimensional space suggests
that the limit theory of such spaces should be accessible independently of norms,
and our earlier theorem that every linear transformation with a
finite-dimensional domain is automatically bounded reinforces this impression. We shall look
into this question in this section. In a sense this effort is irrelevant, since we
can't do without norms completely, and since they are so handy that we use
them even when we don't have to.
Roughly speaking, what we are going to do is to study a vector-valued map F
by studying the whole collection of real-valued maps [I o F : I G V*].
Theorem 12.1. If V is finite-dimensional, then f ^ ~^ f in F (with respect to
any, and so every, norm) if and only if l(^n) ~^ KO ^^ ^ ^^^ esich I in F*.
Proof. If f^ -^ f and I G F*, then /(f^) -^ Z(f), since I is automatically
continuous. Conversely, if l(^n) ~^ KO ^^^ every I in F*, then, choosing a basis
{Pij'l for F, we have ei(f^) —^ ei(^) for each functional ei in the dual basis, and
this implies that fn ~^ f in the associated one-norm, since \\^n ~ ^K =
z'l liiiu - ^m -^ 0. D
Remark. If F is an arbitrary normed linear space, so that V* = Hom(F, U)
is the set of bounded linear functionals, then we say that fn ~^ f iveakly if
K^n) -^ KO f^r each I G V*. The above theorem can therefore be rephrased to
say that in a finite-dimensional spacej iveak convergejice and Jiorm convergence are
equivalent notions.
We shall now see that in a similar way the integration and differentiation of
parametrized arcs can all be thrown back to the standard calculus of real-valued
functions of a real variable by applying functionals from F* and using the
natural isomorphism of F** with F. Thus, if / G e([a, 6], F) and X G F*, then

246 COMPACTNESS AND COMPLETENESS 4.11'
X o / G e([a, 6], U), and so the integral ja ^° f exists from standard calculus. If we
vary X, we can check that the map X »-» J^^ X o / is linear, hence is in F**, and
therefore is given by a uniquely determined vector a G V (by duality; sec
Chapter 2, Theorem 3.2.). That is, there exists a unique a G V such thai
X(q:) = Ja X o / for every X G F*, and we clefme this a to be J^/. Thus integra
tion is defined so as to commute with the application of linear functionals
ja f is that vector such that
X ff^A = f^ X(/(0) dt for all X G F*
r*
Similarly, if all the real-valued functions {X of: x ^ F*} are differentiabic
at Xq, then the mapping X »-» (X o /)'(xo) is linear by the linearity of the derivativ(
in the standard calculus:
((^iXi + C2X2) ofY = (ci(Xi of) + C2(X2 o/))' = Ci(Xi ofY + C2(X2 o/)'.
Therefore, there is again a unique a G F such that
(\ofy(x^) = X(a) for all XgF*,
and if we define this a to be the derivative/'(^o), we have again defined an
operation of the calculus by commutativity with linear functionals:
(Xof'Xxo)= (X°/)'(Xo)-
Now the fundamental theorem of the calculus appears as follows.
If F(x) = j^f, then (X o F)(x) = j^ \ o f hy the weak definition of the
integral. The fundamental theorem of the standard calculus then says that
(X o 7^)' exists and (X o Fy(x) = (X o f)(x) = \(f(x)). By the weak definition ol
the derivative we then have that F' exists and F\x) = f{x).
The one conclusion that we don't get so easily by weak methods is the norm
inequality \\jaf\\ ^ (b — a)||/||oo. This requires a theorem about norms on
finite-dimensional spaces that we shall not prove in this course.
Theorem 12.2. ||a**|| = ||a|! for each a G F.
What is being asserted is that lub |a**(X)|/||X|| = ||a||. Since a**(X) = X(a),
and since |X(a)| < ||X|| • |la|| by the definition of ||X||, we see that
lub |a**(X)|/||X|| < M.
Our problem is therefore to find X g F* with ||X|| = 1 and |X(a)| = ||a||. If we
multiply through by a suitable constant (replacing a by ca, where c = 1/|!q:||),
we can suppose that ||a!| = 1. Then a is on the unit spherical surface, and the
problem is to find a functional X G F* such that the affine subspace (hyperplanc^
where X = 1 touches the unit sphere at a (so that X(a) = 1) and otherwise
lies outside the unit sphere (so that |X(f)| < 1 when ||f|| = 1, and hence
||X|| < 1). It is clear geometrically that such "tangent planes" must exist, but,
we shall drop the matter here.

4.12 WEAK METHODS 247
If we assume this theorem, then, since
1^ (('/)! = ir^(/(^))^^l ^ (b-a)m^x{\\(m)\:tG[a,b]}
I \Ja /\ \Ja I
< (6 - a)||X|| max {||/(0I|} (from |X(a)| < ||X|| • ||a||)
= (6 - a)||X|| • 11/11.,
we get
\\Ja II II \Ja / II I \Ja ' I
the extreme members of which form the desired inequality.

CHAPTER 5
SCALAR PRODUCT SPACKs
In this short chapter we shall look into what is gohig on behind two-norms, aiul
we shall find that a wholly new branch of linear analysis is opened up. TIkm
norms can be characterized abstractly as those arising from scalar product
They are the finite and infinite-dimensional analogues of ordhiary geometrn
length, and they carry with them practically all the concepts of Euclidean
geometry, such as the notion of the angle between two vectors, perpendicular!(\
(orthogonality) and the Pythagorean theorem, and the existence of man\
rigid motions.
The impact of this extra structure is particularly dramatic for infinite
dimensional spaces. Infinite orthogonal bases exist in great profusion and can
be handled about as easily as bases in finite-dimensional spaces, although tin
basis expansion of a vector is now a convergent infinite series, J ^ J^^i ^i^^<
IMany of the most important series expansions in mathematics are examples ol
such orthogonal basis expansions. For example, we shall see in the next chaptci
that the Fourier series expansion of a continuous function / on [0, tt] is the basi^
expansion of / under the two-norm 11/112 = (J-/2)i/2 for the particular ortho-
onal basis [an] "i = [sin nt] ^. If a vector space is complete under a scalar product
norm, it is called a Hilbert space. The more advanced theory of such spaces V
one of the most beautiful parts of mathematics.
1. SCALAR PRODUCTS
A scalar product on a real vector space T^ is a real-valued function from V X i
to [R, its value at the pair < J, r7> ordinarily being designated (J, 77), such that
^) (?5 v) ^^ linear in ^ when 77 is held fixed;
b) (?, v) = (v, 0 (symmetry);
c) (?, 0 > 0 if J p^ 0 (positive definiteness).
If (c) is replaced by the weaker condition
cO (?,?)> 0 for all ? G V,
then (J, 77) is called a semiscalar product.
Two important examples of scalar products are
n
(x, j) = H -UVi when V = W"
I
248

5.1 SCALAR PRODUCTS 249
and
(/, g) = Cm g(t) dt when V = e([a, 6]).
On a complex vector space (b) must be replaced by
bO {^, v) = (Vj 0 (Hermitian symmetry),
where the bar denotes complex conjugation. The corresponding examples are
(z, w) = X!i z{iFi when V = C and (/, g) = j^ fg when V is the space of
continuous complex-valued functions on [a, 6]. We shall study only the real case.
It follows from (a) and (b) that a semiscalar product is also linear in the
second variable when the first variable is held fixed, and therefore is a symmetric
i)ilinear functional whose associated quadratic form g(f) = (^, 0 is positive
definite or positive semidefinite [(c) or (c'); see the last section in Chapter 2].
The definiteness of the form q has far-reaching consequences, as we shall begin
to see at once.
Theorem 1.1. The Schwarz inequality
\(^,v)\<{^,0"Hr„r,y"
is valid for any semiscalar product.
Proof. We have 0 < (f - ^77, f - ^77) = (f, f) - 2tU, 77) + t^(rj, rj) for every
t G U. Since this quadratic in t is never negative, it cannot have distinct roots,
and the usual (6^ — 4ac)-formula implies that 4(f, 77)^ — 4(f, ^)(77, 77) < 0,
which is equivalent to the Schwarz inequality. D
We can also proceed directly. If (77, 77) > 0, and if we set t = (^, 'n)/('n) v)
in the quadratic inequality in the first line of the proof, then the resulting
expression simplifies to the Schwarz inequality. If (77, 77) = 0, then (f, 77) must
also be 0 (or else the beginning inequality is clearly false for some t), and now
the Schwarz inequality holds trivially.
Corollary. If (f, 77) is a scalar product, then ||f|| = (f, ^)^'^ is a norm.
Proof
= ll^f + 2(^,7;)+ ||7;f<||^f +211^11 11,11+ ||,f (by Schwarz)
= (IUII + NII)^
proving the triangle inequality. Also, ||cf|| = (c^,c^)^'^= (c^(f, f))^^^ =
\c\Ul n
Note that the Schwarz inequality |(f, 77)! < ||f|| II77II is now just the
statement that the bilinear functional (f, 77) is bounded by one with respect to the
scalar product norm.
A normed linear space V in which the norm is a scalar product norm is
called a pre-Hilbert space. If V is complete in this norm, it is a Hilbert space.

250 SCALAR PRODUCT SPACES 5.1
The two examples of scalar products mentioned earlier give us the real
explanation of our two-norms for the first time:
/ n \ 1/2
= (e4)
/2
l|x||2 = ( U ^n for X
and
h „xl/2
- if.'')
for /Geaa,6])
are scalar product norms.
Since the scalar product norm on R"" becomes Euclidean length under a
Cartesian coordinate correspondence with Euclidean n-space, it is conventional
to call W^ itself Euclidean n-space E^ when we want it understood that th('
scalar product norm is being used.
Any finite-dimensional space F is a Hilbert space with respect to any scalar
product norm, because its finite dimensionality guarantees its completeness.
On the other hand, we shall see in Exercise 1.10 that C([a, 6]) is incomplete in th('
two-norm, and is therefore a pre-Hilbert space but not a Hilbert space in this
norm. (Remember, however, that C([a, 6]) is complete in the uniform norm
ll/lloo.) It is important to the real uses of Hilbert spaces in mathematics that any
pre-Hilbert space can be completed to a Hilbert space, but the theory of infinites-
dimensional Hilbert spaces is for the most part beyond the scope of this book.
Scalar product norms have in some sense the smoothest possible unil
spheres, because these spheres are quadratic surfaces.
It is orthogonality that gives the theory of pre-Hilbert spaces its special
flavor. Two vectors a and p are said to be orthogonal^ written a J_ /?, if (a, p) = 0.
This definition gets its inspiration from geometry; we noted in Chapter 1 that
two geometric vectors are perpendicular if and only if their coordinate triples x
and y satisfy (x, y) = 0. It is an interesting problem to go further and to show
from the law of cosines (c^ = a^ + 6^ — 2ah cos 6) that the angle 6 between two
geometric vectors is given by (x, y) = ||x|| ||y|| cos 6. This would motivate us
to define the angle 6 between two vectors f and 77 in a pre-Hilbert space by
(f; ^) = llfll Nil cos 6j but we shall have no use for this more general
formulation.
We say that two subsets A and B are orthogonal, and we write A ^- B,\{
a J_ /? for every am A and p in 5; for any subset A we set A^ = {p ElV: p ^- A].
Lemma I.I. If/? is orthogonal to the set A, then p is orthogonal to L{A), t\\v
closure of the linear span oi A. It follows that B^ is a closed subspace for
every subset B.
Proof, The first assertion depends on the linearity and continuity of the scalar
product in one of its variables; it will be left to the reader. As for A = B^j il
includes the closure of its own linear span, by the first part, and so is a closed
subspace. D

5.1 SCALAR PRODUCTS 251
Lemma 1.2. In any pre-Hilbert space we have the parallelogram law,
||a + 0f+||a-/3f =2(||af+||0f),
and the Pythagorean theorem,
a _L /3 if and only if ||a + /3f - \\a\\^ + \\0\\^,
If {aj 1 is a (pairwise) orthogonal collection of vectors, then
1
1
Proof. Since ||a + /3f= ||a||2 + 2(a,/3) + ||/3||2, by the bilinearity of the
scalar product ||a + |(3||^, we see that ||a + /^H^ = ||a||2 + ||i(3||2 if and only if
(a, /?) = 0, which is the Pythagorean theorem. Writing down the similar
expansion of ||a — j(3||^ and adding, we have the parallelogram law. The last
statement follows from the Pythagorean theorem and Lemma 1.1 by induction.
Or we can obtain this statement directly by expanding the scalar product on the
left and noticing that all "mixed terms" drop out by orthogonality. D
The reader will notice that the Schwarz inequality has not been used in this
lemma, but it would have been silly to state the lemma before proving that
U\\ = (f, O'^'isanorm.
If {aji are orthogonal and nonzero, then the identity ||Z!i Xi«i||^ =
YJi xfWaiW^ shows that Xi ^i«i can be zero only if all the coefficients Xi are zero.
Thus,
Corollary. A finite collection of (pairwise) orthogonal nonzero vectors is
independent. Similarly, a finite collection of orthogonal subspaces is
independent.
EXERCISES
1.1 Complete the second proof of Theorem 1.1.
1.2 Reexamine the proof of Theorem 1.1 and show that if f and 77 are independent,
then the Schwarz inequality is strict.
1.3 Continuing the above exercise, now show that the triangle inequality is strict
it" f and 77 are independent.
1.4 a) Show that the sum of two semiscalar products is a semiscalar product.
b) Show that if (ju, v) is a semiscalar product on a vector space W and if T is a
linear transformation from a vector space V to W, then [f, 77] = (Tju, Tv) is a
semiscalar product on V.
c) Deduce from (a) and (b) that
U,g) =mg{a)+ C f{t)g\t)dt
J a
is a semiscalar product on F = C^([a, 5]). Prove that it is a scalar product.

252 SCALAR PRODUCT SPACES T) .'
1.5 If a is held fixed, we know that/(f) = (f, a) is continuous. Why? Prove mom
generally that (f, 77) is continuous as a map from F X F to R.
1.6 Let V be a two-dimensional Hilbert space, and let {ai, 0:2] be an}' basis for 1
Show that a scalar product (f, rj) has the form
(f, 77) = axiiji + h(xiy2 + X2Z/i) + cx2y2,
where h^ < ac. Here, of course, f = xiai + 0:20:2, 77 = ijiai + ?/2«2.
1.7 Prove that if a;(x, y) = axijji + b(xiy2 + 0:22/1) + cx2y2 and 6^ < ac, then «
is a scalar product on R^.
1.8 Let a)(^, rj) be any symmetric bilinear functional on a finite-dimensional vectoi
space T^ and let q{^) = a;(f, f) be its associated quadratic form. Show that for aii\
choice of a basis for T^ the equation q{^) = 1 becomes a quadratic equation in 11 k
coordinates [xi] of J.
1.9 Prove in detail that if a vector /? is orthogonal to a set ^ in a pre-Hilbert space
then /? is orthogonal to L{A).
1.10 We know from the last chapter that the Riemann integral is defined for the s(>l
8 of uniform limits of real-valued step functions on [0, 1] and that 8 includes all tlu
continuous functions. Given that k is the step function whose value is 1 on [0, ^] and
0 on [^, 1], show that ||/ — k\\2 > 0 for any continuous function/. Show, howevci,
that there is a sequence of continuous functions {fn] such that \\fn — k\\2 —> 0. Show,
therefore, that ©([0, 1]) is incomplete in the two-norm, b}' showing that the abo\ c
sequence [fn] is Cauchy but not convergent in C([0, 1]).
2. ORTHOGONAL PROJECTION
One of the most important devices in geometric reasoning is "dropping a
perpendicular" from a point to a line or a plane and then using right triangle
arguments. This device is equally important in pre-Hilbert space theory. If M
is a subspace and a is any element in V, then by "the foot of the perpendicular
dropped from a to M" we mean that vector fx in M such that (a — ju) J_ 71/,
if such a ju exists. (See Fig. 5.L) Writing a as ju + (a — ju), we see that the
existence of the "foot" fx in M for each a in V is
equivalent to the direct sum decomposition V = M ® M^.
Now it is precisely this direct sum decomposition
that the completeness of a Hilbert space guarantees,
as we shall shortly see. We start by proving the
geometrically intuitive fact that ju is the foot of the
perpendicular dropped from a to M if and only if ju
is the point in M closest to a. ^^*
Lemma 2.1. If ju is in the subspace M, then (a — ju) J_ M if and only if ix is
the unique point in M closest to a, that is, ju is the "best approximation" to
a in M.
Proof. If (a — ju) J- M and f is any other point in M, then ||a — f||^ =^
||(a - m) + (m - f)||' == \W - mII' + IIm - ff > \W - Mf. Thus M is the

5.2 ORTHOGONAL PROJECTION 253
unique point in M closest to a. Conversely, suppose that ju is a point in M closest
to a, and let f be any nonzero vector in M, Then ||a — mII^ ^ II (« ~" m) + t^Vj
which becomes 0 < 2t{a — Mj f) + ^^11 ?||^ when the right-hand scalar product is
expanded. This can hold for all t only if (a — ju, f) = 0 (otherwise let t = ?).
Therefore, {a — fx) ± M, U
On the basis of this lemma it is clear that a way to look for ju is to take a
sequence jUn in M such that ||a — Mnll —^ p(«j M) and to hope to define jjl as its
limit. Here is the crux of the matter: We can prove that such a sequence {fin} is
always Cauchy, but its limit may not exist if M is not complete!
Lemma 2.2. If {fjLn} is a sequence in the subspace M whose distance from
some vector a converges to the distance p from a to M, then (ju^} is Cauchy.
Proof, By the parallelogram law, \\fjLn — Mm||^ = ||(a — Mn) — (a — fJLm)\\^ =
2(\\a — MnT + \W — MmT) — ||2a: — {iXn + Mm)r. Since the first term on the
right converges to 4p^ as n, m—> oo, and since the second term is always
< —4p^ (factor out the 2), we see that ||/Xn — MmP —> 0 as n,m —> oo. D
Theorem 2.1. If M is a complete subspace of a pre-Hilbert space F, then
V = M ® M\ In particular, this is true for any finite-dimensional sub-
space of a pre-Hilbert space and for any closed subspace of a Hilbert space.
Proof. This follows at once from the last two lemmas, since now ju = lim ju^
exists, ||a — ju|| = p(a, M), and so (a — ju) ± M. D
If F = M 0 M-^, then the projection on M along M^ is called the orthogonal
projection on M, or simply the projection on M, since among all the projections on
M associated with the various complements of M, the orthogonal projection is
distinguished. Thus, if M is a complete subspace of F, and if P is the projection
on Mj then P(f) is at once the foot of the perpendicular dropped from ^ to M
(which is where the word "projection" comes from) and also the best
approximation to f in M (Lemma 2.1).
Lemma 2.3. If {MJ i is a finite collection of complete, pairwise orthogonal
subspaces, and if for a vector a in F, a^ is the projection of a on Mi for
i = 1, . . . , n, then 2Z i a^ is the projection of a on 0i Mi.
Proof. We have to show that a — 2Zi ai is orthogonal to 0i My, and it is
sufficient to show it orthogonal to each Mj separately. But if f G Mj, then
(a — 2Zi aij f) = (a — ajj f), since (a^, f) = 0 for i 9^ j, and (a — ay, f) = 0
because ay is the projection of a on Mj. Thus (a — 2Zi a^, f) = 0. D
Lemma 2.4. The projection of f on the one-dimensional span of a single
nonzero vector 77 is ((f, ^)/||^||^)^.
Proof. Here ju must be of the form x-q. But (f — X77) ± 77 if and only if
0 = ($ - Xt;, 7;) = (f, 7;) - x||7;f, or X= (f, 7;)/||7;f. D

254 SCALAR PRODUCT SPACES 5.2
We call the number (f, t;)/!!?;!!^ the -q-Fourier coefficient of f. If 77 is a unit
(normalized) vector, then this Fourier coefficient is just (f, 77). It follows from
Lemma 2.3 that if {ipi\\ is an orthogonal collection of nonzero vectors, and if
{x^]\ are the corresponding Fourier coefficients of a vector f, then YJ\ Xi<Pi is th('
projection of f on the subspace M spanned by {(pi} j. Therefore, f — J^l ^i<Pi J- ^^,
and (Lemma 2.1) J^^ xnpi is the best approximation to f in M. If f is in M, then
both of these statements say that f = YJi xnpi, (This can of course be verified
directly, by letting f = YJ\ (^i^i be the b^sis expansion of f and computinjz;
(fj ^j) = Zi «i(<y^i, <y^y) = (^jW^PjV')
If an orthogonal set of vectors {ipi} is also normalized (||<^i|| = 1), then we
call the set orthonormal.
Theorem 2.2. If {(pi] "^ is an infinite orthonormal sequence, and if {xi} "^ ar(»
the corresponding Fourier coefficients of a vector f, then
00
^ x^ < \\^\f (Bessel's inequality),
1
and f = J^"^ xnpi if and only if Y.^! xf = \\ f p (Parseval's equation).
Proof. Setting an = 2Zi Xi(pi and f = (f — o*^) +0*^, and remembering that
^ — (Tn -L (Tn, we have
1
Therefore, 2Zi a;f < || f ||^ for all n, proving Bessel's inequality, ando-n —> f (that
is, IIf — cr^ll —> 0) if and only if J^l xf —> ||$||^, proving Parseval's identity. D
We call the formal series 2Z Xi(pi the Fourier series of f (with respect to the
orthonormal set {<Pi}). The Parseval condition says that the Fourier series of J
converges to f if and only if || f ||^ = JZT ^f-
An infinite orthonormal sequence {<pi} "^ is called a 6asis for a pre-Hilbert
space F if every element in V is the sum of its Fourier series.
Theorem 2.3. An infinite orthonormal sequence {(pi] "J^ is a basis for a pre-
Hilbert space V if (and only if) its linear span is dense in V.
Proof. Let f be any element of F, and let {xi} be its sequence of Fourier
coefficients. Since the linear span of {(pi} is dense in F, given any e, there is a
finite linear combination 2Z^ yi(pi which approximates f to within e. But
2ZT ^i^i is the best approximation to $ in the span of {<Pi}^, by Lemmas 2.3 and
2.1, and so
^-z
Xi(Pi
1
< e for any m > n.
That is, f = ZT xm. U
Corollary. If F is a Hilbert space, then the orthonormal sequence {(pij't
is a basis if and only if {ipi} -^ = {0}.

5.2 ORTHOGONAL PROJECTION 255
Proof, Let M be the closure of the linear span of {<^i}^. Since V = M -\- M-^,
and since M-^ = {<pi}^, by Lemma 1.1, we see that {<pi}^ = {0} if and only if
V = M, and, by the theorem, this holds if and only if {(pi} is a basis. D
Note that when orthogonal bases only are being used, the coefficient of a
vector J at a basis element /? is always the Fourier coefficient (J, j(3)/|||(3||^.
Thus the /^-coefficient of J depends only on /? and is independent of the choice
of the rest of the basis. However, we know from Chapter 2 that when an
arbitrary basis containing /? is being used, then the /^-coefficient of J varies with the
basis. This partly explains the favored position of orthogonal bases.
We often obtain an orthonormal sequence by "orthogonalizing" some given
sequence.
Lemma 2.5. If {aj is a finite or infinite sequence of independent vectors,
then there is an orthonormal sequence {(pi] such that {ai}^ and {<^i}i,have
the same linear span for all n.
Proof, Since normalizing is trivial, we shall only orthogonalize. Suppose, to be
definit>e, that the sequence is infinite, and let Mn be the linear span of
{«!, . . . , an}. Let fjLn be the orthogonal projection of an on Mn-i, and set
<Pn = oin — fJin (and <pi = ai). This is our sequence. We have (pi G Mi C Mn-i
if i < n, and (pn -L Mn-i, so that the vectors (pi are mutually orthogonal.
Also, ipn 9^ 0, since an is not in Mn-i- Thus {<pi}\ is an independent subset of
the n-dimensional vector space Mn, by the corollary of Lemma 1.2, and so
{iPi\\ spans Mn. □
The actual calculation of the orthogonalized sequence {<pn} can be carried
out recursively, starting with <^i = «i, by noticing that since ju^ is the projection
of an on the span of <y^i, . . . , <Pn—i, it must be the vector ^\~^ cupi^ where Ci is
the Fourier coefficient of an with respect to <pi.
Consider, for example, the sequence {x^}o in e([0, 1]). We have <p\ =
«! = 1. Next, <^2 = «2 — M2 = ^ ~" c • 1, where
C=(«2,^l)/|klf = /o^-l//o (l)' = i
Then <pz = ol^ — {c2<P2 + Ci<P\), where
Ci = /o^^-l/Jo(l)^=*
and
C2 = Jo X\X - i)//; (x - 1)=^ = (i - i)/(^) = 1.
Thus the first three terms in the orthogonalization of {x^}o in e([0, 1]) are 1,
X — ^, x'^ — {x — ^) — ^ = x'^ — X -\- ^, This process is completely
elementary, but the calculations obviously become burdensome after only a few terms.
We remember from general bilinear theory that if for ^ in F we define
(9^: F -> R by d^{^) = (J, /3), then 6^ G F* and (9: ^S j-^ (9^ is a finear mapping
from V to F*. If (f, rj) is a scalar product, then 6^(13) = \\l3\\^ > 0 if /3 ?^ 0, and
so 6 is injective. Actually, 6 is an isometry, as we shall ask the reader to show in

256 SCALAR PRODUCT SPACES 5.2
an exercise. If V is finite-dimensional, the injectivity of 6 implies that 6 is an
isomorphism. But we have a much more startling result:
Theorem 2.4. 6 is an isomorphism if and only if F is a Hilbert space.
Proof. Suppose first that F is a Hilbert space. We have to show that 6 is sur-
jective, i.e., that every nonzero F in F* is of the form 6^. Given such an F, let A^
be its null space, let a be a vector orthogonal to N (Theorem 2.1), and consider
/? = ca, where c is to be determined later. Every vector f in F is uniquely a sum
f = Xj(3 + 77, where rj is in N. [This only says that V/N is one-dimensional, which
presumably we know, but we can check it directly by applying F and seeing that
F(^ — xl3) = 0 if and only if x = F(^)/F(l3).] But now the equations
F(^) = F(x^ + v) = xF{0) = xcF(a)
and
e^ii) = ($, /3) = (x0 + 77, /3) = xMl^ = xc^aW^
show that dp = F if we take c = 7^(a)/||a||^
Conversely, if 6 is surjective (and assuming that it is an isometry), then it is
an isomorphism in Hom(F, F*), and since F* is complete by Theorem 7.6,
Chapter 4, it follows that F is complete by Theorem 7.3 of the same chapter.
We are finished. D
EXERCISES
2.1 In the proof of Lemma 2.1, if (a — ju, f) 9^ 0, what value of t will contradict
the inequality 0 < 2t(a — ju, 9 + t^^\\^?
2.2 Prove the "only if" part of Theorem 2.3.
2.3 Let {Mi} be an orthogonal sequence of complete subspaces of a pre-Hilbert
space V, and let Pi be the (orthogonal) projection on Mi. Prove that {Pi^] is Cauchy
for any J in V.
2.4 Show that the functions {sin nt] n=i form an orthogonal collection of elements in
the pre-Hilbert space ©([0, tt]) with respect to the standard scalar product (/, g) =
So fit) git) dt. Show also that ||sin nt\\2 = VV^.
2.5 Compute the Fourier coefficients of the function f{t) = t in C([0, tt]) with
respect to the above orthogonal set. What then is the best two-norm approximation
to t in the two-dimensional space spanned by sin t and sin 2^? Sketch the graph of this
approximating function, indicating its salient features in the usual manner of calculus
curve sketching.
2.6 The "step" function / defined by f(t) = 7r/2 on [0,7r/2] and f{t) = 0 on (7r/2, tt]
is of course discontinuous at 7r/2. Nevertheless, calculate its Fourier coefficients with
respect to {sinn^]n=i in ©([0, tt]) and graph its best approximation in the span of
{sin nt] 1.
2.7 Show that the functions {sin nt] n=i U {cos nt] n=o form an orthogonal collection
of elements in the pre-Hilbert space ©([—tt, tt]) with respect to the standard scalar
product (/, fir) = Sl^f{t)g{t)dt.

5.3 SELF-ADJOINT TRANSFORMATIONS 257
2.8 Calculate the first three terms in the orthogonalization of {a:"}o in ©([—1, 1]).
2.9 Use the definition of the norm of a bounded linear transformation and the
Schwarz inequality to show that ||^^|| < \\^\\ [where 6p(^) = (J,/3)]. In order to
conclude that /?»—> ^^s is an isometry, we also need the opposite inequality, ||^^|| > \\I3\\,
Prove this by using a special value of ?.
2.10 Show that if V is an incomplete pre-Hilbert space, then V has a proper closed
subspace M such that M-^ = {0}. [Hint: There must exist F G V* not of the form
^(?) = (?7 «)•] Together with Theorem 2.1, this shows that a pre-Hilbert space F is a
Hilbert space if and only if F = M 0 M^ for every closed subspace M.
2.11 The isometry ^:af—> 6a [where daiO = (^, «)] imbeds the pre-Hilbert space V
in its conjugate space F*. We know that F* is complete. Why? The closure of F as
a subspace of F* is therefore complete, and we can hence complete F as a Banach
space. Let H be its completion. It is a Banach space including (the isometric image of)
F as a dense subspace. Show that the scalar product on F extends uniquely to H and
that the norm on H is the extended scalar product norm, so that i/ is a Hilbert space.
2.12 Show that under the isometric imbedding a»—> ^^ of a pre-Hilbert space F into
F* orthogonality is equivalent to annihilation as discussed in Section 2.3. Discuss the
connection between the properties of the annihilator A^ and Lemma LI of this chapter.
2.13 Prove that if C is a nonempty complete convex subset of a pre-Hilbert space F,
and if a is any vector not in C, then there is a unique n G C closest to a. (Examine the
proof of Lemma 2.2.)
3. SELF-ADJOINT TRANSFORMATIONS
Definition. If F is a pre-Hilbert space, then T in Horn V is self-adjoint if
(Ta, 13) = (a, Tl3) for every a, /? G F. The set of all self-adjoint
transformations will be designated SA.
Self-adjointness suggests that T ought to become its own adjoint under the
injection ^ of F into F*. We check this now. Since (a, /?) = 6^(0), we can rewrite
the equation {Ta, /?) = (a, T0) as d^{Ta) = Bt^^ol), and again as (r*((9^))(a) =
O'l^a^ by the definition of T*. This holds for all a and /3 if and only if r*((9^) =
6t^ for all /? G F, or r* o ^ = ^ o T'j which is the asserted identification.
Lemma 3.1. If F is a finite-dimensional Hilbert space and {<pi} \ is an
orthonormal basis for F, then T E. Hom(F) is self-adjoint if and only if the
matrix {tij] of T with respect to {<pi) is symmetric (t = t*).
Proof. If we substitute the basis expansions of a and /? and expand, we see that
(a, r/5) = {Ta, 13) for all a and /3 if and only if (<^^, T<pj) = {T<pi, <pj) for all i
and j. But T<pi = 2Zfc=i tkWk, and when this is substituted in these last scalar
products, the equation becomes Uj = tji. That is, T is self-adjoint if and only if
t = t*. D
A self-adjoint T is said to be nonnegative if (T'f, f) > 0 for all f. Then
l^j v] = (T^j rj) is Si semiscalar product!

258 SCALAR PRODUCT SPACES 5/^
Lemma 3.2. If T is a nonnegative self-adjoint transformation, then
\\T{0\\ < \\T\\"\T^, ^Y'^ for all f. Therefore, if (Tf, f) = 0, then
T^ = 0, and, more generally, if (Tf^, f^) -> 0, then T{^n) -> 0.
Proof. If T is nonnegative as well as self-adjoint, then [f, ??] = (Tf, rj) is a
semiscalar product, and so, by Schwarz's inequality.
Taking rj = T^, the factor on the right becomes {T{T^), T^Y'^, which is less
than or equal to ||r|| ^^^HTf ||, by Schwarz and the definition of ||!r||. Dividing by
II^^11J we get the inequality of the lemma. D
If a p^ 0 and T{a) = ca for some c, then a is called an eigenvector (proper
vector, characteristic vector) of 7", and c is the associated eigenvalue (proper
value, characteristic value).
Theorem 3.1. If F is a finite-dimensional Hilbert space and T is a self-
adjoint element of Hom F, then V has an orthonormal basis consisting
entirely of eigenvectors of T.
Proof. Consider the function (T^, $). It is a continuous real-valued function
of ^, and on the unit sphere S = {^ : \\^\\ = 1} it is bounded above by ||T||
(by Schwarz). Set m = lub {(T^, ^) : ||^|| = 1}. Since S is compact (being
bounded and closed), (T^, ^) assumes the value m at some point a on S. Now
m — 7" is a nonnegative self-adjoint transformation (Check this!), and {Ta, a) =
m is equivalent to ((m — T)a, a) = 0. Therefore, (m — T)a = 0 by Lemma
3.2, and Ta = ma. We have thus found one eigenvector for T. Now set Fi = F,
«! = a, and mi — m, and let V2 be {ai}^. Then T[V2] C F2, for if ^ J_ ai,
then {T^, a,) = ($, Ta,) = m($, a,) = 0.
We can therefore repeat the above argument for the restriction of T to the
Hilbert space V2 and find 0:2 in F2 such that ||a2|| = 1 and ^(0:2) = ^220:2?
where mo = lub {{T^, ^) : ||^|| = 1 and ^ G V2] • Clearly, m2 < m,. We then
set F3 = {«i, 0:2}"^ and continue, arriving finally at an orthonormal basis
{ai} 1 of eigenvectors of T. D
Now let Xi, . . . , Xr be the distinct values in the list mi, ... , m^, and let il/y
be the linear span of those basis vectors ai for which m^- = \j. Then the sub-
spaces Mj are orthogonal to each other, F = 0\ il/y, each Mj is T-invariant,
and the restriction of T to Mj is X^ times the identity. Since all the nonzero
vectors in Mj are eigenvectors with eigenvalue X^, if the aiS spanning Mj are
replaced by any other orthonormal basis for Mj, then we still have an ortho-
normal basis of eigenvectors. The a/s are therefore not in general uniquely
determined. But the suhspaces Mj and the eigenvalues Xy are unique. This will
follow if we show that every eigenvector is in an AIj.
Lemma 3.3. In the context of the above discussion, if ^ 7^ 0 and T(^) = x^
for some a; in R, then ^ G Mj (and so x = \j) for some j.

5.3 SELF-ADJOINT TRANSFORMATIONS 259
Proof. Since V = @\ Mj, we have J = Y^'i ^i with ^i G Mi. Then
r r r r
J2 x^i = x^= T{^) = Z T{^i) = Z \i^, and T. (x - X,)f, = 0.
1 11 1
Since the subspaces Mi are independent, every component {x — \i) ^i is 0. But
some ^j 9^ 0, since ^ 9^ 0. Therefore, x = Xy, ^i = 0 for i 9^ j, and
^ = $y G My. D
We have thus proved the following theorem.
Theorem 3.2. If F is a finite-dimensional Hilbert space and T is a self-
adjoint element of Hom V, then there are uniquely determined subspaces
{Vi}\ of Vj and distinct scalars {\i}\, such that {Vi} is an orthogonal
family whose sum is V and the restriction of T to Vi is \i times the identity.
If F is a finite-dimensional vector space and we are given T G Hom V, then
we know how to compute related mappings such as T^ and T~^ (if it exists)
and vectors Ta, T~^a, etc., by choosing a basis for V and then computing matrix
products, inverses (when they exist), and so on. Some of these computations,
particularly those related to inverses, can be quite arduous. One enormous
advantage of a basis consisting of eigenvectors for T is that it trivializes all of
these calculations.
To see this, let {l3n} be a basis of V consisting entirely of eigenvectors for T,
and let {r„} be the corresponding eigenvalues. To compute T^, we write down
the basis expansion for f, f = Z!i Xil3i, and then T^ = J^l ^i^ifii- T^ has the
same eigenvectors, but with eigenvalues {rf}. Thus T^a = 2Zi rfxi^i. T~^
exists if and only if no Vi = 0, in which case it has the same eigenvectors with
eigenvalues {1/r,}. Thus T''^^ = Li (Vn)/3i- If ^^(0 = Zo (^nt"" is any
polynomial, then P{T) takes 0i into P(rO/3i. Thus P{T)^= Zo ^O^O^A-
By now the point should be amply clear.
The additional value of orthonormality in a basis is already clear fom the
last section. Basically, it enables us to compute the coefficients {xi} of f by
scalar products: Xi = (f, I3i).
This is a good place to say a few words about the general eigenvalue problem
in finite-dimensional theory. Our complete analysis above was made possible by
the self-adjointness of T (or the symmetry of the matrix t). What we can say
about an arbitrary T in Hom V is much less satisfactory.
We first note that the eigenvalues of T can be determined algebraically, for X
is an eigenvalue if and only if T — \I is not infective, or, equivalently, is
singular, and we know that T — X/issingularifandonly if its determinant A(r — X7)
is 0. If we choose any basis for F, the determinant of T — \I is the determinant
of its matrix t — Xe, and our later formula in Chapter 7 shows that this is a
polynomial of degree n in X. It is easy to see that this polynomial is independent
of the basis; it is called the characteristic polynomial of T. Thus the eigenvalues
of T are exactly the roots of the characteristic polynomial of T.

260 SCALAR PRODUCT SPACES 5M
However, T need not have any eigenvectors! Consider, for example, a 90°
rotation in the Cartesian plane. This is the map T: <x,y> »—> <—y^x>.
Thus T{b^) = b^ and T{b^) = -b\ so the matrix of T is
P -']
1 0
Then the matrix of T — X is
X -1
1 X
and the characteristic polynomial of T is the determinant of this matrix: X^ + 1.
Since this polynomial is irreducible over U, there are no eigenvalues.
Note how different the outcome is if we consider the transformation with the
same matrix on complex 2-space C^. Here the scalar field is the complex number
system, and T is the map <Zi, Z2> >—> < —Z2, ^i^ from C^ to C^. But now
X^ + 1 = (X + ^)(X — ^), and T has eigenvalues zt^! To find the eigenvectors
for ^, we solve T{z) = ^z, which is the equation < —Z2, Zi> = <izi, iz2>, or
Z2 — —izi. Thus <1, —i> (or ^<l, —i> = <^, 1>) is the unique
eigenvector for i to within a scalar multiple.
We return to our real theory, li T E. Horn V and n = d{V)j so that
c/(Hom V) = n^, then the set of n^ + 1 vectors {T^}f in Hom V is dependent.
But this is exactly the same as saying that p(T) = 0 ior some polynomial p of
degree <?i^. That is, any T in Hom V satisfies a polynomial identity p{T) = 0.
Now suppose that r is an eigenvalue of T and that T{^) = r^. Then p{T){^) ^
p{r)^ ^ 0, and so p{r) = 0. That is, every eigenvalue of 7" is a root of the
polynomial p. Conversely, if p{r) = 0, then we know from the remainder
theorem of algebra that ^ — r is a factor of the polynomial p{t)j and therefore
{t — vY^ will be one of the relatively prime factors of p. Now suppose that p is
the minimal polynomial of T (see Exercise 3.5). Theorem 5.5, Chapter 1, tells
us that {T — rl)"^ is zero on a corresponding subspace A^ of F and therefore, in
particular, that T — rl is not injective when restricted to A^. That is, r is an
eigenvalue. We have proved:
Theorem 3.3. The eigenvalues of T are zeros (roots) of every polynomial
p{t) such that p{T) = 0, and are exactly the roots of the minimal polynomial.
EXERCISES
3.1 Use the defining identity (T^, t)) = (^, Tt)) to show that the set SA of all self-
adjoint elements of Hom T^ is a subspace. Prove similarly that if S and T are self-
adjoint, then ST is self-adjoint if and only if ST = TS. Conclude that if T is
self-adjoint, then so is p{T) for any polynomial p.
3.2 Show that if T is self-adjoint, then aS = T^ is nonnegative. Show, therefore,
that if T is self-adjoint and a > 0, then T'^ + al cannot be the zero transformation.
3.3 Let p{t) = ^^ + 5^ + c be an irreducible quadratic polynomial (5^ < 4c), and
let 7" be a self-adjoint transformation. Show that p{T) 9^ 0. (Complete the square
and apply earlier exercises.)

5.3 SELF-ADJOINT TRANSFORMATIONS 261
3.4 Let T be self-adjoint and nilpotent (T" = 0 for some n). Prove that T = 0.
This can be done in various ways. One method is to show it first for n = 2 and then
for n = 2"^ by induction. Finally, any n can be bracketed by powers of 2, 2^ <
n < 2"*+! .
3.5 Let V be any vector space, and let T be an element of Hom V. Suppose that there
is a polynomial q such that q{T) = 0, and let p be such a polynomial of minimum
degree. Show that p is unique (to within a constant multiple). It is called the minimal
polynomial of T. Show that if we apply Theorem 5.5 of Chapter 1 to the minimal
polynomial p of T, then the subspaces Ni must both be nontrivial.
3.6 It is a corollary of the fundamental theorem of algebra that a polynomial with
real coefficients can be factored into a product of linear factors {t — r) and irreducible
quadratic factors' (^^ + ht-\- c). Let T be a self-adjoint transformation on a finite-
dimensional Hilbert space, and let p{t) be its minimal polynomial. Deduce a new
proof of Theorem 3.1 by applying to p{t) the above remark. Theorem 5.5 of Chapter 1,
and Exercises 3.1 through 3.4.
3.7 Prove that if T is a self-adjoint transformation on a pre-Hilbert space V, then
its null space is the orthogonal complement of its range: iV(r) = (.R{T))\ Conclude
that if y is a Hilbert space, then a self-adjoint T is injective if and only if its range is
dense (in V).
3.8 Assuming the above exercise, show that if F is a Hilbert space and T is a self-
adjoint element of Hom V that is hounded below (as well as bounded), then T is sur-
jective.
3.9 Let T be self-adjoint and nonnegative, and set m = lub {{T^, ^) : ||^|| = 1}.
Use the Schwarz inequality and the inequality of Lemma 3.2 to show that m = \\T\\.
3.10 Let y be a Hilbert space, let T be a self-adjoint element of Hom V, and set
m = lub {(T^, ^) : ||^|| = 1}. Show that if a > m, then a — T { = al — T) is in-
vertibleand ||(a — T)~^|| < l/(a — m). (Apply the Schwarz inequality, the definition
of m, and Exercise 3.8.)
3.11 Let P be a bounded linear transformation on a pre-Hilbert space V that is a
projection in the sense of Chapter 1. Prove that if P is self-adjoint, then P is an
orthogonal projection. Now prove the converse.
3.12 Let y be a finite-dimensional Hilbert space, let T in Hom V be self-adjoint,
and suppose that S in Hom V commutes with T. Show that the subspaces Mj of
Theorem 3.1 and Lemma 3.3 are invariant under S.
3.13 A self-adjoint transformation T on a finite-dimensional Hilbert space V is said
to have a simple spectrum if all its eigenvalues are distinct. By this we mean that all
the subspaces Mj are one-dimensional. Suppose that T is a self-adjoint transformation
with a simple spectrum, and suppose that S commutes with T. Show that S is also
self-adjoint. (Apply the above exercise.)
3.14 Let H he 3. Hilbert space, and let a;[^, rj] be a bounded bilinear form on H X H.
That is, there is a constant h such that
H^,v]\<bU\\M for all |, t, G//.
Show that there is a unique T in Hom V such that a;[^, r;] = (^, Tr]). Show that T is
self-adjoint if and only if a; is symmetric.

262 SCALAR PRODUCT SPACES 5.4
4. ORTHOGONAL TRANSFORMATIONS
Assuming that F is a Hilbert space and that therefore 6: V -^ F* is an
isomorphism, we can of course replace the adjoint T* G Hom V* of any T g Hom V
by the corresponding transformation ^"^0^*0^^ Hom V. In Hilbert space
theory it is this mapping that is called the adjoint of T and is designated T*,
Then, exactly as in our discussion of a self-adjoint T, we see that
(Ta, p) - (a, T*p) for all a, /? G 7
and that T* is uniquely defined by this identity. Finally, T is self-adjoint if and
only liT = T*,
Although it really amounts to the above way of introducing T* into Hom F,
we can make a direct definition as follows. For each rj the mapping f »—> (T^, r;)
is linear and bounded, and so is an elemenfcof F*, which, by Theorem 2.4, is
given by a unique element 13^ in V according to the formula (T^j rj) = {^j /3^).
Now we check that r;»—> /3^ is linear and bounded and is therefore an element of
Hom V which we call T*, etc.
The matrix calculations of Lemma 3.1 generalize verbatim to show that the
matrix of T* in Hom V is the transpose t* of the matrix t of T,
Another very important type of transformation on a Hilbert space is one
that preserves the scalar product.
Definition. A transformation T G Hom V is orthogonal if (Ta, TjS) =
(a, 13) for all a, /? G 7.
By the basic adjoint identity above this is entirely equivalent to (a, T*Tp) =
(a, p)j for all a, p, and hence to T*T = I. An orthogonal T is injective, since
IITall^ = \\a\\^, and is therefore invertible if 7 is finite-dimensional. Whether 7
is finite-dimensional or not, if T is invertible, then the above condition becomes
T* = T~^
If T G Hom R^j the matrix form of the equation T*T = / is of course
t*t = e, and if this is written out, it becomes
n
Z) ikitkj = 8} for all i, j,
k=i
which simply says that the columns of t form an orthonormal set (and hence a
basis) in U^, We thus have:
Theorem 4.1. A transformation T G Hom U^ is orthogonal if and only if
the image of the standard basis {5*} 1 under T is another orthonormal basis
(with respect to the standard scalar product).
The necessity of this condition is, of course, obvious from the scalar-product-
preserving definition of orthogonality, and the sufficiency can also be checked
directly using the basis expansions of a and p.
We can now state the eigenbasis theorem in different terms. By a diagonal
matrix we mean a matrix which is zero everywhere except on the main diagonal.

5.4 ORTHOGONAL TRANSFORMATIONS 263
Theorem 4.2. Let t = {tij} be a symmetric n X n matrix. Then there
exists an orthogonal n X n matrix h such that b~Hb is a diagonal matrix.
Proof, Since the transformation T G Hom U^ defined by t is self-adjoint, there
exists an orthonormal basis {b*} i of eigenvectors of Tj with corresponding
eigenvalues {rj^. Let B be the orthogonal transformation defined by B{8^) =
h^\ j = 1, . . . , n. (The n-tuples b^ are the columns of the matrix b = {hij}
of B,) Then (B'^ o T o B)(8') = Vjb', Since {B'^ o T o B){b') is the jth
column of b~Hb, we see that s = b~^tb is diagonal, with Sjj = rj. D
For later applications we are also going to want the following result.
Theorem 4.3. Any invertible T G Hom F on a finite-dimensional Hilbert
space V can be expressed in the form T = RS, where R is orthogonal and aS
is self-adjoint and positive.
Proof, For any T, T*T is self-adjoint, since {T*T)* = T*T** = T*T, Let
{(Pi}\ be an orthonormal eigenbasis, and let {rJ ^ be the corresponding
eigenvalues of T*T, Then 0 < \\T<pi\\^ = {T*T<pi, <pi) = {rm, <pi) = n for each i.
Since all the eigenvalues ofT*T are thus positive, we can define a positive square
root aS = (T*Ty'^ by Scpi = (r^)^'^<Pi, z = 1, 2, . . . , n. It is clear that aS^ =
T*T and that aS is self-adjoint.
Then A = ST''' is orthogonal, for (ST-'a, ST-''p) = {T'^a, S^T-^p) =
{T-^a, T*TT-''l3) = {T-^a, T*I3) = {Tf-^a, p) = (a, /?). Since T = A'^aS,
we set 72 = A~^ and have the theorem. D
It is not hard to see that the above factorization of T is unique. Also, by
starting with TT*jWe can express T in the form T = SRj where aS is self-adjoint
and positive and R is orthogonal.
We call these factorizations the polar decompositions of T, since they
function somewhat like the polar coordinate factorization z = re^^ of a complex
number.
Corollary. Any nonsingular n X n matrix t can be expressed as t = udv,
where u and v are orthogonal and d is diagonal.
Proof. From the theorem we have t = rs, where r is orthogonal and s is
symmetric. By Theorem 4.2, s = bdb~\ where d is diagonal and b is orthogonal.
Thus t = rs = (rb)db~^ = udv, where u = rb and v = b~^ are both
orthogonal. D
EXERCISES
4.1 Let y be a Hilbert space, and suppose that S and T in Hom V satisfy
{T^,V) = {^,Srj) for all $,77.
Write out the proof of the identity S -^ 6''^ o T* o $.

264 SCALAR PRODUCT SPACES 5.5
4.2 Write out the analogue of the proof of Lemma 3.1 which shows that the matrix
of T* is the transpose of the matrix of T.
4.3 Once again show that if (^, rj) = (^, f) for all ^, then 77 = f. Conclude that if
S, T in Hom V are such that (^, Trj) = (^, Srj) for all 77, then T = S,
4.4 Let {a, b} be an orthonormal basis for R^, and let t be the 2X2 matrix whoso
columns are a and b. Show by direct calculation that the rows of t are also ortho-
normal.
4.5 State again why it is that if V is finite-dimensional, and if S and T in Hom V
satisfy S o T = 7, then T is invertible and S = T~^. Now let 7 be a finite-dimensional
Hilbert space, and let T be an orthogonal transformation in Hom V. Show that T* is
also orthogonal.
4.6 Let t be an n X n matrix whose columns form an orthonormal basis for R".
Prove that the rows of t also form an orthonormal basis. (Apply the above exercise.)
4.7 Show that a nonnegative self-adjoint transformation aS on a finite-dimensional
Hilbert space has a uniquely determined nonnegative self-adjoint square root.
4.8 Prove that if F is a finite-dimensional Hilbert space and T G Hom F, then the
^^polar decomposition" of T, T = RS^ of Theorem 4.3 is unique. (Apply the above
exercise.)
5. COMPACT TRANSFORMATIONS
Theorem 3.1 breaks down when V is an infinite-dimensional Hilbert space.
A self-adjoint transformation T does not in general have enough eigenvectors
to form a basis for V, and a more sophisticated analysis, allowing for a
"continuous spectrum" as well as a "discrete spectrum", is necessary. This
enriched situation is the reason for the need for further study of Hilbert space
theory at the graduate level, and is one of the sources of complexity in the
mathematical structure of quantum mechanics.
However, there is one very important special case in which the eigenbasis
theorem is available, and which will have a startling application in the next
chapter.
Definition. Let V and W be any normed linear spaces, and let aS be the unit
ball in V. A transformation T in Hom(7, W) is compact if the closure of
T[S] in W is sequentially compact.
Theorem 5.1. Let V be any pre-Hilbert space, and let T g Hom V be self-
adjoint and compact. Then the pre-Hilbert space R = range (T) has an
orthonormal basis {(pi} consisting entirely of eigenvectors of T, and the
corresponding sequence of eigenvalues {r^} converges to 0 (or is finite).
Proof. The proof is just like that of Theorem 3.1 except that we have to start a
little differently. Set m = \\T\\ = lub {\\T{0\\ : U\\ = 1}, and choose a
sequence {fn} such that IIfnil = 1 for all n and 11^(^^)11 -^ ^- Then

5.5 COMPACT TRANSFORMATIONS 265
and since m^ — T^ is a nonnegative self-adjoint transformation, Lemma 3.2
tells that (m^ — T^){^n) -^ 0. But since T is compact, we can suppose (passing
to a subsequence if necessary) that {T^n} converges, say to p. Then T^^n -^ Tpj
and so m^^n -^ Tp also. Thus U -^ Tp/m^ and p = \imTU= T^{p)/m^
Since ||/3|| = lim HT'C^n)!! = ^, we have a nonzero vector jS such that T^{l3) =
m^p. Set a = p/\\l3\\.
Wehavethusfounda vectorasuch that ||a|| = landO = (m^ — T^){a) =
(m - T){m+ T){a), Then either (m + T){a) = 0, in which case T{a) =
—ma, or {m + T)(a) = 7 9^ 0 and (m — r)7 = 0, in which case Tl = ml.
Thus there exists a vector (pi {a or t/||7||) such that ||<^i|| = 1 and T{(pi) =
ricpi, where |ri| = m. We now proceed just as in Theorem 3.1.
For notational consistency we set mi = m, Vi = V, and now set V2 =
{iPi}-^, Then ^Fa] C Fg, since if a ± <pi, then (Ta, <pi) = (a, T<p2) =
^i{oi, <pi) = 0. Thus T t 72 is compact and self-adjoint, and if m2 = \\T \ V2\\,
there exists (P2 with ||<^2|| = 1 and T{(p2) = r2(P2j where |r2| = m2. We continue
inductively, obtaining an orthonormal sequence {(Pn} C V and a sequence
{vn} C U such that Tcpn = Tnifn and \rn\ = \\T t 7^11, where
We suppose for the moment, since this is the most interesting case, that
Vn 9^ 0 for all nr Then we claim that \rn\ -^ 0. For \rn\ is decreasing in any case,
and if it does not converge to 0, then there exists a 6 > 0 such that |r^| > 6
for all n. Then \\T(<pd - r(^y)f = ||r,^, - rj<p,r = llw^f + Ik.wf =
^i + ^/ ^ 26^ for all i 9^ i, and the sequence {T{<p{)} can have no convergent
subsequence, contradicting the compactness of T, Therefore \rn\ i 0.
Finally, we have to show that the orthonormal sequence {cpi} is a basis for R.
li 13 = T{a), and if {bn} and {an} are the Fourier coefficients of p and a,
then we expect that bn = rndn, and this is easy to check: bn = (/?, <Pn) =
{T(a), iPn) = (a, T{iPn)) = (a, Vn^Pn) = ^n(«, (fn) = Tnan^ TMs is jUSt Saying
that T{an(Pn) = bn<Pn, and therefore p — YJi bnpi = T(a — Yn ca^i)- Now
a — X^i anpi is orthogonal to {(pi}\ and therefore is an element of 7n+i, and the
norm of T on Vn+\ is kn+i|. Moreover, ||a — YJ\ «i<^i|| < ||«IL by the
Pythagorean theorem. Altogether we can conclude that
/5 — S ^i'Pi
< kn + ll
and since Vn^i -^0, this implies that P = J^t ^i^i- Thus {(pi) is a basis for
R{T). Also, since T is self-adjoint, N{T) = R{T)^ = fe}-^ = (iT 7^.
If some Vi = 0, then there is a first n such that r^ = 0. In this
case \\T \ Vn\\ = Kl = 0, so that Vn C N{T), But ipi G /?(r) if i < n, since
then <pi = T{<pi)/ri, and so N{T) = R{T)^ C {<^i, . . . , <^n-i}-^ = 7^.
Therefore, N{T) = Vn and R{T) is the span of {ip^]T\ D

CHAPTER 6
DIFFERENTIAL EQUATIONS
This chapter is not a small differential equations textbook; we leave out far too
much. We are principally concerned with some of the theory of the subject,
although we shall say one or two practical things. Our first goal is the
fundamental existence and uniqueness theorem of ordinary differential equations,
which we prove as an elegant application of the fixed-point theorem. Next wo
look at the linear theory, where we make vital use of material from the first two
chapters and get quite specific about the process of actually finding solutions.
So far our development is linked to the initial-value problem, concerning th(^
existence of, and in some cases the ways of finding, a unique solution passing
through some initially prescribed point in the space containing the solution
curves. However, some of the most important aspects of the subject relate to
what are called boundary-value problems, and our last and most sophisticated
effort will be directed toward making a first step into this large area. This will
involve us in the theory of Chapter 5, for we shall find ourselves studying self-
adjoint operators. In fact, the basic theorem about Fourier series expansions
will come out of recognizing a certain right inverse of a differential operator to bo
a compact self-adjoint operator.
1. THE FUNDAMENTAL THEOREM
Let A be an open subset of a Banach space W, let / be an open interval in U, and
let F: I X A -^ W he continuous. We want to study the differential equation
da/dt = F(t, a).
A solution of this equation is a function f:J-^A, where J is an open subinterval
of /, such that/'(0 exists and
f(t) = F(t,f{t))
for every t in J. Note that a solution / has to be continuously differentiate, for
the existence of /' implies the continuity of /, and then f\t) = F(tjf(t)) is
continuous by the continuity of F.
We are going to see that if F has a continuous second partial differential,
then there exists a uniquely determined "local" solution through any point
<toy ao> G I X A,
266

6.1 THE FUNDAMENTAL THEOREM 267
In saying that the solution / goes through <toy ao> y we mean, of course, that
«o == /(^o)- The requirement that the solution / have the value ao when t = to
is called an initial condition.
The existence and continuity of dF%t,a> implies, via the mean-value
theorem, that F(t, a) is locally uniformly Lipschitz in a. By this we mean that
for any point <to, ao> in / X A there is a neighborhood M X N and a constant
6 such that \\F(t, 0 — F{t, 7))\\ < h\\i — 7)\\ for all t in M and all f, t) in N.
To see this we simply choose balls M and N about ^o and ao such that dF%t,a> is
bounded, say by 6, on M X Nj and apply Theorem 7.4 of Chapter 3. This is the
condition that we actually use below.
Theorem 1.1. Let A be an open subset of a Banach space W, let / be an
open interval in U, and let F be a continuous mapping from I X A to W
which is locally uniformly Lipschitz in its second variable. Then for any
point <to, ao> in / X A, for some neighborhood U of ao and for any
sufficiently small interval J containing to, there is a unique function / from
J to U which is a solution of the differential equation passing through the
point <to, ao>.
Proof, If / is a solution on J through <^o, oio^ j then an integration gives
fit) -f{to)= f'F(s,f{s))ds,
ho
so that
fit) = cco+ f'F(sJ{s))ds
Jto
for t gJ. Conversely, if/ satisfies this "integral equation", then the
fundamental theorem of the calculus implies that f'(t) exists and equals F(tjf{t))
on J J so that/is a solution of the differential equation which clearly goes through
< toj ao^ ' Now for any continuous f: J -^ A we can define g:J-^Why
git) = ao+ f'F(sJis))ds,
Jto
and our argument above shows that / is a solution of the differential equation if
and only if / is a fixed point of the mapping K\f^-^g. This suggests that we try
to show that K is a contraction, so that we can apply the fixed-point theorem.
We start by choosing a neighborhood L X C/ of <^o; olq> on which Fit, a)
is bounded and Lipschitz in a uniformly over t. Let J be some open subinterval
of L containing ^o; and let V be the Banach space (Be(J, W) of bounded
continuous functions from J to W. Our later calculation will show how small we
have to take J. We assume that the neighborhood [/ is a ball about a^ of radius
r, and we consider the ball of functions 11 = Briao) in F, where olq is the
constant function with value a^. Then any/in 11 has its range in [/, so that F(^, fit))
is defined, bounded, and continuous. That is, K as defined earlier maps the ball
11 into 7.

268 DIFFERENTIAL EQUATIONS 6.1
We now calculate. Let F be bounded by m on L X U and let b be the length
of J. Then
||K(ao) - ^olU = lub III C F{s, ao) ^s|| :tGj\ < 8m (1)
by the norm inequality for integrals (see Section 10 of Chapter 4). Also, if/i
and /2 are in 11, and if c is a Lipschitz constant for F on L X U, then
\\KUx) - K(f2)\U = \uh{\\rF(s,Ms)) - F(sj2(s))\\\
iWJto 11'
< 8\uh{\\F{sJds)) -F{s,Ms))\\}
< 8c\uh{\\h{s) -/2(s)||}
= 8c\\fi -/2II00. (2)
From (2) we see that K is a contraction with constant C = 8c ii 8c < 1, and
from (1) we see that K moves the center ao of the ball 11 a distance less than
(1 — C)r if 8m < (1 — 8c)r, This double requirement on 8 is equivalent to
r
8 <
m -\- cr
and with any such 8 the theorem follows from a corollary of the fixed-point
theorem (Corollary 2 of Theorem 9.1, Chapter 4). D
Corollary. The theorem holds ii F: I X A -^ W is continuous and has a
continuous second partial differential.
We next show that any two solutions through <tQj q:o> must agree on the
intersection of their domains (under the hypotheses of Theorem 1.1).
Lemma 1.1. Let Qi and ^2 be any two solutions of da/dt = Fit, a) through
<^0; (^Q^' Then ^i(^) = ^2(0 ^^ all t in the intersection J = Ji n J2 of
their domains.
Proof. Otherwise there is a point s in J such that gx{s) 9^ ^2(5)- Suppose that
s > ^0; and set C = {^: ^ > ^0 and ^i(^) ^ ^2(0} and x = gib C. The set C
is open, since gi and ^2 are continuous, and therefore x is not in C, That is,
^i(^) = Q2{^)' Call this common value a and apply the theorem to <x, a>.
With r such that Br{oL) CA, we choose 8 small enough so that the differential
equation has a unique solution g from (x — 5, x + 5) to Br{oL) passing through
<x, a>, and we also take 5 small enough so that the restrictions of gi and ^2 to
{x — 5, a; + 5) have ranges in Br (a). But then gi = g2 = g on this interval
by the uniqueness of ^, and this contradicts the definition of x. Therefore,
9i = ^2 on the intersection of their domains. D
This lemma allows us to remove the restriction on the range of / in the
theorem.
Theorem 1.2. Let A, /, and F be as in Theorem 1.1. Then for any point
<toy ao> in I X A and any sufficiently small interval neighborhood J of ^o;
there is a unique solution from J to A passing through <^o; ^0^ •

6.1
THE FUNDAMENTAL THEOREM
269
Fig. 6.1
Global solutions. The solutions we have found for the differential equation
da/dt = F{tj a) are defined only in sufficiently small neighborhoods of the initial
point ^0 and are accordingly called local solutions. Now if we run along to a
point <^i, ai > near the end of such a local solution and then consider the local
solution about <^i, ai > , first of all it will have to agree with our first solution
on the intersection of the two domains, and secondly it will in general extend
farther beyond ti than the first solution, so the two local solutions will fit together
to make a solution on a larger ^interval than either gives separately. We can
continue in this way to extend our original solution to what might be called a
global solution, made up of a patchwork of matching local solutions. These
notions are somewhat vague as described above, and we now turn to a more
precise construction of a global solution.
Given <^o, «o> Cl X A, let ^ be the family of all solutions through
< to, ao >. Thus g G^if and only if (/ is a solution on an interval J Cl,to gJ,
and g{to) = ao- Lemma 1.1 shows exactly that the unionf / of all the functions g
in ?F is itself a function, for if <^i,a:i> Ggi and <^i,a:2> G ^2, then ai =
gdt) = 02(t) = 0:2-
Moreover, / is a solution, because around any x in its domain / agrees with
some g G^. By the way / was defined we see that / is the unique maximum
solution through <to, aQ>, We have thus proved the following theorem.
Theorem 1.3. Let F: I X A -^ V he Si. function satisfying the hypotheses of
Theorem 1.1. Then through each <^o, «o> in / X A there is a uniquely
determined maximal solution to the differential equation da/dt = F{t, a).
In general, we would have to expect a maximal solution to "run into the
boundary of A " and therefore to have a domain interval J properly included in J,
as Fig. 6.1 suggests.
t Remember that we are taking a function to be a set of ordered pairs, so that the
union of a family of functions makes precise sense.

270 DIFFERENTIAL EQUATIONS 6.1
However, if A is the whole space W, and if F{t, a) is Lipsohitz in a for each t,
with a Lipschitz bound c{t) that is continuous in tj then we can show that each
maximal solution is over the whole of /. We shall shortly see that this condition
is a natural one for the linear equation.
Theorem 1.4. Let W be a Banach space, and let / be an open interval in U.
Let F: I X W —^ W he continuous, and suppose that there is a continuous
function c: / —> K such that
\\F{t,a,)-F(t,a2)\\ <c(<)||ai - aall
for all t in / and all ai, 0:2 in W. Then each maximal solution to the
differential equation da/dt = F(t, a) has the whole of / for its domain.
Proof. Suppose, on the contrary, that (/ is a maximal solution whose domain
interval J has right-hand endpoint 6 less than that of /. We choose a finite open
interval L containing 6 and such that L C I (see Fig. 6.2). Since L is compact,
the continuous function c{t) has a maximum value c on L. We choose any ti
in L n J close enough to 6 so that 6 — ^1 < 1/c, and we set ai = g{ti) and
m = max \\F(t, ai)\\ on L. With these values of c and m, and with any r, the
proof of Theorem 1.1 gives us a local solution / through <^i, ai > with domain
(^1 — ^yti + 5) for any 8 less than r/{m + re) = l/(c + (m/r)). Since we
now have no restriction on r (because A = W)j this bound on d becomes
1/c, and since we chose ti so that ^1 + (1/c) > 6, we can now choose d so that
ti + d > b. But this gives us a contradiction; the maximal solution g through
<^i,ai> includes the local solution /, so that, in particular, ti + 8 < b.
We have thus proved the theorem. D
I I 1" I
L Fig. 6.2
Going back to our original situation, we can conclude that if the Lipschitz
control of F is of the stronger type assumed above, and if the domain J of some
maximal solution g is less than 7, then the open set A cannot be the whole of W.
It is in fact true that the distance from g{t) to the boundary of A approaches
zero as t approaches an endpoint 6 of J which is interior to I. That is, it is now a
theorem that p(f{t)j A') —> 0 as ^ —> 6. The proof is more complicated than our
argument above, and we leave it as a set of exercises for the interested reader.
The nth-order equation. Let Ai, A2, . . . , A^ be open subsets of a Banach
space T^, let 7 be an open interval in R, and let G:7xAiXA2X---xA^—>TF
be continuous. We consider the differential equation
d^'a/dt'' = G{t, a, da/dt, . . . , d^'-^a/de'^).

6.1 THE FUNDAMENTAL THEOREM 271
A function /: J -^ T7 is a solution to this equation if J is an open subinterval of
/, / has continuous derivatives on J up to the nth order, /^*~^^[J] C Aij i =
1, . . . , n, and
for t G J. An initial condition is now given by a point
The basic theorem is almost the same as before. To simplify our notation, let a
be the n-tuple <ai, a2, . . . , oin> in W^ = V, and set A = Hi Ai. Also let
yl/ be the mapping f \-^ </,/',•-' ,/^''~^^>. Then the solution equation
becomes/^^^(O - G(t,ypf{t)).
Theorem 1.5. Let G: / X A -^ T7 be as above and suppose, in addition,
that G{tj a) is locally uniformly Lipschitz in a. Then for any <toy fi> in
/ X A and for any sufficiently small open interval J containing ^o; there is a
unique function/from J to TF such that/is a solution to the above nth-order
equation satisfying the initial condition ^/(^o) = P-
Proof. There is an ancient and standard device for reducing a single nth-order
equation to a system of first-order equations. The idea is to replace the single
equation
d^'a/dr = Git, a, da/dt, . . . , d^'-^a/de'^)
by the system of equations
dai/dt = a2,
da2/dt = asj
dan—i/dt = anj
dan/dt = G{tj ai, 0:2, • • • , ^n),
and then to recognize this system as equivalent to a single first-order equation
on a different space. In fact, if we define the mapping F = <F^j , . . j F^>
from / X A to 7 = W^ by setting F\tj a) = ai^i for ^ = 1, . . . , n — 1, and
F^(tj ot) = G{tj ol)j then the above system becomes the single equation
dot/dt = F{t,oL),
where F is clearly locally uniformly Lipschitz in a. Now a function f =
<fii ' ' • ,fn^ from J to 7 is a solution of this equation if and only if
/l = /2j
J2 =" fs}
Jn—l ^^ Jri)
fn = G{tj fij . . . J fn)j

272 DIFFERENTIAL EQUATIONS 6.1
that is, if and only if /i has derivatives up to order n, ypifi) = f and fi^\t) =
G(ij^fi{i))- The n-tuplet initial condition \l/f{to) = fi is now just f(^o) = P-
Thus the nth-order theorem for G has turned into the first-order theorem for F^
and so follows from Theorems 1.1 and 1.2 D
The local solution through <to, fi> extends to a imique maximal solution
by Theorem 1.3 applied to our first-order problem da/dt = F{t,oL)j and the
domain of the maximal solution is the whole of / if G{tj a) is Lipschitz in a with
a bound c{t) that is continuous and if A = W^, as in Theorem 1.4.
EXERCISES
1.1 Consider the equation da/dt = F{t, a) in the special case where W — R^.
Write out the equation as a pair of equations involving real-valued functions and real
variables.
1.2 Consider the system of differential equations
dx/dt = t-\- x'^ -\- y^, dy/dt = cos xy.
Define the function F:U^ -^ U^ so that the above system becomes
da/dt = F{t,a),
where a = <x, y>.
1.3 In the above exercise show that F is uniformly Lipschitz in a on R X ^, where
A is any bounded open set in U^. Is F uniformly Lipschitz on R X R^?
1.4 Write out the above system in terms of a solution function / = </i,/2>'.
Write out for this system the integrated form used in proving Theorem 1.1.
1.5 The fixed-point theorem iteration sequence that we used in proving Theorem 1.1
starts off with /o as the constant function ao and then proceeds by
fn{t) = ao+ f F{sjn-i{s)) ds.
Jo
Compute this sequence as far as /4 for the differential equation
dx/dt == t+x lf{t) = t + f{t)]
with the initial condition/(O) = 0. That is, take/o = 0 and compute/i,/2,/a, and/4
from the formula. Now guess the solution / and verify it.
1.6 Compute the iterates/o,/i,/2, and/a for the initial-value problem
dy/dx = x+ y^j y{0) = 0.
Supposing that the solution / has a power series expansion about 0, what are its first
three nonzero terms?
1.7 Make the computation in the above exercise for the initial condition/(O) = — 1.
1.8 Do the same for/(0) = +1.

6.1 THE FUNDAMENTAL THEOREM 273
1.9 Suppose that TF is a Banach space and that F and G are functions from R X W^
to W satisfying suitable Lipschitz conditions. Show how the second-order system
would be brought under our standard theory by making it into a single second-order
equation.
1.10 Answer the above exercise by converting it to a first-order system and then to a
single first-order equation.
1.11 Let ^ be a nonnegative, continuous, real-valued function defined on an interval
[0, a] C K, and suppose that there are constants h and c > 0 such that
d{x) < cf d{t) dt + bx for all x G [0, a],
Jo
a) Prove by induction that ii m = II^IU, then
m(cx)'' . b A (cxY ,
^(^) < -, r- 1^ —TT- for every n.
nl c^^ j\
b) Then prove that
e(x) < - (e'"" - 1) for all x.
c
1.12 Let T^ be a Banach space, let / be an interval in U, and let F he a, continuous
mapping from I X W to W. Suppose that \\F{tj ao)\\ < b for alU G / and that
\\F{t,a)-F{t,m<c\\a-p\\
for all t in / and all a, p in W, Let / be the global solution through <^o, olq>, and
set e{x) = 11/(^0 + x) — ao||. Prove that
e{x) < r d{t)dt+bx
Jo
for a: > 0 and ^o + ^ in /. Then use the result in the above exercise to derive a much
stronger bound than we have in the text on the growth of the solution f{t) as t goes
away from ^o-
1.13 With the hypotheses on F as in the above exercise, show that the iteration
sequence for the solution through <^o, oio> converges on the whole of / by showing
inductively that if /o = ao and
fn(t) =ao+ f F({s)Jn-l{s)) ds,
Jo
then
|/n(0-/n-l(OI<^^-
From these inequalities prove directly that the solution/ through <to,ao> satisfies
11/(0 - aoll < 7 («'""'"' - !)•

274 DIFFERENTIAL EQUATIONS 6.2
2. DIFFERENTIABLE DEPENDENCE ON PARAMETERS
It is exceedingly important in some applications to know how the solution to the
system
fit) = G{t,m), f{h) =«!
varies with the initial point <tij ai> . In order to state the problem precisely,
we fix an open interval J, set 11 = Br{a.Q) CV= (Be(J, W) as in the previous
section, and require a solution in 11 passing through <^i, ai > , where <^i, ai >
is near <toj ao>. Supposing that a unique solution / exists, we then have a
mapping < ^i, ai > »-^ /, and it is the continuity and differentiability of this map
that we wish to study.
Theorem 2.1. Let L X C/ be a neighborhood of <toy ao> in the Banach
space U X Wy and let F{tj a) be a bounded continuous mapping from
L X UtoW which is Lipschitz in a uniformly over t. Then there is a
neighborhood J X N oi <toj ao> with the following property. For any <tij ai>
inJxN there is a unique function / from J to U which is a solution of the
differential equation da/dt — F(t,a) passing through <^i,ai>, and the
mapping <ti, ai > f-^/ from J X A^ to F is continuous.
Proof. We simply reexamine the calculation of Theorem 1.1 and take b a little
smaller. Let Kiti, ai,f) be the mapping of that theorem but with initial point
<ti, ai >, so that g = K{ti, a\, f) if and only if g(t) = a^ + jt[ F(s,f(s)) ds for
all t in J. Clearly K is continuous in <ti, ai> for each fixed /.
If N is the ball Br/2(oio)j then the inequality (1) in the proof shows that
||K(^i, ai, ao) — aoll ^ ||«i — aoll + dm < r/2 + 8m, The second inequality
remains unchanged. Therefore, / f-^ K{tij ai, /) is a map from 11 to 7 which is a
contraction with constant C — 8cU 8c < Ij and which moves the center ao of 11
a distance less than (1 — C)r if r/2 -\- 8m < {1 — 8c)r. This new double
requirement on 8 is equivalent to
8 <
2(m + cr)
which is just half the old value. With J of length 5, we can now apply Theorem
9.2 of Chapter 4 to the map K{tij ai,/) from (J X iV) X 11 to F, and so have
our theorem. D
If we want the map <^i, ai > »—>/ to be differentiate, it is sufficient, by
Theorem 9.4 of Chapter 4, to know in addition to the above that
K: (J X iV) X 11 -^ F
is continuously differentiable. And to deduce this, it is sufficient to suppose that
dF exists and is uniformly continuous on L X U.
Theorem 2.2. Let L X C/ be a neighborhood of < to, ao ^ ^^ the Banach
space U xWy and let F{tj a) be a bounded mapping from L X UtoW such
that dF exists, is bounded, and is imiformly continuous on L X U, Then, in

6.2 DIFFERENTIABLE DEPENDENCE ON PARAMETERS 275
the context of the above theorem, the solution / is a continuously differ-
entiable function of the initial value <ti, ai > .
Proof. We have to show that the map K{ti, ai, /) from (J X N) X'UtoV is
continuously differentiable, after which we can apply Theorem 9.4 of Chapter 4, as
we remarked above. Now the mapping h ^-^ k defined by k(t) = jt[ h(s) ds is a
bounded linear mapping from F to 7 which clearly depends continuously on ^i,
and by Theorem 14.3 of Chapter 3 the integrand map f ^-^ h defined by h(s) =
F{sj /(s)) is continuously difTerentiable on 11. Composing these two maps we see
that dK%ti,aiJ> exists and is continuous on J X iV X 11. Now
so that dK^ = I, and AKl^t„a,j>(h) = -Jt['^^ F(sJ(s)) ds, from which it
follows easily that dKl^ti,ai,f>(h) =—hF{ti,f(t)). The three partial differentials
dK^j dK^j and dK^ thus exist and are continuous on J X iV X 11, and it follows
from Theorem 8.3 of Chapter 3 that K(tij ai,/) is continuously difTerentiable
there. D
Corollary. If s is any point in J, then the value f(s) of a solution at s is a
differentiable function of its value at ^o-
Proof. Let /« be the solution through <to, a>. By the theorem, a i-^ /« is a
continuously difTerentiable map from N to the function space V = 6^e(Jj W).
Eut TTs'f^-^fis) is a bounded linear mapping and thus trivially continuously
difTerentiable. Composing these two maps, we see that a »—> /«(«) is continuously
difTerentiable on N. D
It is also possible to make the continuous and difTerentiable dependence of
the solution on its initial value < to, ao ^ ^^^^ ^ global afTair. The following is
the theorem. We shall not go into its proof here.
Theorem 2.3. Let /be the maximal solution through < to, ao > with domain
J, and let [a, h] be any finite closed subinterval of J containing ^o- Then there
exists an e > 0 such that for every <ti, ai> G Be(<toj «o^) the domain
of the global solution through <ti, ai> includes [a, h], and the restriction
of this solution to [a, h] is a continuous function of <^i, ai > . If F satisfies
the hypotheses of Theorem 2.2, then this dependence is continuously
difTerentiable.
Finally, suppose that F depends continuously (or continuously difTer-
entiably) on a parameter X, so that we have F(\, t, a) on M X I X A. Now the
solution / to the initial-value problem
fit) = Fit, fit)), f(t,) = a,
depends on the parameter X as well as on the initial condition/(^i) = ai, and if
the reader has fully understood our arguments above, he will see that we can
show in the same way that the dependence of / on X is also continuous
(continuously differentiable). We shall not go into these details here.

276 DIFFERENTIAL EQUATIONS 6.3
3. THE LINEAR EQUATION
We now suppose that the function F of Section 1 is from I X W to W and
continuous, and that F{t, a) is linear in a for each fixed t. It is not hard to see that
we then automatically have the strong Lipschitz hypothesis of Theorem 1.4,
which we shall in any case now assume. Here this is a boundedness condition
on a linear map: we are assuming that F(tj a) = Tt(a)j where Tt G Hom Wj
and that \\Tt\\ < c(t) for all t, where c(t) is continuous on /.
As one might expect, in this situation the existence and uniqueness theory of
Section 1 makes contact with general linear theory. Let Xq be the vector space
e(/, W) of all continuous functions from / to W, and let Xi be its subspace
e^(/, W) of all functions having continuous first derivatives. Norms will play
no role in our theorem.
Theorem 3.1. The mapping SiXi-^Xq defined by setting g = Sf U
g(t) = f'{t) — F(tjf(t)) is a surjective linear mapping. The set N of global
solutions of the differential equation da/dt = F(tj a) is the null space of aS,
and is therefore, in particular, a vector space. For each to E: I the restriction
to N of the coordinate (evaluation) mapping irt^'-f^-^ f{to) is an isomorphism
from N to W. The null space M of ttiq is therefore a complement of iV in Xi,
and so determines a right inverse R of aS. The mapping / j-^ < aS/, f{to) >
is an isomorphism from Xi to Xq X W, and this fact is equivalent to all
the above assertions.
Proof, For any fixed g in Xq we set G{ty a) = F{t, a) + g{t) and consider the
(nonlinear) equation da/dt = G{tj a). By Theorems 1.3 and 1.4 it has a unique
maximal solution / through any initial point < toj ao >, and the domain of / is
the whole of /. That is, for each pair <gj a> in Xq XW there is a unique/in
Xi such that <Sfjf(to)> = <g,a>. The mapping
<S,irt,>:f^<SfJ{t,)>
is thus bijective, and since it is clearly linear, it is an isomorphism. In particular,
aS is surjective. The null space AT of aS is the inverse image of {0} X W under the
above isomorphism; that is, ttiq t iV is an isomorphism from N to W,
Finally, the null space M oi ttiq is the inverse image of Xq X {0} under
<aS, 7riQ>, and the direct sum decomposition Xi = M ^ N simply reflects the
decomposition XqXW = (Xq X {0}) 0 ({0} X W) under the inverse
isomorphism. This finishes the proof of the theorem. D
The problem of finding, for a given g in Xq and a given ao ^^ W, the unique/
in Xi such that aS(/) = g and/(^o) = «o is called the initial-value problem. At
the theoretical level, the problem is solved by the above theorem, which states
that the uniquely determined / exists. At the practical level of computation, the
problem remains important.
The fact that M = Mt^ is a complement of N breaks down the initial-value
problem into two independent subproblems. The right inverse R associated with

6.3 THE LINEAR EQUATION 277
MtQ finds h in Xi such that SQi) = g and h{to) = 0. The inverse of the
isomorphism f^-^f{to) from N to W selects that k in Xi such that aS(A;) = 0 and
k{to) = ao- Then / = h + k. The first subproblem is the problem of "solving
the inhomogeneous equation with homogeneous initial data", and the second is
the problem of "solving the homogeneous equation with inhomogeneous initial
data". In a certain sense the initial-value problem is the "direct sum" of these
two independent problems.
We shall now study the homogeneous equation da/dt = Tt{a) more closely.
As we saw above, its solution space N is isomorphic to W imder each projection
map TTf = / f-^ f{i). Let (pt be this isomorphism (so that <pt = T^t \ N). We now
choose some fixed ^o in /—we may as well suppose that / contains 0 and take
^0 = 0—and set Kt = (ft "^ <^^^ Then {Kt} is a one-parameter family of linear
isomorphisms of W with itself, and if we set ffi{t) = Kt{ff)j then/|3 is the solution
of da/dt = Tt{a) passing through <0, p> . We call Kt 2i fundamental solution of
the homogeneous equation da/dt = Tt{a).
Since f^(i) = Tt{U(i)), we see that d(Kt)/dt = Tt ^ Kt in the sense that
the equation is true at each ^ in W. However, the derivative d(Kt)/dt does not
necessarily exist as a norm limit in Hom W. This is because our hypotheses on Tt
do not imply that the mapping t^-^ Tt'is continuous from / to Hom W. If this
mapping is continuous, then the mapping <tj A> j-^ T^ o A is continuous from
/ X Hom W to Hom Wy and the initial-value problem
dA/dt = Tto A, Ao= I
has a unique solution A; in e ^(/, Hom W). Because evaluation at jS is a boimded
linear mapping from Hom W to Wj At{^) is a dififerentiable function of t and
dAtm/dt= {dAt/dtW) = Tt{At{P)).
This implies that At{ff) = Ki(^) for all ^, so Kt = At. In particular, the
fundamental solution t^-^Kt is now a differentiable map into Hom Wj and
dKt/dt = Tt "^ Kt. We have proved the following theorem.
Theorem 3.2. Let ^»—> T; be a continuous map from an interval
neighborhood / of 0 to Hom W. Then the fundamental solution t^-^ Kt of the
differential equation da/dt = Tt{a) is the parametrized arc from / to
Hom W that is the solution of the initial-value problem dA/dt = Tt "^ A,
Ao= I.
In terms of the isomorphisms Kt = K(t), we can now obtain an explicit
solution for the inhomogeneous equation da/dt = Tt{a) + g{t). We want/such
that
fit) - Ttim) = g(t).
Now K\t) = TtoK(t), so that Tt = K'{t) o K{ty\ and it follows from
Exercise 8.12 of Chapter 4 and the general product rule for differentiation

278 DIFFERENTIAL EQUATIONS G..'i
(Theorem 8.4 of Chapter 3) that the left side of the equation above is exactly
/:«) (I [/^a)-'(/«))])•
The equation we have to solve can thus be rewritten as
We therefore have an obvious solution, and even if the reader has found our
paotivating argument too technical, he should be able to check the solution by
differentiating.
Theorem 3.3. In the context of Theorem 3.2, the function
m = Kt[f^Kr\g{s))ds]
is the solution of the inhomogeneous initial-value problem
da/dt = Tt{a) + git), /(O) = 0.
This therefore is a formula for the right inverse 72 of aS determined by th(»
complement Mq of the null space N of aS.
The special case of the constant coefficient equation, where the "coefficient"
operator Tt is a fixed T in Hom W, is extremely important. The first new fact,
to be observed is that if / is a solution of da/dt = T{a)j then so is /'. For the
equation/'(O = T{f{t)) has a differentiable right-hand side, and differentiating,
wegetr'W = T{f{t)), That is:
Lemma 3.1. The solution space N of the constant coefficient equation
da/dt = T{a) is invariant under the derivative operator D.
Moreover, we see from the differential equation that the operator D on N
is just composition with T, More precisely, the equation/'(^) = T(f{t)) can be
rewritten wt "^ D = T o ttu and since the restriction of Wt to N is the
isomorphism <pt from N to Wj this equation can be solved for T. We thus have the
following lemma.
Lemma 3.2. For each fixed t the isomorphism (ft from N to W takes the
derivative operator D on N to the operator T on W. That is,
T = <pt o J) o (p^ .
The equation for the fundamental solution Kt is now dS/dt = TS. In the
elementary calculus this is the equation for the exponential function, which leads
us to expect and immediately check that Kt = €*^. (See the end of Section 8 of
Chapter 4.) The solution of da/dt = T{a) through <0^ ^> is thus the function
0 J'

6.3 THE LINEAR EQUATION 279
If T satisfies a polynomial equation 7)(!r) = 0, as we know it must if W is
finite-dimensional, then our analysis can be carried significantly further. Suppose
for now that p has only real roots, so that its relatively prime factorization is
pW = n?(^ - >^*)'^'. Then we know from Theorem 5.5 of Chapter 1 that W
is the direct sum W = @\ Wi of the null spaces Wi of the transformations
{T — \i)^\ and that each Wi is invariant under T. This gives us a much simpler
form for the solution curve e^'^a if the point a is in one of the null spaces Wi.
Taking such a subspace Wi itself as W for the moment, we have {T — X/)^ = 0,
so that T = \I + R, where R"^ = 0, and the factorization e*^ = e'^e*^,
together with the now finite series expansion of e*^, gives us
J1' J^\ \ 4-Df \ \ I ^m-i R^~ (q^)
e a= e I a + tR{a) H h ^ (m — 1) ij *
Note that the number of terms on the right is the degree of the factor {t — X)"*
in the polynomial j){t).
In the general situation where W = @\ Wi, we have a = J^\ ai, e*^(a)^ =
E^i e^'^{a^), and each e^^(a,) of the above form. The solution of f{t) = T(f{t))
through the general point <0, a> is thus a finite sum of terms of the form
Ve^^^Pijj the number of terms being the degree of the polynomial p.
If TF is a complex Banach space, then the restriction that p have only real
roots is superfluous. We get exactly the same formula but with complex values
of X. This introduces more variety into the behavior of the solution curves
since an outside exponential factor e^^ = e^^e^^^ now has a periodic factor if
Altogether we have proved the following theorem.
Theorem 3.4. If TF is a real or complex Banach space and T G Hom TF,
then the solution curve in W of the initial-value problem f'{t) = T{f{t)),
m = ^, is
0 J'
If T satisfies a polynomial equation (T — X)^ = 0, then
m
= e'^ [^ + tRifi) + ■■■ + ^J"_\y R^'-'iff)] ,
where R = T — \I. li T satisfies a polynomial equation p{T) = 0 and p
has the relatively prime factorization p{x) = Hi (^ ~ ^i)*"*; then f{t) is a
sum of k terms of the above type, and so has the form
fit) = E t'V^%i,
where the number of terms on the right is the degree of the polynomial p,
and each fiij is a fixed (constant) vector.

280 DIFFERENTIAL EQUATIONS Q.',]
It is important to notice how the asymptotic behavior of f{t) as ^ -^ + oo is
controlled by the polynomial roots X^-. We first restrict ourselves to the solution
through a vector a in one of the subspaces Wiy which amounts to supposing that
{T — X)"^ = 0. Then if X has a positive real part, so that e^^ = e^f'e'^' with
jjL > Oj then 11/(011 -^ 00 in exponential fashion. If X has a negative real pari,
then f{t) approaches zero as ^ -^ oo (but its norm becomes infinite exponentially
fast as t -^ — oo). If the real part of X is zero, then \\f{t)\\ -^ oo like f^'^ if
m > 1. Thus the only way for / to be bounded on the whole of U is for the real
part of X to be zero and m = 1, in which case / is periodic. Similarly, in tho
general case where p{T) = ]J\ {T — X^)"^" = 0, it will be true that all the
solution curves are bounded on the whole of U if and only if the roots X^ are all
pure imaginary and all the multiplicities m^ are 1.
EXERCISES
3.1 Let / be an open interval in U, and let W he a. normed linear space. Let F{t, a)
be a continuous function from / X W to W which is linear in a for each fixed t. Prove
that there is a function c(t) which is bounded on every closed interval [a, b] included
in / and such that \\F{t, a)\\ < c(0||a:|| for all a and t. Then show that c can be made
continuous. (You may want to use the Heine-Borel property: If [a, h] is covered by n
collection of open intervals, then some finite subcollection already covers [a, b].)
3.2 In the text we omitted checking that /i-^/(^> — G{tJJ', . . . ,/^^"i) is sur-
jective from Xn to Xq. Prove that this is so by tracking down the surjectivity through
the reduction to a first-order system.
3.3 Suppose that the coefficients ai{t) in the operator
0
are all themselves in C^. Show that the null space A^ of T is a subspace of 6"*+^. State
a generalization of this theorem and indicate roughly why it is true.
3.4 Suppose that W is a Banach space, T G Hom W, and fi is an eigenvector of T
with eigenvalue r. Show that the solution of the constant coefficient equation da/dt =
T{a) through <0,i9> is/(0 = e'^fi.
3.5 Suppose next that W is finite-dimensional and has a basis {Pi}\ consisting of
eigenvectors of T, with corresponding eigenvalues r,. Find a formula for the solution
through <0, «> in terms of the basis expansion of a.
3.6 A very important special case of the linear equation da/dt — Tt{a) is when the
operator function Tt is periodic. Suppose, for example, that T<+i = Tt for all i.
Show that then Kt^n = Kt{KiY for all t and n.
Assume next that K\ has a logarithm, and so can be written K\ = e"^ for some A
in Hom W. (We know from Exercise 11.19 of Chapter 4 that this is always possible
if W is finite-dimensional.) Show that now Kt can be written in the form
where B{t) is periodic with period 1.

6.4 THE nTH-ORDER LINEAR EQUATION 281
3.7 Continuing the above exercise, suppose now that TF is a finite-dimensional
complex vector space. Using the analysis of e^"^^ given in the text, show that the
differential equation da/dt = Tt(a) has a periodic solution (with any period) only if
Ki has an eigenvalue of absolute value 1. Show also that if Ki has an nth root of
unity as an eigenvalue, then the differential equation has a periodic solution with
period n.
3.8 Write out the special form that the formula of Theorem 3.3 takes in the constant
coefficient situation.
3.9 It is interesting to look at the facts of Theorem 3.1 from the point of view of
Theorem 5.3 of Chapter 1. Assume that *S: Xi -^ Xo is surjective and that its null
space N is isomorphic to W under the coordinate (evaluation) map wtQ. Prove that if M
is the nullspace of wtQ in Xi, then S f M is an isomorphism onto Xo by applying this
theorem.
4. THE nTH-ORDER LINEAR EQUATION
The nth-order linear differential equation is the equation
d^'a/dr = Git, a, da/dt, . . . , d'^-^a/dt''-^),
where G{t, a) = G(t, ai, . . . , a^) is now linear from V = W^ to W for each t in /.
We convert this to a first-order equation dot/dt = Fit, ot) just as before, where
now F is a map from I XV ioV that is linear in its second variable a, F(t, ot) =
Our proof of Theorem 1.5 showed that a function/ in 6^"^^/, W) is a solution
of the nth-order equation d^a/df^ = G(t, a, . . . , d^~^a/dt^~^) if and only if
the n-tuplet xl/f = </,/',.-., /^^~'^^> is a solution of the first-order equation
doi/dt = F{tjCt) = Ttipt). We know that the latter solutions form a vector
subspace N of e^{I, W), and since the map yp\f^ </,/',..- ,/^''~^^> is
linear from e^(/, W) to e^{I, W), it follows that the set N of solutions of the
nth-order equation is a subspace of ©"^(Z, W) and yp f iV is an isomorphism from
TV to N. Since the coordinate evaluation ,^^ = tt^ f N is an isomorphism from N
to W^ for each t (Theorem 3.1), it follows that the map
takes N isomorphically to W^. Its null space Mt is a complement of N in C^, as
before. Here Mt is the set of functions / in ©"^(Z, W) such that f(t) = - - - =
j(n-l)(^) = 0.
We now consider the special case W = U. For each fixed tj G is now a linear
map from U^ to U, that is, an element of (R^)*, and its coordinate set with
respect to the standard basis is an n-tuple k= </ci,...,/c^>. Since the
linear map varies continuously with t, the n-tuple k varies continuously with t.
Thus, when we take ^ into account, we have an n-tuple/c(^) = <ki{t)j... , kn{t)>
of continuous real-valued functions on I such that
G{t, Xi, . . . , Xn) = 22 ki{t)Xi.

282 DIFFERENTIAL EQUATIONS 6.1
The solution space N of the nth-order differential equation
d^'a/dr = G(a, . . . , d''-^a/dt''-\ t)
is just the null space of the linear transformation L\Q'^{I,R)—^ Q^{I,R)
defined by
(L/)(0 = /'"'(O - k„it)f"-'\t) fci(<)/(<)•
If we shift indices to coincide with the order of the derivative, and if we let/^''^
also have a coefficient function, then our nth-order linear differential operator L
appears as
(Lm) = a,(t)f^\t) + '-' + ao(t)m.
Giving/^'^^ a coefficient function a„ changes nothing provided an(t) is never
zero, since then it can be divided out to give the form we have studied. This is
called the regular case. The singular case, where a„(0 is zero for some t, requires
further study, and we shall not go into it here.
We recapitulate what our general linear theory tells us about this situation.
Theorem 4.1. L is a surjective linear transformation from the space Q^(I)
of all real-valued functions on / having continuous derivatives througli
order n to the space 6^(7) = 6(7) of continuous functions on 7. Its null
space N is the solution space of our original differential equation. For
each ^0 in ^ the restriction to N of the mapping <Pto ° "^'-f *-^ ^/(^o)? • • • >
f^^~^\to)> is an isomorphism from N to [R^, and the set Mt^ of functions
f in Q^ such that f(to) = - - • = f'^^~^\to) = 0 is therefore a complement
of A" in e'^(7), and determines a Hnear right inverse of L.
The practical problem of "solving" the differential equation L(/) = ^ for/
when g is given falls into two parts. First we have to find the null space N of L,
that is, we have to solve the homogeneous equation L(f) = 0. Since N is an
n-dimensional vector space, the problem of delineating it is equivalent to findin<!;
a basis, and this is clearly the efficient way to proceed. Our first problem
therefore is to find n linearly independent solutions {ui}^ of L(/) = 0. Our second
problem is to find a right inverse of L, that is, a linear way of picking one f such
that L(f) = g for each g. Here the obvious thing to do is to try to make the
formula of Theorem 3.3 into a practical computation. If v is one solution of
L(f) = g, then of course the set of all solutions is the affine subspace N -\- v.
We shall start with the first problem, that of finding a basis {ui}\ of solutions
to L(f) = 0. Unfortunately, there is no general method available, and we have
to be content with partial success. We shall see that we can easily solve the
first-order equation directly, and that if we can find one solution of the nth-
order equation, then we can reduce the problem to solving an equation of order
n — 1. Moreover, in the very important special case of an operator L with
constant coefficients. Theorem 3.4 gives a complete explicit solution.
The first-order homogeneous linear equation can be written in the form
y' + a{i)y = 0, where the coefficient of y' has been divided out. Dividing by //

6.4 THE nTH-ORDER LINEAR EQUATION 283
and remembering that y'jy = (log ?/)', we see that, formally at least, a solution
is given by log ?/ = — Ja(0 dt or y = e~^^^^^^\ and we can check it by
inspection. Thus the equation y' -\- y/t = Q has a solution y = e~^^^ ^ = \/t, as the
reader might have noticed directly.
Suppose now that L is an nth-order operator and that we know one solution
u oi Lf = 0. Our problem then is to find n — 1 solutions Vij . . . jVn—i
independent of each other and of u. It might even be reasonable to guess that these
could be determined as solutions of an equation of order n — 1. We try to find
a second solution v(t) in the form c(t)u{t)j where c(t) is an unknown function.
Our motivation, in part, is that such a solution would automatically be
independent of u unless c{i) turns out to be a constant.
Now if v(t) = c(t)u(t), then v^ = cu' + c'u, and generally
1=0 V/
If we write down L{v) = Yi% aj(t)v^^\t) and collect those terms involving c(t)j
we get
n
L(v) = c(i) ^ ajU^^^ + terms involving c', . . . , c^""^
0
= cL(u) + S(c') = aS(cO,
where aS is a certain linear differential operator of order n — 1 which can be
explicitly computed from the above formulas. We claim that solving S{f) = 0
solves our original problem. For suppose that {gi}1~^ is a basis for the null
space of aS, and set Ci(t) = Jo Qi. Then L(ciu) = S(Ci) = S(gi) = 0 for i =
1, . . . , n — 1. Moreover, u, CiU, . . . , c^-iu are independent, for if u =
El"' k^c^u, then 1 = Zi"' te(0 and 0 = Li"' ^i^KO = Zi"' kigi(t),
contradicting the independence of the set {gi}.
We have thus shown that if we can find one solution u of the nth-order
equation L/ = 0, then its complete solution is reduced to solving an equation
Sf = 0 of order n — 1 (although our independence argument was a little
sketchy).
This reduction procedure does not combine with the solution of the first-
order equation to build up a sequence of independent solutions of the nth-order
equation because, roughly speaking, it "works off the top instead of off the
bottom". For the combination to be successful, we would have to be able to
find from a given nth-order operator a first-order operator aS such that N(S) C
N(L)j and we can't do this in general. However, we can do it when the coefficient
functions in L are all constants, although we shall in fact proceed differently.
Meanwhile it is valuable to note that a seconc^-order equation Lf=0 can
be solved completely if we can find one solution u, since the above argument
reduces the remaining problem to a first-order equation which can then be solved
by an integration, as we saw earlier. Consider, for instance, the equation
2/" — 2y/t^ = 0 over any interval / not containing 0, so that ao(t) = l/t^ is
continuous on /. We see by inspection that u(i) = t^ is one solution. Then we

284 DIFFERENTIAL EQUATIONS G. 1
know that we can find a solution v(t) independent of u(t) in the form v(t) = t^c{t)
and that the problem will become a first-order problem for c'. We have, in fact,
v' = ec' + 2tc and v" = th" + W + 2c, so that L{v) = v" ~ 2v/t^ -
t^c" + 4^c', and L{v) = 0 if and only if (c')' + (4/Oc' = 0. Thus
(to within a scalar multiple; we only want a basis!), and v = t^c{t) = 1//.
(The reader may wish to check that this is the promised solution.) The null
space of the operator L(/) = /" — 2//^^ is thus the linear span of {i^, l/t].
We now turn to an important tractable case, the differential operator
Lf = aj'^' + an-if""-'' + • • • + ao/,
where the coefficients ai are constants and a„ might as well be taken to 1. What,
makes this case accessible is that now L is a polynomial in the derivative operatoi'
D. That is, if Df = /', so that D'f = f'\ then L = p(D), where p(x) -
The most elegant, but not the most elementary, way to handle this equation
is to go over to the equivalent first-order system dx/dt = T(x) on U^ and to
apply the relevant theory from the last section.
Theorem 4.2. If p(t) = (t — h)^, then the solution space N of the
constant coefficient nth-order equation p(D)f = 0 has the basis
{e'\te'\...,t^-'e''}.
If p(t) is a polynomial which has a relatively prime factorization p(t) =^
Ua Viif) with each p^{t) of the above form, then the solution space of thc^
constant coefficient equation p{D)f = 0 has the basis \JB^, where Bi is th(;
above basis for the solution space Ni of pi(D)f = 0.
Proof. We know that the mapping ^:/»—> <f,f'j • • • ,/^'^~"^^> is an
isomorphism from the null space N of p(D) to the null space N of dx/dt — T{x). It is
clear that yp commutes with differentiation, \p{Df) = </', . . . ,f'^^> = D\p(f),
and since we know that N is invariant under D by Lemma 3.1, it follows (and can
easily be checked directly) that N is invariant under D. By Lemma 3.2 we hav(^
T = (ft o D o (p~^^ which simply says that the isomorphism <pt-^ -^ ff^^ takes
the operator D on N into the operator T on U^. Altogether (pt "^ ^P takes D
on N into T on W", and since p(D) = 0 onN, it follows that p(T) = 0 on R^.
We saw in Theorem 3.4 that if p(T) = 0 and p = (^ — by, then the
solution space N of dx/dt = T{x) is spanned by vectors of the form
e^'x, . . . , r-^e^'x.
The first coordinates of the n-tuple-valued functions g in N form the space N
(under the isomorphism / = ^~^g), and we therefore see that N is spanned by
the functions e^\ . . . , t^~^e^K Since N is n-dimensional, and since there are ?t
of these functions, the spanning set forms a basis.

6.4 THE nTH-ORDER LINEAR EQUATION 285
The remainder of the theorem can be viewed as the combination of the
above and the direct application of Theorem 5.5 of Chapter 1 to the equation
p(Z)) = 0 onNj or as the carry-over to N under the isomorphism \l/~^ of the facts
already established for N in the last section. D
If the roots of the polynomial p are not all real, then we have to resort to
the complexification theory that we developed in the exercises of Section 11,
Chapter 4. Except for one final step, the results are the same. The one extra
fact that has to be applied is that the null space of a real operator T acting on a
real vector space Y is exactly the intersection with Y of the null space of the
complexification S of T acting on the complexification Z = Y ® lY oi Y.
This implies that if p{t) is a polynomial with real coefficients, then we get the
real solutions of p(D)f = 0 as the real parts of the complex solutions. In order
to see exactly what this means, suppose that q(x) = {x^ — 26a; + c)"^ is one of
the relatively prime factors of p(x) over U, with x^ — 26a; + c irreducible over U.
Over C, q(x) factors into (x — \)'^(x — \)'^, where X = 6 + zco and o:^ = c — h^.
It follows from our general theory above that the complex 2m-dimensional
null space of q(D) is the complex span of
{e^\ te^\ . . . , r-'e^\ e^\ te^\ . . . , f^-^^'}.
The real parts of the complex linear combinations of these 2m functions is a
2m-dimensional real vector space spanned by the real parts of the above functions
and the real parts of i times the above functions. That is, the null space of the
real operator q{D) is a 2m-dimensional real space spanned by
{e^^ cos co^, te^^ cos co^, . . . , r~^e^^ cos co^; e^^ sin co^, . . . , r~^e^^ sin co^.
Since there are 2m of these functions, they must be independent and must form
a basis for the real solution space of q(D)f = 0. Thus,
Theorem 4.3. If p(t) = (t^ + 2ht + c)'^ and h^ < c, then the solution space
of the constant coefficient 2mth-order equation p{D)f = 0 has the basis
{t e cos o:t} i=o ^ {t e sm o)t} i=o ?
where o)^ = c — h^. For any polynomial p(t) with real coefficients, if p(t) ==
III Pi(i) is its relatively prime factorization into powers of linear factors and
powers of irreducible quadratic factors, then the solution space N of
p(D)f = 0 has the basis (J i ^i? where Bi is the basis for the null space of
Pi(D) that we displayed above if pi(t) is a power of an irreducible quadratic,
and Bi is the basis of Theorem 4.2 if pi(t) is a power of a linear factor.
Suppose, for example, that we want to find a basis for the null space of
D^ - 1 = 0. Here p{x) = x^ - 1 = (x - l)(x + l)(x - i){x + ^). The
basis for the complex solution space is therefore {e\ e~\ e'^\ e~^^}. Since e'^^ =
cos t + i sin i, the basis for the real solution space is {e\ e~\ cos t, sin i).

286 DIFFERENTIAL EQUATIONS 6.4
The same problem for Z)^ — 1 = 0 gives us
p{x) = x^ - 1= {x- l){x^ + X + 1)
so that the basis for the complex solution space is
and the basis for the real solution space is
{e^ e-'l^ cos WZt/2), e'"^ sin (V^t/2)}.
*Our results above suggest that the collection Gi of all real-valued
solutions of constant coefficient homogeneous linear differential equations
contains the functions f^, e^\ cos co^, sin ojt for all z, r, and co, and is closed under
addition and multiplication, and is in fact the algebra generated by these functions.
We can easily prove this conjecture. We first consider sums. Suppose that
^(/) = 0 and that aS(^) = 0, where aS and T are two such constant coefficient
operators. Then / + ^ is in the null space oi S o T because S and T commute:
{S o T)(f+g) = (S o T)(f) + (S o T){g) = S(Tf) + T{Sg) = 0 + 0 = 0. We
know that aS and T commute because they are both polynomials in D.
In order to treat products, we first have to recognize that the linear span of
all the trigonometric functions sin at, cos bt is an algebra. In other words, any
finite product of such functions is a linear combination of such functions. This
is the role of a certain class of trigonometric identities, such as 2 sin x cos y =
sin (x + 2/) + sin (x — y), which the reader has undoubtedly had to struggle
with. (And again the mystery disappears when we are allowed to treat them
as complex exponentials.) Then we observe that any function in the algebra d is
a finite sum of terms each of which is of the form fe*"^ sin cot or fe*"^ cos cot for
some i, r, and co. We can exhibit an operator T having such a function in its
null space, and our finite sum of such terms will then be in the null space of the
composition of these operators T by our first argument.
We are tempted to say one more thing. The functions f, e^\ sin cot, cos cot,
and sums of their products can be shown to be exactly the continuous functions
/: R —> R such that the set of translates of / has a finite-dimensional span. That
is, if we define translation through x, K^, by (Kxf)(t) = f(t — x), then for
exactly the above functions/the linear span of {Kxf, x G U} is finite-dimensional.
This second characterization of exactly the same class of functions cannot be
accidental. Part of the secret lies in the fact that the constant coefficient
operators T are exactly those linear differential operators that commute with
translation. That is, if r is a linear differential operator, then T o Kx = K^ o T ior
all X if and only if T has constant coefficients. Now we have noted in an early
chapter that U T o S = S o T, then the null space of T is invariant under S.
Therefore, the null space iV of a constant coefficient operator T is invariant under
all translations: Kx[N] C N for all x. Now we know that N is finite-dimensional

6.4 THE nTH-ORDER LINEAR EQUATION 287
from our differential equation theory. Therefore, the functions in N are such
that their translates have a finite-dimensional span!
This device of gaining additional information about the null space A^ of a
linear operator T by finding operators aS that commute with T, so that N is
AS-invariant, is much used in advanced mathematics. It is especially important
when we have a group of commuting operators aS, as we do in the above case with
the operators aS = Kx.
What we have not shown is that if a continuous function / is such that its
translation generates a finite-dimensional vector space, then / is in the null space
of some constant coefficient operator p{D). This is delicate, and it depends on
showing that if {Kt} is a one-parameter family of linear transformations on a
finite-dimensional space such that Ks-\-t = Kg "^ Kt and K^ —> / as ^ —> 0, then
there is an aS in Hom V such that Kt = e^^.^
EXERCISES
Find solutions for the following equations.
4.2 x" +2x' — Sx = 0
4.4 x" + 2a:' + a: = 0
a: = 0 4.6 x'" — a: = 0
4.8 x''' = 0
4.10 Solve the initial-value problem x'' + 4x' — 5a: = 0, a:(0) = 1, x\0) = 2.
4.11 Solve the initial-value problem a:'" + a;' = 0, a:(0) = 0, a:'(0) = — 1, a:"(0) = 1.
4.12 Find one solution u of the equation 4^^a:" + a: = 0 by trying u{t) = r, and then
find a second solution as in the text by setting v{t) = c{t)u{t).
4.13 Solve t^x'" — Stx' + 3a: = 0 by trying u(t) = r.
4.14 Solve ^a:" + a:' = 0.
4.15 Solve ^(a:'" + a:') + 2(a:" + x) = 0.
4.16 Knowing that e~^^ cos ojt and e~^^ sin ojt are solutions of a second-order linear
differential equation, and observing that their values at 0 are 1 and 0, we know that
they are independent. Why?
4.17 Find constant coefficient differential equations of which the following functions
are solutions: t^, sin t, t^ sin t.
4.18 If / and g are independent solutions of a second-order linear differential equation
w" + aiw' + a2U = 0 with continuous coefficient functions, then we know that the
vectors <f{x), f\x)> and "^gix), g'{x)> are independent at every point x. Show
conversely that if two functions have this latter property, then they are solutions
of a second-order differential equation.
4.19 Solve the equation (D — a)^/ = 0 by applying the order-reducing procedure
discussed in the text starting with the obvious solution e"'K
4.1
4.3
4.5
4.7
4.9
x" — 3a:' + 2a: = 0
x" + 2a:' + 3a: = 0
a:'" - 3a:" + 3a:' -
a:(6) - x" = 0
x'" - a:" = 0

288 DIFFERENTIAL EQUATIONS 6.5
5. SOLVING THE INHOMOGENEOUS EQUATION
We come now to the problem of solving the inhomogeneous equation L(/) = g.
We shall briefly describe a practical method which works easily some of the time
and a theoretical method which works all the time, but which may be hard to
apply. The latter is just the translation of Theorem 3.3 into matrix language.
We first consider the constant coefficient equation L(/) = ^ in the special
case where g itself is in the null space of a constant coefficient operator aS. A
simple example is y^ — ay = e^^ (or y^ — ay = sin ht), where g(t) = e^^ is in the
null space of S = (D — h). In such a situation a solution / must be in the
null space of aS o L, for aS q L(/) = S(g) = 0. We know what all these functions
are, and our problem is to select / among them such that L(/) is the given g.
For the moment suppose that the polynomials L and aS (polynomials in D)
have no factors in common. Then we know that L is an isomorphism on the
null space Ns of aS and therefore that there exists an / in Ns such that Lf = g.
Since we have a basis for Ns, we could construct the matrix for the action of L on
Ns and find / by solving a matrix equation, but the simplest thing to do is take
a general linear combination of the basis, with unknown coefficients, let L act
on it, and see what the coefficients must be to give g.
For example, to solve y' — ay = e^\ we try f(t) = ce^^ and apply
L:(D - a)(ce^') = (h - a)ce^' I e^\
and we see that c = 1/(6 — a).
Again, to solve y' — ay =^ cos ht, we observe that cos ht is in the null space
of aS= Z)^ + 6^ and that this null space has the basis {sin ht, cos ht]. We
therefore set f{t) = ci sin 6^ + C2 cos ht and solve (D — a)f = cos ht, getting
(—aci — hc2) sin ht + (6ci — ac2) cos ht = cos ht,
—aci — hc2 = 0,
hci — ac2 = I,
and
h a
a^ + V a^ + h^
When L and aS do have factors in common, the situation is more complicated,
but a similar procedure can be proved to work. Now an extra factor t'^ must be
introduced, where i is the number of occurrences of the common factor in L.
For example, in solving (D — r)^f = e^\ we have S o L = (D — r)^, and so
we must set f(t) = ct^e^K Our equation then becomes
(D - r)hth'' = 2ce'' I e\
and so c = \.
For (D^ + 1)/ = sin t we have to set /(O = ^(ci sin t + C2 cos i), and after
we work it out we find that ci = 0 and C2 = —\, so that j =^ —\t cos i.

6.5 SOLVING THE INHOMOGENEOUS EQUATION 289
This procedure, called, naturally, the method of undetermined coefficients,
violates our philosophy about a solution process being a linear right inverse. Indeed,
it is not a single process, applicable to any g occurring on the right, but varies
with the operator aS. However, when it is available, it is the easiest way to
compute explicit solutions.
We describe next a general theoretical method, called variation of parameters,
that is a right inverse to L and does therefore apply to every g. Moreover, it
inverts the general (variable coefficient) linear nth-order operator L:
mm = i: ai{t)f\t).
0
We are assuming that we know the null space N of L; that is, we assume
known n linearly independent solutions {ui}\ of the homogeneous equation
Lf = 0. What we are going to do is to translate into this context our formula
Kt Jo K~^{g{s)) ds for the solution to da/dt = Tt{a) + g(t). Since
is an isomorphism from the solution space N of the nth-order equation L(/) = 0
to the solution space N of the equivalent first-order system dx/dt = Tt{\)j it
follows that if we have a basis {ui}\ for N, then the columns of the matrix
Wij = vij~ ^^ form a basis for N.
Let w(0 be the matrix Wij(}) = Uj'^~^\i), Since evaluation at t is the
isomorphism (ft from N to R"^, the columns of w{t) form a basis for W^, for each t.
But Kt{(x.) is the value at t of the solution of dx/dt = Tt{\) through the initial
point < 0, a > , and it follows that the linear transformation Kt takes the columns
of the matrix w(0) to the corresponding columns of w{t). The matrix for Kt is
therefore w(0 • w(0)~\ and the matrix form of our formula
m = Ktf\Ksr'(g(s))ds
Jo
is therefore
f(^) = w(0 • w(0)-^ • r w(0) • w(s)-^ • g(s) ds,
Jo
Moreover, since integration commutes with the application of a constant linear
transformation (here multiplication by a constant matrix), the middle w(0)
factors cancel, and we have the result that
{{t)=w{t)' r w(sr'' g{s) ds
Jo
is the solution of dx/dt = Tt{x) + g{t) which passes through <0, 0> . Finally,
set k(s) = w(s)~^ • g(s)j so that this solution formula splits into the pair
m = Mt) ■ C k(s) ds, w(«) ■ k(s) = g(«).
-'0

290 DIFFERENTIAL EQUATIONS 6.5
Now we want to solve the inhomogeneous nth-order equation L(/) = g, and
this means solving the first-order system with g = <0, , , , ,0, g>, Therefore,
the second equation above is equivalent to
^ Wij{s)kj{s) = 0, i < n,
3
X; Wnj{.s)kj{s) = g{s).
3
Moreover, the solution / of the nth-order equation is the first component of the
n-tuple f (that is, / = ^"^f), and so we end up with
n C n
f(t) = 2 '^ii(0 / kj(s) ds = J^ Uj{t)cj{t),
j=i Jo 1
where Ci{t) is the antiderivative Jq hi{s) ds. Any other antiderivative would do
as well, since the difference between the two resulting formulas is of the form
2Z^ aiUi(t), a solution of the homogeneous equation L(f) = 0. We have proved
the following theorem.
Theorem 5.1. If {ui(t)}^i is a basis for the solution space of the homogeneous
equation L(h) = 0, and if f(t) = J2\ Ci(t)ui(t), where the derivatives cl(t)
are determined as the solutions of the equations
E c'S)u[^\t) = 0, i = 0, . . . , n - 2,
i
i: c'i{t)uT-'\t) = g{t),
i
then L(/) = g.
We now consider a simple example of this method. The equation y" -\- y =
sec X has constant coefficients, and we can therefore easily find the null space of
the homogeneous equation 2/" + 2/. A basis for it is {sin x, cos x}. But we can't
use the method of undetermined coefficients, because sec x is not a solution of a
constant coefficient equation. We therefore try for a solution
v{x) = Ci{x) sin X + C2(x) cos x.
Our system of equations to be solved is
c'l sin X + C2 cos X = 0,
c'l cos X — C2 sin X = sec x.
Thus C2 = — Ci tan x and cJ(cos a: + sin x tan x) = sec x, giving
c[ = 1, C2 = —tan X,
Ci = X, C2 = log cos X,
and
v{x) = X sin X + (log cos x) cos x.
(Check it!)

6.5
SOLVING THE INHOMOGENEOUS EQUATION
291
This is all we shall say about the process of finding solutions. In cases where
everything works we have complete control of the solutions of L(/) = g^ and
we can then solve the initial-value problem. If L has order ^^, then we know that
the null space N is ^^-dimensional, and if for a given g the function v is one
solution of the inhomogeneous equation L(/) = g, then the set of all solutions is
the ^^-dimensional plane (affine subspace) M = N -\- v. If we have found a
basis {uiW for iV, then every solution of L{f) = g is of the form / = X! i CiUi + v.
The initial-value problem is the problem of finding / such that L(/) = g and
fito) = air (to) = al,., j'^'-'^to) = al where <a?, . . . , a^> = a« is the
given initial value. We can now find this unique / by using these n conditions
to determine the n coeflficients Ci in / = X! ^i^i + ^- We get n equations in the n
unknowns Ci, Our ability to solve this problem uniquely again comes back to the
fact that the matrix Wij{to) = Uj''~^\to) is nonsingular, as did our success in
carrying out the variation of parameters process.
We conclude this section by discussing a very simple and important example.
When a perfectly elastic spring is stretched or compressed, it resists with a
"restoring" force proportional to its deformation. If we picture a coiled spring
lying along the x-axis, with one end fixed and the free end at the origin when
undisturbed (Fig. 6.3), then when the coil is stretched a distance x (compression
being negative stretching), the force it exerts is —ex, where c is a constant
representing the stiffness, or elasticity, of the spring, and the minus sign shows
that the force is in the direction opposite to the displacement. This is Hookers
law.
Fig. 6.3
Suppose that we attach a point mass m to the free end of the spring, pull the
spring out to an initial position Xq = a, and let go. The reader knows perfectly
well that the system will then oscillate, and we want to describe its vibration
explicitly. We disregard the mass of the spring itself (which amounts to
adjusting m), and for the moment we suppose that friction is zero, so that the
system will oscillate forever. Newton's law says that if the force F is applied to
the mass m, then the particle will accelerate according to the equation
d^x ^
Here F = —ex, so the equation combining the laws of Newton and Hooke is
dt^x . -
rn— + cx = Q.

292 DIFFERENTIAL EQUATIONS 6.5
This is almost the simplest constant coefficient equation, and we know that the
general solution is
X = ci sin Qt + C2 cos Q^,
where Q = \^c/m. Our initial condition was that x = a and x^ = 0 when t = 0.
Thus C2 = a and Ci = 0, so x = a cos Qt, The particle oscillates forever
between x = —a and x = a. The maximum displacement a is called the
amplitude A of the oscillation. The number of complete oscillations per unit time
is called the frequency /, so / = Q/27r = \/c/27r\/m. This is the quantitative
expression of the intuitively clear fact that the frequency will increase with the
stiffness c and decrease as the mass m increases. Other initial conditions are
equally reasonable. We might consider the system originally at rest and strike
it, so that we start with an initial velocity v and an initial displacement 0 at
time t = 0. Now C2 = 0 and x = ci sin ^t. In order to evaluate ci, we
remember that dx/dt = V Sit t = 0, and since dx/dt = ciQ cos ^t, we have v = ciQ
and ci = v/^, the amplitude for this motion. In general, the initial condition
would hex = a and x^ = v when t = 0, and the unique solution thus determined
would involve both terms of the general solution, with amplitude to be calculated.
The situation is both more realistic and more interesting when friction is
taken into account. Frictional resistance is ideally a force proportional to the
velocity dx/dt but again with a negative sign, since its direction is opposite to
that of the motion. Our new equation is thus
'^lP + ^Tt + ''' = ^'
and we know that the system will act in quite different ways depending on the
relationship among the constants m, /c, and c. The reader will be asked to explore
these equations further in the exercises.
It is extraordinary that exactly the same equation governs a freely oscillating
electric circuit. It is now written
J. d^x . j^dx . I ^
where L, R, and C are the inductance, resistance, and capacitance of the circuit,
respectively, and dx/dt is the current. However, the ordinary operation of such
a circuit involves forced rather than free oscillation. An alternating (sinusoidal)
voltage is applied as an extra, external, "force" term, and the equation is now
■f (J/ X . j-K tlX . X . ,
This shows the most interesting behavior of all. Using the method of
undetermined coefficients, we find that the solution contains transient terms that
die away, contributed by the homogeneous equation, and a permanent part of
frequency co/27r, arising from the inhomogeneous term a sin o)t. New phenomena
called phase and resonance now appear, as the reader will discover in the exercises.

5.1
5.4
5.6
5.7
x" — X = t"^ 5.2 x'' — X = sin
x^^ -\- X = sin t 5.5 y^^ — y^ = x^
y" -y' = e-
Consider the equation y" -^ y = sec x thj
6.5 SOLVING THE INHOMOGENEOUS EQUATION 293
EXERCISES
Find particular solutions of the following equations.
5.3 x'' — X = sin t + t"^
(Here y^ = dy/dx.)
sec X that was solved in the text. To what
interval / must we limit our discussion? Check that the particular solution found in
the text is correct. Solve the initial-value problem for
f\x)+f{x) = sec:c, /(O) = 1, f{0) = -1.
Solve the following equations by variation of parameters.
5.8 x" +x = tan t 5.9 x'" -^ x' = t 5.10 y" -^ y = I
5.11 2/^^^ — 2/ = cos X 5.12 y" +42/ = sec 2x 5.13 y" + 4^/ = sec x
5.14 Show that the general solution
Ci sin ^t + C2 cos ^t
of the frictionless elastic equation m(dH/dt^) -\- ex = 0 can be rewritten in the form
A sin {^t — a).
(Remember that sin (x — y) = sin x cos y — cos x sin y.) This type of motion along
a line is called simple harmonic motion.
5.15 In the above exercise express A and a in terms of the initial values dx/dt = v
and X = a when t = 0.
5.16 Consider now the freely vibrating system with friction taken into account, and
therefore having the equation
m(dH/dt^) + k (dx/dt) + ex = 0,
all coefficients being positive. Show that if k^ < 4mc, then the system oscillates forever,
but with amplitude decreasing exponentially. Determine the frequency of oscillation.
Use Exercise 5.14 to simplify the solution, and sketch its graph.
5.17 Show that if the frictional force is sufficiently large (k^ > 4mc), then a freely
vibrating system does not in fact vibrate. Taking the simplest case k^ = 4mc, sketch
the behavior of the system for the initial condition dx/dt = 0 and x = a when ^ = 0.
Do the same for the initial condition dx/dt = v and a; = 0 when ^ = 0.
5.18 Use the method of undetermined coefficients to find a particular solution of the
equation of the driven electric circuit
d X dx X
L-Z-Z-+ R-^+- = asinco^.
dt^ dt c
Assuming that R > 0, show by a general argument that your particular solution is in
fact the steady-state part (the part without exponential decay) of the general solution.

294 DIFFERENTIAL EQUATIONS 6.0
5.19 In the above exercise show that the "current" dx/dt for your solution can b(^
written in the form
dx a
— = sin (o)t — a),
where X = Lo) — l/coC Here a is called the phase angle.
5.20 Continuing our discussion, show that the current flowing in the circuit will have
a maximum amplitude when the frequency of the "impressed voltage" a sin cot is
l/27r\/LC. This is the phenomenon of resonance. Show also that the current is in
phase with the impressed voltage (i.e., that a: = 0) if and only ii L = C = 0.
5.21 What is the condition that the phase a be approximately 90°? —90°?
5.22 In the theory of a stable equilibrium point in a dynamical system we end up with
two scalar products (?, rj) and ((?, r;)) on a finite-dimensional vector space F, the
quadratic form q(^) = ^((?, ?)) being the potential energy and p(?0 = ^(?', ?0 being the
kinetic energy. Now we know that dqaiO = ((<^j ?)) ^^^ similarly for p, and because of
this fact it can be shown that the Lagrangian equations can be written
dt
(|..)^ («..)).
Prove that a basis {l3i] ^ can be found for V such that this vector equation becomes the
system of second-order equations
d Xi .
—- = XiXi, I = 1, . . . ,n,
where the constants X^ are positive. Show therefore that the motion of the system is the
sum of n linearly independent simple harmonic motions.
6. THE BOUNDARY-VALUE PROBLEM
We now turn to a problem which seems to be like the initial-value problem but
which turns out to be of a wholly different character. Suppose that T' is a second-
order operator, which we consider over a closed interval [a, b]. Some of the most
important problems in physics require us to find solutions to T{f) = g such that
/ has given values at a and 6, instead of / and /' having given values at a single
point ^0- This new problem is called a boundary-value problem, because {a, b} is
the boundary of the domain / = [a, b]. The boundary-value problem, like the
initial-value problem, breaks neatly into two subproblems if the set
M= {/ee^([a,6]):/(a)=/(5) = 0}
turns out to be a complement of the null space N of T, However, if the reader
will consider this general question for a moment, he will realize that he doesn't
have a clue to it from our initial-value development, and, in fact, wholly new
tools have to be devised.
Our procedure will be to forget that we are trying to solve the boundary-
value problem and instead to speculate on the nature of a linear differential

6.6 THE BOUNDARY-VALUE PROBLEM 295
operator T from the point of view of scalar products and the theory of self-
adjoint operators. That is, our present study of T will be by means of the scalar
product (/, g) = ja fif)g(t) dt, the general problem being the usual one of
solving Tf = ghy finding a right inverse S of T. Also, as usual, S may be
determined by finding a complement M of N{T). Now, however, it turns out that if T
is "formally self-adjoint", then suitable choices of M will make the associated
right inverses S self-adjoint and compact^ and the eigenvectors of aS, computed as
those solutions of the homogeneous equation Tf — rf = 0 which lie in M, then
allow (relatively) the same easy handling of S, by virtue of Theorem 5.1 of
Chapter 5, that they gave us earlier in the finite-dimensional situation.
We first consider the notion of "formal adjoint" for an nth-order linear
differential operator T. The ordinary formula for integration by parts.
[''f'9 = f9]' - ['fg',
Ja -ia Ja
allows the derivatives of/ occurring in the scalar product {Tf^ g) to be shifted
one at a time to g. At the end, / is undifferentiated and g is acted on by a certain
^^th-order linear differential operator R. The endpoint evaluations, like the
above fgfa, that accumulate step by step can be described as
0<i+j<n
where the coefficient functions kij{x) are linear combinations of the coefficient
functions ai{x) and their derivatives. Thus
(Tf, g) = (/, Rg) + B{f, g)t
The operator R is called the formal adjoint of T, and if R = T, we say that T is
formally self-adjoint.
Every application of the integration by parts formula introduces a sign
change, and the reader may be able to see that the leading coefficient of R is
(—1)^ times the leading coefficient of T. Assuming this, we see that a necessary
condition for formal self-adjointness is that ^^ be even, so that R and T have the
same first terms.
Supposing that T is formally self-adjoint, we seek a complement M of the
null space iV of 7" in e^([a, h\) with the further property that aS, the associated
right inverse of T, is self-adjoint as a mapping from the pre-Hilbert space
e^([a, 6]) to itself. Let us see what this further requirement amounts to. For
any Uj v G (3^, set f = Su and g = Sv^ so that / and g are in M and u = Tf,
V = Tg. Then (u, Sv) = (Tf, g) = (/, Tg) + B{f, g)t = (Su, v) + B{S, g)t
We thus have:
Lemma 6.1. If T' is a formally self-adjoint differential operator and M is a
complement of the null space of 7", then the right inverse of T determined
by M is self-adjoint if and only if
f,geM =^ B{f, g)ta = 0.

296 DIFFERENTIATi EQUATIONS 6.6
From now on we sliall consider only the second-order case* Howe¥er, almost
everything tliat we are going to do works perfectly well for the general case, the
price of generality being only additional notational complexity.
We start by computing the formal adjoint of the second-order operator
Tf = C2r + ci/' + Col We have
{Tig) = f%2fV-- ['c.^fg f f'cofg,
J a -J a J a
I' cj'g ^ Cifgf ~ I'ficigy,
J a Ja J a
r C2f"g = C2f'gt - [''f'(c2gy
J a Jn Ja
Ja Ja
giving
and
Ba g) =- e^if'g - g[n + id - c^fg.
Thus Eg = ctg'' + (2c.^ - Ci)g' + « - ci + co)g and ^ -= f if and only if
2c2 — ci =■ ct (and €2 -- cj == 0)^ that is, C2 =^ ci. We have proved:
Lemma 6,2. The second-order differential operator T is formally self-
adjoint if and onl^^ if
Tf = cj" -i cy + c^f ^ (c^y + Co/,
in which case
B{f, g) = c^iJ'g - .97).
A constant coefficient operator is thus formally self-adjoint if and only
if ci = 0.
Supposing that T is formally self-ad joint, we now try to find a complement M
of its null space N such that /, g e M ^^ B(/, g)]l = 0. Since N is two-dimen-
sional; any complement M can be described as the intersection of the null space
of two linear functional h and I2 on X2 ■= C^{{a,!)]), For example^ the "one
point" complement Mt^ that we had earlier in connection with the initial-value
problem is the intersection of the null spaces of the two functionals hif) = /(io)
and IsC/) = f^ih)* Here^ however, the vanishing of li and I2 for two functions
/ and g must imply that ii(/, g)]l = C2{f'g — //)]« = 0, and the functionals
li(/) must therefore involve the values of / and /^ at a and at fc. We would
naturally guessj and it can be proved^ that each of li and 12 must be of the form,
m - ktfia) + k2fid) h hm + hfih)^
Our problem can therefore be restated as follows. We must find two linear
functionals li and I3 of the above general form such that if M is the intersection

6.6 THE BOUNDAMY-VALUE PHOBLEM 297
of their null spaces^ then
a) M is a complement of N^ and
b) f,geM=^ C2{f'g - o'!)t = 0,
in which case we call the boundary condition li(/) -^ mf) ^ 0 self-adjoint.
Lemiiia 6.3. We can replace (a) by
a') T is injective on ill.
Proof, If T is injective on M, then M n N = {0} j so that the map
/-> <hif),h(f)>
is injective on N, and thereforej because N is two-dimensional, is an isomorphism
from F to R^ (by the corollary of Theorem 2.4, Chapter 2). Then M is a
complement of N by Theorem 5.3 of Chapter 1. D
Now we can easily write down various pairs li and /2 that form a self-adjoint
boundary condition. We list some below.
1) / CE M ^ m = fib) = 0 Ithat is, h(J) = M and hif) = Kb)].
2) fsM^fia) =r{b) -0.
3) ]\lore generally, f'{a) — kf{d), f\b) == cf{h), (Tn fact^ li can be any I that
depends only on the values at a, and 12 can be any I that depends only on b.
Thus hif) = kjia) + t2/'(a), and if hif) - hiy) =- 0, then the pairs
</(a),/'(a)> and <gia), c/ia)> are dependent, since both lie in the one-
dimensioTml mill space of h, ^^^^ ^^^ /V 9JU = 0- The same holds for I2
and 5j so that this split pair of endpoint conditions makes fe(/, g)fa = 0 by
making the values of B at a and at b separately 0,)
4) If C2ia) = C2(5), then take/ e M ^ fia) - fib) and/'(a) - fib). That
is, hif) = fia) - fib) and hif) = fia) ^ fib).
We now show that in every case but (3) the condition (a') also holds if wc
replace Thy T -- X for a suitable A. This is true also for case (3), but the proof is
harder, and we shall omit it.
Lemma 6.4. Suppose that M is defined by one of the self-adjoint boundary
conditions (1), (2), or (4) above, that €^(0 S: w > 0 on [a, b], and that
X > co(0 + 1 on [a, b]. Then
2 I 11/112
1((7^-X)/J)| > m\:f'\'i-\ 11/
2
for all / G. .1/. In particular, M is a complement of the null space of T -™- X
and licence defines a self-adjoint right inverse of T — \.
Proof, We have
((X - T)f,f) = [' icjyf] f (X - Co)f
Ja Ja
i5 fb
C2f'f] + [ C,{ff + [ (X --. C,)f
Ja Ja Ja

298 DIFFERENTIAL EQUATIONS 6.G
Under any of conditions (1), (2), or (4), C2fJ]t = 0, and the two integral terms
are clearly bounded below by m||/'||2 and ||/||2, respectively. Lemma 6.3 then
implies that M is a complement of the null space of T — X. D
We come now to our main theorem. It says that the right inverse S oi T — X
determined by the subspace M above is a compact self-adjoint mapping of the
pre-Hilbert space e^([a, 6]) into itself, and is therefore endowed with all the rich
eigenvalue structures of Theorem 5.1 of the last chapter. First we present some
classical terminology. A Sturm-Liouville system on [a, 6] is a formally self-adjoint
second-order differential operator Tf = (C2/O' + Cof defined over the closed
interval [a, h], together with a self-adjoint boundary condition ^i(/) = hif) = 0
for that interval. If C2{t) is never zero on [a, b], the system is called regular. If
02(0) or 02(6) is zero, or if the interval [a, b] is replaced by an infinite interval
such as [a, 00], then the system is called singular.
Theorem 6.1. li T: li, I2 is Si regular Sturm-Liouville system on [a, b], with
C2 positive, then the subspace M defined by the homogeneous boundary
condition is a complement of A^(T' — X) if X is taken sufficiently large, and
the right inverse of T' — X thus determined by M is a compact self-adjoint
mapping of the pre-Hilbert space e^([a, 6]) into itself.
Proof. The proof depends on the inequality of the above lemma. Since we have
proved this inequality only for boundary conditions (1), (2), and^(4), our proof
will be complete only for those cases.
Set^ g= (T- X)f. Since ||r7||2||/||2 > l((^ - A)/,/)i by the Schwarz
inequality, we see from the lemma first that II/II2 < ||{/||2||/||2) so that
11/1I2 < hh,
and then that m||/'III < WghWfh < ||<7||1, so that
ll/'IU < Wgh/Vm.
We have already checked that the right inverse S of the formally self-
adjoint T — \ defined by M is self-adjoint, and it remains for us to show that the
set S[U] = {/: ||g||2 < 1} has compact closure. For any such / the Schwarz
inequality and the above inequality imply that
\Ky) - f{x)\ < / l/'l = / l/'l • 1 < WS'Uy - x\"' < ^^
1/2
\/m
Thus S{U] is uniformly equicontinuous. Since the common domain of the
functions in S[U] is the compact set [a, 6], we will be able to conclude from
Theorem 6.1 of Chapter 4 that the set S[U] is totally bounded if we can show
that there is a constant C such that all the functions in S[U] have their ranges in
[—C, C]. Taking y and x in the last inequality where |/| assumes its maximum
and minimum values, we have ||/||oo — min |/| < (6 — a)^^^/Vm. But

6.6 THE BOUNDARY-VALUE PROBLEM 299
(min |/|)(6 - a)''' < II/II2 < hh < 1, and therefore
ll/IU <c = 1/(6 - a)"' + ib- ayyv^.
Thus AS[t/] is a uniformly equicontinuous set of functions mapping the
compact set [a, b] into the compact set [—C, C], and is therefore totally bounded in
the uniform norm. Since C{[a, h]) is complete in the uniform norm, every sequence
in S[U] has a subsequence uniformly converging to some / G C, and since
II/II2 ^ {b — a)^^^||/||oo, this subsequence also converges to / in the two-norm.
We have thus shown that if H is the pre-Hilbert space G{[a, h]) under the
standard scalar product, then the image S[U] of the unit ball U C H under S has
the property that every sequence in S[U] has a subsequence converging in H.
This is the property we actually used in proving Theorem 5.1 of Chapter 5, but
it is not quite the definition of the compactness of S, which requires us to show
that the closure S[U] is compact in H. However, if {g^} is any sequence in this
closure, then we can choose {f^} in S[U] so that \\^n — tn\\ < V^- The
sequence {fn} has a convergent subsequence {^n{m)}m as above, and then {^n{m)}m
converges to the same limit. Thus aS is a compact operator. D
Theorem 6.2. There exists an orthonormal sequence {cpn} consisting entirely
of eigenvectors of T and forming a basis for M. Moreover, the Fourier
expansion of any f G M with respect to the basis {cpn} converges uniformly
to / (as well as in the two-norm).
Proof. By Theorem 5.1 of Chapter 5 there exist an eigenbasis for the range of S,
which is M. Since Scpn = r^ipn for some nonzero r^, we have {T — \){rn^n) =
(Pn and Tcpn = ((1 + ^Tn)/rn)(Pn' The uniformity of the series convergence
comes out of the following general consideration.
Lemma 6.5. Suppose that T' is a self-adjoint operator on a pre-Hilbert
space V and that T is compact as a mapping from F to <V, q>, where g is a
second norm on V that dominates the scalar product norm p (q > cp).
Then T is compact (from ptop), and the eigenbasis expansion X! ^n^n of an
element 13 in the range of T converges to 13 in both norms.
Proof. Let U be the unit ball of V in the scalar product norm. By the hypothesis
of the lemma, the g-closure B of T[U] is compact. B is then also p-compact, for
any sequence in it has a g-convergent subsequence which also p-converges to the
same limit, because p < cq. We can therefore apply the eigenbasis theorem.
Now let a and ^ = T{a) have the Fourier series X! (^i^i and X! ^i^i, and let
T{(Pi) = rupi. Then 6^- = r^-, because hi = {T{a), ipi) = (a, T{(pi)) =
(a, rupi) = ri{a, cpi) = ritti. Since the sequence of partial sums X!i aicpi is
p-bounded (Bessel's inequality), the sequence {2]i hicpi} = {T(^1 anpi)] is
totally g-bounded. Any subsequence of it therefore has a subsubsequence
g-converging to some element 7 in V. Since it then p-converges to T, 7 must be fi.
Thus every subsequence has a subsubsequence g-converging to fi^ and so
{Si ^i^i\ itself g-converges to fi by Lemma 4.1 of Chapter 4. D

300 DIFFERENTIAL EQUATIONS 6.6
EXERCISES
6a Given that Tf{x) = xf'{x)+f{x) and Sf{x) = f\x), compute T o SsmdSo T.
6.2 Show that the differential operators T = aD and S = bD commute if and
only if the functions a{x) and b{x) are proportional.
6.3 Show that the differential operators T = aD^ and S = bD commute if and
only if b(x) is a first-degree polynomial b{x) = ex -\- d and a{x) = k(b{x))^.
6.4 Compute the formal adjoint aS of T if
a) Tf = r, b) Tf = r, c) Tf = r, d) iTf)(x) = xf\x),
e) {Tf){x) = x^f\x).
6.5 Let S and T be linear differential operators of orders vi and n, respectively.
What are the coefficient conditions for aS o T to be a linear differential operator of
order m + n?
6.6 Let T be the second-order linear differential operator
(T/)(0 = a2{t)f"{t) + ai{t)f{t) + aoiOm.
What are the conditions on its coefficient functions for its formal adjoint to exist?
What are these conditions for T of order n?
6.7 Let S and T be linear differential operators of order m and n, respectively, and
suppose that all coefficients are C°°-functions (infinitely differentiable). Prove that
SoT — ToSisoi order < m+n — 1.
6.8 A 5-blip is a continuous nonnegative function (p such
that ip = 0 outside of [—8, 8] and fl^cp = 1 (Fig. 6.4). We
assume that there exists an infinitely differentiable 1-bHp (p.
Show that there exists an infinitely differentiable 5-blip for
every 5 > 0. Define what you would mean by a 5-blip centered
at X, and show that one exists.
Fig. 6.4
6.9 Let/ be a continuous function on [a, b] such that (/, g) = fa fg = 0 whenever
g is an infinitely differentiable function which vanishes near a and b. Show that / = 0.
(Use the above exercise.)
6.10 Let e'°([a, b]) be the vector space of infinitely differentiable functions on [a, b],
and let T be a second-order linear differential operator with coefficients in 6°°;
(T/)(0 = a2{t)f\t) + aiit)f{t) + aoit)m.
Let aS be a linear operator on ©'^([a, b]) such that
{Tf, g) - (/, Sg) = K(f, g)
is a bilinear functional depending only on the values of /, g, f', and g' at a and b.
Prove that S is the formal adjoint of T. [Hint: Take / to be a 5-bHp centered at .r.
Then K{f, g) = 0. Now try to work the assertion to be proved into a form to which
the above exercise can be applied.]

6.7 FOURIER SERIES 301
6.11 Prove an nth-order generalization of the above exercise.
6.12 Let X be the space of Hnear differential operators with e°°-coefficients, and let At
be the formal adjoint of T. Prove that T -^ At is an isomorphism from X to X.
Prove that ^(to^) = As° At.
7. FOURIER SERIES
There are not many regular Sturm-Liouville systems whose associated ortho-
normal eigenbases have proved to be important in actual calculations. Most
orthonormal bases that are used, such as those due to Bessel, Legendre, Hermite,
and Laguerre, arise from singular Sturm-Liouville systems and are therefore
beyond the limitations we have set for this discussion. However, the most well-
known example, Fourier series, is available to us.
We shall consider the constant coefficient operator Tf = Z)^/, which is clearly
both formally self-adjoint and regular, and either the boundary condition
/(O) = /(tt) = 0 on [0, tt] (type 1) or the periodic boundary condition/(—tt) =
/(tt), n-T) = f\ir) on [-TT, tt] (type 4).
To solve the first problem, we have to find the solutions of f" — \f = 0
which satisfy /(O) = /(tt) = 0. If X > 0, then we know that the
two-dimensional solution space is spanned by {e'^^, e~^^}, where r = X^^^. But if Cie^^ +
C2e~^^ is 0 at both 0 and tt, then ci = C2 = 0 (because the pairs < 1, 1 > and
<e^^, e~^^> are independent). Therefore, there are no solutions satisfying the
boundary conditions when X > 0. If X = 0, then f{x) = Cix -\- Cq and again
ci = Co = 0.
If X < 0, then the solution space is spanned by {sin rx, cos rx], where
r = (—X)^^^. Now if Ci sin rx + C2 cos rx is 0 at x = 0 and x = tt, we get,
first, that C2 = 0 and, second, that rw = utt for some integer n. Thus the
eigenfunctions for the first system form the set {sin nx} ^, and the corresponding
eigenvalues of D^ are {—n^}i.
At the end of this section we shall prove that the functions in e^([a, 6])
that are zero near a and 6 are dense in e([a, 6]) in the two-norm. Assuming this,
it follows from Theorem 2.3 of Chapter 5 that a basis for M is a basis for C^, and
we now have the following corollary of the Sturm-Liouville theorem.
Theorem 7.1. The sequence {smnx}i is an orthogonal basis for the pre-
Hilbert space e^[0, tt]). If / g e\[0, tt]) and /(O) = /(tt) = 0, then the
Fourier series for / converges uniformly to /.
We now consider the second boundary problem. The computations are a
little more complicated, but again if/(x) = cie^^ + C2e~^^, and if/(—tt) = /(tt)
and /'(—tt) = /'(tt), then / = 0. For now we have
cie""^ + C2e"^ = cie"^ + C2e-"^
giving ci = C2, and
Cire~^'^ — C2re^^ = Cire^^ — C2re~^'^,

302 DIFFERENTIAL EQUATIONS 6.7
giving Ciie^"^ — e~^'^) = 0, and so Ci = 0. Again/(a:) = CiX + Cq is ruled out
Finally, if/(x) = Ci sin rx + C2 cos rx, our boundary conditions become
2c 1 sin rw = 0 and 2rc2 sin ttt = 0,
so that again r = n, but this time the full solution space of (D^ + n^)f = 0
satisfies the boundary condition.
Theorem 7.2. The set {sin nx} ^ U {cos ?ix} 0 forms an orthogonal basis for
the pre-Hilbert space e^([-7r, tt]). If / G e2([-7r, w]) and /(-tt) = /(tt),
/'(—tt) = /'(tt), then the Fourier series for / converges to / uniformly on
[ — TT, tt].
Remaining "proof. This theorem follows from our general Sturm-Liouvilie dis-
cussion except for the orthogonality of sin nx and cos nx. We have
/TT
sin nt cos nt dt
-IT
/TT
sin 2nt dt
-IT
= — (l/4n) cos 2na:]!L7r
= 0.
Or we can simply remark that the first integrand is an odd function and therefore
its integral over any symmetric interval [—a, a] is necessarily zero.
The orthogonality of eigenvectors having different eigenvalues follows of
course, as in the proof of Theorem 3.1 of Chapter 5. D
Finally, we prove the density theorem we needed above. There are very
slick ways of doing this, but they require more machinery than we have
available, and rather than taking the time to make the machines, we shall prove the
theorem with our bare hands.
It is standard notation to let a subscript zero on a symbol denoting a class
of functions pick out those functions in the class that are zero "on the boundary"
in some suitable sense. Here eo([^) ^]) will denote the functions in e([a, b]) that
are zero in neighborhoods of a and b, and similarly for GlUa, b]).
Theorem 7.3. (3^([a, b]) is dense in e([a, b]) in the uniform norm, and
Go(K b]) is dense in e([a, b]) in the two-norm.
Proof. We first approximate / G e([a, b]) to within e by a piecewise "linear"
function g by drawing straight line segments between the adjacent points on the
graph of / lying over a subdivision a = Xq < Xi < - - • < Xn = b of [a,b\.
If/ varies by less than e on each interval (x^_i, Xi), then ||/ — g\\^ < e. Now
g\t) is a step function which is constant on the intervals of the above
subdivision. We now alter g\t) slightly near each jump in such a way that the new
function h{t) is continuous there. If we do it as sketched in Fig. 6.5, the total

6.7
FOURIER SERIES
303
Fig. 6.5
Fig. 6.6
integral error at the Jump is zero, J^^.lt/ (h — ^0 — 0, and the maximum error
Jj^.l_3 is 6A/4. This will be less than e if we take d = e/||^'||oo, since A < 2||^'||oo.
We now have a continuous function h such that \ja h{t) dt — {f{x) — /(a)) | < 2e.
In other words, we have approximated / uniformly by a continuously differ-
entiable function.
Now choose g and h in e^([a, 6]) so that first ||/ — ^||oo < e/2 and then
\\c/ - h\U < e/2(h - a). Then
m) - g(o)
JO
h\ < e/2,
and so H{x) = jo h + g{0) is a twice continuously differentiate function such
that 11/ — H\\ao < €. In other words, e^([a, h]) is dense in e([a, 6]) in the
uniform norm. It is then also dense in the two-norm, since
ii/iu = (£f)
1/2
<
.(/:o
1/2
(6-a)
1/211
But now we can do something which we couldn't do for the uniform norm:
we can alter the approximating function to one that is zero on neighborhoods of a
and 5, and keep the two-norm approximation good. Given 6, let e{t) be a non-
negative function on [a, b] such that e{t) = 1 on [a + 26, 6 — 2^], e{t) = 0 on
fa, a+ d] and on [6 — 8, b], e" is continuous, and ||6i|oo = 1. Such an e{t) clearly
exists, since we can draw it. We leave it as an interesting exercise to actually
define e(^). Here is a hint: Show somehow that there is a fifth-degree polynomial
p(0 having a graph between 0 and 1 as shown in Fig. 6.6, with a zero second
derivative at 0 and at 1, and then use a piece of the graph, suitably translated,
compressed, rotated, etc., to help patch together e{i).
Anyway, then \g — eg\2 ^ il(7lU(46)^^^ for any g on e([a, 6]), and if g has
continuous derivatives up to order 2, then so does eg. Thus, if we start with/ in C
and approximate it by g in C^, and then approximate g by eg, we have altogether
the second approximation of the theorem. D

304 DIFFERENTIAL EQUATIONS 6.7
EXERCISES
7.1 Convert the orthogonal basis {sm nx]"^ for the pre-Hilbert space (3([0, tt]) to
an orthonormal basis.
7.2 Do the same for the orthogonal basis {sin nx]i U {cos nx] o foi' G([—tt, tt]).
7.3 Show that {sin nx] i is an orthogonal basis for the vector space V of all odd
continuous functions on [—TT, tt]. (Be clever. Do not calculate from scratch.)
Normalize the above basis.
7.4 State and prove the corresponding theorem for the even functions on [—tt, tt].
7.5 Prove that the derivative of an odd function is even, and conversely.
7.6 We now want to prove the following stronger theorem about the uniform
convergence of Fourier series.
Theorem. Let / have a continuous derivative on [—tt, tt], and suppose that
/(—tt) = /(tt). Then the Fourier series for/ converges to / uniformly.
Assume for convenience that / is even. (This only cuts down the number of
calculations.) Show first that the Fourier scries for /' is the series obtained from the
Fourier series for / by term-by-term differentiation. Apply the above exercises here.
Next show from the two-norm convergence of its Fourier series to /' and the Schwarz
inequality that the Fourier series for/ converges uniformly.
7.7 Prove that (cos nx] o is an orthonormal basis for the space 71/ of 6^-functions
on [0, tt] such that/'(0) = /'(tt) = 0.
7.8 Find a fifth-degree polynomial p{x) such that
p{0) = p\0) = p'\0) = 0, p\l) = p"(l) = 0, p{l) = 1.
(Forget the last condition until the end.) Sketch the graph of p.
7.9 Use a "piece" of the above polynomial p to construct a function e{x) such that e^
and e^^ exist and are continuous, e{x) = 0 when x < a ^ 3 and x > b — 3, e{x) = 1
on [a+ 23, b — 23], and ||e||oo = 1.
7.10 Prove the AVcierstrass theorem given below on [0, tt] in the following stei)s. We
know that / can be uniformly ai)i)roximated by a 6^-function g.
1) Show that c and d can be found and that g{t) — cit) — c? is 0 at 0 and tt.
2) Use the Fourier series ex})ansion of this function and the ]\Iaclaurin series for
the functions sin nx to show that the polynomial p{x) can be found.
Theorem {The Weierstrass approximation theorem). The polynomials are dense
in 6([a, b]) in the uniform norm. That is, given any continuous function / on
[a, b] and any e, there is a polynomial p such that \j{x) — p{x)\ < e for all x
in [a, b].

CHAPTER 7
MULTILINEAR FUNCTIONALS
This chapter is principally for reference. Although most of the proofs will be
included, the reader is not expected to study them. Our goal is a collection of
basic theorems about alternating multilinear functional, or exterior forms, and
the determinant function is one of our rewards.
1. BILINEAR FUNCTIONALS
We have already studied various aspects of bilinear functionals. We looked at
their duality implications in Section 6, Chapter 1, we considered the "canonical
forms" of symmetric bilinear functionals and their equivalent quadratic forms
in Section 7, Chapter 2, and, of course, the whole scalar product theory of
Chapter 5 is the theory of a still more special kind of bilinear functional. In this
chapter we shall restrict ourselves to bilinear and multilinear functionals over
finite-dimensional spaces, and our concerns are purely algebraic.
We begin with some material related to our earlier algebra. If V and W are
finite-dimensional vector spaces, then the set of all bilinear functionals on
V X W is pretty clearly a vector space. We designate it F* ® TF* and call it
the tensor product of F* and TF*. Our first theorem simply states something
that was implicit in Theorem 6.1 of Chapter 1.
Theorem 1.1. The vector spaces 7* ® TF*, Hom(y, TF*), and Hom(TF, 7*)
are naturally isomorphic.
Proof. We saw in Theorem 6.1 of Chapter 1 that each/in F* (x) TF* determines
a linear mapping a i-^ /« from TF to F*, where/«(^) = /(^, a), and we also noted
that this correspondence from F* ® TF* to Hom(TF, F*) is bijective. All that
the present theorem adds is that this bijective correspondence is linear and so
constitutes a natural isomorphism, as does the similar one from F* ® TF* to
Hom(F, TF*). To see this, let fr be the bilinear functional corresponding to T in
Hom(F, TF*). Then f.r+S) = It + fs. for f,T+S){a, ^) = ({T + S){a)){^) =
{T{a) + S{a))(^) = (T{a)){P) + {S{a))(^) = Ma, ^) + fs(a, ^). We can
do the same for homogeneity.
The isomorphism of F* ® TF* with Hom(TF, F*) follows in exactly the
same way by reversing the roles of the variables. We are thus finished with the
proof. D
305

306 MULTILINEAR FUNCTIONALS 7.2
Before looking for bases in F* ® TF*, we define a bilinear functional 7 (x) X
for any two functional 7 G 7* and X G TT* by (7 ® X)(^, rj) = y(^)Hv)'
We call 7 ® X the tensor product of the functional 7 and X and call any bilinear
functional having this form elementary. It is not too hard to see that/G F* ® TF*
is elementary if and only if the corresponding T G Hom(y, PF*) is a dyad.
If V and W are finite-dimensional, with dimensions m and n, respectively,
then the above isomorphism of F* (x) TF* with Hom(y, TF*) shows that the
dimension of F* ® W* is mn. We now describe the basis determined by given
bases in V and W.
Theorem 1.2. Let {ai}"^ and {Pj}1 be any bases for V and TF, and let their
dual bases in F* and TF* be {jUi}? and {^y}?. Then the mn elementary
bilinear functional {m ® Vj} form the corresponding basis for F* ® TF*.
Proof. Since jUi ® ^y(^, rj) = iJLi{^)Pj{rj) = XiVj, the matrix expansion/(^, rj) =
Hij tijXiyj becomes/(^, v) = Hij Ujifii ® Pj)(^, v) or
id
The set {^li ® ^y} thus spans F* ® TF*. Since it contains the same number of
elements {mn) as the dimension of F* ® TF*, it forms a basis. D
Of course, independence can also be checked directly. If Ylij kjifJ^i ® ^i) =
0, then for every pair <k,l>,tki= lli,j Uj{^i ® Vj){au, fii) = 0.
We should also remark that this theorem is entirely equivalent to our
discussion of the basis for Hom(F, TF) at the end of Section 4, Chapter 2.
2. MULTILINEAR FUNCTIONALS
All the above considerations generalize to multilinear functionals
/: Fi X • • • X Fn -^ R.
We change notation, just as we do in replacing the traditional <x, y> G H^^ by
X = <xi, . . . J Xn> G U^. Thus we write f{ai, . . . , a-n) = f{oi), where
oi = <ai, . . . , an> G Fi X • • • X F^. Our requirement now is that
/(«!, . . , ,an)
be a linear functional of aj when ai is held fixed for all i 9^ j. The set of all such
functionals is a vector space, called the tensor product of the dual spaces
Ft, ... , F*, and is designated F? ® • • • ® F*.
As before, there are natural isomorphims between these tensor product
spaces and various Hom spaces. For example,
Vt®"'®V* and Hom(Fi, F| ® • • • ® F*)
are naturally isomorphic. Also, there are additional isomorphisms of a variety

7.2 MULTILINEAR FUNCTIONALS 307
not encountered in the bilinear case. However, it will not be necessary for us to
look into these questions.
We define elementary multilinear fiinctionals as before. If X^ G Vf^ i =
1, . . . , n, and f = < ^i, - - - , ^n>, then
(Xl®---®Xn)(€) = Xl(h)---Xn(U.
To keep our notation as simple as possible, and also because it is the case of
most interest to us, we shall consider the question of bases only when Vi =
V2 = '" = Vn= V. In this case (7*)® = 7* ® • • • ® 7* (in factors) is
called the space of covariant tensors of order n (over V).
If {ay} 7 is a basis for V and /g (7*)®, then we can expand the value
f(^) = fi^i) ' ' ' ) ^n) with respect to the basis expansions of the vectors ^i just
as we did when / was bilinear, but now the result is notationally more complex.
If we set ^i = J^Y=i ^]^3 for z = 1, . . . , n (so that the coordinate set of ^i is
X* = {x'j} j) and use the linearity of the/(^i,. .. , ^n) in its separate variables one
variable at a time, we get
/v5l) • • • ) ?nj ^^ 2^ Xp^Xp^ ' ' ' Xp^J[ap^j Oip^, ' ' ' ) ^Pn))
where the sum is taken over all n-tuples p= <Pi,.-.,Pn^ such that I < Pi <
m for each i from 1 to m. The set of all these n-tuples is just the set of all
functions from {1, . . . , n} to {1, . . . , m}. We have designated this set m^, using the
notation n = {1, . . . , n}, and the scope of the above sum can thus be indicated
in the formula as follows:
j{kl) ' ' ' ) knj "= 2^_^Pi ' ' ' XpjioLp^^ ' • ' ) ^PrJ-
A strict proof of this formula would require an induction on n, and is left to the
interested reader. At the inductive step he will have to rewrite a double sum
UpG;^ IZyGm_as the single sum l]qGm^i+i using the fact that an ordered pair
<p, j> in m^ X m is equivalent to an (n + l)-tuplet q G m^+^ where qi = pi
for i = 1, . . . , n and ^n+i = J-
If {jUz}T is the dual basis for 7* and q G m^, let jUq be the elementary
functional ju^^ ® • • • ® fJiq^' Thus/;tq(«pi, • • • , oLpJ = IIiM^/^p,) = 0 unless p = q,
in which case its value is 1. More generally.
Therefore, if we set Cq = /(«3i, - • • ? ««„), the general expansion now appears as
K^ly ' ' ' , ^n) = 1]_ CpMp(^1, • . • , ^n)
or / = X Cp/Xp, which is the same formula we obtained in the bilinear case, but
with more sophisticated notation. The functionals {jUp : p G m^} thus span
(7*)®. They are also independent. For, if X) ^pMp = 0? then for each q,
Cq = 2] CpjUp(a^j, . . . , aqj = 0. We have proved the following theorem.

308 MULTILINEAR FUNCTIONALS 7..'^
Theorem 2.1. The set {jUp : p G m^} is a basis for (7*)®. For any / in
(F*)® its coordinate function {Cp} is defined by Cp = Roip^, . . . , oip^).
Thus/ = Z CpMp and/(^i, . . . , ^n) = Z CpjUp(^i, • • • , ^n) = Z Cp x^^ • • • x^^
for any / G (7*)® and any < h, . . . , ^^> G V.
Corollary. The dimension of (F*)® is m^.
Proof. There are m^ functions in m^, so the basis {jUp : p G m^} has m^
elements. D
3. PERMUTATIONS
A permutation on a set aS is a bijection f:S-^S. If §>{S) is the set of all
permutations on S, then S = S(aS) is closed under composition (a, p G $ =^ <t o p E:S)
and inversion (a G S =^ a~^ G S). Also, the identity map / is in S, and, of course,
the composition operation is associative. Together these statements say exactly
that S is a group under composition. The simplest kind of permutation other
than / is one which interchanges a pair of elements of S and leaves every other
element fixed. Such a permuation is called a transposition.
We now take S to be the finite set n = {1, . . . , n} and set Sn = S(ri).
It is not hard to see that then any permutation can be expressed as a product of
transpositions, and in more than one way.
A more elementary fact that we shall need is that if p is a fixed element of §>n,
then the mapping (7 i-^ (7 o p is a bijection S^ "-^ ^n- It is surjective because
any d' can be written d' = (d' o p~i) o p, and it is injective because ai o p =
(72 o p =» {ci o p) o p-i = ((72 o p) o p-^ =f^ q-^ = (j.^. Similarly, the mapping
a i-^ p o a {p fixed) is bijective.
We also need the fact that there are n\ elements in S^. This is the
elementary count from secondary school algebra. In defining an element a G §>n,
(7(1) can be chosen in n ways. For each of these choices (7(2) can be chosen in
n — 1 ways, so that <(7(1), (7(2) > can be chosen in n{n — 1) ways. For each of
these choices (7(3) can be chosen in n — 2 ways, etc. Altogether (7 can be chosen
in n{n — l)(n — 2) • • • 1 = n! ways.
In the sequel we shall often write ^pa^ instead of ^p ° (t\ just as we
occasionally wrote ^ST' instead of 'S o T^ for the composition of linear maps.
If 5 = < ^1, • • • , ^n> G y^ and (7 G Sn, then we can "apply (7 to f", or
"permute the elements of < ^i, . . . , ^n^ through a". We mean, of course, that
we can replace < ^i, . . . , ^n^ by < ^^(d, . . . , ^cr(n)^) that is, we can replace {
by 5 o (7.
Permuting the variables changes a functional /g (F*)® into a new such
functional. Specifically, given / G (F*)® and a G Sn? we define f by
r(€) = M ° ^"') = /(^cr-l(l), . . . , ^cr-l(n)).
The reason for using a~^ instead of a is, in part, that it gives us the following
formula.

7.4 THE SIGN OF A PERMUTATION 309
Lemma 3.1. /"^i^^^ = {f^Y^
Proof, f^'^'ii) = /({o (<ri oa^)-') = /({ o (cr^ i o<j-,')) = ^ ^ a^') o a',') =
fi(fo<r2^)= (r'P({)- □
Theorem 3.1. For each (7 in S|2. the mapping T^ defined, by /1—> f^ is a linear
isomorphism of (F*)® onto itself. The mapping (7 i-^ 7"^ is an antihomo-
morphism from the group S^ to the group of nonsingular elements of
Hom((y*)®).
Proof. Permuting the variables does not alter the property of multilinearity, so
T^ maps (7*)© into itself. It is linear, since {af+hg)'' = af + he/. And
Tpa = TffO Tp, because/'^ = (fy. Thuscr i-^ T^ preserves products, but in the
reverse order. This is why it is called an an^zhomomorphism. Finally,
7^(^-1) o T, = 7^(,,-i) = Ti = I,
so that Tff is invertible (nonsingular, an isomorphism). D
The mapping (T »-> 7"^ is a representation (really an an^irepresentation) of the
group §>n by linear transformations on (F*)®.
Lemma 3.2. Each T'^ carries the basis {jUp} into itself, and so is a
permutation on the basis.
Proof. We have (n^Y^^) = Mp(€ o a-') = UU fJ^pM^'Hi))- Setting^ = a-\i)
and so having i = <t(j), this product can be rewritten 11?=! l^Pa(j)(^j) = Mpo(r(€)-
Thus
(Mp)"" = Mpo
poff)
and since p »-^ p o (j is a permutation on m^, we are done. D
4. THE SIGN OF A PERMUTATION
We consider now the special polynomial EonW^ defined by
E{x) = E(xi, . . . ^Xn) = n (^* — ^j)'
This is the product over all pairs <i,j> ^n X n such that i < j. This set of
ordered pairs is in one-to-one correspondence with the collection P of all pair
sets {if j} C n such that i 9^ j, the ordered pair being obtained from the
unordered pair by putting it in its natural order. Now it is clear that for any
permutation (7 G Sn, the mapping {z, j) •—^ \^(j)) ^U)} is a permutation of P.
This means that the factors in the polynomial £"^(x) = E{x o a) are exactly the
same as in the polynomial E{x) except for the changes of sign that occur when a
reverses the order of a pair. Therefore, if n is the number of these reversals, we
have E'' = {—l^E. The mapping a i-^ (—1)^ is designated 'sgn' (and called
"sign"). Thus sgn is a function from §>n to {1, —1} such that E^ = {sgn(T)E,

310 MULTILINEAR FUNCTIONALS 7.5
for all (7 G Sm. It follows that
sgnpo- = (sgnp) (sgno-),
for {sgnpa)E = E^'' = (E^y = {sgna)EP = (sgnp) {sgna)E, and we can
evaluate E at any n-tuple x such that E(x) 9^ 0 and cancel the factor £'(x).
Also
sgn (7 = — 1 if (7 is a transposition.
This is clear if g interchanges adjacent numbers because it then changes thc^
sign of just one factor in £'(x); we leave the general case as an exercise for the
interested reader.
5. THE SUBSPACE QJ" OF ALTERNATING TENSORS
Definition. A covariant tensor /g (F*)® is symmetric if f^ = f for all
o-G Sn-
If / is bilinear [/ g (F*)®], this is just the condition /(^, rj) = f{rj, Q for
all ^,^ G F.
Definition. A covariant tensor / G (F*)® is antisymmetric or alternating if
f = (sgn a)f for all a G Sn-
Since each (T is a product of transpositions, this can also be expressed as the
fact that / just changes sign if two of its arguments are interchanged. In the
case of a bilinear functional it is the condition/(^, rj) = —/{v, 0 for all ^jV ^V.
It is important to note that if/is alternating, then/({) = 0 whenever the n-tuple
5 = < ^1, • • • , ^n^ is not injective {^i = ^j for some i 9^ j). The set of all
symmetric elements of (F*)® is clearly a subspace, as is also the (for us) more
important set d^ of all alternating elements. There is an important linear
projection from (F*)® to a^ which we now describe.
Theorem 5.1. The mapping /h-> (l/n!)X^^Es^ (sgn (t)/"^ is a projection il
from (F*)® to a^.
Proof. We first check that Q/G a^ for every / in (F*)®. We have (Q/)^ -
{l/n\)J2<x (sgnd)/"^^. Now sgncr = (sgndp)(sgnp). Setting a' = a o p and
remembering that (7 1-^ cr' is a bijection, we thus have
(SlfY = ^^E (sgn or'= (sgnp)(Q/).
Hence Q/ G a^.
If / is already in a"^, then f = (sgncr)/ and Q/ = (l/^!)Zl(rGs„/- Since S„
has n! elements, Q/ = /. Thus Q is a projection from (F*)® to d^. U
Lemma 5.1. Q(/0 = (sgn p)Q/.
Proof. The formula for 12(/^) is the same as that for (12/)^ except that pa replaces
ap. The proof is thus the same as the one for the theorem above. D

7.5 THE SUBSPACE O/^ OF ALTERNATING TENSORS 311
Theorem 5.2. The vector space d^ of alternating n-linear functionals over
the m-dimensional vector space V has dimension (^).
Proof. If / G (1^ and / = Y.p CpjUp, then since f = (sgn a)/, we have
Up CpjUpocr = Up (sgn(7)CpjUp for any a in Sn- Setting p o (j = q, the left sum
becomes Xlq Cqo(r-iMq) and since the basis expansion is unique, we must have
Cqo(r-i = sgn (TCq or Cp = (sgn(7)Cpocr for all p E m^. Working backward, we see,
conversely, that this condition implies that f = (sgncr)/. Thus / G (2,^ if and
only if its coordinate function Cp satisfies the identity
<^p = (sgn(7)Cpocr for all p Grrf and all a G Sn-
This has many consequences. For one thing, 0^=0 unless p is one-to-one
(injective). For if pi = pj and a is the transposition interchanging i and j\ then
p OCT = p, Cp = (sgn(7)Cpocr = —Cp, and so Cp = 0. Since no p can be injective
if n > m, we see that in this case the only element of d^ is the zero functional.
Thus n > m=^ dim a"" = 0.
Now suppose that n < m. For any injective p, the set {p o a :a G Sn}
consists of all the (injective) n-tuples with the same range set as p. There are
clearly n\ of them. Exactly one q = p ° o" counts off the range set in its natural
order, i.e., satisfies qi < q2 < ' ' ' < ^n- We select this unique q as the
representative of all the elements p o a having this range. The collection C of these
canonical (representative) g's is thus in one-to-one correspondence with the
collection of all (range) subsets of m = {1, . . . , m} of size n.
Each injective p G m"" is uniquely expressible as p = q o o- for some q G C,
a G Sn- Thus each / in a"" is the sum Y^c^gc JlaG§>^ ^qo^rMqc^r- Since t^^a = (sgn a)tq,
this sum can be rewritten Y^^gc t^ Y.<x (sgn(7)jUqo(r = ZlqGC ^q^q, where we have
set Vq = Lcr (sgno-)/Xqocr = n\Q{fjLq).
We are just about done. Each v^ is alternating, since it is in the range of 12,
and the expansion
f=^ Mq
qGC
which we have just found to be valid for every f G Gi^ shows that the set
{j^q : q G C} spans a"". It is also independent, since Z!qGC ^q^q = UpGW" ^pMp
and the set {jUp} is independent. It is therefore a basis for 6i^.
Now the total number of injective mappings p from n = {1, . . . , n} to
rn = {1, . . . , m} is m(m — 1) • • • (m — n + 1)? for the first element can be
chosen in m ways, the second in m — 1 ways, and so on down through n choices,
the last element having m — (n— l) = m — n~}~l possibilities. We have seen
above that the number of these p's with a given range is n\. Therefore, the
number of different range sets is
, ^. m —n+1 ^! /mX
m{m — 1) 1 = —j7 TT =1 I •
ni nl{m — n)l \nj
And this is the number of elements q G C D

312 MULTILINEAR FUNCTIONALS 7.()
The case n = m is very important. Now C contains only one element, the
identity / in S^, so that
/ = CiVi = CiJ2 (Sgn <T)fla
and
/(^l, . . . , ^m) = C/ X) (sgn(T)iUcr(l)(h) * * * Mcr(m)(^m)
<T
= ClU (Sgn (T)xi(i) • • • X^cm).
This is essentially the formula for the determinant, as we shall see.
6. THE DETERMINANT
We saw in Section 5 that the dimension of the space d^ of alternating m-forms
over an m-dimensional F is O = 1. Thus, to within scalar multiples there is only
one alternating m-linear functional D over V = W^, and we can adjust thc^
constant so that Z)(5\ . . . , 8^) = 1. This uniquely determined m-form is thc^
determinant functional, and its value Z)(x\ . • • > x^) at the m-tuple <x\ ..., x^>
is the determinant of the matrix x = {xij} whose jth column is x-^ for,/ = 1,. .., ni.
Lemma 6.1. D(t\ . . . , t^) = T^aGS^ (sgno-)^^(i),i • • • ^,r(m),m.
Proof. This is just the last remark of the last section, with the constant cj = 1,
since i)(5\ . . . , 8^) = 1, and with the notation changed to the usual matrix
form tij. D
Corollary 1. i)(t*) = D(t).
Proof. If we reorder the factors of the product ^^^,1 * * * ^cr^.m in the order of the*
values ai, the product becomes ti,p^ • • • tm,p^, where p = a~^. Since
is a bijection from S^ to §>n, and since sgn((7~^) = sgncr, the sum in the lemma
can be rewritten as ^pG§>^ (sgnp) ti,p^ • • • tm,p^' But this is
E(sgnpK^,i---^p^,^=i)r). □
Corollary 2. D{t) is an alternating m-linear functional of the rows of t.
Now let dim V = m, and let / be any nonzero alternating m-form on V, For
any T in Hom V the functional/r defined by/7^(^i, . . . , ^n) = /(^h, • • • , ^^n)
also belongs to d'^. Since d^ is one-dimensional, fr = ^rf for some constant hr.
Moreover, Ut is independent of/, since if Qt = kr^g and g = c/, we must havc^
cfr = /cy^'c/and kr^ = kr- This unique constant is called the determinant of T]
we shall designate it A(T'). Note that A(T') is defined independently of any basis
for 7.
Theorem 6.1. A(aS o T) = A{S) A{T).

7.6 THE DETERMINANT 313
Proof
A{SoT)f(^„ . . . , ^^)=f(^SoT){^,), . . . , (^o 7^)(^^))
= f{S(T{^,)),...,S{T{U)))
= A{S)f{m,), . . . , T{U)) = A{T) A(^)/(^i, . . . , U.
Now divide out /. D
Theorem 6.2. If 6 is an isomorphism from V to W, and if 7" G Hom V and
S = doT o d-\ then A(aS) = A(7^).
Proof. If / is any nonzero alternating m-form on W^ and if we define g by
f/(^i) • • • , ?n) = fi^iii ' ' ' , Q^n)' Then ^ is a nonzero alternating m-form on V.
Now f{S o d^u ...,So dU) = A(S)f{d^, . . . , d^n) = HS)g{^i,. . • , ?n),
and also/(^ o e^u • • • , ^ ° ^^n) = fiS o T^^i,. . . , ^ o T^) = g{T^,, . . . , T^U =
HT)g(^i, . . . , ^n). Thus A{S)g = A{T)g and A(^) = A{T). D
The reader will expect the two notions of determinant we have introduced
to agree; we prove this now.
Corollary 1. If t is the matrix of T with respect to some basis in V, then
D{t) = A{T).
Proof. If 6 is the coordinate isomorphism, then T = 6 o T o e~^ is in Hom W^
and A{T) = A{T) by the theorem. Also, the columns of t are the m-tuple
T{8_^), . . . , ?(5^). Thus_i)(t) = D{t\ . . . , t^) = D{T{8^), ..., ?(5^)) =
A{T) D{8\ . . . , 5^) = A{T). Altogether we have D{t) = A{T). D
Corollary 2. If s and t are mX m matrices, then Z)(s • t) = Z)(s) D{t).
Proof. D{s • t) = A(^ o T) = A{S) A{T) = D{s) D{t). D
Corollary 3. D{t) = 0 if and only if t is singular.
Proof. If t is nonsingular, then t~^ exists and D{t) D{t~^) = D{tt^^) =
D{I) = 1. In particular, D{t) f^ 0. If t is singular, some column, say ti, is a
linear combination of the others, ti = Zl^ c^•t^, and Z>(t], . . . , t^) =
Z^ CiD{ti, t2, . . . , t^) = 0, since each term in the sum evaluates D at an
//i-tuple having two identical elements, and so is 0 by the alternating property. D
We still have to show that A has all the properties we ascribed to it in
Chapter 2. Some of them are in hand. We know that A{S o T) = A{S) A{T),
and the one- and two-dimensional properties are trivial. Thus, if T interchanges
independent vectors ai and 0:2 in a two-dimensional space, then its matrix with
respect to them as a basis is t = [? ol? ^^^ so A(T) = D{t) = —1.
The following lemma will complete the job.
Lemma 6.2. Consider Z)(t) = Z)(t\ . . . , t"^) under the special assumption
that t"^ = d"^. If s is the (m — 1) X (m — 1) matrix obtained from the
m X m matrix t by deleting its last row and last column, then Z)(s) = ^(t).

314 MULTILINEAR FUNCTIONALS 7.(')
Proof. This can be made to follow from an inspection of the formula of Lemma
6.1, but we shall argue directly.
If t has 5"^ also as itsjth column for soniej f^ m, then of course Z)(t) = 0
by the alternating property. This means that Z)(t) is unchanged if the jth column
is altered in the mth place, and therefore D{t) depends only on the values Uj in
the rows i 9^ m. That is, Z)(t) depends only on s. Now t 1-^ s is clearly a sur-
jective mapping to O^^-^x^-i^ and, as a function of s, Z)(t) is alternatinj^:
(m — 1)-linear. It therefore is a constant multiple of the determinant D
on IR^'^~^^^^'^~^\ To see what the constant is, we evaluate at
s= <5S..., 5^-i>.
Then Z)(s) = 1 = Z)(t) for this special choice, and so Z)(s) = Z)(t) in general. 11
In order to get a hold on the remaining two properties, we consider an
m X m matrix t whose last m — n columns are 5^"^^ . . . , 5^, and we apply
the above lemma repeatedly. We have, first, D{t) = i)((t)^^), where (t)'"'"
is the (m— l)X(m— 1) matrix obtained from t by deleting the last row and
the last column. Since this matrix has 8^~^ as its last column (5^~^ being now
an {m — 1)-tuple), the same argument shows that its determinant is the same
as that of the (m — 2) X (m — 2) matrix obtained from it in the same way.
We can keep on going as long as the 5-columns last, and thus see that D{i) is tin•
determinant of the nX n matrix that is the upper left corner of t. If we interprc^l
this in terms of transformations, we have the following lemma.
Lemma 6.3. Suppose that V is m-dimensional and that T in Hom V is
the identity on an (m — n)-dimensional subspace X. Let F be a
complement of X, and let p be the projection on Y along X. Then p o (T \ Y) can
be considered an element of Hom Y and A{T) = Ay(p ° {T \ F)).
Proof. Let on, . . . , o:^ be a basis for F, and let an-\-i) . . . , o:^ be a basis for A'.
Then {a:,} 7 is a basis for V, and since T{ai) = ai for i = n + 1, . . . , m, the
matrix for T has 8^ as its ith column for i = n + 1^ . . . ^ m. The lemma will
therefore follow from our above discussion if we can show that the matrix of
p o {T t F) in Hom F is the nX n upper left corner of t. The student should
be able to see this if he visualizes what vector (p o T) (ai) is for i < n. D
Corollary. In the above situation if F is also invariant under T', thcni
A(r) = AyiT \ Y).
Proof. The proof follows immediately since now p ° {T \ Y) = T \ Y. II
If the roles of X and F are interchanged, both being invariant under T and
T being the identity on F, then this same lemma tells us that A(T') = Ax{T \ X).
If we only know that X and F are T'-invariant, then we can factor T into u
commuting product T = Ti o T2 = T'2 ° ^1, where Ti and T2 are of the two
more special types discussed above, and so have the rule A{T) = A(T'i) A{T2) =^
Ax{T \ X) Ay{T \ Y), another of our properties listed in Chapter 2.

7.6 THE DETERMINANT 315
The final rule is also a consequence of the above lemma. If T is the identity
on X and also on F/X, then it isn't too hard to see that p ° {T \ Y) is the
identity, as an element of Hom F, and so A (7") = 1 by the lemma.
We now prove the theorem concerning "expansion by minors (or cofactors)".
Let t be an m X m matrix, and let (t)^^ be the {m — 1) X {m — 1) sub matrix
obtained from t by deleting the pth row and ?'th column. Then,
Theorem 6.3. D{t) = Y.T=i {-iT^'tir i>((t)^0- That is, we can evaluate
^(t) by going down the rth column, multiplying each element by the
determinant of the (m— l)X(m— 1) matrix associated with it, and
adding. The two occurrences of 'D^ in the theorem are of course over
dimensions m and m — 1, respectively.
Proof. Consider D{t) = D(t^, . . . , t^) under the special assumption that
t^ = 8^. Since Z)(t) is an alternating Hnear functional both of the columns of t
and of the rows of t, we can move the ?'th column and pth row to the right-
bottom border, and apply Lemma 6.2. Thus
D(t) = (-i)"-^(-i)"-^i)(tn = (-ir^'D(tn,
assuming that the rth column of t is 8^. In general, t^ = JlT=i Ur 8\ and if we
expand Z>(t ^ . . . , t^) with respect to this sum in the ?^th place, and if we use the
above evaluation of the separate terms of the resulting sum, we get D{t) =
Corollary 1. If s 9^ r, then ZT=i (-ly^'tis i>((t)^"0 = 0.
Proof. We now have the expansion of the theorem for a matrix with identical sth
and rth columns, and the determinant of this matrix is zero by the alternating
property. D
For simpler notation, set Cij = (—l)'"^-^ Z)((t)''^). This is called the cofactor
of the element Uj in t. Our two results together say that
m
In particular, if D{t) 9^ 0, then the matrix s whose entries are Sri = Cir/D(t) is
the inverse of t. This observation gives us a neat way to express the solution of
a system of linear equations. We want to solve t • x = y for x in terms of y,
supposing that D(t) 9^ 0. Since s is the inverse of t, we have x = s • y. That is,
^i = L^i ^jiVi = (IlT=i yiCij)/D{t) for j = 1, . . . , m. According to our
expansion theorem, the numerator in this expression is exactly the determinant
dj of the matrix obtained from t by replacing its jih column bj^ the m-tuple y.
Hence, with dj defined this way, the solution to t • x = y is the m-tuple
This is Cramer'8 rule. It was stated in slightly different notation in Section 2.5.

316 MULTILINEAR FUNCTIONALS 7.7
7. THE EXTERIOR ALGEBRA
Our final job is to introduce a multiplication operation between alternating
?2-linear functionals (also now called exterior n-forms). We first extend the tensor
product operation that we have used to fashion elementary covariant tensors out
of functionals.
Definition. If / G (7*)® and g G (7*)®, then f®gh that element of
(y*)(SP defined as follows:
We naturally ask how this operation combines with the projection Q of
(y*)^SB> onto a^+^
Theorem 7.1. Q(/® (/) = ^{f ® ^O) = ^(^/® O)-
Proof. We have
^(/ ® %) = (^^^ Z (sgn (7)(/ ® Q^)^
:p-^ Z (sgn 0-) (f ® ^ X) (sgn p)g']
(n +
We can regard p as acting on the full n + I places of / ® (7 by taking it as thc^
identity on the first n places. Then (/ ® g^Y = {f ® 9^- Set pa = a\ For
each (t' there are exactly /! pairs <Pja> with per = cr', namely, the pairs
{ <p, p~ V' > : p G Sz} • Thus the above sum is
Z(sgn(TO(/®(7r = 12(/®(7).
The proof for Q(Q/ ® (7) is essentially the same. D
Definition. If / G a"^ and g G a\ then f A g = Ct^)^{f®g)'
Lemma 7.1. /i A /2 A ' ' ' A //c = (n!/^i!n2! • • • ^/c!)^(/i ®"'® fk),
where rii is the order of/^•, z = 1, . . . , /c, and n = J2\
'hiproof. This is simply an induction, using the definition of the wedge operation A
and the above theorem. D
Corollary. If X,- G F*, z = 1, . . . , n, then
Xi A • • • A An = n!Q(Xi ® • • • ® X^).
In particular, if gi < • • • < g^ and {iu^}7 is a basis for F*, then
fJ'qi A ' ' ' A iJLq^ = n!Q(jUq) = the basis element Vq of d^.

7.7 THE EXTERIOR ALGEBRA 317
Theorem 7.2. If / G a^ and g G a\ then g A f= (-1)^7 A g. In
particular, X A X = 0 for X G F*.
Proof. We have g ® f = (/ ® gY, where a is the permutation moving each of
the last I places over each of the first n places. Thus a is the product of In
transpositions, (sgncr) = ( — 1)^^, and
^{g®f) = ^{f®gr= (sgn(7)Q(/®(7)= (-iy^n{f®g).
We multiply by (^J^) and have the theorem. Q
Corollary. If {X,}^ C F*, then Xi A • • • A X^ = 0 if and only if the
sequence {Xj^ is dependent.
Proof. If {\i} is independent, it can be extended to a basis for F*, and then
Xi A • • • A Xn is some basis vector v^ of a^ by the above corollary. In
particular, Xi A • • • A X^ 7^ 0.
If {X^} is dependent, then one of its elements, say Xi, is a linear combination of
the rest, Xi = Y.2 (^i^i and Xi A X2 A • • • A X^ = Y.i=2 (^i^i A (X2 A • • • A X^).
The ith. of these terms repeats X^-, and so is 0 by the lemma and the above
corollary. D
Lemma 7.2. The mapping <f,g> 1-^ / A (7 is a bilinear mapping from
a"" X a^ to a'^'^K
Proof. This follows at once from the obvious bilinearity of / ® (7. D
We conclude with an important extension theorem.
Theorem 7.3. Let 6 be the alternating n-linear map
<Xi, . . . , \n> i-^ Xi A • • • A X^
from (F*)^ to d^. Then for any alternating n-linear functional /^(Xi,. .., X^)
on (F*)^, there is a uniquely determined linear functional G on a^ such that
F = God, The mapping G ^-^ F is thus a canonical isomorphism from
(a^)*toa^(F*).
Proof. The straightforward way to prove this is to define G by establishing its
necessary values on a basis, using the equation F = God, and then to show
from the linearity of G, the alternating multilinearity of
<Xi, . . . , X^> ^-^ Xi A • • • A X^,
and the alternating multilinearity of F that the identity F = G o d holds
everywhere. This computation becomes notationally complex. Instead, we shall
be devious. We shall see that by proving more than the theorem asserts we get
a shorter proof of the theorem.
We consider the space (2,^(F*) of all alternating n-linear functions on (F*)^.
We know from Theorem 5.2 that (l(a^(F*)) = Q, since (l(F*) = d(V) = m.

318 MULTILINEAR FUNCTIONALS 7.7
Now for each functional G in (d^)*, the functional G o ^ is alternating and n-
linear, and soG»-^F=Go^isa mapping from (Q,^)* to (2,^(7*) which is
clearly linear. Moreover it is injective, for if G ?^ 0, then /^(^u^d), . . . , jU(z(n)) =
G{Vq) 9^ 0 for some basis vector z^^ = ^u^d) A • • • A jU(z(n) of (2,^(7*). Since
d{a'^{V*)) = (!^) = dia"^) = d((a'')*), the mapping is an isomorphism (by
the corollary of Theorem 2.4, Chapter 2). In particular, every F in a^(7*) is
of the form G o d. D
It can be shown further that the property asserted in the above theorem is
an abstract characterization of (2^. By this we mean the following. Suppose thai
a vector space X and an alternating mapping (p from (7*)^ to X are given, and
suppose that every alternating functional F on (F*)^ extends uniquely to a
linear functional G on X (that is, F = G o cp). Then X is isomorphic to (2^, and
in such a way that <p becomes 6.
To see this we simply note that the hypothesis of unique extensibility is
exactly the hypothesis that ^: G \~^ F = G ^ cp is am isomorphism from X* to
a^(7*). The theorem gave an isomorphism 0 from (a^)* to (2^(7*), and the
adjoint ($~^ o ©)* is thus an isomorphism from X** to (a^)**, that is, from X
to d^. We won't check that cp "becomes" 6.
By virtue of Corollary 1 of Theorem 6.2, the identity Z)(t) = Z)(t*) is the
matrix form of the more general identity A{T) = A(T'*), and it is interesting to
note the "coordinate free" proof of this equation. Here, of course, T ^ Hom V.
We first note that the identity (T'*X)(^) = \(T^) carries through the
definitions of ® and A to give
7^*Xi A • • • A 7^*X,(^i, . . . , ^,) = Xi A • • • A X,(7^^i, . . . , T^U- (*)
Also, ev^: Xi A • • • A X^ i-^ Xi A • • • A Xn(^i, • • • , ^n) is an alternating
n-linear functional on an(7*) for each g G V^. The left member of (*) is thus
eVg(7'*Xi, . . . , 7'*X^), and, if n = dim 7, this is A(7'*)eVg(Xi, . . . , X^) by the
definition of A. By the same definition the right side of (*) becomes
A(7^)[Xi A • • • A X,(^i, . . . , U] = A(7^)ev^(Xi, . . . , X,).
Thus (*) implies the identity A(7'*)eVg = HT)ev^. Since ev^ 9^ 0 if ^ =
{^i}1 is independent, we have proved that A(T'*) = A(T').
We call a wedge product Xi A • • • A X^ of functionals X^- G 7* a multi-
vector. We saw above that Xi A • • • A X^ ?^ 0 if and only if {X,}^ is
independent, in which case {Xj^ spans an n-dimensional subspace of 7*. The
following lemma shows that this geometric connection is not accidental.
Lemma 7.3. Two independent n-tuples {X^}^ and {fJLi}\ in 7* have the
same linear span if and only if jUi A • • • A Mn = ^(^1 A • • • A Xn) for
some /c.
Proof. If {juy} 1 C L({X^} i), then each jjlj is a linear combination of the X/s, and
if we expand jUi A • • • A jUn according to these basis expansions, we get
k{\i A • • • A X^). If, furthermore, {fiij^ is independent, then k cannot be zero.

7.8 EXTERIOR POWERS OF SCALAR PRODUCT SPACES 319
Now suppose, conversely, that /jli A • • • A iJLn = k(\i A • • • A X^), where
k 7^ 0. This implies first that {juj i is independent, and then that
My A (Ai A • • • A \n) = 0
for each j, so that each /jlj is dependent on {X^}^. Together, these two
consequences imply that the set {jUz} i has the same linear span as {Xj^. □
This lemma shows that a multivector has a relationship to the subspace it
determines like that of a single vector to its span.
8. EXTERIOR POWERS OF SCALAR PRODUCT SPACES
Let y be a finite-dimensional vector space, and let ( , ) be a nondegenerate
(nonsingular) symmetric bilinear form on V. In this and the next section we shall
call any such bilinear form a scalar product, even though it may not be
positive definite. We know that the bilinear form ( , ) induces an isomorphism
of V with y* sending ?/ G F into y G F*, where y{x) — (x, "y) = (x, y) for all
X gV. We then get a nondegenerate form (scalar product), which we shall
continue to denote by ( , ), on F* by setting (i7, v) = (t6, v). We also obtain a
nondegenerate scalar product on (2,^ by setting
{ui A ' ' ' A Uq, vi A ' ' ' A Vq) = det {{Ui, vj)). (8.1)
To check that (8.1) makes sense, we first remark that for fixed zJi, . . . , y^ G F*,
the right-hand side of (8.1) is an antisymmetric multilinear function of the
vectors Ui, . . . jUq, and therefore extends to a linear function on (2^(F*) by
Theorem 7.3. Similarly, holding the u's fixed determines a linear function on
(^^(F*), and (8.1) is well defined and extends to a bilinear function on (2^(F*).
The right-hand side of (8.1) is clearly symmetric in u and z;, so that the bilinear
form we get is indeed symmetric. To see that it is nondegenerate, let us choose a
basis Uij . . . J Un so that
{ui, Uj) = ±. 8ij. (8.2)
(We can always find such a basis by Theorem 7.1 of Chapter 2.) We know that
{ui} = {ui^ A • • • A Ui] forms a basis for (2,^, where i = <iij . . . ^ iq> ranges
over all g-tuplets of integers such that I < ii < - - - < iq < rij and we claim
that
(i7i, i7j) = ±5ij. (8.3)
In fact, if i 5^ j, then ir 9^ js for some value of r between 1 and q and for all s.
In this case one whole row of the matrix (ui^, Uj^) vanishes, namely, the rth row.
Thus (8.1) gives zero in this case. If i = j, then (8.2) says that the matrix has
±1 down the diagonal and zeros elsewhere, establishing (8.3), and thus the fact
that ( , ) is nondegenerate on (2^. In particular, we have
{ui A ' ' ' A Un, ui A ' ' ' A Un) = (—1)*, (8.4)
where # is the number of minus signs occurring in (8.3).

320 MULTILINEAR FUNCTIONALS 7.9
9. THE STAR OPERATOR
Let y be a finite-dimensional vector space endowed with a nondegenerate scalar
product as in Section 8. The space (2,^ is one-dimensional if n is the dimension of
V. The induced scalar product on (2^ is nondegenerate, so that (u, u) is either
always positive or always negative for all nonzero i7 G (2^. In particular, there
are exactly two tI's in (2,^ with {%u) = ±1. Let us choose one of them and hold
it fixed for the remainder of this section. Geometrically, this amounts to choosing
an orientation on V. We thus have picked a
i7 G a^ with (i7, u) = (-1)^. (9.1)
Let V be some fixed element of d^. Then for any y G (2^~^, v A y G (2^, and so
we can write v A y = fv{y)u, where fy{y) depends linearly on y. Since the
induced scalar product ( , ) on Gi^~^ is nondegenerate, there is a unique
element *v G (i^~^ such that {y, *v) = My). To repeat, we have assigned a
*2J G (2^~^ to each y G (2^ by setting
(^, ^v)u = V A y. (9.2)
We have thus defined a map, *, from d^ to a^~^. It is clear from (9.2) that this
map is linear. Let t^i, . . . , t6^ be a basis for V satisfying (8.2) and also u =
Ui A ' ' ' A Unf and construct the corresponding bases for the spaces (2^ and
Q^n-3 Then Ui A Uj = 0 if any ii occurring in the g-tuplet i also occurs in j.
If no ii occurs in j then
Ui A Uj ^ ^i^...i^j^...j^_qUj
where e^ = sgn k.
If we compare this with (9.2) and (8.3), we see that
*27i = ±€,i...,y^...y^_^l7j, (9.3)
where the sign is the same as that occurring in (8.3), i.e., the sign is positive or
negative according as the number of ji with {Uj^^ Uj^ = — 1 which appear in j
is even or odd. Applying * to (9.3), we see that
**2j = (-l)«(^-«)+#z; for all v^a^. (9.4)
Let V and lo be elements of (2^. Then
(*z;, ^w)u = V A "^w = { — iy^^~^^^w A V = {—iy^^~^\^^w, v)u.
If we apply (9.4), we see that
(*z;^ ^w) = (—l)*(y, w). (9.5)

CHAPTER 8
INTEGRATION
1. INTRODUCTION
In this chapter we shall present a theory of integration in n-dimensional
Euclidean space E^, which the reader will remember is simply Cartesian n-space
W^ together with the standard scalar product. Our main item of business is to
introduce a notion of size for subsets of E^ (area in two dimensions, volume in
three . . .). Before proceeding to the formal definitions, let us see what properties
we would like our notion of size to have. We are looking for a function (jl which
assigns a number fjL{A) to bounded subsets A C E^.
i) We would like ijl{A) to be a nonnegative real number.
ii) If Ac B, we would expect to have iJi(A) < ii{B).
iii) If A and B are disjoint (that is, A n 5 = 0), then we would expect to
have ii{A D B) = ii{A) + fx{B).
iv) Let T be any Euclidean motion. * For any set A let TA be the set of all
points of the form T'x, where x ^ A. We then would expect to have
fjL{TA) = ijl{A). (Thus we want "congruent" sets to have the same size.)
v) We would expect a "lower-dimensional set" (where this is suitably
defined) to have zero size. Thus points in the line, curves in the plane,
surfaces in three-space, etc., should all have zero size.
vi) By the same token, we would expect open sets to have positive size.
In the above discussion we did not specify what kind of sets we were talking
about. One might be ambitious and try to assign a size to every subset of E^.
This proves to be impossible, however, for the following reason: Let U and V
be any two bounded open subsets of E^. It can be shownj that we can find
* Recall that a Euclidean motion is an isometry of E^ and can thus be represented as
the composition of the translation and an orthogonal transformation,
t S. Banach and A. Tarski, Sur la decomposition des ensembles de pointes en par tie
respectivement congruentes. Fund. Math. 6, 244-277 (1924). R. ]\L Robinson, On the
decomposition of spheres, Fund. Math. 34, 246-260 (1947).
321

322 INTEGRATION 8.2
decompositions
k k
U= \J Ui and F = U Vi
i=l i=l
with Ui n Uj = 0 = Vi n Vj for i 9^ j, and Euclidean motions Ti with
TiTJi = F^ In other words, we can break up TJ into finitely many pieces, move
these pieces around, and then recombine them to get F. Needless to say, the
sets JJi will have to look very bad. A moment's reflection shows that if we wish
to assign a size to all subsets (including those like C/^), we cannot satisfy (ii),
(iii), (iv), and (vi). In fact, (iii) [repeated (/c — 1) times] implies that
m(c/) = E M(C'^f).
and (iv) implies that [x(JJi) = ^(Fi). Thus ijl{U) = ju(F), or the size of any two
open sets would coincide. Since any open set contains two disjoint open sets,
this implies, by (ii), that fjL{U) > 2ijl{U)j so fjL{U) = 0.
We are thus faced with a choice. Either we dispense with some of
requirements (i) through (vi) above, or we do not assign a size to every subset of E'^.
Since our requirements are reasonable, we prefer the second alternative. This
means, of course, that now, in addition to introducing a notion of size, we must
describe the class of "good" sets we wish to admit.
We shall proceed axiomatically, listing some "reasonable" axioms for a class
of subsets and a function fi.
2. AXIOMS
Our axioms will concern a class 3) of subsets of E'^ and a function (jl defined on 3).
(That is, iJi{A) is defined if A is a subset of E'^ belonging to our collection 3).)
I. 3) is a collection of subsets of E'^ such that:
3)1. If A G 3) and 5 G 3), then AuBe^, AnBe^, and A — 5 G 3).
3)2. If A G 3) and 7" is a translation, then TA G 3).
3)3. The set [Jl = {x :0 < x' < 1} belongs to 3).
II. The real-valued function fx has the following properties:
ill. fjL(A) > 0 for all A G 3).
m2. If a g 3), 5 G 3), and a n 5 = 0, then fjL{A \J B) = fjL(A) + fi(B).
/aS. For any ^4 G 3) and any translation T', we have ijl{TA) = ijl{A).
m4. M(nJ) = 1.
Before proceeding, some remarks about our axioms are in order. Axiom 3)1
will allow us to perform elementary set-theoretical operations with the elements
of 3). Note that in Axioms 3)2 and fiS we are only allowing translations, but in

8.2
AXIOMS
323
our list of desired properties we wanted proper behavior with respect to all
Euclidean motions in (iv). The reason for this is that we shall show that for
"good" choices of 3), the axioms, as they stand, imiquely determine ju. It will
then turn out that ijl actually satisfies the stronger condition (iv), while we
assume the weaker condition ju3 as an axiom.
Fig. 8.1
Axiom 5)3 guarantees that our theory is not completely trivial, i.e., the
collection 33 is not empty. Axiom ju4 has the effect of normalizing jjl. Without it,
any (jl satisfying jul, /i2, and ju3 could be multiplied by any nonnegative real
number, and the new function ju' so obtained would still satisfy our axioms.
In particular, ju4 guarantees that we do not choose fx to be the trivial fimction
assigning to each A the value zero.
Fig. 8.2
Our program for the next few sections is to make some reasonable choices for
3> and to show that for the given 2) there exists a unique (jl satisfying jjlI through />t4.
An important elementary consequence of the 3), ju-axioms that we shall
frequently use without comment is:
iDiS. If J. C Ui ^i and all the sets are in 2), then /i(4) < J^\ fxiAi).
Our beginning w^ork wdll be largely combinational. We will first consider
(generalized) rectangles, which are just Cartesian products of intervals, and the
way in which a point inside a rectangle determines a splitting of the rectangle
into a collection of smaller rectangles, as indicated in Fig. 8.1. This is associated
with the fact that the intersection of any two rectangles is a rectangle and the
difference of two rectangles is a finite disjoint union df rectangles (see Fig. 8.2).

324
INTEGRATION
8.3
Fig. 8.3
We call a set A paved if it can be expressed as the union of a finite disjoint
collection /> of rectangles (a paving of ^4). It will follow from our combinational
considerations that the collection 3)min of all the paved sets satisfies Axioms 3)1
through 3)3 and is the smallest family that does: any other collection 3) satisfying
the axioms includes a)min- It will then follow that if /jl satisfies /jlI through />t4 on
3)min, then it must have the natural value (the product of the lengths of the sides)
for a rectangle. This implies that /a is uniquely defined on 3)miii by requirements
julI through />t4, since the value /Ji{A) for any paved set A must be the sum of the
natural values for the rectangles in a paving of A. The existence of /jl on a)inin
thus depends on the crucial lemma that two different pavings of the set A give
the same sum. (See Fig. 8.3.)
This comes down to the fact that the "intersection" of two pavings of A is
a third paving "finer" than either, and the fact that when a single rectangle is
broken up, the natural values of ja for the pieces add up to /a for the fragmented
rectangle.
All these considerations are elementary but exceedingly messy in detail. We
give the proofs below for the reader to refer to in case of doubt, but he may
prefer to study only the definitions and statements of results and then to proceed
to Section 6.
3. RECTANGLES AND PAVED SETS
We first introduce some notation and terminology. Let a = <a\...,a^>
and b = <6\ . . . , 6^> be elements of E^. By the rectangle \Z}^ we shall mean
the set of all X = <xS . . . , x^> in E^ with a' < x' < h\ Thus
Da = {x : a^ < x^ < 6\ z = 1, . . . , n}. (3.1)
Note that in order for Oa to be nonempty, we must have a' < 6' for all i.
In other words.
Da = 0 if a^ > 6^ for some i. (3.2)
In the plane {n = 2) for instance, our rectangles Oa correspond to ordinary
Euclidean rectangles whose sides are parallel to the axes. (We should perhaps use
an additional adjective and call our sets level rectangles, braced rectangles, or
something else, but for simplicity we shall just call them rectangles.) Note that
in the plane our rectangles include the left-hand and lower edges but not the
right-hand and upper ones (see Fig. 8.4).

8.3
RECTANGLES AND PAVED SETS
325
a:,=0
-X2 = 0
Fig. 8.4
For general n, if we set 1 = < 1, . . . , 1 > , then our notation coincides with
that of 3)3.
We now collect some elementary facts about rectangles. It follows
immediately from the definition (3.1) that if a = <a^, , . , , a^> ,h = < 6\ . . . , 6"" > ,
etc.. then i. ^ ^
Da n nf = DL (3.3)
where
e' = max {a\ &) and /' = min (6', d"), z = 1, . . . , n.
(The reader should draw various different instances of this equation in the plane
to get the correct geometrical feeling.) Note that the case
where Da nnf= 0
is included in (3.3) by (3.2). Another immediate consequence of the definition
(3.1) is
^ Da = n?a for any translation T. (3.4)
We will now establish some elementary results which will imply that any 3)
satisfying Axioms 3)1 through 3)3 must contain all rectangles.
Lemma 3.1. Any rectangle Qa can be written as the disjoint union
k
LJa ^ U LJa^, where b^ — a^ ^ LJo.
r = l
(What this says is that any "big" rectangle can be written as a finite union
of "small" ones.)
Proof. We may assume that Oa 7^ 0 (otherwise take /c = 0 in the union).
Thus 6^ > a^ In particular, if we choose the integer m sufficiently large,
(1/2"^) (b - a) will lie in nJ.
By induction, it therefore suffices to prove that we can decompose Oa into
the disjoint union
271
Da = U Do: with ds - c. = i(b - a).
(3.5)
(For then we can continue to subdivide until the rectangles we get are small
enough.)
We get this subdivision in the obvious way by choosing the vertex "in the
middle" of the rectangle and considering all rectangles obtained by cutting
□a through this point by coordinate hyperplanes. To write down an explicit

326
INTEGRATION
8.3
a[i,2] =b0
a{2]
1 ^a{2]
1 ^^^
1 1^0
\ ^b|i,2}
1 ^''^'
1 ^ari]
b = bn
b[i]
Fig. 8.5
formula, it will be convenient to use the set of all subsets of {1, . . . , n} as an
indexing set, rather than the integers 1, . . . , 2^. Let J denote an arbitrary
subset of {1, 2, . . . , n}. Let sij = Ka], . . . ^a!j> and b/ = <6j, . . . , 6}> be
given by
and h} = L_l^ .f .^^_
a}
|a' +
if i G J,
if i^J
Then any x G Oa lies in one and only one
Da;;. In other words, Hl'j n Da^ =
0UJ 9^ Ksmd Uaii/ Daj = Da- (The case where n = 2 is shown in Fig. 8.5.)
Since bj — a^^ = ^(b — a) for all J, we have proved the lemma. D
We now observe that for any c G Do we have, by (3.3),
ns = ns n Dc-i. (3.6)
Let Ty denote translation through the vector v. Then Dc-i = ^c-i Do by
(3.4). Thus by Axioms 3)2 and 3)3 the rectangle Dc-i must belong to 3).
By (3.6) and Axiom 3)1 we conclude that Do ^ 3) for any c G Do-
Observe that TJOy^ = Da by (3.4). Thus
Da e 3) whenever b — a G Do-
(3.7)
If we now apply Lemma 3.1 we conclude that
Da e 3) for all a and b.
We make the following definition.
Definition 3.1. A subset aS C E^ will be called a paved set if S is the disjoint
union of finitely many rectangles.
We can then assert:
Proposition 3.1. Any 3) satisfying Axioms 3)1 through 3)3 must contain all
paved sets. Let 3)miii denote the collection of all finite unions of rectangles;
then 3)miii satisfies Axioms 3)1 through 3)3.
Proof. We have already proved the first part of this proposition. We leave the
second part as an exercise for the reader.

8.4
THE MINIMAL THEORY
327
4. THE MINIMAL THEORY
We are now going to see how far /x is determined by Axioms /jlI through /xS.
In fact, we are going to show the mCDh) is what it should be; i.e., if
a=<aS...,a^> and b = <6\ . . . , 6^> ,
then we must have
MCDa) =
0
if D^
(6i - ai) • • • {bn - an) if Da ^
(4.1)
Axiom/X4 says that (4.1) holds for the special case a = 0, b = 1. Examining
the proof of Lemma 3.1 shows that OJ can be written as the disjoint union of 2^
rectangles, all congruent (via translation) to Do^^j where ^ = (^, . . . , J).
Axioms ju2 and ju3 then imply that
M(nn = ^-
Repeating this argument inductively shows that
(4.2)
We shall now use (4.2) to verify (4.1). The idea is
to approximate any rectangle by unions of
translates of cubes r
LJo •
Fig. 8.6
Observe that in proving (4.1) we need to consider only rectangles of the form
□S- In fact, we take c = b — a and observe that
r-aCQa) = as,
SO Axiom ju3 implies that ju(na) = KlUS)) and by definition c^ - - - c^ =
(6^ — a^)" ' (6^^ — a""). If Do = 0, then (4.1) is trivially true (from Axiom
fx2). Suppose that Do ^ 0- Then c = <c^ . . . , c^> with c^ > 0 for all i.
For each r there are n integers N^, , , , , N^ such that (Fig. 8.6)
N'/2' < c' < {N' + \)I2\ (4.3)
In what follows, let k = </c\ . . . , /c^>, L = <Z\ . . . , r>, etc., denote
vectors with integral coordinates (i.e., the /c/s are integers). Let us write
k < Lif /c,- < li for all i, U N = <Ni, . . . , Nn>, then it follows from (4.3)
and the definitions that
□;i
(l/2^)k+l/2' p-|c
/20k
whenever
For any k and L,
Since
r-^(l/20k+l/2'^ r-^(l/20L+l/2'^
L_J(i/20k ' ' L_J(i/20L
p-l(l/20k+l/2'^
^LJ(i/20k
0
2nr
k < N.
if k F^ 1.

328 INTEGRATION
by (4.2) (and Axiom ^t2) and
I I r-i(l/2^)k+l/2'^ r-nc
U LJ(i/20k ^ LJo,
k<N
we conclude that
M(nS) ^ 2^ X (the number of k satisfying 0 < k < N).
It is easy to see that there are Ni • N2 • . . . • Nn such k, so that
MDS) > ~X (A^x • • • Ar„) = (|-l) . . . (^|l^ .
According to (4.3), N,/2'' > c, - 1/2'', so we have
M(nS)>(c^-^.)---(c"-^)- (4.1)
nsc u Di;;
Similarly,
l(l/2^)k+l/2^
1/2 ^)k ?
1 1 1 k<N+2
and we conclude that
MDS) <(c^ + |)---fc- + |:^) (4.r,)
2^
Letting r —> (X) in (4.4) and (4.5) proves (4.1).
In deriving (4.1) we made use of Axiom ^t4. Examining our argument shows
that if iJL^ satisfied ijl2 and ^t3 but 7iot ijl4:, we could argue in the same mannc^r
except that we would have to multiply everything by the fixed constant m'IDo)-
To sum up, we have proved:
Proposition 4.1. If ^t satisfies Axioms ^tl through ^t4, then the value of ^i
or any rectangle is uniquely determined and is given by (4.1). If fx' satisfi(\s
^tl through idS, then for any rectangle Gaj
M'(na) = KfjiiBl) where K = ^'(nJ).
5. THE MINIMAL THEORY (Continued)
We will now show that formula (4.1) extends to give a unique ijl defined on SDmi,,
so as to satisfy Axioms fil through ju4. We must establish essentially two facts.
1) Every union of rectangles can he ivritten as a disjoint union of rectangles
This will then allow us to use Axiom ^tl to determine ^t(A) for every A E 3)miti,
if A is the disjoint union of the [I]a-- Since A might be written in another way
as a disjoint union of rectangles, this formula is not well defined until we
establish that:
2) If A = UlZla- = UDc/ ^^6 i^^o re'presentations of A as a disjoint union
of rectangles, then
i:M(na;) = Ziu(n^).

8.5
THE MINIMAL THEORY (cONTINUEd) 329
Fig. 8.7
<c
<c
<c
<c
ci>
, cl>\
ci>
<cl 4>
<4' 4>
1
1
1
1
1
1 a
<cl, cl>\
<cl, C2>
_ —4
<4' ^4>
<4, C2>
-1
1
<4,4> 1
Fig. 8.8
c\, cl>
<c\, c^>
cl cl>
<cl ci>
<c\, c\>
<c\, cf>
We first introduce some notation.
Definition 5.1. A paving /> of E^ is a finite collection of mutually disjoint
rectangles. The floor of this paving, denoted by |/>|, is the union of all
rectangles belonging to p.
lip = {G^;} and T' is a translation, we set Tp = {T^Da:}-
If p and g. are two pavings, we say that ^ is finer than p (and write g < p)
if every rectangle of /> is a union of rectangles of g. It is clear that U p < v
and g < Pf then g < v. Note also that g < P implies \p\ C |^|.
Proposition 5.1. Let p and g be any two pavings,
paving V such that v < p and v < g.
There exists a third
Proof, The idea of the proof is very simple. Each rectangle in p or m g
determines 2^ hyperplanes (each hyperplane containing a face of the rectangle).
If we collect all these hyperplanes, they will "enclose" a number of rectangles.
We let V consist of those rectangles in this collection which do not contain any
smaller rectangle. Figure 8.7 shows the case (for n = 2) where p and g each
contain one rectangle. Here v contains nine rectangles.
We now fill in the details of this argument. Let ci = <c}, . . . , c^ >, . . . ,
c/c = <c^, . . . , c^> be all the vectors that occur in the description of the
rectangles of p and g. (In other words, if Qa E /> or G ^, then a and b are among
the c's.) Let di, . . . , d^" be the vectors of the form < c/^, . . . , c^^ >, where the z*/s
range independently from 1 to /c, (so that there are ¥^ of them). (See Fig. 8.8 for
the case where n = 2 and p and g consist of one rectangle each.) For each di
there is at most one smallest dy(i) such that di < dy(i). In fact, if
di
<ct
cl>.
then set dy(i) = <c/j, . . . , c7„>, where
cy, = mm c,
^31
.>'\i

330 INTEGRATION 8.5
Let V = {Ddf'^}- Then v is finer than p and ^. In fact, if Da ^ />, say, then
Da = Dd^ for suitable a and ^ and
uz= U n^f^ (5.1)
da<di
dy(i)<d^
To see this observe that if x G EUdf) then d^ < x < A^. Choose a largest
di < X. Then di < x < dy(o, so x G Odf'^- This proves the proposition. W(^
will later want to use the particular form of the v we constructed to find
additional information. D
We can now prove (1) and (2).
Lemma 5.1. Let />i, . . . , />z be pavings. Then there exists a paving ^
such that 1^1 = |/>i| U • • • U \pi\.
Proof. By repeated applications of Proposition 5.1 we can choose a paving 'i^
which is finer than all the />/s. Then each \pi\ is the union of suitable rectangles
of ?^. Let g be the collection of all these rectangles occurring in all the />/s.
Then |^| = |/>i| U- • • U \pk\- □
In particular, we have proved (1). More generally, we have shown that every
A G 3)min is of the form A = \p\ for a suitable paving />. We now wish to turn
our attention to (2).
Lemma 5.2. Let c} < • • • < c^\, c\ < - • - < c^^j . . . , c^ < • • • < c^^he v
sequences of numbers. Then
Proof, In fact, c". — Cj = c\ — c\ + cl — c^ + - - - + c\. — c'._i, so that th(^
lemma follows from (4.1) when we multiply out all the factors. D
We now prove (2). Let /> = {Da-} and ^ = {Det)? where A = \p\ = \g\.
Let V be the paving we constructed in the proof of Proposition 5.1. Let ^ ^
(Ddr) be the collection of those rectangles Ddf of v such that Ddf C \p\ = \g\.
Then to prove (2) it suffices to show that
i: mcd^f) = r iu(n^:) = e mcd^^). (5.2)
Now each rectangle CHaJ is decomposed into rectangles Ddf^^^ according to (5.1),
that is, 3ii = da, hi = d/?, etc.
By construction of the d's, this is exactly a decomposition of the typc^
described in Lemma 5.2. Thus (5.1) implies that
da<d;
dy(i)<dp

8.6 CONTENTED SETS 331
Summing over all \Z\^i (and doing the same for \Z\t0 proves (5.2). We can thus
state:
Theorem 5.1. Every A G a)min can be written SiS A = |/>|. The number
Ijl{A) = J^DGpfJ'GU) does not depend on the choice of />. We thus get a
well-defined function (jl on 3)min- It satisfies Axioms fxl through ju4. If fi^ is
any other function on SDmin satisfying ju2 and ju3, then fx^A) = Kijl{A),
where K = fx'iBl)-
Proof. The proof of the last two assertions of the theorem is easy and is left as an
exercise for the reader.
6. CONTENTED SETS
Theorem 5.1 shows that our axioms are not vacuous. It does not provide us
with a satisfactory theory, however, because 3)min contains far too few sets.
In particular, it does not fulfill requirement (iii), since 3)min is not invariant
under rotations, except under very special ones. We are now going to remedy
this by repeating the arguments of Section 4; we are going to try to approximate
more general sets by sets whose (jl^s we know, i.e., by sets contained in 3)min-
This idea goes back to Archimedes, who used it to find the areas of figures in
the plane.
Definition 6.1. Let A be any subset of E^. We say that /> is an inner paving
of A if |/>| C A, We say that g is an outer paving of A if A C |^|.
We list several obvious facts.
If \MCAC \g\, then m(/>) < iu(^). (6.1)
If \P\CAC \gl then \Tp\ cTAc \Tg\, (6.2)
U Ai n A2= 0 and |/>i| C ^i, |/>2| C ^2,
then />i U />2 is an inner paving of Ai U A2. (6.3)
Definition 6.2. For any bounded subset A of E^ let
M*(^) = lub m(|/>I)
He A
be called the inner content of A and let
il(A)= gib Ml^l)
be called the outer content of A.
Note that since A is bounded, there exists a g with A C |^|. This shows that
pi{A) is defined. This together with (6.1) shows that ju*(^) is defined and that
„*{A) < niA). (6.4)

332 INTEGRATION 8.6
Definition 6.3. A set A will be called contented if ju*(^) = m(^)- We call
jU*(A) = iJi{A) the content of A and denote it by ju(A).
Observe that every A G SDmin is contented. In fact, if A = |?/|, then v is
both an inner and an outer paving of A, Thus ju*(^) = m(^) = m(I^I)) and the
new definition of ijl{A) coincides with the old one.
Our next immediate objective is to show that the collection of all contented
sets fulfills Axioms 3)1 through 3)3.
Proposition 6.1. A set A is Contented if and only if its boundary is
contented and has content zero.
Proof. Suppose A is contented. For any 5 > 0 we can find an inner paving p
and an outer paving g such that ju(^) — ju(/>) < 5/2. We want to replace /> by
a close paving />' with |/>'| C int A. To do this, we choose a small number rj
and replace each rectangle Q^ of /> by na+^cb-a!- We let />,, be the collection
of all these rectangles. Then |/>^| C int |/>|, so |/>^| C int A. Furthermore,
l^(\Pv\) = (1 ~ 277)V(|/>|), since the factor (1 — 277) is the decrease of each side
of each rectangle of />. Similarly, we replace ^ by a slightly larger ^^, with
A C int gr] and ^(^7?) < (1 + 27]Yii{g). By choosing 77 sufficiently small, wo
can thus arrange that mC^t?) — ^(/^tj) < 5. Let ?^ be a paving which is finer
than gy^ and />^, with \v\ = |^^|. Let ^ Ct/ consist of those rectangles of v
lying in int A, Then \%/\ = \grj\ D \^\ D |/>^|, so ju(|?^|) — M(kl) < 5. Bui
dA C\t^ — ^1, so that ll{dA) < iji(\v — a.\) = ju(l^l) — M(kl) < 5. In other
words, il{dA) = 0.
Conversely, suppose that dA has content zero. Let ^ be an outer paving of
dA with ju(l^l) < ^' Let ?^ be a paving finer than ^ and such that A C |?^|.
Let p dv consist of those rectangles contained in A. Let g dv consist of those
rectangles lying in \p\ U U|. Thenju(|^|) < ju(|/>l) +M(kl) < m(|/>|) + e-
Furthermore, A C |^|. In fact, let x G A. Then x G D for some D ^ ^- If D H ^A 7^
0, then □ n 1^1 F^ 0, so n C |-4.|, since ?/ is a refinement of ^. If □ n ^A = 0,
then every point of □ must lie in A, so that □ C |/>|. We have thus constructed
p and g with \p\ C A C \g\ and ju(^) — ju(/>) < e. Since we can do this for any e,
this implies that A is contented. D
Proposition 6.2. The union of any finite number of sets with content zero
has content zero. If A C B and B has content zero, then so does A.
Proof. The proof is obvious.
Theorem 6.1. Let 3)con denote the collection of all contented sets. Then
3)con satisfies Axioms 3)1 through 3)3, and the fx given in Definition 6.3
satisfies jul through ju4. If ju' is any other function on a)con satisfying jjlI througli
ju3, then ju' = KfXj where K = ju'CDo)-
Proof. Let us verify the axioms.
3)1. For any A and 5,
d{A UB) CdAudB and a(A n S) C ^A U dB.

8.7
WHEN IS A SET CONTENTED?
333
By Proposition 6.1, if ^4 and B are contented, then ^^4 and dB have content
zero. Thus so do dA U dB, d{A U B), and ^(^4 n B), by Proposition 6.2.
Hence A U B and A n B are contented.
3)2. Follows immediately from (6.1).
3)3. Is obvious.
/jl2. If AI and A 2 are contented, we can find inner pavings />i and />2 such
that)u(-4i) -)u(|/>i|) < e/2andM(-42) - )u(|/>2|) < e/2. If .410^42 = 0,
then />! U />2 is an inner paving of i4i U ^42, and so
fi{Ai U A2) > ^i{Ai) +fi{A2).
On the other hand, let ^1 and ^2 be outer pavings of A i and ^42, respectively,
with m(^i) < KAi) + e/2 and ^(^2) < f^{A2) + e/2.
Let ?^ be a paving with \v\ = \gi\ U |^2|- Then %^ is an outer paving of
^1 U ^2 and m(|?^|) < K\3i\) + m(I^2|). Thus ^i{Ai U A^) < m(|?^|) <
ju(i4i) + ju(-42) + e, or ju(i4i U ^42) < ju(-4i) + m(-42). These two
inequalities together give ju2.
/aI. Is obvious.
/jl3. Follows from (6.2) and Definition 6.3.
fii. We already know.
The second part of the theorem follows from Theorem 5.1 and Definition 6.3.
In fact, we know that ju'(|/>|) = Kijl{\p\)j and (6.1) together with Axiom ju2
implies that ju'(|/>l) ^ m'(-4) < ju'(l^l)- Since we can choose /> and g to be
arbitrarily close approximations to A (relative to ju), we are done. D
Remark. It is useful to note that we have actually proved a little more than
what is stated in Theorem 6.1. We have proved, namely, that if 3) is any
collection of sets satisfying 3)1 through 3)3, such that a)min C 3) C 3)con, and if ju': 3) —> IR
satisfies jul through ju3, then ijl\A) = Kijl{A) for all A in 3), where K = ^'(no)-
7. WHEN IS A SET CONTENTED?
We will now establish some useful criteria for deciding whether a given set is
contented.
Recall that a closed ball 5^ with center x and
radius r is given by
B:,= {y:\\y-x\\<r}. (7.1)
Note that
■Bi C nx±^i+'^' for any e > 0, (7.2)
and
(See Fig. 8.9.)
U^licB;:
-sjn r
(7.3)
x+rl
Fig. 8.9

334 INTEGRATION 8.7
If we combine (7.2) and (7.3), we see that any cube G lies in a ball B such
that ix{B) < 2^(Vn)V(C) and that any ball B lies in a cube C such that ju(C) <
S^iV^rniB),
Lemma 7.1. Let A he Si subset of E^. Then A has content zero if and only
if for every e > 0 there exist a finite number of balls {Bi} covering A
with E iJi{Bi) < e.
Proof. If we have such a collection of covering balls, then by the above remark
we can enlarge each ball to a rectangle to get a paving /> such that ^ C |/>| and
m(I/>I) < S'^{Vn)'^e. Therefore, il{A) = 0 if we can always find the {Bi}.
Conversely, suppose A has content 0. Then for any 8 we can find an outer
paving /> with ju(|/^l) < ^- I^'or each rectangle □ in the paving we can, by the
arguments of Section 4, find a finite number of cubes which cover Q and whose
total content is as close as we like to ijl{\Z}), say <2ju(n). By doing this for each
□ E />, we have a finite number of cubes {\Z}i} covering A with total content
less than 25. Then by our remark before the lemma each cube \Z\t lies in a ball
Bi such that ju(^^) < 2^(\/^)^m(^z), and so we have a covering of A by balls Bi
such that E iJi{Bi) < 2^+i(\/ri)^5. If we take 8 = e/2^+H\/^)^ we have the
desired collection of balls, proving the lemma. D
Recall that a map (^ of ^ C E^ —> E^ is said to satisfy a Lipschitz condition
if there is a constant K (called the Lipschitz constant) such that
y{y) - <p{x)\\ < K\\y - x\\. (7.4)
Proposition 7.1. Let ^ be a set of content zero with A C U^ and let
(p: U —^ E^ satisfy a Lipschitz condition. Then (p{A) has content zero.
Proof. The proof consists of applying both parts of Lemma 7.1. Since A has
content zero, for any e > 0 we can find a finite number of balls covering A whose
total outer content is less than e/K'^, By (7.4), (p{Bl.) C B^[^), so that the images
of the balls covering A cover (p(A) and have a total volume less than e. D
Recall that if (^ is a (continuously) differentiable map of an open set U into
E^, then cp satisfies a Lipschitz condition on any compact subset of U.
As a consequence of Proposition 7.1, we can thus state:
Proposition 7.2. Let cp he Si continuously differentiable map defined on an
open set U, and let ^ be a bounded set of content zero with A C U. Then
(p{A) has content zero.
Let A he any compact subset of E^ lying entirely in the subspace given by
x^ = 0. Then A has content zero. In fact, for some sufficiently large fixed r,
the set A is contained in the rectangle
n<™-r,o)> for any 6 > 0,
which has arbitrarily small volume.

8.8 BEHAVIOR UNDER LINEAR DISTORTIONS 335
Now let ^: V C E^~^ —> E^ be a continuously differentiable map given by
Let B be any bounded subset of E^~^ with B CV. We can then write ^(5) =
(p{A), where A is the set of points in E^ of the form (2/, 0), where y E: B, and
where ^ is a differentiable map such that
cp{x\ . . . , X-) = <i.\x\ . . . , X--'), . . . , r{x\ . . . , x--')>.
By Proposition 7.2 we see that fi{yp(B)) = 0. Thus,
Proposition 7.3. Let xp he Si differentiable map of F C E^~^ into E^, and
let 5 be a bounded set such that B CV. Then \1^{B) has content zero.
We have thus recovered requirement (v) of Section 1.
An immediate consequence of Propositions 7.3 and 6.1 is:
Proposition 7.4. Let il C E^ be such that ^^4 C \J^i{Bi) where each ^^ and
Bi is as in Proposition 7.3. Then A is contented.
This shows that every set "we can draw" is contented.
Exercise. Show that every ball is contented.
8. BEHAVIOR UNDER LINEAR DISTORTIONS
We shall continue to derive consequences of Proposition 7.L
Proposition 8.1. Let ^ be a one-to-one map of C7 —> E^ which satisfies a
Lipschitz condition and is such that (p~^ is continuous. If A C U is
contented, then so is (p{A).
Proof, Since A is contented, ^^4 has content zero. By the conditions on cp,
we know that d(p(A) = (p{dA). Thus d(p{A) has content zero, and so (p{A) is
contented. D
An immediate consequence of Proposition 8.1 is:
Proposition 8.2. Let L be a linear transformation of E^. Then LA is
contented whenever A is contented.
Proof. If L is nonsingular. Proposition 8.1 applies. If L is singular, it maps all
of E^ onto a proper subspace. Any such subspace is contained in the image of
{x : x^ = 0} by a suitable linear transformation, and so ijl{LA) = 0 for any
contented A. D
Theorem 8.1. Let L be a linear transformation of E^. Then for any
contented A we have /t a\ i i ^ t i / ^ x /o ix
fi{LA) = \det L\fx{A), (8.1)
Proof. We can restrict our attention to nonsingular L, since we have already
checked Eq. (8.1) for det L = 0. If L is nonsingular, then L carries the class of

336 INTEGRATION 8.9
contented sets into itself. Let us define ijl' by ix'{A) = iJi{LA) for each A G 3)con-
We claim that ju' satisfies Axioms jul through ju3 on SDcon-
In fact, ^tl and ju2 are obviously true; ijlS follows from the fact that for any
translation T'^, we have Tl^L = LT^, so that
H\T,A) = fjL{LT,A) = h{TlvLA) - n{LA) = n'{A),
By Theorem 5.2 we thus conclude that
ll' = kLfJL,
where /cl is some constant depending on L. We must show that /c^ = [det L\.
We first observe that if 0 is an orthogonal transformation, then
fxiOA) = fjiiA).
In fact, we know that fJi{OA) = koii{A). If we take A to be the unit ball 5j,
then OBl = Bl, so ko = 1.
Next we observe that ix{LiL2A) = kL^fJi{L2A) = /cli/clsMC^)? so that
kLiL2 ^ ^Li^L2-
Now we recall that any nonsingular L can be written as L = POj where P
is a positive self-adjoint operator and 0 is orthogonal. Thus /c^ = kp and
|det L\ = |det P\ |det 0\ = |det P|, so we need only verify (8.1) for positive self-
adjoint linear transformations. Any such P can be written as P = OiDO~[^j
where Oi is orthogonal and D is diagonal. Since P is positive, all the eigenvalues
of D are positive. Since det P = det D and kp = ko, we need only verify (8.1)
for the case where L is given by a diagonal matrix with positive eigenvalues
Xi, . . . , X^. But then LQj = Q<^i--^n>^ go l^j^at
/(DJ) = mOo^'^'-''^^) = Xi • • • An = |det L|,
verifying (8.1). D
Exercise. Let vi, . . . , Vn be vectors of E''. B}^ the parallelepiped s])anne(l by vi,. .., Vn
we mean the set of all vectors of the form 2]?=i ^'^i, where 0 < a;' < 1. Show that its
content is |det ((\i, vy))|^^^.
9. AXIOMS FOR INTEGRATION
So far we have shown that there is a unique (jl defined for a large collection of
sets in E^. However, we do not have an effective way to compute ^t, except in
very special cases. To remedy this we must introduce a theory of integration.
We first introduce some notation.
Definition 9.1. Let / be any real-valued function on E^. By the support of
/, denoted by supp /, we shall mean the closure of the set where / is not zero;
that is,
supp/= {x:/(x) ^0}.

8.9 AXIOMS FOR INTEGRATION 337
Observe that
supp (/ + (7) C supp / U supp g (9.1)
and
supp fg C supp / n supp g. (9.2)
We shall say that / has compact support if supp/ is compact. Equation (9.1)
[and Eq. (9.2) applied to constant g] shows that the set of all functions with
compact support form a vector space.
Let T be any one-to-one transformation of E^ onto itself. For any fimction /
we denote by Tf the functions given by
(TfKx) = f{T-'x). (9.3)
Observe that if T and T~^ are continuous, then
supp 7^/= 7^ supp/. (9.4)
Definition 9.2. Let A he Si subset of E^. By the characteristic function of A,
denoted by Ca, we shall mean the function given by
, . _ fl if x^A,
eAW - 1^ if X g ^. (9.5)
Note that
^AiHAa = ^Ai • eA2, (9.6)
6A1UA2 = ^Ai + 6A2 — ^AinA2) (9.7)
supp ca = A, (9.8)
TeA = eTA (9.9)
and
for any one-to-one map T of E^ onto itself.
By a theory of integration on E^ we shall mean a collection ^ of functions
and a rule J which assigns a real number J/ to each / E 9^, subject to the
following axioms:
$F1. 9^ is a vector subspace of the space of all bounded functions of compact
support.
9^2. If / G 3" and 7" is a translation, then Tf G 9^.
$F3. 6n belongs to 9^ for any rectangle □.
Jl. / is a linear function on 3^.
J2. ^Tf= J/for any translation T,
J3. If/> 0, then J/> 0.
Note that the axioms imply that 3^ contains all functions of the form
^Di + ^02 + • • • + ^Da f^^ ^^y rectangles Db • • • > D/c- In particular, for any
paving />, the function e [^[ must belong to 3^.

338 INTEGRATION 8.10
Also note that from J3 we have at once the stronger version:
J3'. f < g=^ jf < jg, since then g - f > 0.
Proposition 9.1. Let 9^, J be a system satisfying Axioms ^1 through 9^3 and
Jl through J4. Then
^eA = fJi{A) (9.10)
f'
for every contented set A such that Ca E 9^, and
J5. IJ/I < ||/||ocM(supp/) forevery/G9^.
Proof. The axioms guarantee that e^ E 9^ for every A g SDmin and that ^'(^4) =
J^A satisfies fil through ^t4. Therefore, Jca = m(-4) for every A G SDmin by the
uniqueness of (jl (Proposition 4.1). It follows that if ^4 is a contented set such
that Ca G 9^, and if /> and g are inner and outer pavings of A, then
m(|/>I) = /^i^i < /^A < /^i^i = m(I^I).
Therefore, J^a lies between jU*(-4) and /l(i4), and so equals iJi(A). For any
/g9^ and any ^4 G SDmin such that supp/^i4, we have — ||/||oo6a </^
WfllooCA, and therefore |J/| < ||/||ooju(^) by Js' and (9.10). Taking the greatest
lower bound of the right side over all such sets ^4, we have J5. D
10. INTEGRATION OF CONTENTED FUNCTIONS
We will now proceed to deal with Axioms 9^ and J in the same way we dealt
with Axioms 33 and ju. We will construct a "minimal" theory and then get a "big"
one by approximating. According to Proposition 9.1, the class 9^ must contain
the function 6|^| for any paving />. By ^1 it must therefore contain all linear
combinations of such.
Definition 10.1. By a paved function we shall mean a function f = fp
given by
/= Z Ciea, (10.1)
DiGp
for some paving />.
It is easy to see that the collection of all paved fimctions satisfies Axioms 9^1
through 9^3. Furthermore, by Proposition 9.1 and Axiom Jl the integral, J, is
uniquely determined on the class of all paved functions by
// = L CifjiiUi) (10.2)
if/is given by (10.1).
The reader should verify that if we let ^p be the class of all paved functions
and let J be given by (10.2), then all our axioms are satisfied. Don't forget to
show that J is well defined: if/is expressed as in (10.1) in two ways, then the
sums given by (10.2) are equal.

8.10 INTEGRATION OF CONTENTED FUNCTIONS 339
The paved functions obviously form too small a collection of functions.
We would like to have an 9^ including all continuous functions with compact
support and all characteristic functions of the form Ca with A contented, for
example.
Definition 10.2. A bounded function / with compact support is said to be
contented if for any e > 0 and 5 > 0 there exists a paved function g = g^^^
and a contented set ^4 = Ae,8 such that
|/(x) - g{x)\ < e for all x^ A (10.3)
and
fi{A) < 8. (10.4)
The pair <g, A> will be called a paved e, 5-approximation to /.
Let us verify that the collection of all contented functions, 9^coii, satisfies
Axioms 9^1 through 9^3. It is clear that if/is contented, so is a/for any constant a.
If/i and /2 are contented, let <(7i, Ai> and <g2, A2> be paved e,5-approxi-
mations to /i and /2, respectively. Then
|(/i+/2)(x) - ((7i + (72)(x)| < 2e for all x^A.uA^,
and
fx(Ai U A2) < 25.
Thus <(7i + (72, ^1 U A2> gives a paved 2e, 25-approximation to /i + /2.
To verify ^2 we simply observe that U <g, A> isa. paved e, 5-approximation
to /, then < Tg, TA > is one to Tf.
A similar argument establishes the analogous result for multiplication:
Proposition 10.1. Let /i and /2 be two contented functions. Then /i/2 is
contented.
Proof. Let M be such that |/i(x) < M and |/2(x)| < M for all x. Recall that
the product of two paved functions is a paved function. Using the same notation
as before, we have
|/i/2(x) - (7i(x)(72(x)| < |/i(x)| |/2(x) - (72(x)| + |(72(x)| |/i(x) - (7i(x)|
<Me+(M + e)e for all x^AiUA2.
Thus <gig2, Ai U A2> is a paved (2M + e)e, 25-approximation to /1/2. □
As for 9^3, it is immediate that a stronger statement is true:
Proposition 10.2. If 5 is a contented set, then es is a contented function.
Proof. In fact, let /> be an mner paving of B with ijl(B) — ju(|/^I) < 5- Then
6b(x) - e|^,(x) = 0 if x^B-\p\,
^""^ fJiiB - \p\) < 8,
so 6[^|,|/>| is a paved e,5-approximation to es for any e > 0. D

340 INTEGRATION 8.10
We now establish a useful alternative characterization of a contented
function.
Proposition 10.3. A function / is contented if and only if for every e there
are paved functions h and k such that h < f < k and j{k — h) < e.
Proof. If/is contented, let E be a rectangle including supp /. Let <g, A> he an
e, 5-approximation to /. Let P be a paved set including A = A^^^ such thai-
ju(P) < 8, and let m be a bound of |/|. Then g — e{eR) — mep < f < g ^-
^{^r) + mep, where the outside functions are clearly paved and the differences
of their integrals is less than 2e^t(i?) + 2m b. Since e and 5 are arbitrary, we have
our h and k. Conversely, if h and k are paved functions such that h < f < k
and j{k — h) < a, then the set where k — h > a^^^ is a paved set A.
Furthermore, a^'^fji{A) < jeA{k — h) < J(/c — h) < a, so thatAt(^) < a^'^- Given e
and 8, we only have to choose a < min (e^, 8^) and take g as either /c or /i to see
that / is contented. D
Corollary. A function / is contented if for every e there are contented
functions/i and/2 such that /i < / < /2 and J(/2 — /i) < e.
Proof. For then we can find paved functions h < fi and k > f^ such thai
J(/i — h) < e and J(/c — /2) < e and end up with h <f < k and j{k — h) < 3e. D
Theorem 10.1. Let 9^ be a class of functions satisfying Axioms ^1 through ^3
and such that 9^p C 9^ C 9^con- Then there exists a unique J satisfying Axioms
Jl through J4 on ^.
Proof. If J is any integral on 9^ satisfying Axioms Jl through J4, then we must
have J/ simultaneously equal to lub jh for h paved and </ and equal to gib jk
for k paved and >/, by Proposition 10.3. The integral is thus uniquely
determined on ^. Moreover, it is easy to see that if the integral on ^ is defined by
J/ = lub jh = gib jk, then Axioms Jl through J4 follow from the fact that
they hold for the uniquely determined integral on the paved functions. D
Exercise 10.1. Let/ and Q be contented functions such that f(x) = g{x) for x^ A,
where }j^{A) = 0. Then // = fg. (This shows that for the purpose of integration
we need to know a function only as to a set of content zero.)
Definition 10.3. Let / be a contented function and A a contented set.
We call jeAf the integral off over A and denote it by J a/. Thus
fj=f^Af- (10-5)
An immediate consequence of Axiom Jl and (9.7) is
f f= f f+ f f- f /• (10.6)
JAiI)A2 J Ai J A2 JAinA2
An immediate consequence of Exercise 10.1 is
\[ f\< sup \f{x)\KA).
\Ja I xEA

8.10
INTEGRATION OF CONTENTED FUNCTIONS
341
We close this section by giving another useful characterization of contented
functions.
Proposition 10.4. Let / be a bounded function with compact support.
Then / is contented if and only if to every e > 0 and 5 > 0 we can find an
77 > 0 and a contented set A^ such that (liA^) < 8 and
|/(x) —/(y)| < e whenever ||x — j\\ < v and x, y g A^. (10.7)
Proof. Suppose that for every e, 8 we can find rj and A^. Let p = {□^} be a
paving such that
i) supp/c|/>|;
ii) if X, y G {Ji, then ||x — y\\ < v]
iii) if ^ = {[Ji G />: Dz n As F^ 0}, then m(|^|) < 25.
Then let fe,28(^) = /(xf) when x G {Zli, where x^ is some point of Qi. By (ii)
and (iii), we see that /e,25)1^1 is a paved e, 25-approximation to /. Thus / is
contented.
Conversely, suppose that / is contented,
and let/e/2,5/2, A^/2,8/2 be a paved
approximation to /.
Let /> = {□i} be the paving associated
with fe/2,8/2' Replace each □i by the
rectangle IZii obtained by contracting {Zii about
its center by a factor (1 — ^). (See Fig. 8.10.)
Thus M(nO = (1 - Oy(ay For any x,
y G ua, if
l|x — y|| < V,
where rj is sufficiently small, then x and y belong
to the same Q-. If
X, y G U^^^ X, y g A,i2,8/2,
and
Fig. 8.10
y|| < V,
then
|/(X) -/(y)| < |/(X) -/e/2,5/2(x)| + |/(y) -/e/2,5/2(y)|
+ |/6/2.5/2(x) — /e/2,5/2(y)|.
But the third term vanishes, so that |/(x) — /(y)| < €. Now by first choosing ^
sufficiently small, we can arrange that ju(|/>| ~ UCIIO < ^/2- Then we can
choose 7] so small that \\x — y\\ < rj implies that x, y belong to the same {Zli if
X, y G UD^ For this rj and for As = ^e/2.5/2 U (|/>| - UDO. Eq. (10.7) holds,
SindfxiAs) < 8. D
In particular, a bounded function which is continuous except at a set of
content zero and has compact support is contented.

342 INTEGRATION 8.11
EXERCISES
10.2 Show that for any bounded set ^, e^ is a contented function if and only if A is
a contented set.
10.3 Let/be a contented function whose support is contained in a cubeQ. For each b
let ps = {[Hi.sltGis be a paving with \ps\ = [H ^^^ whose cubes have diameter less
than 8. Let Xi,8 be some point of \Zli.8- The expression
is called a Riemann 5-approximating sum for/. Show that for any € > 0 there exists :i
5o[ = 5o(/)] > 0 such that
(f- Z/Cx.XDm)! < e
whenever 5 < 80.
11. THE CHANGE OF VARIABLES FORMULA
This section will be devoted to the proof of the following theorem, which is of
fundamental importance.
Theorem 11.1. Let U and V be bounded open sets in IR", and let 9 be a
continuously difFerentiable one-to-one map of U onto V with 9 ~ ^ difFerentiable
Let / be a contented function with supp f CV. Then (/ ° 9) is a contented function,
and
/^/=/^(/°^)|det/^|. (11.1)
Recall that if the map cp is given hy y^ = (p^(x^, . . . , x'^), then J^ is thc'
linear transformation whose matrix is [dcpi/dxj].
Note that if (^ is a nonsingular linear transformation (so that J^ is just cp),
then Theorem 11.1 is an easy consequence of Theorem 8.1. In fact, for functions
of the form eA we observe that ca ° (f ^ e^—iA, and Eq. (11.1) reduces, in this
case, to (8.1). By linearity, (ILl) is valid for all paved functions.
Furthermore, /o (y^ is contented. Suppose |/(x) —/(y)| < €when||x —y|| <
(f and X, y g A, with ijl(A) < 5. Then |/o ,^(u) — f° <^(v)| < € when
||u - v|| < rj/M and u, V g ^"^A),
withfi(cp~^A) < 8/\detcp\.
Now let ge,8, ^€,5 be an approximating family of paved functions for/. Then
\fo(p(x) — (7e,5°<^(x)| < eforxg (p~\A,^8) Sindfi((p~^A,^3) < 8/\det <p\. Thus
j{0e,8 ° <^)|det (p\ —> J(/o (^)|det (p\, and Eq. (11.1) is valid for all contented/.
The proof of Theorem 11.1 for nonlinear maps is a bit more tricky. It
consists essentially of approximating cp locally by linear maps, and we shall do it in
several steps. We shall use the uniform norm ||x||oo = max |x'| on W^. This is
convenient because a ball in this norm is actually a cube, although this nicety
isn't really necessary.

8.11 THE CHANGE OF VARIABLES FORMULA 343
Let ^ be a (continuously) differentiable map defined on a convex open set U.
If the cube D = Dp^ri lies in U, then the mean-value theorem (Section 7,
Chapter 3) implies that for any y E □,
ll^(y) - ^(p)lloo < ||y - pIIoo sup ||/^(z)||.
Thus
^O) C Uf^tl'l where K = sup |I/^(z) ||.
zGD
Thus
mCiAO)) < (sup ||/^(z)||)"M(n). (11.2)
zGD
Lemma 11.1. Let (p be as in Theorem 11.1. Then for any contented set A
with il C C/ we have
m(^(^)) < I |det/^|. (11.3)
Proof. Let us apply Eq. (11.2) to the map yp = L^^cp^ where L is a linear
transformation. Then
idetL|-v(^(n)) = iuCi'-vo)) < (sup ii/L-v(z)i!)V(n).
zGD
Since Jl—^^p — L^^J^p, we get
M(^(n)) < |detL|(sup |iL-V^(z)||)V(n) (11-4)
zGD
for any □ contained in the domain of the definition of (p and for any linear
transformation L.
For any € > 0, let 8 be so small that ||/^(x)~^/^(y)|| < 1 + € for
||x - ylloo < 8
for all X, y in a compact neighborhood of A. (It is possible to choose such a 8,
since J(x) is a uniformly continuous function of x, so that t/^(x)~V^(y) is close
to the identity matrix when x is close to y; see Section 8, Chapter 4.)
Choose an outer paving g^ = {\Z}i} of Aj where the Qt ^re cubes all having
edges of length less than 8. Let x^- be a point of \Z}i' Then applying (11.4) to
each [^i taking L = /^(x^), we get
^(<p{A)) < m(^(UI)) = ZMBd < Z |clet/^(xi)|(l + e)>(ni).
We can also suppose 8 to have been taken small enough so that
|det/^(z)| > (1 — €)|det/^(x^)| for all z G Di and all i.
Then we have
[ Idet/^l > (1 - €)|det/^(x,-)|M(ni),
and so /•
m(^U)) < j-^ (1 + erj^^ \detJ^\.
Since e is arbitrary and s is an arbitrary outer paving of A, we get (11.3). D

344 INTEGRATION 8.1 1
We can now conclude that / o <^ is contented for any contented / witli
supp f CV. In fact, let K be chosen so large that it is a Lipschitz constant for if
on ip~^(suppf), and so large that K > |det J^—i(u)| for u G supp/. Now given
€ and 8, we can find an rj such that
|/(u) — /(v)| < e if ||u — y\\ < V and u,y ^ A3 with fiiA^) < 8.
But this implies that
1/ o ^(x) - / o ^(y)| < e if ||x - y|| < v/K and x, y g ^-'(A^),
where iJi(ip~^(A3)) < K3, by (11.3). Since K was chosen independently of e
and 8, this shows that / o (^ is contented.
Lemma 11.2. Let (p, U, and V be as in Theorem 11.1. Let / be a nonnega-
tive contented function with supp f CV. Then
f^f<f^{fo^)\detJ,\. (11.5)
Proof. Let <g, A> be a paved e, 5-approximation to / with g(u.) < /(u) for all u.
If p = m^} is the paving associated with g, we may assume that supp f C\p\
Then
f(7= E oM^^iUi) <i:gM[ ^ IdetJ.I <E f , (/o<,)|detJ,|
= f ^ (fo^)\detJ,\= f{fo^)\detJ,\.
yu^-i(n,) J
Since we can choose g so that jg —> jf, we obtain (11.5). D
Lemma 11.3. Let (p, U, V, and / be as in Theorem 11.1. Let / be a non-
negative function. Then Eq. (11.1) holds.
Proof. Let us apply (11.5) to the map <p~^ and the function (f o (p)\detJ^\.
Since J^(x) o J^_i((p(x)) = id, we obtain
f(fo^)\detJ^\ < f[(fo^) o ^-'](\detJ^\ o cp-i)|det J^-i|
Combining this with (11.5) proves the lemma. D
Completion of the proof of Theorem 11.1. Any real-valued contented function can
be written as the difference of two positive contented functions. If for all \,
fix) > —M for some large M, we write /= (f-\- Mea) — Me^, where
supp / C D- Since we have verified Eq. (11.1) for nonnegative functions, and sincc^
both sides of (11.1) are linear in/, we are done. Similarly, any bounded complex-
valued contented function / can be written as / = /i + if2, where /i and /2 are
bounded real-valued contented functions. D

8.11
THE CHANGE OF VARIABLES FORMULA
345
In practice, we sometimes may apply Eq. (11.1) to a situation where the
hypotheses of Theorem 11.1 are not, strictly speaking, verified. For instance, in
IR^ we may want to introduce "polar coordinates". That is, we let r, 6 be
coordinates on W^] if S is the set 0 < ^ < 27r, 0 < r, we consider the map (p: S —^W^
given by X = r cos 6j y = r sin 6, where x, y are coordinates on a second copy
of U^. Now this map is one-to-one and has positive Jacobian for r > 0. If we
consider the open sets U C S given by 0 < r, 0 < ^ < 27r and V C^^ given by
V = U^ — {xy y :y = Of X > 0}, the hypotheses of Theorem 11.1 are fulfilled,
and we can write (since det J^ = r)
ff= fif°<p)r
(11.6)
if supp f CV. However, Eq. (11.6) is valid without the restriction supp f CV.
In fact, if De is a strip of width e about the ray 2/ = 0, x > 0, then / = fe^^ +
fef,n-D^ and ^feo^ —> 0 as e —> 0 (Fig. 8.11). Similarly, J(/o (p)(r o (p)eD^ o v? —> 0,
so that (11.6) is valid for all contented / by this simple limit argument.
D,
<p-\D,)
Fig. 8.11
We will not state a general theorem covering all such useful extensions of
Theorem 11.1. In each case the limit argument is usually quite straightforward
and will be left to the reader.
EXERCISES
11.1 By the parallelepiped spanned by v^, . . ., v"" we mean the set of all a: = ^ J^v^ +
. . . _|_ ^riyn^ where 0 < ^* < 1. Show that the content of this parallelepiped is given by
|det((vi,v,))|i/2.
11.2 Express the content of the ellipsoid
(xV
^<l)
in terms of the content of the unit ball.
11.3 Compute the Jacobian determinant of the map -Kr, 6> ^-^ <x, y>, where
X = r cos 6, y = r sin 6.
11.4 Compute the Jacobian determinant of the-map ^r, 6, <p> ^-^ <x, y, z>,
where x = r cos (f sinS, y = r sin ^ sin ^, 2: = r cos 6.
11.5 Compute the Jacobian determinant of the map -Kr, 6, z> ^-^ <x,y, z>,
where x = r cos 6, y = r sin ^, 2; = z.

346 INTEGRATION 8.12
12. SUCCESSIVE INTEGRATION
In the case of one variable, i.e., the theory of integration on IR^, the fundamental
theorem of the calculus reduces the computation of the integral of a function to
the computation of its antiderivative. The generalization of this theorem to
n dimensions will be presented in a later chapter. In this section we will show
how, in many cases, the computation of an n-dimensional integral can be reduced
to n successive one-dimensional integrations.
Suppose we regard IR^, in some fixed way, as the direct product W^ =
U^ X U^. We shall write every z G IR"" as z = < x, y >, where x gU^ and y G UK
Definition 12.1. We say that a contented function / is contented relative tci
the decomposition K"* = R^ X U^ if there exists a set A/ C K^ of content
zero (in U^) such that
i) for each fixed x G IR^, x g A/, the function /(x, •) is a contented fimctiori
on U^;
ii) the function JrZ/ which assigns to x the number Jr^/(x, •) is a contented
function on U^,
It is easy to see that the set of all such functions satisfies Axioms 3^1 througli
3^3. (The only axiom that is not immediate is 3^2. But this is an easy consequence
of the fact that any translation T can be rewritten as T1T2 ,where Ti is a
translation in U^ and T2 is a translation in UK)
It is equally easy to verify that the rule which assigns to any such / the
number
satisfies Axioms Jl through J4. The only one which isn't immediately obvioun
is J3. However, if p is any paving with supp / C \p\j then
/< ll/lkw
and
smce
for any rectangle (direct verification). Thus, by the uniqueness part of Theorem
10.1, we have
Lnf=ULlf(-'-))- (12.1)
Note, in particular, that if / is also contented relative to the decomposition
R^ = R^ X R^, then
In particular, for such / the double integration is independent of the order.

8.12
SUCCESSIVE INTEGRATION
347
In practice, all the functions that we shall come across will be contented
relative to any decomposition of W^. In particular, writing IR^ = R^ X • • • X H^^,
we have
In terms of the rectangular coordinates x^,. . . , x^, this last expression is usually
written as
/(•••(//(^\...,^")^^')---)^^".
For this reason, the expression on the left-hand side of (12.2) is frequently
written as
f^J= f ■■■ jf{x\...,x'')dx'---dx''.
Let us work out some simple examples illustrating the methods of
integration given in the previous sections.
Example I. Compute the volume of the intersection of the solid cone with vertex
angle a {vertex at 0) with the spherical shell 1 < r < 2 (Fig. 8.12). By a Euclidean
motion we may assume that the axis of the cone is the ^-axis. If we introduce
polar coordinates, we see that the set in question is the image of the set
I—l<i,o,o> f i :^ r <. z,
in the <r, ^, ^>-space (Fig. 8.13).
0 < ^ < 27r, 0 < d < a/2
Fig. 8.12
Fig. 8.13
By the change of variables formula and Exercise 11.4 we see that the volume
in question is given by
/ f2 /.27r /.a/2
r^ sine = / r^ sin 6 dO d(p dr
/•2 /.a/2
= 27r/ / r^ sm Odd dr
Ji Jo
= 2TrC [1 - cos {a/2)y dr
= 27r[l - cos (a/2)](| - ^).

348 INTEGRATION 8.12
Example 2. Let 5 be a contented set in the plane, and let /i and j^ be two con
tented functions defined on B, Let A be the set of all <x, y, z> G E^ such thai
<x, y> G 5 and fi(x, y) < z < f2ix, y). If G is any contented function on A,
we can express the integral J^ G as
rf2^^^y^
[ G = [ i I ' G{x, y, z) dz\ dx dy.
J A J A Vfiix,y) f
x'^ir
For example, compute the integral ja z, where
A is the set of all points in the unit ball lying
above the surface z = x^ + y^ (Fig. 8.14). '^'^-^^...--^ Fig. 8.14
Thus
A = {<x, y,z>: x^ + y^ + z^ < l,z > x^ + y^}.
We must have x^ + y^ < a, where a^ + a = 1 [so that a = (\/5 — l)/2], in
order for <x, y, z> to belong to A. Then fi{x, y) = x^ + y^, /2(x, y) ^
Vl — (^2 + 7/2), and
/,
zdz=Ul- ix' + y')- {x' + y')\
Jfl(x,y)
so that, using polar coordinates in the plane (and Exercise ILS),
f z=i[ [I- {x' + y') - {x' + y'f] = irf^\{l - r'~ r') dr.
Ja Jx^-\-y^<a Jo
As we saw in the last example, part of the problem of computing an integral
as an iterated integral is to determine a good description of the domain of
integration in terms of the decomposition of the vector space. It is usually a
great help in visualizing the situation to draw a figure.
Example 3« Compute the volume enclosed by a surface of revolution. Here we arc
given a function / of one variable, and we consider the surface obtained by
rotating the curve x = f{z), zi < z < Z2, around the 2-axis (Fig. 8.15). Wc*
thus wish to compute iJi{A), where
A = {<x, y, z> : x^ + y^ < fiz), zi < z < Z2}.
Here it is obviously convenient to use cylindrical coordinates, and we sec
that A is the image of the set
B = {<r, d, z> :r < f{z), 0 < 6 < 2ir}
in the <r, 6, 2>-space. By Exercise 11.5, we wish to compute
2/(0)'
jr= I f rdrdzde=2-wl (f r dr) dz = 2t "'-^^dz.
JB Jo Jill Jo Jzi \Jo / Jzi
Thus
J Zi
Uz.

8.12
SUCCESSIVE INTEGRATION 349
Fig. e.l5
EXERCISES
12.1 Compute the volume of the region between the surfaces 2 = a;^ -f y^ a^^d
z ^ x + y.
12.2 Find the volume of the region in E^ bounded by the plane 2: = 0, the cylinder
JJ.2 _j_ ^2 __ 2x, and the cone z = -\-\/x^ + y^.
12.3 Compute Ja (x^ + y^)^ dx dy dz, where A is the region bounded by the plane
z = 2 and the surface x^ ~\- y^ = 2z.
12.4 Compute Ja x, where
A - {<x,y,z>:x^+y^ + z^< a^, x > 0,y > 0, z > 0}.
12.5 Compute
Aw
.2 .2 ^2\l/2
X , y
+k+-^
62
ly '
wheri^ A is the region bounded by the ellipsoid
2 2 ^2
a2^52^c2
iet p be a nonnegative function (to be called the density of mass in the following
discussion) defined on a domain D in E^. The total mass of <Dj p> is defined as
M
[ p{x)
JD
dx.
If ikf 7^ 0, the center of gravity of <D,p> is the point C ^ <Ci,C2,Cz>, where
C2 = -^ / X2p(x) dx,
X3p(x) dx.
'^ = hL'-

350 INTEGRATION 8.12
12.6 A homogeneous solid (where p is constant) is given by a:i > 0, 0:2 >: 0, 0:3 > 0,
and
222
q2 52 q2 — '
Find its center of gravity.
12.7 The unit cube has density p(x) = 0:10:3. Find its total mass and its center of
gravity.
12.8 Find the center of mass of the homogeneous body bounded by the surfaces
^^ + 2/^ + 2^ = a^ and x'^ -\- y'^ = ax.
The notion of center of mass can, of course, be defined for a region in a Euclidean space
of any dimension. Thus, for a region D in the plane with density p, the center of mass
will be the point -<a:o, 2/0 > , where
Jd xp Sd yp
xo = -7i and 2/0 = -f
JDP JDP
12.9 Let D he Si region in the x^-plane which lies entirely in the half-plane a: > 0.
Let A be the solid in E^ obtained by rotating D about the ^-axis. Show that m(^) =
2tt dfx(D), where d is the distance of the center of mass of the region D (with uniform
density) from the z-axis. (Use cylindrical coordinates.) This is known as Guldin^s rule.
Observe that in the definition of center of gravity we obtain a vector (i.e., a point
in E^) as the answer by integrating each of its coordinates. This suggests the following
definition: Let F be a finite-dimensional vector space, and let ei, . . . , e^; be a basis
for V. Call a map/from E"" to F (/ is a vector-valued function on E"" with values in V)
contented if when we write /(x) = ^fi(^)ei, each of the (real-valued) functions fi
is contented. Define the integral of / over D by
// = ^(//^)-
12.10 Show that the condition that a function be contented and the value of its
integral are independent of the choice of basis ei, . . . , ek.
Let ^ be a point not in the closed domain Z), which has a
mass distribution p. The gravitational force on a particle of
unit mass situated at ^ is defined to be the vector
L
P(x)(x-e,^
Id Wx-rn
(here x — ^ is an E^-valued function on E^). Fig. 8.16
12.11 Let D be the spherical shell bounded by two concentric spheres Si and S2
(Fig. 8.16), with center at the origin. Let p be a mass distribution on D which depends
only on the distance from the center, that is, p(x) = /(||x||). Show that the
gravitational force vanishes at any ^ inside Si.
12.12 <D, p>- is as in Exercise 12.5. Show that the gravitational force on a point
outside S2 is the same as that due to a particle situated at the origin and whose mass
is the total mass of D.

8.13 ABSOLUTELY INTEGRABLE FUNCTIONS 351
13. ABSOLUTELY INTEGRABLE FUNCTIONS
Thus far we have been dealing with bounded functions of compact support. In
practice, we would like to be able to integrate functions which neither are
bounded nor have compact support. Let / be a function defined on E^, let M be
a nonnegative real number, and let ^4 be a (bounded) contented subset of E^.
Let f^ be the function
[O if X g A,
f^(x) - { M if X G ^ and \f{x)\ > M,
[fix) if xG-A and|/(x)| < M.
Thus f^ is a bounded function of compact support. It is obtained from / by
cutting / back to zero outside A and cutting / back to M when \f(x)\ > M.
We say that a function / is absolutely integrable if
i) f^ is a contented function for all M > 0 and contented sets A; and
ii) for any e > 0 there is a bounded contented set A^ such that ^^^ • / is
bounded and for all M > 0 and all B with B n A^ = 0,
/I
Wl < e.
It is easy to check that the sum of two absolutely integrable functions is again
absolutely integrable. Thus the set of absolutely integrable functions forms a
vector space. Note that if/ satisfies condition (i) and \f{x)\ < \g{x)\ for all x,
where g is absolutely integrable, then / is absolutely integrable.
Let / be an absolutely integrable function. Given any e, choose a
corresponding A^. Then for any numbers Mi and M2 > max^^e^^ 1/(^)1 and for any sets
AiD Ae and A2 D A^,
//^/-//f/|</|/^/_..| + /
4M2
JA2-A,
< 2e.
If we let e —> 0 and choose a corresponding family of A^^ then the above
inequality implies that the lim J/a^ is independent of the choice of the A^. We define
this limit to be J/.
We now list some very crude sufficient criteria for a function to be absolutely
integrable. We will consider the two different causes of trouble—nonbounded-
ness and lack of compact support.
Let / be a bounded function with /a contented for any contented set A,
Suppose \f(x)\ < C||x||~^ for large values of \\x\\. Let Br be the ball of radius r
centered at the origin. If ri is large enough so that the inequality holds for
il^ll ^ ^1, then for 7*2 > Vi we have
J J Br2—Bri J ri
-k
where Q^ is some constant depending on n (in fact, it is the "surface area'' of the
unit sphere in E^). If /c > n, this last integral becomes

352 INTEGRATION 8.13
which is < [Cln/{k — n)]r1~^, which tends to zero as ri —> oo if /c > n. Thus
w^e can assert:
Let / be a bounded function such that /^ is contented for any contented
set A. Suppose that |/(x)| —> 0 as ||x|| —> oo in such a way that ||x||^|/(x)| is
bounded for some k > n. Then / is absolutely integrable.
Now let us examine the situation when / is of compact support but
unbounded. Suppose first that there is a point Xq such that / is bounded in the
complement of any neighborhood of xq. Suppose, furthermore, that
|/(x)| < C'\\x - xoll
—k
for some constants C and k. Then if |/(x)| > M, \\x — Xo|| ^ > M/C or
||x - xoll < C/M'^K
Let Bi be the ball of radius CM^^^^ centered at Xq. Then |/(x)| > Mi
implies that x G 5i. Furthermore, for M2 > Mi we have
[ If""'] <&l \\x-x^f + M2ix{B^),
JB^ JB1—B2
where B2 is the ball of radius CM^^^^ centered at Xq. Thus
Ik
-n/k
rCMJ^'^
JBi JCM2
where 12^ and Vn depend only on n. If k < n, the integral on the right becomes
y^n—k y^n—k
J^ /jyr{k—n)lk jij(k—n)/k\ ^ _^ j^{k—n)lk
n — k n — k
Thus
/.
\f'\ < const (Mf-")/' + M^'-"^/"),
which can be made arbitrarily small by choosing Mi large.
Thus if/has compact support and is such that/^ is contented for all M and
|/(x)| < C\\x — Xo||~^ with k < n, then/is absolutely integrable.
More generally, let aS be a bounded subset of an ^dimensional subspace of
E^. Let d{x) denote the distance from x to S. Let / be a function of compact
support with f^ contented for all M. If |/(x)| < C d{x)~^ with k < n — I,
then f is absolutely integrable. The proof is similar to that given above and is left
to the reader.
Let {fk} be a sequence of absolutely integrable functions. Under what
conditions will the sequence jfn —> J/if the sequence/^(x) —> /(x)? Even if the
sequence converges uniformly, there is no guarantee that the integrals converge.
For instance, if/a, = (^/k'^)e{j^, then |/fc(x)| < l//c^, so that fk approaches zero
uniformly. On the other hand, jfk = 1 for all k.

8.13
ABSOLUTELY INTEGRABLE FUNCTIONS
353
We say that a set of functions {fk} is uniformly absolutely integrable if for
any e > 0 there is an A^ which can be chosen independently of k such that
Jb I
< e
for all M
wherever B D A^ = 0.
We frequently verify that {//,} is uniformly absolutely integrable by showing
that there is an absolutely integrable function g such that \fk{x)\ < \g{x)\ for
all k and x.
Let {fk} be a uniformly absolutely integrable sequence of functions. Suppose
that fk—^f uniformly. Suppose in addition that / is absolutely integrable. Then
jfk —> J/. In fact, for any 5 > 0 we can find a ko such that \fk{x) — f{x)\ < 8
for all k > ko and all x. We can also find A^ and M^ such that
ijfk ■— jf\ < \jfk,A^ — jfk\ + IjfAe ~ // + \jfk,A^ — I/aA
< e+e+8fjL{A,),
which can be made arbitrarily small by first choosing e small (which then gives
an Ag) and then choosing 8 small (which means choosing ko large).
The main applications that we shall make of the preceding ideas will be to
the problems of computing iterated integrals and of differentiating under the
integral sign.
Proposition 13.1. Let / be a function on
X
Suppose that the set of
functions {/{x, •)} is uniformly absolutely integrable, where x is restricted
to lie in a bounded contented set K C 11^^. Then the function exx^^' f is
absolutely integrable, and
[ if= [ [if{^,y)dydx= f f f{x,y)dxdy.
Proof. By assumption, for any e > 0 we can find M and A^C^^ such that
li\fA{x,-)\ < € if AnA, = 0. (13.1)
Now for any set B in K",
< ii{K)e if BnKx A,= 0.
This shows that eKx»^f is absolutely integrable on R". Now choose a sufficiently
large d = di X da and an M such that
[ if- [ if'^
and also such that
ffix, ■) - ffMix, •)
< e
< €,
for all X ElK.

354 INTEGRATION 8.13
Then we have
< e
and
Thus
so that
\[ if-1 if'
[ if°- I f,f¥{x,y)dydx.
f J ~~ I f if{x,y)dydx\ < e +fi{K)e,
f if=[ fif{^,y)dydx.
Finally Eq. (13.1) shows that the function F(y) = jx f{', y) is absolutely
integrable. In fact, using the same A and M as in (13.1), we get
f\F¥'\ < KK)e.
Thus we get
[ if""[ f z/(^' y)dydx= L ( f{x, y) dx dy. D
Jkx^ jk Jr'- Jr'' Jk
An extension of the same argument shows the following.
Proposition 13.2. Let / be absolutely integrable on W^ and such that the
functions/(x, •) are uniformly absolutely integrable for each x G U^. Then
jf = IIfix, y) dy dx.
We now turn our attention to the problem of differentiating imder the
integral sign.
Proposition 13.3. Let {t, x) i-^ F{t, x) be a function on / X IR^, where
/ = [a, 6] C H^. Suppose that
i) F and dF/dt are continuous functions on / X W^;
ii) (dF/dt){tj •) is a uniformly absolutely integrable family of functions;
iii) F{tj •) is absolutely integrable for all t G /.
Let f{t) — JF{tj •)• Then / is a differentiable function of t and
f'{t) = f^„{dF/dt){t,-).
Proof. Let G{t) = ^u^ {dF/dt){t^ •)• Then G{t) is continuous; hence we can pass
to the limit under the integral sign of a family of absolutely integrable
fimctions. Furthermore,
£' G{s) ds = l^^l^ {dF/dt){s, •) ds

8.14 PROBLEM set: the FOURIER TRANSFORM 355
by Proposition 13.1. Thus
[' G{s) = /■„ (Fit, •) - F{a, •)) - [„F{t, •) - f ^F{a, •) = fit) - /(«)•
Differentiating this equation with respect to t gives the desired result. D
Finally, let us state the change of variables formula for absolutely integrable
functions.
Let (p: U —^ y be a differentiate one-to-one map with differentiate inverse,
where U and V are two open sets in U^. Let / be an absolutely integrable
function defined on V. Then (/ o <^)|det J^\ is an absolutely integrable
function on U and
f^f= f^(fo^)\detJ^\.
Proof. To show that (/o <^)|det/^| is absolutely integrable, let e > 0 and
choose am A^CV such that (ii) holds. Then A^ is compact, and therefore so is
(p'~^{Ae). In particular, <^~^(i4e) is a bounded contented set and |det/^| is
bounded on it. li B n <^~^(i4e) = 0, where B C U is bounded and contented,
then
f |detJ,|^l(/o^)^|< f \n<e.
JB J(p{B)
This shows that (/o <^)|det /^| is absolutely integrable. The rest of the
proposition then follows from
by letting e —> 0. D
EXERCISES
13.1 Evaluate the integral /_^^ e~^^ dx. [Hint: Compute its square.]
13.2 Evaluate the integral /o° e~^^a:^^ dx.
13.3 Evaluate the volume of the unit ball in an odd-dimensional space. [Hint:
Observe that the Jacobian determinant for "polar" coordinates is of the form r'^~^ X /,
where / is a function of the "angular variables". Thus the volume of the unit ball is of
the form C/o r^~^ dr^ where C is determined by integrating / over the "angular
variables". Evaluate C by computing (/^^ e~^^ dx)"^.]
14. PROBLEM SET: THE FOURIER TRANSFORM
Let a = <q:i, . . . , Q:n> be an n-tuple whose entries are nonnegative integers.
By D" we shall mean the differential operator
Z)«= —
dxV ' ' 'd^n"

356 INTEGRATION 8.14
Let |a:| = ai + • • • + ^n- Let Q{x, D) = Y.\a\<k aa{x)D'^ be the differential
operator where each a^ is a polynomial in x. Thus if / is a C^-function on U^,
we have
iQf)(x) = E a^(x)DJix).
\a\<k
For any / which is C°° on W^ we set
ii/iio = sup mx)\.
We denote by S the space of all f ^C^ such that
ll/llo < ^ (14.1)
for all Q. To see what this means, let us consider those Q with k = 0. Then
(14.1) says that for any polynomial a(-) the function a • / is bounded. In other
words, / vanishes at infinity faster than the inverse of any polynomial; that is,
lim ||x||^/(x) = 0
llxll->oo
for all p. To say that (14.1) holds means that the same is true for any derivative
of / as well.
If / is a C°°-function of compact support, then (14.1) obviously holds, so
/ G S. A more instructive example is provided by the function n given by
Since lim^^oo r^e~^^ = 0 for any p, it follows that limnxn^oo a{x)n{x) = 0. On
the other hand, it is easy to see (by induction) that D^n{x) = Pa{x)n{x) for
some polynomial Pa- Thus Qn{x) = PQ{x)n{x), where Pq is a polynomial.
Thus n ^S.
It is easy to see that the space S is a vector space. We shall introduce a
notion of convergence on this space by saying that /n —^ / if for every fixed Q,
(Note that the space S is not a Banach space in that convergence depends on an
infinity of different norms.)
EXERCISES
14.1 Let <^ be a (7°°-function which grows slowly at infinity. That is, suppose that
for every a there is a polynomial Pa such that
|Z)«<^(a:)| < Pa(x) for all x.
Show that if / G S, then (pf G S. Furthermore, the map of S into itself sending f —^ (pf
is continuous, that is, if/n —>/, then (pfn —> (pf.

8.14 PROBLEM set: the FOURIER TRANSFORM 357
For o: = <x\ . . . , x^'y ^W" smd ^ = < ^S . . . , r> G R''* we denote the value
of ^ at a; by
(x,^) = xi^^H h^c-^-.
Also for any a = <a^, . . . , a^> and any a: E B^^ we let
and similarly ^« = (fi)«i • • • (^")«n, etc.
For any/E S we define its Fourier transform/, which is a function on IR^*, by
ho = fe-'^'''^f(x) dx.
We note that
/(O) = jf and 1/(01 < j\f\.
14.2 Show that/ possesses derivatives of all orders with respect to f and that
in other words,
mm) = 9(^),
where gf(a:) = (—i)'"'a:"/(a:).
14.3 Show that <>
[Hint: Write the integral as an iterated integral and use integration by parts with
respect to the jth variable.]
14.4 Conclude that the map /i-^ / sends S(IR'') into S(IR''*) and that if /n -> 0 in S,
then/„->OinS(IR^*).
14.5 Show that ^^
£/(^) = e-^<-.f>/(Q for any co E U\
Recall that Ta,/(a:) = f(x — co).
14.6 For any / E S define 7 by
fix) = f{-x),
where denotes complex conjugation. Show that
M = M^
14.7 Let n = 1, and let/ be an even real-valued function of x. Show that
/(© = j cos (x, 0 fix) dx.
14.8 Let n(x) = e-(i/2)-r2 ^^^^^ ^ ^ u^i^ gj^^^ ^^^^
and conclude that
logn(^) = ~i^2_^ const,

358 INTEGRATION 8.14
SO that
n(0 = const X e-(i/2)€^
Evaluate this constant as \/27r by setting ^ = 0 and using Exercise 13.1. Thus
14.9 Show that the hmit limc_>o Se^^ (sin x)/x dx exists. Let us call this limit d.
Show that for any i^ > 0, limc_>o Se^^ (sin Rx)/x dx = d.
If / E S, we have seen that/ E S(IR^*). We can therefore consider the function
The purpose of the next few exercises is to show that
/(2/) = (2^/e''^^'*>/(f)df. (14.2)
We first remark that since all integrals involved are absolutely convergent, it
suffices to show that
fiy)
lim . . . lim -i- /" " / ' he, ..., r)e«'''«'+-+^''«'" d^ ■ ■ ■ df.
Substituting the definition of / into this formula and interchanging the order of
integration with respect to x and ^, we get
Um... lim (f)" [["■■■(' fix\ ..., ^'.)e'K«'-')f'+-+(«''-'')f"l a^ ■ ■ ■ d^dx.
i2l->oo i2„->oo \2'r'/ J J—R^ J—Ri
It therefore suffices to evaluate this limit one variable at a time (provided the
convergence is uniform, which will be clear from the proof). We have thus reduced the problem
to functions of one variable. We must show that if/G S(lI^O, then
We shall first show that
fiy) = lim-L // Kx)e'"'-''^d^dx.
R^oo ^T^ J J —R
w that
f{y)= limi. [[ }{x)e'"'-'''dkdx,
where d is given in Exercise 14.9.
14.10 Show that this last integral can be written as
1 r ., X sin^R(y-^ 1 rf(y-u)+f(y+u) Ru
-^ / j{x) dx = - sm — du.
2d J-00 y — X d Jo 2 u
14.11 Let
^(^),/(.-^)+/fa+^)_^(^).

8.14 PROBLEM set: the FOURIER TRANSFORM 359
Show that g{0) =0 and conclude that g(x) = xh{x) ior 0 < x < 1, where hsC^. By
integrating by parts, show that
/ g(u) sin [- / g(u) sin —
lim 1 ("fcli^
jB_>oo d Jo
1
< const -^'
K
)+f(y + u) . Ru , ., ,
sin — du = f{y).
2 u
This proves that
iih
14.12 Using Exercise 14.8, conclude that d = 7r/2.
Let /i E S and /2 E S. Define the function /i * /2 by setting
/i */2W = ffi(x — y)f2(y) dy.
Note that this makes good sense, since the integrand on the right clearly converges for
each fixed value of x. We can be more precise. Since fi E S, we can, for any integer p,
find a Kp such that
SO that
,„„ \hiy)\<T^-
' \\y\\>R 1 -j- /C^
Then
/
J Wn
fa+ \\xr)fi(x ~ y)f2(y) dy= f (1+ \\xr)Mx - y)h{y) dy
J y 111/ll<(l/2) 11x11
+ f {l+\\xr)h{x-y)f2{y)dy.
The first integral is at most
C„(4llx||)"(l + \\x\\Y max \Mz)\ ^^^l^^^^^y '
while the second is at most
ll»^_,■■■l..■.M -^p(ilMI)"
(l+IWr) max!/.(.)! .^_^^,|l^ll^^
By choosing p > q-jr n, we see that both terms go to zero. Thus
lim (1+11x11')/!*/2(x) =0.
11x11—>oo
14.13 Show that
£-.»•«-(i)-'-/.*(i)
Conclude that /i */2 E S.

360 INTEGRATION 8.14
14.14 Show that if ip is any bounded continuous function on IR'*, then
[[(fix+y)/ (x)f2{y) dx dy = j(p(u)(fi */2)(^) du.
14.15 Conclude that
7r^(© = /i(©/2(^).
14.16 Show that
f^f(y) = (£ff\m'e'''^'d^,
14.17 Conclude that for any / G S,
/I'l'=(s)7i'i' <»«
[Hint: Set ?/ = 0 in Exercise 14.16.]
The following exercises use the Fourier transform to develop facts which are useful
in the study of partial differential equations. We will make use of these facts at the
end of the last chapter. The reader may prefer to postpone his study of these problems
until then.
On the space S, define the norm || ||s by setting
11/11^ =(2x)-"|(l+ll?ll')1/(0l'd?,
and the scalar product (/, g)s by
(f,g)s = fii+ufrk&md^.
14.18 Let s = i^ be a nonnegative integer. Show that
ll/ll'= Z ,.j,^ I ., I \Dy(x)fdx,
where al = ail • • • anl [Use the multinomial theorem, a repeated application of
Exercise 14.3, and Eq. (15.3).]
We thus see that ||/||i2 measures the size of/ and its derivatives to order R in
the square integral norm. It is helpful to think of || \\s as a generalization of this notion
of size, where now s can be an arbitrary real number.
Note that
11/11. < WfWt if s<L
For any real s define the operator K* by setting
^fiO = (1 + II ff )'/(?)•
14.19 Show that the operator K = K^ is given by
oxi

8.14 PROBLEM set: the FOURIER TRANSFORM 361
14.20 Show that for any real numbers s and t,
\\K%\t = ||/||,+2.
and
{K^f,g)i = (f,K^9)t= (f,9)s+t.
14.21 Show that K*"""* = K* o K\ so that, in particular, K^ in invertible for all s.
We now define the space Hg to be the completion of S under the norm || ||s. The
space Hs is a Hilbert space with a scalar product ( , )s. We can think of the elements
of Hs as "generalized functions with generalized derivatives up to order s". By
construction, the space S is a dense subspace of Hg in the norm || ||s. We note that
Exercise 14.20 implies that the operator K^ can be extended to an isometric map of Ht into
Ht-2s. We shall also denote this extended map by K\ By Exercise 14.21,
K-^:Ht-2s-^Ht
is the inverse of K% so that K^ is a norm-preserving isomorphism of Ht onto Ht-2s'
14.22 Let uG Hs and v G //_«. Show that
iK^^)oi < II^IMIHI—
Thus we can extend <u,v>-^(u,v)oiosi function on Hg X H^s which is linear in
u and antilinear in v [that is, (u, avi -\- b2V2)o = a(u, vi) -\- b(u, V2)] and satisfies the
above inequality. Thus any v E H^s defines a bounded linear function, I, on Hs by
l(u) = (u, v)o.
14.23 Conversely, let Z be a bounded linear function on Hs. Show that there is a
V E H^s with l(u) = (u, v)o for all u E Hs. [Hint: Consider the hnear form v = K'^w,
where w; is a suitable element of Hs, using Theorem 2.4 of Chapter 5.]
14.24 Show that
II II \(u,v)o\
M-s = sup II II '
uGH„ \\u\\s
(Exercise 14.22 gives an inequality. If z; f^ 0, take u = K'^'v to get
(K-'^h, K-''h)
m\
l|2
in order to get an equality.)
14.25 Let 2s > n (where our functions are defined on W'). Show that for any/G S
we have
sup \f{x)\ < 11/11, r f (1 + ll^f )~'l'^' di (Sobolev's inequality).
(Use Eq. (14.2), Schwarz's inequality, and the fact that the integral on the right of
the inequality is absolutely convergent.)
Sobolev's inequality shows that the injection of S into (7(IR^) extends to a continuous
injection of Hg into C{W')j where (7(IR^) is given the uniform norm. We can thus
regard the elements of Hs as actual functions on IR^ if s > n/2.

362 INTEGRATION 8.14
By induction on \a\ we can assert that for s > n/2, any/G -^ui+s has \a\
continuous derivatives and
sup I D7(x)I < Call/Ill„|+.. (14.4)
14.26 Let 12 be a bounded open subset of R^. Let (^ E S satisfy supp cp C^. Show
that
\m)\ < fJii^y^HAo for all ^
14.27 Let d' = lubxGn |^1, and let d« = (d^)"! • • • (d")«n. Show that
14.28 Show that
and conclude that
l(i+llff)V(?)l<M(Si)Hlf.
14.29 More generally, let ^ be a function in S which satisfies \l/(x) = 1 for all x G^,
and let (^ E S satisfy supp cp C^. Show that
where \l^^(x) = 4/{x) e~'^'''^\ and that
\DiH^)\ < II^IMI^^II-.,
where \l/l(x) = x'^(x)e-'^'^'^\
Let us denote by H^ the completion under || ||s of the space of those functions in cS
whose supports lie in 12. According to Exercise 14.29, any cp E: H^ defines an actual
function <^ of ^ which is differentiable and satisfies
where ||\^^(a:j||_s depends only on 12, a, ^, and —s, and is independent of (p. Further-
more, ll^ll? = S(l + \\m'\H^)\^ d^.
14.30 Let s < t. Then the injection Ht —> i/s is a compact mapping. That is, it
{(pi} is a sequence of elements of //? such that \\(Pi\\t ^ 1 for all i, then we can select a
subsequence {cpij} which converges in || ||s. [Hint: By Exercise 14.29, the sequence ot
functions cpiiO is bounded and equicontinuous on {.f : ||^|| < r} for only fixed r. Wv
can thus choose a subsequence which converges uniformly and therefore a subsubsc-
quence which converges on {^ : ||^|| < r} for all r (the uniformity possibly depending
on r). Then if {cpij} is this subsubsequence,
ikv - 'Pifs = /(I + ufrwijio - >PiM' d^
+ /■ (i+ufy\<Pijm~<PiM'd^
J \\^\\>r
J llf ll<r
+ (i+ufy-'{\\<Pijfi + \\<Pifi}.]

CHAPTER 9
DIFFERENTIABLE MANIFOLDS
Thus far our study of the calculus has been devoted to the study of properties
of and operations on functions defined on (subsets of) a vector space. One of
the ideas used was the approximation of possibly nonlinear functions at each
point by linear functions. In this chapter we shall generalize our notion of space
to include spaces which cannot, in any natural way, be regarded as open subsets
of a vector space. One of the tools we shall use is the "approximation" of such a
space at each point by a linear space.
Suppose we are interested in studying functions on (the surface of) the unit
sphere in E^. The sphere is a two-dimensional object in the sense that we can
describe a neighborhood of every point of the sphere in a bicontinuous way by
two coordinates. On the other hand, we cannot map the sphere in a bicontinuous
one-to-one way onto an open subset of the plane (since the sphere is compact and
an open subset of E^ is not). Thus pieces of the sphere can be described by open
subsets of E^, but the whole sphere cannot. Therefore, if we want to do calculus
on the whole sphere at once, we must introduce a more general class of spaces
and study functions on them.
Even if a space can be regarded as a subset of a vector space, it is conceivable
that it cannot be so regarded in any canonical way. Thus the state of a
(homogeneous ideal) gas in equilibrium is specified when one gives any two of the three
parameters: temperature, pressure, or volume. There is no reason to prefer any
two to the third. The transition from one set of parameters to the other is given
by a one-to-one bidifferentiable map. Thus any function of the states of the
gas which is a differentiable function in terms of one choice of parameters is
differentiable in terms of any other. Thus it makes sense to talk of differentiable
functions on the states of the gas. However, a function which is linear in terms
of one choice of parameters need not be linear in terms of the other. Thus it
doesn't really make sense to talk of linear functions on the states of the gas.
In such a situation we would like to know what properties of functions and what
operations make sense in the space and are not artifacts of the description we
give of the space.
Finally, even in a vector space it is sometimes convenient to introduce
"nonlinear coordinates" for the solution of specific problems: for example, polar
coordinates in Exercises 11.3 and 11.4, Chapter 8. We would therefore like to
know how various objects change when we change coordinates and, if possible,
to introduce notation which is independent of the coordinate system.
363

364
DIFFERENTIA BLE MANIFOLDS
9.1
We will begin our formal discussion with the definition of differentiable
manifolds. The basic idea is similar to the one that is used in everyday life to
describe the surface of the earth. One gives a collection of charts describing small
overlapping portions of the globe. We can piece the whole picture together by
seeing how the charts match up.
<Pi{Ui)
<Pj{Uj)
Fig. 9.1
1. ATLASES
Let M be a set. Let F be a Banach space. (For almost all our applications we
shall take V to be U^ for some integer n.) A V-atlas of class C^ on M is a
collection d of pairs (C/^, (pi) called charts, where Ui is a subset of M and <pi is a bijec-
tive map of Ui onto an open subset of V subject to the following conditions
(Fig. 9.1):
AI. For any (C/^, (pi) G Ct and (Uj, cpj) G d the sets (fiiUi fl Uj) and
(fjiUi n Uj) are open subsets of F, and the maps
(fi o (pj^: (pj{Ui n Uj) —> (fiiUi n Uj)
are differentiable of class C^.
A2. ljUi= M.
The fimctions ifi o ipj ^ are called the transition functions of the atlas ^.
The following are examples of sets with atlases.
Example I. The trivial example. Let M be an open subset of V. If we take ft
to consist of the single element (U^ cp), where U = M and ^: C7 —> 7 is the
identity map, then Axioms Al and A2 are trivially fulfilled.
Example 2. The sphere. Let M = S"" denote the subset of W'^^ given by
(x^)^ + • • • + (x"^"^^)^ = 1. Let the set Ui consist of those points for which
^n+i v^ _-^^ ^^^ Yet U2 consist of those points for which x"^"^^ < 1. Let
cpiiUi-^ IR^

9.1
be given by
.^■o..(.\...,.-^)=^-3^
i= 1,
ATLASES 365
• ,^,
where 2/\ ...,?/ are coordinates on W. Thus the map ipi is given by the
projection from the "south pole", <0, . . . , 0, -1 >, to IR^ regarded as the
equatorial plane (see Fig. 9.2). Similarly, define (^2 by
1 — x^H-l
Then (^i(f/i n U2) = <P2{Ui n U2) =^ {y GW'iy 9^ 0}. Now
.n-\-U2
1 - X'
n + 1
Thus
^ 1 - (x"+^)
(1 + X^ + l)2 ~ 1 -f :c^ + l
or
^2(^) =
^iW
In other words, the map (^2 ° <^r\ defined for all ?/ 5^ 0, is given by
<^2 ° ^1 {y) = -irw^ •
Thus conditions Al and A2 are fulfilled.
Fig. 9.2
Note that the atlas we gave for the sphere contains only two charts (each
given by polar projection). An atlas of the earth usually contains many more
charts. In other words, many different atlases can be used to describe the same
set. We shall return to this point later.

366
DIFFERENTIABLE MANIFOLDS
9.1
Fig. 9.3
Fig. 9.4
Example 3. The circle. The circle fi^Ms a "one-dimensional sphere" and therefore
has an atlas as described in Example 2. We wish to describe a different atlas
on S^. Regard S^ as the imit circle x\-^ xl = Ij and consider the fimction ^i,
defined in a neighborhood of < 1, 0> on the upper semicircle of aS\ which gives
the angle from the point on S^ to < 1, 0> (see Fig. 9.3). As we move
counterclockwise around the circle, this function is well defined imtil we hit <1, 0>
again. We will take, as the first chart in our atlas, (C7i, ^i), where Ui =
S^ — {<1,0>} and (9i is the fimction defined above. LetU2 = S^ — {<0, 1>},
and define $2 to be 7r/2 plus the angle (measured counterclockwise) from
<0, 1> (see Fig. 9.4). Now C7i n C/2 = ^S^ - {<1, 0>, <0, 1>}, and
e,(U,nU2)= (0,27r) - {t/2}.
ei[UinU2)
-h
7r/2
-4-
27r
Also, d2{Ui n C/2) = (7r/2, 27r + 7r/2) - {27r}.
0 7r/2
The map 62 ° B~[^ is given by
-i^:,) = j^ + 2^
62^11002)
-I
e2oe-^\x)
27r 27r + 7r/2
if 0 < X < 7r/2,
if 7r/2 < X < 27r.
Example 4. The product of two atlases. Let d = {(C/^, (fi)} be a Fi-atlas on a set
Mj and let (B = {{Wj, ypj)} be a y2-atlas on a set Nj where Vi and V2 are
Banach spaces. Then the collection e = {{Ui X Fy, (fi X i/j)} is a (Fi X F2)-
atlas onM X N. Here cpi X \l/j{p, q) = <cpiip), ^M)> if <P: Q> ^UiX Wj.
It is easy to check that C satisfies conditions Al and A2. We shall call C the
product of d and (B and write C = a X (B.
For instance, let M = (0, 1) C K^ and N = S^. Then we can regard
M X iV as a cylinder or an annulus. If M = N = S^j then M X iV is a torus.
Cylinder
Annulus
Torus

9.2 FUNCTIONS, CONVERGENCE 367
It is an instructive exercise to write down the atlases and transition functions
explicitly in these cases.
Example 5. As a generalization of our first example, let aS be a submanifold of
an (n + m)-dimensional vector space X, as defined in Section 12 of Chapter 3.
For each neighborhood N defined there, the set S f) Nj together with the map if
which is defined as the projection tti restricted to S, provides a chart with
values in V (where X is viewed as F X W). In such a neighborhood N the set S
is presented as a graph of fimction F. In other words,
SnN = {<x,Fix)> gFX W'.x^TTiiS)},
where F is a smooth map of A = Ti(S f) N) into W. Let N^ be another such
neighborhood with corresponding projection t[ (where now X is identified with
V X W in some other way). Then cp^ o (p~^(x) = 7ri(x, F(x)), which shows
that (^' o (^~"Ms a smooth map. Thus every submanifold in the sense of Chapter 3
possesses an atlas.
Exercise. Let P^ (projective n-space) denote the space of all lines through the origin
in IR^"^^. Any such line is determined by a nonzero vector lying on the line. Two such
vectors, <x^, . . , , x^"^^ > and <y^, . . . , y^~^^ > , determine the same line if and only
if they differ by a factor, that is, y* = \x^ for all i, where X is some (nonzero) real
number. We can thus regard an element of P^ as an equivalence class of nonzero
vectors. For each i between 1 and n + 1, let Ui C P^ be the set of those elements
coming from vectors with x^ 7^ 0. Map
Ui
by sending
.1 n+1. J X X X^ X ^ \
<X , . . . ,X >F->< -r,... , r- , r- , . . . , r- >.
' ' ^X' ' ' X' ' X' ' ' X^ f
Show that the map ai is well defined and that {{Uij ai)} is an atlas on P^.
2. FUNCTIONS, CONVERGENCE
Let a be a F-atlas of class C^ on a set M. Let / be a real-valued function defined
on M. For a chart (C/^-, (pi) we obtain a fimction fi defined on (Pi{Ui) by setting
fi=focpi-\ (2.1)
The function fi can be regarded as the "local expression of /" in terms of the
chart (Uij cpi). In general, the fimctions fi will look quite different from one
another. For example, let M = S^, let GL be the atlas described, and let / be the
fimction on the sphere assigned to the point <x\ . . ., x^~^^ > the value x'^^^.
Then
2
My) =f° fT'iy) = Y+J^2 - ^'
while
f2(y) = f" •P2\y) = 1 - Y+Jyp'
as one can check by solving the equations.

368 DIFFERENTIABLE MANIFOLDS 9.2
Returning to the general discussion, we observe that the functions fi are
not completely independent of one another. In fact, it follows from the
definition (2.1) that we have
/z ° <^i ° ^T^ = fj on (fjiUi n Uj). (2.2)
[Thus in the example cited above we indeed have f2{y) = fi{y/\\y\\^)y as is
required by (2.2).]
We now come to a simple but important observation. Suppose we start
with a collection of fimctions {/J, each/^- defined on (pi{Ui)j and such that (2.2)
holds. Then there exists a imique function f on M such that fi = f° ^pT^- In
fact, define / by setting /(p) = fi{<Pi{p)) if p E Ui. For / to be well defined, we
must be sure that this definition is consistent, i.e., that if p is also in C7y, then
fi{(Piip)) = fj{^j{p))i but this is exactly what (2.2) says.
We can thus think of a real-valued function in two ways: as either
i) an object defined invariantly on M, i.e., a map from M to IR, or
ii) a collection of objects (in this case functions) one defined for each chart
and satisfying certain "transition laws", namely (2.2).
This dual way of looking at objects on M will recur quite frequently in what
follows.
Let M be a set with an atlas of class C^. We will say that a function / is of
class C^ {I < k) if each of the functions fi defined by (2.1) is of class CK Note
that since I < /c, this can happen without any interference from (2.2). If/^ G C^
and (p^^ o cpj ^C^ (k > Q, then fi o (,^^1 o cpj) g CK If I were larger than k,
then in general fi would not be of class C^ if fj were, and there would be very
few functions of class CK
Since we will not wish to constantly specify degrees of differentiability of
our atlas, from now on when we speak of an atlas we shall mean an atlas of class C°°.
Let M be a set with an atlas d. We shall say that a sequence of points
{xi G M} converges to x G M if
i) there exists a chart {Uij cpi) G GL and an integer Ni such that x G Ui and
for all k > Nj Xk ^ Ui;
ii) ^i{xk)k>N converges to ip{x).
Note that if {Uj, cpj) is any other chart with x G Ujy then there exists an Nj
such that (fjixk) G Uj for k > Nj and (pj{xk) —> (Pj{x). In fact, choose Nj so
that (Pi{xk) G (Pi{Ui n Uj) for all k > Nj. (This is possible since (Pj{Ui 0 Uj)
is open by Al.) The fact that the (pj{xjc) converge to (pj{x) follows from the
continuity of (pj ° (fi^. It thus makes good sense to say that {xk\ converges to x.
Warning. It does not make sense to say that a sequence {xk\ is a Cauchy
sequence. Thus, for example, let M = S^ with the atlas described above. If {xk\
is a sequence of points converging to the north pole in aS^, then (pi{xk) —> 0,
while (P2{xk) —^ 00. This example becomes even more sticky if we remove the
north pole, i.e., l^iM = S^ — { < 0, . . . , 0, 1 >} and define the charts as before.

9.3 DIFFERENTIABLE MANIFOLDS 369
Then {xk} has no limit (in M). Clearly, {(pi{xk)} is a Cauchy sequence, while
{^2i^k)} is not.
Once we have a notion of convergence, we can talk about such things as
open sets and closed sets. We could also define them directly. For instance, a set
U is open if (pi{U n Ui) is an open subset of (Pi{Ui) for all charts (C/^, (^i), and so on.
EXERCISES
2.1 Show that the above definition of a set's being open is consistent, i.e., that there
exist nonempty open sets. (In fact, each of the Ui^s is open.)
2.2 Show that a sequence {xa} converges to x if and only if for every open set U
containing x there is an Nu with Xa E: U for a > Nu-
Let a = {{Uij (fi)} be an atlas on M, and let U be an open subset of M relative
to this atlas. Let d \ U he the collection of all pairs {Ui n C/, (Pi \ U). It is
easy to check that d f C7 is an atlas on U. We shall call it the restriction of
a to U.
Let / be a function defined on the open set U. We say that / is of class C^
on U if it is of class C^ relative to the atlas 6i \ U on U. For later convenience
we shall say that a function / defined on a subset of M is of class C^ if
i) the domain of / is some open set U of M, and
ii) / is of class C^ on U.
3. DIFFERENTIABLE MANIFOLDS
In our discussion of the examples in Section 1, the particular choice of atlas that
we made in each case was rather arbitrary. We could equally well have
introduced a different atlas in each case without changing the class of differentiable
functions, or the class of open sets, or convergent sequences, and so on. We
therefore introduce an equivalence relation between atlases on M:
Let a I and (2,2 be atlases on M. We say that they are equivalent if their
union di U (22 is again an atlas on M.
The crucial condition is that Al still hold for the union. This means that
for any charts (C/i, cpi) G (2i and (IFy, i/^y) G (2,2 the sets cpiiUi f] Wj) and
i/j{Ui n Wj) are open and cpi o ypy^ is a differentiable map of \pj(Ui n Wj) onto
iPi{Ui n Wj) with a differentiable inverse.
It is clear that the relation introduced is an equivalence relation.
Furthermore, it is an easy exercise to check that if / is a function of class C^ with respect
to a given atlas, it is of class G^ with respect to any equivalent one. The same is
true for the notions of open set and convergence.

370 DIFFERENTIABLE MANIFOLDS 9.3
Definition 3.1. A set M together with an equivalence class of atlases on M is
called a differentiable manifold if it satisfies the "Hausdorff property": For any
two points jci ^ X2 of M there are open sets Uiand U2 with jci G Ui and JC2G U2 with
c/i n c/2 = 0.
In what follows we shall (by abuse of the language) denote a differentiable
manifold by M, where the equivalence class of atlases is understood. By an
atlas of M we shall then mean an atlas belonging to the given equivalence class,
and by a chart of M we shall mean a chart belonging to some atlas of M.
We shall also adopt the notational convention that V is the Banach space
where the charts on M take their values (and shall say that M is a F-manifold).
If there are several manifolds, Mi, M2, etc., under discussion, we shall denote
the corresponding vector spaces by Fi, V2, etc. If 7 = K", we say that M is
an n-dimensional manifold.
Let Ml and M2 be differentiable manifolds. A map cp: Mi —> M2 is called
continuous if for any open set U2 C M2 the set <^~^(C/2) is an open subset of ilf 1.
Let X2 E M2, and let U2 be any open set containing X2. If (fixi) = X2, then
(p'^^{U2) is an open set containing Xi. If (TF, a) is a chart about Xi, then
TF n <p~^{U2) is an open subset of IF, and a(TF fl (p~^iU2)) is an open set in Vi
containing a{xi). Therefore, there exists an € > 0 such that (p{x) G U2 for all
X G IF, wdth \\a{x) — a:(xi)|| < €. In this sense, all points "close to Xi" are
mapped "close to X2". Note that the choice of e will depend on the chart (TF, a)
as well as on xij X2, C/2, and cp.
If Ml, M2, and M3 are differentiable manifolds, and if cp: Mi —> M2 and
yp: M2 —> M3 are continuous maps, it is easy to see that their composition
yp o (p is a. continuous map from Mi to M3.
Let (^ be a continuous map from Mi to M2. Let (TFi, ai) be a chart on Mi
and (TF2, (X2) a chart on M2. We say that these charts are compatible (under cp)
if (p{Wi) C TF2. If (2'2 is an atlas on M2 and Q-i is an atlas on Mi, we say that di
and (2,2 are compatible under cp if for every (TFi, ai) G di there exists a
(TF2, 0:2) G (2,2 compatible with it, i.e., such that <^(TFi) C TF2. (Note that the
map a2° {(p \ TFi) o q:~Ms then a continuous map of an open subset of Fi into
F2.) Given (22 and (^, we can always find an (2i compatible with (22 under cp.
In fact, let 6i\ be any atlas on Mi, and set
ai = {(TFi n ^-HW2)), a r (iFi n ^-^"^^2))},
where (TFi, a) ranges over all charts of (2'i and (TF2, fi) ranges over all charts
of a2.
Definition 3.2. Let Mi and M2 be differentiable manifolds, and let (^ be a
map: Mi A M2. We say that (p is differentiable if the following hold:
i) (p is continuous.
ii) Let GLi and GL2 be compatible atlases under cp. Then for any compatible
(IFi, ai) G ai and (IF2, a2) ^ (^2, the map
a2 ° (p ° olY : ai(IFi) —» a2(lF'2)
is differentiable (as a map of an open subset of a Banach space into a
Banach space). (See Fig. 9.5.)

9.3
DIFFERENTIABLE MANIFOLDS
371
Fig. 9.5
«2(^^2r
In order to check that a continuous map ip is differentiate, it suffices to
check much less than (ii). Condition (ii) relates to any pair of compatible atlases
and any pair of compatible charts. In fact, we can assert:
Proposition 3.1. Let (f.Mi —» M2 be continuous, and let (^i and GL2 be
compatible atlases under cp. Suppose that for every (TFi, ai) G GLi there
exists a (TF2, a2) ^ ^2 with (p{Wi) C W2 and a2 ° (p ° a7^ differentiate.
Then (p is differentiate.
Proo/. Let (Uiy jSi) and (f72, iS2) be any charts on Mi and If 2 with <^(f7i) C f/2.
We must show that 02 ° ^ ° fiT^ '^^ differentiable. It suffices to show that it is
differentiable in the neighborhood of every point iS(xi), where Xi G Ui. Choose
(TFi, ai) G (Ji with X G TFi, and choose (TF2, ai) G (^2 with <^(TFi) C W2.
Then on 0i{Wi n f/i), we have
02 ° <P ° 0T^ = (i^2 ° «i"^) o (a2 o <^ o aF^) o («i ° ^r^),
so that the left-hand side is differentiable. D
In other words, it suffices to verify differentiability with one pair of atlases.
We have as a consequence:
Proposition 3.2. Let (f'.Mi
Then ^ o <^ is differentiable.
M2 and ^:M2'-^ M^ be differentiable.
Proof. Let (^3 be an atlas on M3. Choose (I2 compatible with (^3 under ^,
and then choose an atlas di on Mi compatible with (J2 under <^. For any
(TFi, ai) G ai choose (TF2, ^2) ^ (X2 and (TF3, a3) G dz with <^(TFi) C TF2 and
#(^^2) C TF3. Then a3 o ^ o <^ o a^^ == (ag 0^0 a^^) o (a2 o <^ o «7^) is
differentiable. D
Exercise 3.1. Let Mi = ^, let M2 = IP**, and let cp: Mi —> M2 be the map sending
each point of the unit sphere into the Kne it determines. (Note that two antipodal

372 DIFFERENTIABLE MANIFOLDS 9.o
points of ^S^ go into the same point of P^.) Construct compatible atlases for cp and
show that if is differentiable.
Note that if / is any function on M with values in a Banach space, then/ is
differentiable as a function (in the sense of Section 2) if and only if it is
differentiable as a map of manifolds. In particular, let cp: Mi —> M2 be a differentiable
map, and let/be a differentiable function on M2 (defined on some open subset,
say t/2). Then/ o (^ is a differentiable function on Mi [defined on the open set
(P~^{U2)]- Thus (f "pulls back" a differentiable function on M2 to Mi. From
this point of view we can say that (p induces a map from the collection of
differentiable functions on M2 to the collection of differentiable functions on Mi. Wo
shall denote this induced map by <^*. Thus
differentiable functions on M2 -^ differentiable functions on Mi
is given by
If;/': M 2 —> Ms is a second differentiable map, then {\l/ o (^)* goes from functions
on M3 to functions on Mi, and we have
(^o^)* = ^*o^* (3.1)
(note the change of order). In fact, for g on M3,
(^ o ,p)^g = g o (xly o cp) = (g oxp) o cp = (^*[;A*[^]].
Observe that if (p is any map from Mi —> ilf 2 and / is any function defined
on a subset S2 of M2, then the "puUback" <^*[/] ^ f ° <^ is a function defined on
(P~^{S2) oi Ml. The fact that cp is continuous allows us to conclude that if S2 is
open, then so is <^~H^2)- The fact that cp is differentiable implies that <^*[/] is
differentiable whenever / is.
The map <^* commutes with all algebraic operations whenever they arc
defined. More precisely, suppose / and g take values in the same vector space
and have domains of definition Ui and f/2. Then / + gr is defined on Ui fi (^■.•
and <^*[/] + <^*[{/] is defined on (p~^{Ui fi U2), and we clearly have
>P*[f+9l = <P*[f] + <P
*r
EXERCISES
3.2 Let M2 be a finite-dimensional manifold, and let cp: 71/i —> M2 be continuoll^
Suppose that <^*[/] is differentiable for any (locally defined) differentiable real-vakuM
function /. Conclude that cp is differentiable.
3.3 Show that if <^ is a bounded linear map between Banach spaces, then (^* :i
defined above is an extension of <^* as defined in Section 3, Chapter 2.

9.4
THE TANGENT SPACE
373
4. THE TANGENT SPACE
In this section we are going to construct an "approximating vector space" to a
differentiable manifold at each point of the manifold. This will allow us to
formulate most of the notions of the differential calculus on manifolds.
Let M be a differentiable manifold, and let x be a point of M (Fig. 9.6).
Let / C K be an interval containing the origin. Let <^ be a differentiable map
of / into M such that <^(0) = x. We will call cp a (differentiable) curve through x.
Let / be any differentiable real-valued function on M defined in a
neighborhood of X. Then <^*[/] is differentiable on IR and we can consider its derivative
at the origin. Define the operator D^ by
DM) =
dAf]
dt
In view of the hnearity of <^*, the map
/-> DM) is linear:
DMf+^9) = aDM) + ^D^{g).
Similarly, we have Leibnitz's rule:
DMg) = Kx)DM) + oix)DM),
Fig. 9.6
which can easily be checked. The functional D^ depends on the curve cp. If ^
is a second curve, then, in general, D^ 9^ D^. If, however, D^ = D^, then we
say that the curves cp and xp are tangent at x, and we write (p ^ \l/. Thus
(f r^ yp if and only if DM) = DM) for all differentiable functions/.
It is easy to check that ^ is an equivalence relation. An equivalence class
of curves through x will be called a tangent vector at x. If ? is a tangent vector
at X and <^ G ?, we say that ? is tangent to <^ at x.
For any differentiable function / defined about x and any tangent vector g,
we set
?(/) = D^U),
where <^ E ?. Thus ^ gives us a functional on differentiable functions defined
about X. We have
i{af+-bg) = am + hm, (4.1)
iUg) = Kx)m + g{x)i{f).
(4.2)
Let us examine what the equivalence relation ^ says in terms of a chart
{W, a) about x. The functional DM) can be written as
dj^ <P
dt
t=o
d{f ° oi ) ° {a o ip)
Jt
t=o

374
DIFFERENTIABLE MANIFOLDS
9.
If we set <l> = ao ip and F ^ f o a \ then <l> is a parametrized curve in a Banac^li
space and /^ is a differentiable function there. We can thus write
D,{J)=dF{^'{0)) = D^>^^,F.
From this expression we see (setting ^ = a: o ;//) that \l/ ^ ip \i and only ii
<l>'(0) = '^'(O). We thus see that in terms of a chart {W, a), every tangeiif
vector g at X corresponds to a unique vector ^a ^ V given by
^a= (ao cpYiO),
where <^ E {.
CoiiA^erselyj given any v G V, there is a tangent vector f with ^^ = v
In fact, define (p b}^ setting (p(t) = a~'^(a(x) + tv). Then (p is defined in a smaJI
enough interval about 0, and (a o ^p)^ = v.
In short, a choice of chart allows us to identify the set of all tangent vector. •
at X with V. Let {U, ^) be a second chart about x. Then
?/3 = (^ ° ^)'(O) = (0 o a-') o (a o ^)^(O).
By the chain rule we thus have
f/3 = J^oa-i{Gt{x))^a, (4.3)
where Jy(])) is the differential dJp of T at p.
Since /,Soa-i(Q:(x)) is a linear map of V into itself, Eq. (4.3) says that tlir
set of ail tangent vectors at x can be identified with V, the identification beiiifi.
determined up to an automorphism of V. In particular, we can make the sef^
of all tangent vectors at x into a vector space by defining
where f is determined by
- hi} -
hi] a
for some chart a. Equation (4.3) shows that this definition is independent of a.
We shall denote the space of tangent vectors at x by T^iM) and shall call if
the tangent space (to M) at x.
Let i/^ be a differentiable map of Mi to M2, and let (^ be a curve passing
through X G Ml (see Fig. 9.7). Then 1// o ^ is a curve passing through 4/{x) G Mo-
It is eas3^ to check that if <^ ---^ ^, then ^ o (^ ^ 1// o ^. Thus the map ^ induces a
mapping of Tr^(Mi) into T^^r^^{M2), which we shall denote by ip^x- To repeal.
Fig. 9.7

9.4
THE TANGENT SPACE
375
Fig. 9.8 Fig. 9.9
if f G Tx(Mi)j then yp^xi^) = -^7 is determined by
yp o tp ^ r] for all (p G ^.
Let ([/, a) be a chart about x, and let (IF, ^) be a chart about ^(x). Then
and
f« = (a o ^)'(O)
^^ = (|8 o i/. o <^)'(0) = {^oyf^o a~^) o (a: o <^)'(0).
By the chain rule we can thus write
This says that if we identify T^iMi) with Vi via a and identify T^(^x){M2) with
F2 via j8, then \(/^x becomes identified with the linear map /^o^oQ;-™i(a:(x)). In
particular, the map \(/^x is 0, continuous linear mapping from Tx{Mi) to T^(^x){M2).
If (p: Ml —> M2 and ^: M2 —> M3 are two differentiate mappings, then it
follows immediately from the definitions that
(4" ° <^)*x = ^^cpix) ° (P*X'
(4.4)
We have seen that the choice of chart (?7, a) identifies Tx{M) with F.
Now suppose that M is actually F itself (or an open subset of F) regarded as a
differentiable manifold. Then M has a distinguished chart, namely (M, id).
Thus on an open subset of F the identity chart gives us a distinguished way of
identifying Tx{M) with F. It is sometimes convenient to picture Tx{M) as a
copy of F whose origin has been translated to x. We would then draw a tangent
vector at x as an arrow originating at x. (See Fig. 9.8.)
Now suppose that M is a general manifold and that 1^ is a differentiable map
of M into a vector space Fi. Then ^^{Tx{M)) is a subspace of T^f^x){Vi)^
If we regard 4/^{Tx{M)) as a subspace of Fi and consider the corresponding
hyperplane through x, we get the "plane tangent to i/'(M) at x" in the intuitive
sense (Fig. 9.9).

376 DIFFERENTIABLE MANIFOLDS O.T)
It is very convenient to think of tangent vectors in this way, that is, to
regard them as vectors tangent to M if M were mapped into a vector space
If / is a real-valued differentiable function defined in a neighborhood C/ ol
of X G M, then we can regard it as a map of the manifold U to the manifold U'.
We therefore get a map/^^,: T^{M) -> Tf^^^{U^), Recall that we identify Ty(W)
with U^ for any y G W^. Therefore, f^^ can be viewed as a map from T^^iJ^l)
to [R^ The reader should check that this map is indeed given by
fU^) = ?(/) for ? G T,(M). (4.r))
In particular, if we take M3 = U and xj/ = f in (4.4), we can assert:
Let i/' be a differentiable map of il/i to M2, and let / be a differentiable fuiic
tion on M2 defined in a neighborhood of tA(^)- Then for any ^ G !r^(il/i),
From now on, we shall frequently drop the subscript x in i/'^^ when it can hr
understood from the context. Thus we would write (4.4) as (xp o cp)^ ^ xp^ o ip,^,.
Some authors call the mapping xp^^ the differential of i/' at x and designate \i chp,.
If Ml and M2 are open subsets of Banach spaces Vi and V2 (and hence arc
differentiable manifolds under their identity charts), then xp^^ as defined above
does reduce to the differential dxpj^ Avhen T:r,(M^) is identified with Vi. Thi,.
reduction does depend on the identification, however.
5. FLOWS AND VECTOR FIELDS
Let Ml and il/2 be differentiable manifolds. A map g from Mi —> M2 is called n
cliffeomovphism if {/ is a differentiable one-to-one map of Mi onto ill2 such thai,
g~^ is also differentiable.
Let M be a differentiable manifold. A map <^: ill X [R —> il/ is called a onc-
parameter group if
i) if is differentiable;
ii) (p{x, 0) = X for all x G M;
iii) (p{(p(x, s), t) = (p(x, s + t) for all x G M and s, ^ G [I^.
We can express conditions (ii) and (iii) a little differently. Let ipt'.M —^ M
be given by
^t{x) = (p(x, t).
For each t G U the map (pt is differentiable. In fact,
(ft = (p o Li,
where it is the differentiable map of M —^MxU given by Lt(x) = (x, t).
Then condition (ii) says that cpo = id. Condition (iii) says that
If we take ^ = — s in this equation, we get ipt ° (f—t = id. Thus for each /
the map cpt is a diffeomorphism and ((Pt)~^ = <P—t'

9.5
FLOWS AND VECTOR FIELDS 377
Fig. 9.10
We now give some examples of one-parameter groups.
Example 1. Let M = F be a vector space, and let w G M. Let cpiVx U —^ V
be given by
(p(Vj t) = V -\- tw.
It is easy to check that (i), (ii), and (iii) are satisfied. (See Fig. 9.10.)
Example 2. Let M = F be a finite-dimensional vector space, and let A be a
linear transformation A:V—^V. Recall that the linear transformation e^^ is
defined by
.2 A 2 .3 A 3
2!
3!
i.e., for any v gV,
j=OJ'
(See Figs. 9.11 and 9.12.) Since the convergence of the series is uniform on any
■Ul]
7=D
'-[::]
Fig. 9.11
Fig. 9.12

ei(<p{x,t)) = ei{x) + ta,
= di{x) + ta-
e2(<pix,t)) = e2{x) + ta,
= ^2(3;) + ta -
-2ir,
-27r,
X
X
X
X
378 DIFFERENTIABLE MANIFOLDS 9.5
compact set of <v, t>, the map cp: M X U —^ M given by
<p{v, t) = e^^v
is easily seen to be differentiable and to satisfy (ii) and (iii) as well.
Examples. Let M be the circle aS^, and let a be any real number. Let (^^ be tho
diffeomorphism consisting of rotation through angle ta. In terms of the atlas
a = {{Ui, di), {U2, O2)}, the map cp is given by
Ui, Biix) < 27r — ta,
Ui, Biix) > 27r — ta,
U2, 02{x) < 27r + 7r/2 - ta,
U2, 02{x) > 27r + 7r/2 - ta.
(Strictly speaking, this doesn't quite define (p for all values of <x,t>. If
X = <1, 0> and ta = 7r/2, then x ^ U\ and ip{x, 7r/2) g 112- This is easily
remedied by the introduction of a third chart.) It is easy to see that (^ is a one-
parameter group.
Example 4. Let M = aS^ X aS^ be the torus, and let a and 5 be real numbers.
Write X G M as X = <xi, X2> , where Xi G S^. Define cp^^^^^ by
cp ""'^ {Xi,X2,t)= <<pUxi),<pt{X2)>,
where cp^ and cp^ are given in Example 3. Then cp^^^^^ is a one-parameter group
and indeed a rather instructive one. The reader should check to see that
essentially different behavior arises according to whether h/a is rational or irrational.
[The construction of Example 4 from Example 3 can be generalized as
follows. If cp: M X U —^ M and \l/: N X U —^ N are one-parameter groups,
then we can construct a one-parameter group on M X N given by (ptX-^f.
The map of M X N X U-^ M X N sending <x, y, t> -> <cpt{x), ^tiy)> is
differentiable because it can be written as the composite {(p X \p) o A, where
(pXxly: MxUxNxU-^MxN,
and
A:MxA^XlR->MxlRxA^XlR
is given by A{x, y, t) = <x, t, y,t>.]
In each of the four preceding examples we started out with an "infinitesimal
generator" to construct the one-parameter group, namely, the vector w in
Example 1, the linear transformation A in Example 2, the real number a in
Example 3, and the pair <a,h> in Example4. We will now show that associated
with any one-parameter group on a manifold, there is a nice object which wv
can regard as the infinitesimal generator of the one-parameter group.
Let (p: M X U —^ M he Si one-parameter group. For each x G M consider
the map cpx of U —^ M given by
<Px{t) = ip{x, t).

9.5
FLOWS AND VECTOR FIELDS
379
In view of condition (ii), we know that (Px{0) = x. Thus cpx is a curve passing
through X (see Fig. 9.13). Let us denote the tangent to this curve by X{x).
We thus get a mapping X which assigns to each x G M a vector X{x) G Tx{M).
Any such map, i.e., any rule assigning to each x G M a vector in Tx{M), will be
called a vector field. We have seen that every one-parameter group gives rise to a
vector field which we shall call the infinitesimal generator of the one-parameter
group.
Fig. 9.13
Let F be a vector field on M, and let (17, a) be a chart on M. For each
X G 17 we get a vector Y{x)a G V. We can regard this as defining a F-valued
function Y^ on a{U):
Ya{v) = Y(a-\v))a for V G a{U).
(5.1)
Let (W, iS) be a second chart, and let Y^ be the corresponding F-valued function
on P(W). If we compare (5.1) with (4.3), we see that
Y^{^oa-\v)) = J^^a-i{v)oY^{v) if z; G ..(17 n TF). (5.2)
Equation (5.1) gives the "local expression" of a vector field with respect to a
chart, and Eq. (5.2) describes the "transition law" from one chart to another.
Conversely, let 6i be an atlas of M, and let Y^ be a F-valued function defined
on a:(17) for each chart (U, a) G d. Suppose that the Y^ satisfy (5.2). Then for
each X G M we can let Y{x) G T^iM) be defined by setting
Y{x)^= Y^{a{x))
for some chart (17, a) about x. It follows from the transition law given by (4.3)
and (5.2) that this definition does not depend on the choice of (17, a).
Observe that J^oa~^ is a C°°-function (linear transformation-valued function)
on a:(i7 n TF). Therefore, if F is a vector field and F^, is a F-valued C°°-function
on a(17), the function F^ will be C°° on j8(i7 fl F). In other words, it is consistent
to require that the functions Ya be of class C°°. We shall therefore say that F

380 DIFFERENTIABLE MANIFOLDS O.T)
is a C°°-vector field if the function Y^ is C°° for every chart (U, a). As in the case
of functions and mappings, in order to verify that Y is C°°, it suffices to check thai
the Ya are C°° for all charts (17, a) belonging to some atlas of M.
Let us check that the infinitesimal generator X of a one-parameter group ^
is a C°°-vector field. In fact, if (17, a) is a chart, then
Xa{v) = {a o ^,y(o),
where (Px{t) = (p{x, t). We can write a o (p^{t) = #a(y, t), where
^aiv, t) = ao (p(a~\v), t).
Let 17' C i7 be a neighborhood of x such that (p{y, t) ^ U ior y ^ U' and |^| < e
Then #« is a differentiable map of ol{IJ') X I ^^ oi{U), where / = {^ : |^| < el
In terms of this representation, we can write
Xa{v) = ^ {V, 0). (5..3)
This shows that X is a C°°-vector field.
If we evaluate (5.3) in the case of Example 1, we get ^id{^, ^) = y + tu\
so that Xid = w. In the case of Example 2 we get X\^{v) = Av.
There are various algebraic operations that can be performed with vector
fields. The set of all vector fields on M forms a vector space in the obvious way.
If X and Y are C°^-vector fields, then so is aX -\-hY (a and 5 are constants),
where
(aX + hY){x) = aX{x) + hY{x), x^M.
Similarly, we can multiply a vector field by a function. If / is a function and X
is a vector field, we define fX by
UX){x)=f{x)X{x), x^M.
It is easy to see that if / and X are differentiable, then so is fX. It is also easy
to check the various associative laws for this multiplication.
We have seen that any one-parameter group defines a smooth vector field.
Let us examine the converse. Does any C°°-vector field define a one-parameter
group? The answer to the question as stated is "no".
In fact, let X = d/dx^ be the vector field corresponding to translation in thc^
X^-direction in W^. Let M = U^ — C, where C is some nonempty closed set,
of IR^. Then if p is any point of M that lies on a line parallel to that x^-axis whicli
intersects C (Fig. 9.14), then (pt{p) will not be defined (will not lie in M) for
every t.
The reader may object that M "has some
points missing" and that is w^hy X does not
generate a one-parameter group. But we can
construct a counterexample on IR^ itself. In p
fact, if we consider the vector field X on IR^
given by
X,a{x\x^)= (1, -(x2)2), Fig. 9.14

9.5 FLOWS AND VECTOR FIELDS 381
then (5.3) shows that if, if defined, satisfies
where # = #id. If we let y^(t^ x) = x^ o #(x, t)^ then
It = '■ »'<»> - '■■
^--(!/V, v'(0) = x^
If x^ 7^ 0, then the unique solution of the second equation is given by
which is not defined for all values of ^. Of course, the trouble is that we only
have a local existence theorem for differential equations.
We must therefore give up on the requirement that (f be defined on all of
M X K.
Definition 5.1. A flow on M is a map (p of an open set 17 C M X IR ^ M
such that
i) M X {0} C [/;
ii) if is differentiable;
iii) ip{x, Q) = X]
iv) ip{ip{x, s),t) = ip{x,s-]-t) whenever both sides of this equation are
defined.
For X fixed, ipx{t) = (p{x, t) is defined for sufficiently small t^ so that (p gives
rise to a vector field X as before. We shall call X the infinitesimal generator
of the flow (f.
As the previous examples show, there may be no ^ 7^ 0 such that (^(x, t) is
defined for all x, and there may be no x such that ip(x, t) is defined for all t.
Proposition 5.1. Let X be a smooth vector field on M. Then there exists a
neighborhood 17 of M X {0} in M X K and a flow cp: U -^ M having X as
its infinitesimal generator.
Proof. We shall first construct the curve ipx{t) for any x G M, and shall then
verify that <x,t> 1-^ ip{x, t) is indeed a flow.
Let X be a point of ilf, and let ([/, a) be a chart about x. Then X^ gives us
an ordinary differential equation in a{U), namely,
I = Xa{v), V G a{U).
By the fundamental existence theorem for ordinary differential equations, there
exists an e > 0, an open set 0 containing a:(x), and a map
$,:0X {t:\t\ < e} -^ a{U)

382 DIFFERENTIABLE MANIFOLDS 9.5
such that
and
d^a(v, t) XT /^ / A\
j^ = Xa{^a{v,t)).
Here the choice of the open set 0 and of e depends on a{x) and a(U). The
uniqueness part of the theorem asserts that #« is uniquely determined up to the
domain of definition; i.e., if ^y is any curve defined for |^| < e' with #^(0) = v
and
^ = X„(*.W), (5.4)
then ^y{t) = ^a{f^) t).
This implies that
whenever both sides are defined. (Just hold s fixed in the equation.)
Consider the curve ^^x^') defined by
^\{t) = a-i(0«(a(x),^)). (5.5)
It is defined for |^| < e, and is a continuous, in fact difFerentiable map of this
interval into M. Furthermore, if we write i|; = c|)"-^() then (5.4) asserts that the
tangent vector to the curve i|;(^ + •) is X{^{t)), the value of the vector field at the
point ^{t). We will write this condition as
^'{t) = Xm))^ (5.6)
Equation (5.6) is the way we would write the ''first order differential equation"
on M corresponding to the vector field X. A differentiable curve i|; satisfying
(5.6) is called an integral curve of X We now can formulate a manifold version
of the uniqueness theorem of differential equations:
Lemma 5.1. Let i|ii:/ -> M and i|i2:/ -^ M be two integral curves of X defined on
the same interval /. If v|ii(s) = i|i2(s) at some point s E:I then i|ii = i|i2, i.e. ^i{t) =
\\f2it) for all t e /.
Proof. We wish to show that the set where \\fi(t) ^ \\f2(t) is empty. Let
A = {t:t > s and \\fi(t) ^ \\f2(t)}.
We wish to show that A is empty, and similarly that the set B = {t:t < s and
\\)i(t) ^ \\f2(t)} is empty. Suppose that A is not empty, and let
t+ = gib A = gib {t:t > s and i|;i(^) ^ \\f2(t)}.
We will derive a contradiction by
i) using the uniqueness theorem for differential equations to show that \\fi(t+)
^\\)2(t + ),and
ii) using the Hausdorff property of manifolds to show that \\fi(t + ) = i|;2(^ + ).

9.6 LIE DERIVATIVES 383
Details:!). Suppose that i|ii(^ + ) = v|i2(^ + ) = j'E M. We can find a coordinate
chart (p, W) about j^, and then p ° i|;i and p ° ^2 are solutions of the same system
of first order ordinary differential equations, and they both take on the value
P(y) at ^ = ^ +. Hence, by uniqueness for differential equations, P ° i|ii and p ° ^2
must be equal in some interval about t +, and hence ^i{t) = \\)2(t) for all t in this
interval. This is impossible since there must be points arbitrarily close to ^ +
where \\fi(t) ^ \\f2(t) by the gib property of ^ +. This proves i). Now suppose that
i|;i(^ + ) ^ i|i2(^ + ). We can find neighborhoods [7iof i|;i(^ + ) and [72of ^2(^ + ) such
that C7i n [72 = 0- But then the continuity of the i|;i imply that \\fi(t) E: Ui and
\\f2(t) E U2 for t close enough to ^ + , and hence that \\)i(t) ^ \\f2(t) for t in some
interval about t + . This once again contradicts the gib property of ^ + , proving
ii). The same argument with gib replaced by lub shows that B is empty proving
Lemma 5.1. The above argument is typical of a ''connectedness argument."
We showed that the set where \\fi(t) = \\)2(t) is both open and closed, and hence
must be the whole interval /.
Lemma 5.1 shows that (5.5) defines a solution curve of X passing through
X at time ^ = 0, and is independent of the choice of chart in any common
interval of definition about 0. In other words it is legitimate to define the curve
^x(') by
which defines c|)x(rt for |^| < e. Unfortunately the e depends not only on x but
also on the choice of chart. We use Lemma 5.1 and extend the definition of c|)x()
as far as possible, much as we did for ordinary differential equations on a vector
space in Chapter 6: For any s with |s | < e we letj' = 4>x(s) and obtain a curve
(^y(') defined for |^| < e\ By Lemma 5.1
(^y(t) = ^jc(s-^t)i{\s-^t\ < €. (5.7)
It may happen that | s | + e' > e. Then there will exist a t with | ^ | < e' and
|s + ^ I > €. Then the right hand side of (5.7) is not defined, but the left is. We
then take (5.7) as the definition of c|)-^(s + ^), extending the domain of definition
of c|):^(). We then continue: Letlx^ denote the set of all s > 0 for which there
exists a finite sequence ofreal numbers So = 0 < si < . . . < Sk = sand points
xq, . . .xjfe_i EMwithxo = x,si in the domain of definition of c|):^(),X2 = ^xi^i)
and, inductively,
Sj +1 in the domain of definition of ^xi(') and xi + i = ^xMi +1).
If s G it, so is s' for 0 < s' < s, and so is s + 77 for sufficiently small
positive 7]. Thus I^ is an interval, half open on the right. By repeated use of
(5.4) we define (Px{s) for s G it- We construct I^ in a similar fashion and
set /^ = /+ u /7- Then (px{s) is defined for s G h, and / is the maximal
interval for which our construction defines cpx- For each x G M we obtain an
open interval Ix in which the curve (Px{-) is defined.
Let U = \Jxgm{x} X Ix' Then U is an open subset of M X /. To verify this,
let (x, s) G U. We must show that there is a neighborhood W oix and an e > 0
such that s G /x for all |s — s| < e and x ^W. By definition, there is a finite

384 DIFFERENTIABLE MANIFOLDS 9.6
sequence of points x = XqjXi, . . . , Xk and charts (Ui, on), . . . , (17^, a^) with
Xi—i G Ui and Xi G Ui and such that
Oii{Xi) = ^ai(^iiXi-l), ti),
where ti -\- - - - -\- t^ = s. It is now clear from the continuity properties of the
^a that if we choose Xq such that a:i(xo) is close enough to a:i(xo), then the
points Xi defined inductively by
will be well defined. [That is, aj{xj-i) will be in the domain of the definition of
^a,( • , ti)'] This means that s G Ixq for all such points Xq. The same argument
shows that s + ^ G /^o for rj sufficiently small and x sufficiently close to x.
This shows that U is open.
Now define cp by setting
(fix, t) = (Px{t) for (x, t) G U.
That if is differentiable near M X {0} follows from the fact that if is given (in
terms of a chart) as the solution of an ordinary differential equation. The
fundamental existence theorem then guarantees the differentiability. Near the
point (^, t) we can write
(^(X, t) = ip{ip{' ' ' ((fix, ti), ^2) • • • )tk)^ t = ti + t2-\ \- h,
and so cp is differentiable because it is the composite of differentiable maps. D
6. LIE DERIVATIVES
Let (f he Si one-parameter group on a manifold M, and let / be a differentiable
function on M. Then for each t the function cpflf] is differentiable, and for t 9^ 0
we can form the function
<p*An - /
t
(6.1)
We claim that the limit of this expression as ^ —> 0 exists. In fact, for any
X e M, (^f [/](x) = j ^ ipt{x) = f o cp^{t) and, therefore,
li^ dllLzf (^) = lin, /-^^(0-/o^x(Q) ^ j^^j ^ xix)f. (6.2)
Here X{x) is a tangent vector at x and we are using the notation introduced in
Section 4. We shall call the limit of (6.1) the derivative of / with respect to the
one-parameter group cp, and shall denote it by Dxf- More generally, for any
smooth vector field X and diflerentiable function / we define Dxf by
Dxfix) = X(x)f for all x G M, (6.3)
and call it the Lie derivative of / with respect to X. In terms of the flow
generated by Z, we can, near any x G M, represent Dxf as the limit of (6.1),

9.6 LIE DERIVATIVES 385
where, in general, (6.1) will only be defined for a sufficiently small neighborhood
of X and for sufficiently small |^|.
Our notation generalizes the notation in Chapter 3 for directional derivative.
In fact, if M is an open subset of V and X is the "constant vector field" of
Example 1
Xiei= w G F,
then
(Z)x/)id = D^f.a,
where Dy^ is the directional derivative with respect to w.
Note that Dxf is linear in X; that is,
Dax+bvf = aDxf + hDyg
if X and Y are vector fields and a and h are constants.
Let tA be a diffeomorphism of Mi onto M2, and let X be a vector field on M2.
We define the "pullback'^ vector field xp'^lX] on Mi by setting
;A*[X](x) = ^-^^Xiypix)) for all x G M. (6.4)
Note that \l/ must be a diffeomorphism for (6.4) to make sense, since \l/~^ enters
into the definition. This is in contrast to the "pullback^' for functions, which
made sense for any differentiable map. Equation (6.4) does indeed define a
vector field, since
^^^ = \^;rx : T^ix){M2) -> T^iMi) and X(^(x)) G 7^^(x)(M2).
Let us check that tA*[^] is a smooth vector field if X is. To this effect, let Cti
and (^2 be compatible atlases on Mi and M2, and let (17, a) G Cti and {W, ^) G (^2
be compatible charts. Then (6.4) says that
xPnXUv) = Jao^-Ur^{fi-^-Oi-\v)) 'X^{fio,poa-\v)) for V^a{U),
which is a differentiable function of v. Since, by the chain rule,
we can rewrite the last expression more simply as
riXUv) = (J^o^oa-i{v))-'Xp(p o ^ o a-\v)) for V G a{U). (6.5)
Thus \p'^[X]a is the product of a smooth Hom(y2) Fi)-valued function and a
smooth F2-valued function, which shows that tA*[^] is a smooth vector field.
Exercise. Let (f be the flow generated by X on il/2. Show that the flow generated by
xp'^lX] is given by
<x,t> ^xly--^ip(xP(x),t). (6.6)
If (f IS Si one-parameter group, then we can write (6.6) as
<x,t> ^xl^-^ o cp^o xp{x). (6.60

386 DIFFERENTIABLE MANIFOLDS
9.6
The vector field X The vector field Y
{Xf=df/dx) {Yf = x{df/dy))
X{u,v)=<l,0>=8i Y{u,v)=<0,u> <ptiY)
""'"^ ^ V ^^^ i'*." ^^, 74
aS|PwM.«j»^'..ig^^
^p^',^"^;.y..^«
(c)
DvF
(riiiKV inde])endent of 0
'/ "': /i^— "'' /^\ v:- :\ /.
my M^' M
"«#«««*«#*»
^^JBSfWMW^IiMISW^^
(f)
v^'/'^-V)
^^^v)--'^'
,.-,i^^'>?P *^?ii;i i?-L' i.'?
^ lEUfMli
«V:$4l9«l<^';t»h«HIVl^4~
(g) (^)
Fig. 9.15
It is easy to check that if i^i: Mi ^^ M^ and ^2* M^2 —^ M'3 are diffeo-
morphisms and F is a vector field on M3, then
(^2 o ^(,i)*F = ^t^lF.

^•^ LIE DERIVATIVES 387
mx)-x _ ^^^
, . ^ ^ , , , If/is a differentiable function on Mo, then
(since independent of t) '
£»**[xi(^*[/]) = riDxf). (6.7)
In fact, by (6.3) and (4.6) we have, for x e Mi,
D^*mr[f]{x) = ^p*[X]{x)^p*[f] by (6.3)
= ^p-'X{^p{x))r[f] by (6.4)
= (^^^-'X(4.{x)))f by (4.6)
= X{Hx))f
= (DxDi^Pix))
Fig. 9.15 (cont.) = ^*i^xf){x).
Let (p be a one-parameter group on M with infinitesimal generator X, and
let F be another smooth vector field on M. For < 5^ 0 we can form the vector
field
^^^^ m)
and investigate its limit as ^ ^ 0, which we shall call DxY. In Fig. 9.15 we
have shown the calculation of DyX and DxY for two very simple fields on the
Cartesian plane Hi The field X is the constant field Zid = 5i, so that Xf -=
df/dx in terms of Cartesian coordinates x, y. The corresponding flow is given
^y <Pt{x,y) = <x + t,y>. Thus ipt^ = \d if we identify the tangent space at
each point of the plane with the plane itself. Then Y ^ ipfY consists of "moving"
the vector field F to the left by t units. Here we have taken F ^ xb^, so that
Yf -= x{df/dy). In Fig. 9.15(c) we have pictured iffY, and have superimposed F
and cpfY in Fig. 9.15(d). Figure 9.15(e) represents <ptY ~ Y and Fig. 9.15(f)
is {l/t){(pfY — Y}, which coincides with its limit, DxY, since the expression is
independent of L The one-parameter group generated by F is xpt where \l/t{x, y) =
<x,y + tx>. Here at any p E K^ we have yl^t^bi = bi + 162, so that xP^X =
^-t^X{xp{x)) = 81 — 182. In Fig. 9.15(g) we have drawn xp^X and in Fig. 9.15(h)
we have superimposed it on X. Note that DxY = —DyX. However, these
two derivatives are nonzero for quite different reasons. The field <pfY varies
with t because the field F is not constant. The field i^f X varies with t because
of "distortion" in the flow ^^ See Fig. 9.15(g) and (h). In the general case,
DxY will result from a superposition of these two effects. We now make the
general calculation.
Let (17, a) be a chart on M, and for v ^ a(U) let 0 be a sufficiently small open
set containing v, and let € > 0 be sufficiently small, so that ^« given by
^aiw, t) = ao (p(a~\w), t)
is defined torw^O and 1^1 < e. Then, for |^1 < e, Eq. (6.5) imphes that
^flYUv) - (/^.(..oW)~'^.(*.(^ 0). (6.9)

388
DIFFERENTIABLE MANIFOLDS
9.6
The right-hand side of this equation is of the form A^^Zt, where At and Zt are
differentiable functions of t with Aq = I. Therefore, its derivative with respect
to t exists and
dt
= lim
Af'zt
^0
= lim At
t
(Ar'zt
zo)
= lim
Zt
t
AtZQ
lim
t
».(^
Zt — Zq
AtZo
zo'
Now in (6.9) Zt
t t )
= 4 — A'qZq.
'o = dY^(^-ff{v,0)^
= dYaiXM)^
Here Ya is a F-valued function, so dYa is its differential at the point ^a{^, 0)-
Hence dYa{Xa{v)) is the value of this differential at Xa{v). The transformation
At = J^a('^,t) = d{^a)(v,t), SO
dA
dt
d d^a
= d
dt
dt
- d{Xa)v.
Thus the derivative of (6.9) at ^ = 0 can be written as
d{Y^),iXM) - d{X^),(YM) = Dx^,v,Y^ - 2)f«(.)X«.
We have thus shown that the limit in (6.8) exists. If we denote it by DxY,
we can write
.F. - 2)f«(.)X,. (6.10)
{DxYUv) = Dx^,v,ia
As before, we can use (6.10) as the definition of DxY for arbitrary vector
fields X and F. Again, this represents the derivative of F with respect to the
flow generated by X, that is, the limit of (6.9) where now (6.8) is only locally
defined.
From (6.10) we derive the surprising result that DxY = —DyX. For this
reason it is convenient to introduce a notation which expresses the
antisymmetry more clearly, and we shall write
DxY = [X, F].

9.6 LIE DERIVATIVES 389
The expression on the right-hand side is called the Lie bracket of X and F.
We have
[X, Y] = -[Y, X]. (6.11)
Let us evaluate the Lie bracket for some of the examples listed in the
beginning of Section 5. Let M = W.
Example 1. If Xia = Wi and Fid = '^2 are "constant" vector fields, then (6.10)
shows that [X, F] = 0.
Example 2. Let X\^{v) = Av, where ^4 is a linear transformation, and let
Fid = w. Then (6.10) says that
[X, YUv) = -Aw,
since the directional derivative of the linear function Av with respect to lu is Aw.
Example 3. Let Xi^{v) = Av and Yi^{v) = Bv, where A and B are linear
transformations. Then by (6.10),
[X, YU{v) = BAv - ABv = (BA - AB)v. (6.12)
Thus in this case [X, Y] again comes from a linear transformation, namely,
BA — AB. In this case the antisymmetry in A and B is quite apparent.
We now return to the general case. Let <^ be a one-parameter group on M,
let F be a smooth vector field on M, and let / be a difTerentiable function on M.
According to (6.7),
Then
^ptiDyf) - DyS _ 2)^r[F](^t[/]) - 2)F(^r[/]) , 2)F(^r/) - Byf
Dr^^^.Y]-Y\^t [/]
'Un[Y]-Y\
-K^O
Since the functions <^f [/] are uniformly difTerentiable, we may take the limit as
^ ^ 0 to obtain
DxiDyf) = DD^Yf+ DriDxf)
= D[x,Y]f+DriDxf).
In other words,
Ax.F]/ = DxiDyf) - DriDxf). (6.13)
In view of its definition as a derivative, it is clear that DxY is linear in F:
DxiaY, + 6F2) = aDxY, + hDxY2
if a and h are constants and X and F are vector fields. By the antisymmetry,
it must therefore also be linear in X. That is,
Dax,+bx,Y = [aXi + 6X2, F] = a[Xu Y] + h[X2, Y] = aDxJ+hDxJ.

390 DIFFERENTIABLE MANIFOLDS 9.7
Let X and Y be vector fields on a manifold ^2, and let ^ be a diffeomorphisni
of Ml onto M2. Then
V.*[Z, F] = [V^*X,V^*F]. (6.14)
In fact, suppose X generates the flow <^. Then
;/.*[X, F] - ;A*^xF = ;A*lim
s(^9
Since t^"^ q (^^ o ;// is the flow generated by lA*^? we conclude that the last limit,
is Z)^*xtA*^) which proves (6.14).
Now let F and Z be smooth vector fields on M, and let X be the infinitesimal
generator of <^. Then
^r[F,ZJ - [F,Z]
Dx[Y,Z] = \{m ^
^.^[<ptY,<ptZ]-[Y,Z]
t=0
= lim
^
*.
V( F - F *„
+
*
F,
(PtZ — Z
t
= [DxY,Z] + [Y,DxZ].
Thus
[Z, [F, Z]] = [[X, F], Z] + [F, [X, Z]]. (6.15)
In view of the antisymmetry of the Lie bracket, Eq. (6.15) can be rewritten as
[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0. (6.16)
Equation (6.15), or (6.16), is known as Jacobi's identity.
7. LINEAR DIFFERENTIAL FORMS
Let M be a differentiable manifold. We have attached to each x ^M a vector space
TxiM). Its dual space, (TxiM))*, is called the cotangent space to M at x, and will be
denoted by T^(M). Thus an element of T|(M) is a continuous linear function on TxiM);
it is called a covector.
Some explanation of the word "continuous" is in order. In the case where
M [and hence T^iM)] is finite-dimensional, all linear functions on Tx{M) are
continuous, so no further comment is necessary. We shall be concerned primarily
with this case. More generally, let I be a linear function on Tx{M). For any

9.7 LINEAR DIFFERENTIAL FORMS 391
chart ([/, a) about x we have identified Tx{M) with V, thus identifying J G Tx{M)
with ^CK G y. Then I determines a linear function l^ on F by
(?a, ^a) = (?, 0- (7.1)
If {W, (3) is a second chart, then
<?^, k) = {Jao^-^{P(x))^^, la).
Since Jao^-^ (i^(^)) is a continuous map of V into F, we see that la is continuous
if and only if l^ is. We shall therefore say that I is continuous if la is continuous
for some (and hence any) a. In this case we see that (7.1) gives us an
identification of Tx{M) with F* sending I into la- The last equation says that the rule
for change of charts is given by
k= {JaorMx))Yla. (7.2)
Let / be a difTerentiable function on M, and let x G M. Then the function
on Tx{M) sending each ^ G T^^M) into ^(/) will be denoted by df{x). Thus
{^.df{x))= ?/.
It is easy to see that dj G T^(M). In fact, in terms of a chart (U, a) about x,
Note that / assigns an element df{x) of T^(I\I) to each x G M. A map which
assigns to each x G M an element of Tx{]\I) will be called a covector field or a
linear differential form. The linear differential form determined by the function /
will be denoted simply by df.
Let CO be a linear differential form. Thus o:(x) G T^iM) for each x G M.
Let a be an atlas of HI. For each (U, a) G M we obtain the F*-valued function
coa on q:([/) defined by
o:a(v)= {c^{a-Hv))]a fov V G a(U). (7.3)
If (W, 13) G a, then (7.2) says that
a;^(^ o a-i(?^)) = (^o/3-i(^ o a-i(z;)))*a;«(z;)
= (^^oa-i(z;))-i*a;,(z;) for z; G a(f/n TF). (7.4)
As before, Eq. (7.4) shows that it makes sense to require that co be smooth.
We say that co is a C^-difTerential form if co^ is a F*-valued C^-function for every
chart ([/, a). By (7.4) it suffices to check this for all charts in an atlas. Also, if
we are given F*-valued functions co^, each defined on a(U), (U, a) G Gi, and
satisfying (7.4), then they define a linear differential form co on M via (7.3).
If CO is a differential form and / is a function, we define the form /co by
/co(:r) = f{x)cx:{x). Similarly, we define coi + ^2 by
(coi + co2)(:r) = coi(:r) + co2(:r).

392 DIFFERENTIABLE MANIFOLDS 9.7
Let Ml and M2 be differentiable manifolds, and let <^: Mi —> M2 be a
differentiable map. For any :r G Mi wehavethemap <^:jcx- ^x(Mi) —> T'^(^)(M2).
It therefore defines a dual map
(^*x)*:T*(,)(M2)^r,*(Mi).
(The reader can check that if I G T'*(^)(M2), then J —> (<^*(^), I) is a continuous
linear function of ^, by verification in terms of a chart.)
Now let CO be a differential form on M2. It assigns a;(<^(:r)) G T'*(3.)(M2) to
<^(:r), and thus assigns an element (<^*x)*(^(<^(^))) G 2'*(Mi) to :r G Mi. Wo
thus "pull back" the form co to obtain a form on Mi which we shall call <^*a;.
Thus
<^*a;(:r) = (<^^^)*a;(<^(:r)). (7.5)
Note that <^* is defined for any differentiable map as in the case of
functions, not only for diffeomorphisms (and in contrast to the situation for vector
fields).
It is easy to give the expression for <^* in terms of compatible charts {U, a)
of Ml and (W, (3) of M2. In fact, from the local expression for <^* we see that
(^Ma(?^) = {J^o^oa-Kv)Yo^^{& o ^ ° oc-\v)), V G a(U), (7.6)
From (7.6) we see that <^*a; is smooth if co is. It is clear that <^* preserves algebraic,
operations:
<^*(^i + ^2) = <^*^i + <^*co2 (7.7)
and
^*(/co) = ^*[/]^*(co). (7.8)
If <^: Ml —> M2 and \p\ M2 —> M3 are differentiable maps, then (4.4) and
(7.5) show that
(^o <^)*a; = <^V*co. (7.9)
Let ^: Ml —> M2 be a differentiable map, and let / be a differentiable
function on 71/2. Then (4.6) and the definition dj show that
dirm) = r dj. (7.10)
Let <^ be a flow on M with infinitesimal generator X, and let co be a smootli
linear differential form on M. Then the form <^f co is locally defined and, as in the
case of functions and vector fields, the limit as ^ —> 0 of
(ft 0^ — CO
t
exists. We can verify this by using (7.6) and proceeding as we did in the case;
of vector fields. The limit will be a smooth covector field which we shall call
Dxco. We could give an expression for Dx^ in terms of a chart, just as we
did for vector fields.

9.8 COMPUTATIONS WITH COORDINATES 393
If / is a differentiable function, co a smooth differential form, and X the
infinitesimal generator of <^, then
DxC/co) = {Dxf)c^ + fDxc^. (7.11)
In fact,
DAfo^) = lim '^1^^^
= hm(^^^r(co)+/^^^
= (2)x/)a;+/2)xco.
If ^ is a differentiable function on M, then
(ft dg — dg _ dcpfig] — dg _ . /cptlg] — g^
t t
^p^)
An easy verification in terms of a chart shows that the limit of this last expression
exists and is indeed d{Dx(p)- Thus
Dx{df) = d{Dxf). (7.12)
Equations (7.11) and (7.12) show that if
c^ = fidgi-\ \-fkdgk,
then
Dxco = (Dxfi) dg, + --- + {Dxh) dg^ + /i ^(^^x^i) + • • • + A d{Dxgk)-
(7.120
Let CO be a smooth linear differential form, and let X be a smooth vector field.
Then (X, co) is a smooth function given by
{X, ^){x) = {X{x), ^{x)).
Note that (X, co) is linear in both X and co. Also observe that for any smooth
function / we have
(X, df) = Dxf. (7.13)
8. COMPUTATIONS WITH COORDINATES
For the remainder of this chapter we shall assume that our manifolds are finite-
dimensional. Let M be a differentiable manifold whose V = W^. If {U, a) is a
chart of M, then we define the function xl, on U by setting
Xa{x) = ith coordinate of a(x). (8.1)
If/ is any differentiable function on C/, then we can write Eq. (2.1) as
/(^) = fa{Xa(x), . . . , X'^(x))y

394 DIFFERENTIABLE MANIFOLDS 9.8
which we shall write as
/ = fai^a, • • • J ^a)- (8.2)
We define the vector field d/dxl, on U by
(^) (^) = Sii=<0,...,l,...,0». (8.3)
\ a/ a *^li position
If X is any vector field on U, then we have
^=^i,-^ + ---+^.%i' (8.4)
a a
where the functions XI, are defined by
{X)a{a{x)) = <XUx), ..., X:(x) > . (8.5)
Equation (8.4) allows us to regard the vector field X as a "differential operator".
In fact, it follows from the definitions that
a a
Since Xq, is a differentiable function on U, dxl, is a differential form on U and
{dxi)a{v) = 8i for all v eU. (8.7)
In particular,
£., dxiS = di (8.8)
If CO is a differential form on U, then
CO = aia dXa + • • • + ana dXaj (8.9)
where the functions
O/icx. a/re defined by
c^a{ci{x)) = <aic,{x), . . . , ana{x)>G W". (8.10)
It then follows from the definitions that
df = ^^dx^ + --- + ^^dx:. (8.11)
a a
Equation (8.11) has built into it the transition law for differential forms under Ji
change of charts. In fact, if (W, (3) is a second chart, then on [/ fi TF we have,
by (8.11),
dxj = g da;i + • • • + g dxl (8.12)

9.8 COMPUTATIONS WITH COORDINATES 395
If we write co = ai^ dx^ + • • • + a^^ dx^ and substitute (8.12), we get
Zdx^^
Now
dxl
dx3
is the matrix J^oa~'^- If we compare with (8.10), we see that we have recovered
(7.4)._
Since the subscripts a, 13, etc., clutter up the formulas, we shall frequently
use the following notational conventions: Instead of writing xl, we shall write x\
and instead of writing x^^ we shall write y\ Thus
i i i i k k 1
X = Xa, y = Xfij z = Xy, etc.
Similarly, we shall write X^ for X^, F* for X^, a^ for a^^, hi for a^-^, and so on.
Then Eqs. (8.1) through (8.12) can be written as
x'^{x) = iih coordinate of a{x)j (8.1')
f=Ux\...,x^), (8.2')
(^);^) = K (8.30
^=^'^+--- + ^"^' (8.40
(Z)„ («(a;)) = < X\x), ..., X^{x) >, (8.50
i)x/=X'gf+--- + ^%^' (8.60
(dxXiv) = 5i, (8.70
d
^^.,dx^)=Si (8.80
w= aidx^ -] \- a„ dx", (8.90
co„(a(a;)) = <ai{x), . . . , a„(a;)>, (8.100
df='S''^' + ---+fxl''''' (8.110
dy^='£dx^ + --. + '£dx^. (8.120
The formulas for "pullback" also take a simple form. Let ^: Mi —> M2 be a
differentiable map, and suppose that Mi is m-dimensional and M2 is n-dimen-

396 DIFFERENTIABLE MANIFOLDS 9.8
sional. Let {U, a) and (W, (3) be compatible charts. Then the map
is given by
vKHx)) = y\x\ . . . , xn, i=l,...,n, (8.13)
that is, by n functions of m real variables. If / is a function on M2 with
/ =/^(2/\ • • • , 2/") on W,
then
r[n = L{x\...,xn on u,
where
Ux\ ..., xn = U{y\x\ ..., xn...., y''{x\ ..., xn)- (8.14)
The rule for "pulling back" a differential form is also very easy. In fact, if
CO = tti dy^ + • • • + dn dy'^ on W,
then ^*a; has the same form on [/, where we now regard the a's and 2/'s as
functions of the :r's and expand by using (8.12). Thus
^*a; = J^di^dx',
where a^- = a,(2/H:^\ • • • , ^'^), • • • , v'^ix^ . . . , :r^)).
Let xgU. Then
^*(^)(^) = i:g(^)4,(^(^)) (8.15)
gives the formula for ^^x-
EXERCISES
8.1 Let X and ?/ be rectangular coordinates on E^, and let (r, 6) be polar "coordinates"
on E^ — {0}. Express the vector fields d/dr and d/dd in terms of rectangular
coordinates. Express d/dx and d/dy in terms of polar coordinates.
8.2 Let X, y, z be rectangular coordinates on E^. Let
X = y Z--1 Y = z^ ^T"' and Z = x- y^-
oz ay ox oz oy ox
Compute [Z, F], [Z, Z], and [Y, Z]. Note that Z represents the infinitesimal generator
of the one-parameter group of rotations about the a;-axis. We sometimes call Z the
"infinitesimal rotation about the a;-axis". We can do the same for Y and Z.
8.3 Let
oz oy oz ax oy ax
Compute [A, B], [A, C], and [B, C]. Show that Af = Bf = Cf = 0 'li f(x, y, z) =
^2 _j_ ^2 — ^2 Sketch the integral curves of each of the vector fields A, B, and C.

9.9 RIEMANN METRICS 397
8.4 Let
D = U — + V — y E = u- V — ) and /^ = u :^ v — •
ov ou ov ou ou ov
Compute [D, E], [D, F], and [E, F].
8.5 Let Pi, . . . , Pn be polynomials in a;\ . . . , a;" with no constant term, that is,
Pi(0, ...,0) = 0.
Let
^ ^ ^1 _^ I I ^n d
and
dx^ dx
Show that
[/, X] = 0
if and only if the P^'s are linear. [Hint: Consider the expansion of the P^'s into
homogeneous terms.]
8.6 Let X and the Pi's be as in Exercise 8.5, and suppose that the Pi's are linear. Let
and suppose that
\i 9^ Xj for I 9^ j.
Show that [v4, Z] = 0 if and only if Pi = ijiix\ that is,
= fjLix — H ^ finx — for some /x , . . . , /x .
8.7 Let A be as in Exercise 8.6, and suppose, in addition, that
\i 9^ \j + \r for any i, j, r.
Show that if the P/s are at most quadratic, then
[A,X] = 0
if and only if Pi = fiix^. Generalize this result to the case where Pi can be a polynomial
of degree at most m.
9. RIEMANN METRICS
Let M be a finite-dimensional differentiable manifold. A Riemann metric, m,
on M is a rule which assigns a positive definite scalar product ( , )m.x to the
vector space Tx{M) for each x G M. We shall usually drop the subscripts m and
X when they are understood from the context. Thus if m is a Riemann metric
on My X G My and ?, r? G Tx{M), we shall write the scalar product of ^ and r) as

398 DIFFERENTIABLE MANIFOLDS 9.9
Let ([/, a) be a chart of M. Define the functions gij on U by setting
^.V(x)=(|,(.),^-.(x)), (9.1)
SO that Qij = (jjt. If ^, 77 G T:c(M) with
^=Zr£lW and .= Z.%^(A
then
Since f/x^Cr), . . . , dx^ix) is the basis of T^\]\I) dual I0 the basis
we have
^^•_ {^,dx\x)\ V- (^,M^^)),
so that the last equation can be written as
(?, V)m,. = Z g^J(x){^, dx'Xv, dx^}. (9.2)
Equation (9.2) is usually written more succinctly as
m \ U = J: g^j{x) dx'dx'. (9.3)
[Here (9.3) is to be interpreted as a short way of writing (9.2).]
Let {W, 13) be a second chart with
h'^'i^) = (a~. (•-)' ifi w) , X e w,
that is,
m r TF = E hn di/ dij\ (9.4)
Then for x G [/ fi TF, we have
so
9iM) = (^ (X), ir, W) = Z |1 (..) g (..)/^u(.),
that is,
.J^ n, ^
Note that (9.5) is the answer we would get if we formally substituted (8.12) for
the dij^ in (9.4) and collected the coefficients of dx^ dx\
In any event, it is clear from (9.5) that if the hij are all smooth functions
on Wj then the cjij are smooth on IJ fi W. In view of this we shall say that a

9.9 RIEMANN METRICS 399
Riemann metric is smooth if the functions cjij given by (9.3) are smooth for any
chart ([/, a) belonging to an atlas d of M. Also, if we are given functions
dij = Qji defined for each {U, a) G G, such that
i) Z gijix)^"^' > 0 unless J = 0 for all xgU,
ii) the transition law (9.5) holds,
then the Qij define a Riemann metric on M. In the following discussion we shall
assume that our Riemann metrics are smooth.
Let ^: Ml —> M2 be a differentiable map, and let m be a Riemann metric
on M2. For any x G Mi define ( , ),/.*(m).x on T:c(Mi) by
(?j V)^p''(n^),x = (^*(?), ^*(^))m,,/.(x)- (9.6)
Note that this defines a symmetric bilinear function of J and rj. It is not
necessarily positive definite, however, since it is conceivable that ^*(?) = 0
with J 7^ 0. Thus, in general, (9.6) does not define a Riemann metric on Mi.
For certain ^ it does.
A differentiable map ^: Mi —> M2 is called an immersion if ^^^ is an injection
(i.e., is one-to-one) for all :r G M^
If ^: Ml —> M2 is an immersion and m is a Riemann metric on M2, then we
define the Riemann metric ^*(m) on Mi by (9.6).
Let ([/, a) and (TF, 13) be compatible charts of Mi and M2, and let
Then
where
^*(m) \ U = Y. Qij dx^ dx\
fxn,\p{x)
= (*. {^ w), *. {h m)).
which is just (9.5) again (with a different interpretation). Or, more succinctly,
^*(m) \U=Z rQiu)r{dy^)r{dy').
Let us give some examples of these formulas. If M = W', then the identity
chart induces a Riemann metric on IR'^ given by
{dx^Y-] + {dx"")^.
Let us see what this looks like in terms of polar coordinates in R^ and R^.

400 DIFFERENTIABLE MANIFOLDS 9.9
In U^ if we write
then
so
10 = r cos dy X = r sin By
dx^ = cos ^ dr — r sin ^ ddy
dx^ = sin 6 dr -\- r cos 6 ddy
{dxT + {dx^r = dr^ + r^ dd\ (9.7)
Note that (9.7) holds whenever the forms dr and dd are defined, i.e., on all of
U^ — {0}. (Even though the function d is not well defined on all of W^ — {0},
the form dS is. In fact, we can write
de = ^Lt! 7 fjf •) (9.8)
In U^ we introduce
Then
Thus
(:ri)2+ {x^)2
x^ ^ r cos cp sin dy
x^ = r sin cp sin dy
x^ = r cos 6.
dx^ = cos (p sinS dr — r sin <p sin ^ d<p + ?" cos <p cos ^ d^,
(i:r^ = sin cp sin ^ dr + ^ cos cp sin 6 dcp -\- r sin <p cos 6 ddy
dx^ = cos 6 dr — r sin 6 dd.
(dx^ + (dxy + (dx^y = dr^ + r^ sin^ d dcp^ + r^ dd\ (9.9)
Again, (9.9) is valid wherever the forms on the right are defined, which this time
means when (x^)^ + (x^)^ 9^ 0.
Let us consider the map l of the unit sphere S^ —> R^, which consists of
regarding a point of >S^ as a point of U^. We then get an induced Riemann metric
on>S2.
Let us set
dd = t* dd and d(p = t* d<p,
so the forms dd and dip are defined on U = S^ — {<0, 0, 1>, <0, 0, —!>}.
Then on U we can write (since r = 1 on S^)
L'^iidxy + (dxy + {dxY) = ^^ + sin'^ d'dUp^, (9.10)
We now return to general considerations. Let M be a differentiable manifold
and let (7: 7 —> M be a differentiable map, where I is an interval in R^ Let t
denote the coordinate of the identity chart on I, We shall set
=^* (!)(«)'
C'{s) = C* [^^J is), SG I,
SO that (7'(s) G Tc(8){M) is the tangent vector to the curve C at s. If (C/, a) is
a chart on M and X , • • . , X are the coordinate functions of (f7, a), then if

9.9 RIEMANN METRICS 401
C(7') C U for some I' C /,
aoC= <X^ oC,...,x'' oC>,
so that
n').=^^^ ^^(0.
When there is no possibility of confusion, we shall omit the o C and simply write
a o C(t) = <x\t), ..., x''(t)> and C\t)a = <x'\t), ..., x''\t)>.
Now let m be a Riemann metric on M. Then ||C'(0|| = (C\t), C\t)) ^'^ is a
continuous function. In fact, in terms of a chart, we can write
\\C'm = VT,gij(C(t))xi'(t)xr{t).
The integral
f^wcmdt (9.11)
is called the length of the curve C. It will be defined if ||C(Q || is integrable over /.
This will certainly be the case if / and ||C(^)|| are both bounded, for instance.
Note that the length is independent of the parametrization. More precisely,
let <p: / —> / be a one-to-one differentiable map, and let Ci = C o <p. Then at
any r G / we have
that is,
Thus
dt ^
dr
C'lir) = §C'it).
ri(T)ii = ruT))ii|^|- (9.12)
On the other hand, by the law for change of variable in an integral we have
l\\c'm = [wc'um
dip
dr
= /^ \\C'i{-)\\ by (9.12).
More generally, we say that a curve C defined on an interval / is piecewise
differentiable if
i) C is continuous;
ii) I = Ii \J '-' U Ir and C, on each /y, is the restriction of a differentiable
curve defined on some interval /y strictly containing /y.

402 DIFFERENTIABLE MANIFOLDS 9.9
(Thus a piecewise differentiable curve is a curve with a finite number of
"corners".) If C is piecewise differentiable, then ||C'(^)|| is defined and continuous
except at a finite number of ^'s, where it may have a jump discontinuity. In
particular, the integral (9.11) will exist and the curve will have a length.
Exercise. Let C be a curve mapping I onto a straight line segment in W^ in a one-to-
one manner. Show that the length of C is the same as the length of the segment.
Let C: [0, 1] -> R^ be a curve with C(0) = 0 and C(l) = v^R^. If we use
the expression (9.7), we see that
j\\C{t)\\ dt = IV(r\t))^+ {r{t)e\t)ydt
dt
> I r'iOdt
Jo
= ii^^ii,
with equality holding if and only if ^' = 0 and r^ > 0. Thus among all curves
joining 0 to v, the straight line segment has shortest length.
Similarly, on the sphere, let C be any curve C: [0, 1] —> S^ with C(0) =
(0, 0, 1) and C(l) = p 9^ {0, 0, -1), and let di = d{C{l)). Then
f\\C\t)\\ dt = fV(d\t))^ + sm^d(cp\t)ydt
> C w{t)\dt.
Jo
If we let ^1 denote the first point in [0, 1] where 6 = ^i, then
f\\C\t)\\ dt > C \d'{t) I dt > C' \d'{t)\ dt > C' d'{t) dt - ^1,
J Jo Jo Jo
with equality only if <p' = 0 and ^i = L Thus the shortest curve joining any
two points on S^ is the great circle joining them.
In both examples above we were aided by a very fortuitous choice of
coordinates (polar coordinates in the plane and a kind of polar coordinates on the
sphere). We shall see in Section 11, Chapter 13, that this is not accidental. We
shall see that on any Riemann manifold one can introduce local coordinates in
terms of which it is easy to describe the curves that locally minimize length.

CHAPTER 10
THE INTEGRAL CALCULUS ON MANIFOLDS
In this chapter we shall study integration on manifolds. In order to develop
the integral calculus, we shall have to restrict the class of manifolds under
consideration. In this chapter we shall assume that all manifolds M that arise
satisfy the following two conditions:
1) M is finite-dimensional.
2) M possesses an atlas Gi containing (at most) a countable number of
charts; that is, d = {(Ui, a:^•)}^=l.2.... •
Before getting down to the business of integration, there are several technical
facts to be established. The first two sections will be devoted to this task.
1. COMPACTNESS
A subset A of a manifold M is said to be compact if it has the following property:
i) If {Ul} is any collection of open sets with
there exist finitely many of the Ul, say Ul^j . . . , Ul^, such that
A cUl.U" ' U Ul^.
Alternatively, we can say:
ii) A set A is compact if and only if for any family {Ft} of closed sets
such that
AnriFL= 0,
L
there exist finitely many of the Fl such that
AnFL^n-" nF,^= 0.
The equivalence of (i) and (ii) can be seen by taking Ul equal to the
complement of Fi.
In Section 5 of Chapter 4 we established that if M = U is an open subset of
IR'^, then A C U is compact if and only if A is a closed bounded subset of U^.
403

404 THE INTEGRAL CALCULUS ON MANIFOLDS 10.1
We make some further trivial remarks about compactness:
iii) li Ai, . . . , Ar are compact, so is Ai U • • • U Ar.
In fact, if {Ul} covers Ai \J - - - U Ar, it certainly covers each Aj. We can
thus choose for each j a finite subcollection which covers Aj. The union of these
subcollections forms a finite subcollection covering Ai \J - - - U Ar.
iv) If ^: Ml —> M2 is continuous and A C Mi is compact, then ^[A] is
compact.
In fact, if {Ul} covers ^[A], then {<P~^{Ul)} covers A. If the Ul are open,
so are the ^~^{Ul), since yp is continuous. We can thus choose ti, . . . , t^ so that
ACyl^-\UL,)U"'U^-\ULX
which implies that ^[A] C f/t^ U • • • U f/t^.
We see from this that if A = Ai \J - - • U Arj where each Aj is contained
in some T^^•, where (Wi, (3i) is a chart, and (3i(Aj) is a compact subset of W^,
then A is compact. In particular, the manifold M itself may be compact. For
instance, we can write S^ as the union of the upper and lower hemispheres:
S"" = {x : x'''^^ > 0} U {:r : x'''^^ < 0}. Each hemisphere is compact. In fact,
the upper hemisphere is mapped onto {2/ : II2/II ^ 1} by the map <pi of Section 8.1,
and the lower hemisphere is mapped onto the same set by <p2- Thus the sphere
is compact.
On the other hand, an open subset of IR'^ is not compact. However, it can
be written as a countable union of compact sets. In fact, if C/ C IR'^ is an open
set, let
An = {x eU : \\x\\ < n and p{x, dU) > 1/n}.
It is easy to check that A^ is compact and that
{}An= U.
In view of condition (2), we can say the same for any manifold M under
consideration:
Proposition 1.1. Any manifold M satisfying (1) and (2) can be written as
M = U Ai,
i=l
where each AiC M is compact.
Proof. In fact, by (2)
M= 0 Uj,
and by the preceding discussion each Uj can be written as the countable union
of compact sets. Since the countable union of a countable union is still
countable, we obtain Proposition 1.1. D

10.2 PARTITIONS OF UNITY 405
An immediate corollary is:
Proposition 1.2. Let M be a manifold [satisfying (1) and (2)], and let {Ul}
be an open covering of M. Then we can select a countable subcollection
{Uj} such that
\JUj'= M.
Proof. Write M = \jAr, where Ar'is compact. For each r we can choose finitely
many C/r.i, Ur,2, • • • j Ur,k^ so that
ArC Ur,l U- • • U Ur,k^.
The collection
{Ur,k}r=l 00
A;=l kr
is a countable subcollection covering M. D
2. PARTITIONS OF UNITY
In the following discussion it will be convenient for us to have a method of
"breaking up" functions, vector fields, etc., into "little pieces". For this purpose
we introduce the following notation:
Definition 2.1. A collection {qj} of (7°°-functions is said to be a partition of
unity if
i) 9i ^ 0 for all i;
ii) supp gi'\ is compact for all i;
iii) each x G M has a neighborhood Vx such that Vx H supp Qi = 0
for all but a finite number of i; and
iv) Z; gi{x) = 1 for all x G M.
Note that in view of (iii) the sum occurring in (iv) is actually finite, since
for any x all but a finite number of the gi(x) vanish. Note also that:
Proposition 2.1. If A is a compact set and {qj} is a partition of unity, then
A n supp gi = 0
for all but a finite number of i.
Proof. In fact, each x G A has a neighborhood Vx given by (iii). The sets
{Vx}x^A form an open covering of A. Since A is compact, we can select a finite
subcollection {Fi, . . . , Vr) with A C Fi U • • • U Fr- Since each Vk has a
nonempty intersection with only finitely many of the supp gi, so does their
union, and so a fortiori does A. D
t Recall that supp g is the closure of the set {x: g{x) 9^ 0}.

406 THE INTEGRAL CALCULUS ON MANIFOLDS 10.2
Oelifiitioii 2.2. Let {Ui} be an open covering of Jl, and let {gj} he a
partition of unity. We say that {gj} is subordinate to {Ut} if for every j there
exists an t(j) such that
suppgjC Ihay (2.1)
Thewem 2.1. LM. {Ih} be any open covering of M. There exists a partition
of unity {^j} subordinate to {Ui] .
The proof that wc shall present below is due to Bonie and Franiptou.f
First we introduce some preliminary notions.
The function / on K defined by
., . Ur'^'" if w > 0,
^^^)"lo if «<o
is C^, For w 7^ 0 it is clear that / has derivatives of all orders. To check that / is
C^ at 0, it suffices to show that f^^^^{u) -^ 0 as m -* 0 from the right. Bui
f^^\u) ^ Pk{l/u)e'^^'^-% where Pk is a polynomial of degree 2fc, So
Mm f^\u) = Mm Pkis)e'^' = 0,
since e* goes to infinity faster than any poljaiomial.
Note that /(m) > 0 if and only if u > 0. Now consider the function g^ on 1
defined by
gtix) r= fix ^ u)fib ^ x)^
Then gl is C'^ and nonnegative and
glix) 7^ 0 if and only if o < z < h.
More eener ally, if a -- <a\ . . . , a^> and h = <b\ . . . , 5*>, define the
function (/« on R* by setting
where x = <x\ . . , , a:*>. Then g^ > 0, g^ E C*, and
g^(x) > 0 if and only if a^ < x^ < b\ ^ . . , a^ < x^ < h^ (2.2)
Lemma. Let fij ■ ■ - ,fk be C'*-functions on a manifold If, and let W =
{z : a^ < fi(x) < b\ . . . , a^' < /&(x) < 6*}. There exists a nonnegative
C^-function g such that W = {x : gix) > 0},
In fact, if we define g by
gix)'^gUM^).'^'Jki^))^
then it is clear that g has the desired properties.
t Smooth functions on Banaeh manifolds, J. Math and 3l€ch. 15, 877-898 (1%6).

10.2 PARTITIONS OF UNITY 407
We now tiirri to the proof of Theorem 2.1.
Proof. For each x E M choose a f/t containing x and a chart ([/, a) about :i:.
Then a{U n Ui) is an open set containing a{x) in R'". Choose a and b such that
«Ct) e int Da and ft C a(U n l/.).
Let W^ = a^^''\mt \jt)' Then
W.T C f.\ and W^. is compact. (2.3)
Also if x^ . . . , x" are the coordinates given by a,
W, = {y : a' < x\y) < h\ . . . , a'" < x"(-//) < 6^^^,
By our lemma we can find a nonnegative C^'"^"fnnction j\ such that
w,= {y.JM >o;.
Since a: E Wx^ the {W^x} cover M. By Proposition 1.2 we can select a countable
subcovering {W^}. Let ns denote the corresponding functions by Jf, that is,
if Wi = W,., wc set /, = U
Let
Fi- lfi= {x:/i(x) >0},
^2= k:/2(x) >0,/i(x) < i),
Vr = (x :/.(x) > 0,/i(x) < I/);, , , . ,/V-^i(x) < 1/r},
It is clear that Vj is open and tliat Vj C Wj, m that^ by (2.3),
Vj is compact and Fj C f/t (2.4)
for some t = t(J).
For each x G M let g(x) denote the first integer q for which /g(x) > 0.
Thus/p(x) = 0 if p < 5(x) and/^(,,j(x) > 0.
Letl'^-^ {y:f^(x){ij) > ¥q(x)ix)], Since/y(x)(x) > 0, it follows that x E F^
and Vx is open. Furtliennore,
V, nTr= 0 if r > ^(x) and 1/r < M^ix-M)* (2-5)
According to the lemma, each set F/ can be given as Ff = {x : pt(x) > 0},
where p^ is a suitable C^^'-function, Let g = Yl Vi- I" view of (2.5) this i> really
a finite sum in the neighborhood of any x. Thus g is C*. Now fg(x)Cr) > 0,
since x G Fg(j;). Thus {j > 0. Set
We claim that {(jj} is the desired partition of unity. In fact, (i) holds by our
construction, (ii) and (2.1) follow from (2.4), (iii) follows from (2.5), and (iv)
holds by construction, D

408 THE INTEGRAL CALCULUS ON MANIFOLDS 10.3
3. DENSITIES
If we regard U^ as a differentiable manifold, then the law for change of variables
for an integral shows that the integrand does not have the same transition law
as that of a function under change of chart. For this reason we cannot expect
to integrate functions on a manifold. We now introduce the type of object that
we can integrate.
Definition 3.1. A density p is a rule which assigns to each chart (U, a) of
M a function pa defined on a(U) subject to the following transition law:
If {W, 13) is a second chart of M, then
Pa{v) = p^(0 o a-\v))\detJpoa-'(v)\ for v G a(U n W). (3.1)
If G, is an atlas of M and functions pa. are given for all (Ui, ai) G d satisfying
(3.1), then the Pa^ define a density p on M. In fact, if ([/, a) is any chart of M
(not necessarily belonging to d), define pa by
Pa{v) = Paii^i ° oi~^{v))\det Juioa'^iv)] U V G a{U D Ui).
This definition is consistent: If v G a{U n Ui) n a{U n Uj), then by (3.1),
p«/a:y o a~^{v))\det Jocjoa-'(v)\
= poci(^i° Oif^ioLjO a~^{v)))\deiJaioaJ^{oLjOa~^{v))\\detJa^oa-^{v)\
= pocX^i o a~\v))\det Joc-oa-'{v)\
by the chain rule and the multiplicative property of determinants.
In view of (3.1) it makes sense to talk about local smoothness properties of
densities. We will say that a density p is C^ if for any chart ([/, a) the function
Pa is C^. As usual, it suffices to verify this for all charts ([/, a) belonging to some
atlas. Similarly, we say that a density p is locally absolutely integrable if for
any chart ([/, a) the function pa is absolutely integrable. By the last proposition
of Chapter 8 this is again independent of the choice of atlases.
Let p be a density on M, and let :r be a point of M. It does not make sense
to talk about the value of p at x. However, (3.1) shows that it does make sense
to talk about the sign of p at x, More precisely, we say that
p > 0 at :r if pa{a{x)) > 0 (3.2)
for a chart ([/, a) about x. Equation (3.1) shows that if pa{a{x)) > 0, then
p^{^(x)) > 0 for any other chart (TF, 13) about x. Similarly, it makes sense to
say that p < 0 at :r, p > 0 at :r, or p 5^ 0 at :r.
Definition 3.2. Let p be a density on M. By the support of p, denoted by
supp p, we shall mean the closure of the set of points of M at which p does
not vanish. That is,
supp p = {x : p 7^ 0 Sit x}.

10.3 DENSITIES 409
Let pi and p2 be densities. We define their sum by setting
(Pl + P2)a = Pla + P2a (3.3)
for any chart (f/, a). It is immediate that the right-hand side of (3.3) satisfies
the transition law (3.1), and so defines density on M.
Let p be a density, and let / be a function. We define the density jp by
(/P)« = UPa. (3.4)
Again, the verification of (3.1) is immediate in view of the transition laws for
functions.
It is clear that
supp (pi + P2) C supp pi U supp p2 (3.5)
and
supp (/p) = supp/ n supp p. (3.6)
We shall write
Pi ^ P2 at :r if p2 — Pi > 0 at :r
and
Pi ^ P2 if Pi ^ P2 at all X G M.
Let P denote the space of locally absolutely integrable densities of compact
support. We observe that P is a vector space and that the product fp belongs
to P if / is a (bounded) locally contented function and p G P.
Theorem 3.1. There exists a unique linear function J on P satisfying the
following condition: If p G P is such that supp p C U, where ([/, a) is a
chart of M, then
fp= f p«. (3.7)
J Ja{U)
Proof. We first show that there is at most one linear function satisfying (3.7).
Let a be an atlas of M, and let {gj} be a partition of unity subordinate to d.
For each j choose an i{j) so that
supp gj C f/^•,
ijy
Write p = 1 ' p = ^ gj' P' Since supp p is compact, only finitely many
of the terms gjp are not identically zero. Thus the sum is finite. Since J is linear,
jp= jH QjP = Z {qj-P-
By (3.7),
[qjP = f (gjP)oci^jy
J •^«f(y)(^i(y))
Thus
fp=T. I (gjP)a,,jy (3.8)
Thus J, if it exists, must be given by (3.8). To establish the existence of J,

410 THE INTEGRAL CALCULUS ON MANIFOLDS 10.)^
we must show that (3.8) defines a linear function on P satisfying (3.7). The
linearity is obvious; we must verify (3.7).
Suppose supp p C U for some chart {U, a). We must show that
Since p = X (JjP and therefore Pa ^ Yl {OjP)a, it suffices to show that
f {gjp)a = f {9jp)ai, (3.9)
Ja(U) JotiiUi)
where supp (jjp C f/ D Ui. By (3.1),
SO that (3.9) holds by the transformation law for integrals in IR^. D
We can derive a number of useful properties of the integral from the
formula (3.8):
if Pi < P2y then fpi < jp2. (3.10)
In fact, since Qj > 0, we have {gjPi)a ^ ((IjP2)a for any chart (U, a).
Thus (3.10) follows from the corresponding fact on IR"" if we use (3.8).
Let us say that a set A has content zero ifAcAiU---uAp where eacli
Ai is compact, A^ C Ui for some chart {Ui, ai)j and ai{A^) has content zero in
IR^. It is easy to see that the union of any finite number of sets of content zero
has content zero. It is also clear that the function ba is contented.
Let us call a set J5 C M contented if the function es is contented. For any
p G P we define Jb P by
f^p= jenp. (3.11)
It follows from (3.8) that
f p-0
J A
for any p G P if A has content zero. We can thus ignore sets of content zero for
the purpose of integration. In practice, one usually takes advantage of this when
computing integrals, rather than using (3.8). For instance, in computing an
integral over aS^, we can "ignore" any meridian: for example, if
A = {:r G ^" : :r - (^, 0, . . . , ztVl - t^) G B^^+'},
then
f p = f p for any p.
This means that we can compute jsn p by introducing polar coordinates
(Fig. 10.1) and expressing p in terms of them. Thus in S^, ii U = S^ — A and
a is the polar coordinate chart on [/, then
f p = f I Pads dip.
Js^ Jo Jo

10.4
VOLUME DENSITY OF A RIEMANN METRIC
411
Fig. 10.1
It is worth observing that if iV is a differentiable manifold of dimension less
than dim M and ^ is a differentiable map oi N -^ M, then Proposition 7.3 of
Chapter 8 implies that if A is any compact subset of N, then 4^{A) has content
zero in M. In this sense, one can ignore "lower-dimensional sets" when
integrating on M.
4. VOLUME DENSITY OF A RIEMANN METRIC
Let M be a differentiable manifold with a Riemann metric a. We define the
density cr [=o-(a)] as follows. For each chart (U, a) with coordinates x^, . . . ,x'^
let
11/2
(Toc{ol{x)) =
^^H(^^^^'ali^^))]
|det {gijix))\
1/2
Here
is the matrix whose ijth entry is the scalar product of the vectors
(4.1)
^^^^
and ^-7 (x),
dx^ '
so that (in view of Exercise 8.1 of Chapter 8)
o'a(oi(x)) = volume of the parallelepiped spanned by (d/dx^){x) with
respect to the Euclidean metric ( , )a,x on Tx{M).
It is easy to see that (4.1) actually defines a density. Let (TF, ^) be a second
chart about x with coordinates y^, . . . , y^. Then
so that
(^^(Kx)) =
det
l>-w-^4\"'-

412
THE INTEGRAL CALCULUS ON MANIFOLDS
10.4
Now
\dy^ ' dy^J ~~ X'jdyk' dy^ \dx^ ' dx^J
for all A; and l. We can write this as the matrix equation
Wdy^ ' dy^/\ ~ \_dy^\ \_\dx^ ' dyjj
'dx[
Idy^]
so that
(T^iKx))
det
det
det
1/2
dx^
ldy^_
det
det
dx3
Idy^
1/2
L^2/^J
== aa(a{x))
det
W.
If M is an open subset of Euclidean space with the Euclidean metric, then
the volume density, when integrated over any contented set, yields the ordinary
Euclidean volume of that set. In fact, if X ,. . . , ^ are orthonormal coordinates
corresponding to the identity chart, then gij{x) = 0 ii i 9^ j and ga == 1, so
that Cid ^ 1 and thus
Ja J a
1
mU).
J\lore generally, let cp be an immersion of a A;-dimensional manifold M into
W^ such that (p{M) is an open subset of a A;-dimensional hyperplane in W, and
let m be the Riemann metric induced on M by cp. Then, if c denotes the
corresponding volume density, J4 a is the A;-dimensional Euclidean volume of (p{A),
In fact, by a Euclidean motion, we may assume that cp maps M into R^ C ll^^^
Then, since cp is an immersion and M is A;-dimensional, we can use :r\ . . . , :r*
as coordinates on M and conclude, as before, that a is given by the function in
terms of these coordinates, and hence that JaO" = m(<^(^))-
Now let (pi and <^2 be two immersions of M
Let (U, a) be a coordinate
chart on M with coordinates y
by (Pi, then
y . If mi is the Riemann metric induced
and
dy3jmi
dy^/m2
^ fd<pi d<pi\
\dy^ ' dy^'J
/d<p2 ^ d<p2\
\dy^ ' dyjj
where the scalar product on the right is the Euclidean scalar product. Let at

10.4
VOLUME ©13NSITY OF A RIEMANN METRIC
413
Fig. 10.2
and 0-2 be the volume densities corresponding to mi and m2. Then
and
In particular, given an L > 0, there is a K = K{k, n, L) such that if
< L and \\-fj\\ < L
then I by the mean-value theorem,
|0-i« - 0-2a| < K ^^ - —,
for all ^ = 1, . . . , k,
+ '" +
dyk dy4/
Roughly speaking, this means that if <pi and <P2 are close, in the sense that their
derivatives are close, then the densities they induce are close.
We apply this remark to the following situation. We let <pi be an immersion
of M into W and let (TF, a) be some chart of M with coordinates ^^ . . . , /.
We let [/ = Tf — C = Ul/z, where C is some closed set of content zero and
such that UinUv -= 0 U I 9^ V, For each I let zi be a point of Ui whose
coordinates are < 2//,.
i^sZ^Ui. (See Fig. 10.2.)
yi>, and for 2 - < 2/S • • • » /> define (^2 by setting,

414 THE INTEGRAL CALCULUS ON MANIFOLDS 10.4
If the Uis are sufficiently small, then
I dy^ dy^ I
will be small. More generally, we could choose <p2 to be any affine linear map
approximating <pi on each Ui. We thus see that the volume of W in terms of the
Riemann metric induced by (p is the limit of the (surface) volume of polyhedra
approximating (p{W). Here the approximation must be in the sense of slope
(i.e., the derivatives must be close) and not merely in the sense of position.
The construction of the volume density can be generalized and suggests an
alternative definition of the notion of density. In fact, let p be a rule which
assigns to each x in M a function, px, on 7i tangent vectors in TxiM) subject to
the rule
p,(A?i, . . . , Ain) = |det A|p,(?i, . . . , U, (4.2)
where ?^• G Tx{M) and A: Tx{M) -> T^iM) is a linear transformation. Then
we see that p determines a density by setting
p„(«(x))=p(^(x),...,^(x)) (4.3)
if ([/, a) is a chart with coordinates u^, . . . , u'^. The fact that (4.3) defines a
density follows immediately from (4.2) and the transformation law for the
d/du'^ under change of coordinates.
Conversely, given a density p in terms of the p^, define p{d/du^, . . . , d/du'^)
by (4.3). Since the vectors {^/^t^*}i=i,...,n form a basis at each x in Uj any
Ji, . . . , J^ in TxiM) can be written as
where .B is a linear transformation of Tx{M) into itself. Then (4.2) determines
pUu . . . , ?n) as
P(?i, ..., U = \detB\p^(a(x)), (4.4)
That this definition is consistent (i.e., doesn't depend on a) follows from (4.2)
and the transformation law (3.1) for densities.
EXERCISES
4.1 LetM = S^XS^ be the torus, and let <p: M -> R^ be given by
X^ o (p(dl, 62) = COS dl,
x^ o (p(di, 62) = sin dl,
x^ o (p(di, 62) = 2 cos ^2,
x'^o ^{$1,62) = 2 sin (92,

10.4
VOLUME DENSITY OF A RIEMANN METRIC 415
and 6^, 6^ are angular coordi-
where x^, . . . , x^ are the rectangular coordinates on
nates on M.
a) Express the Riemann metric induced on if by <p (from the Euclidean metric
on R^) in terms of the coordinates 6^, 6^. [That is, compute the gij{d^, 6^).]
b) What is the volume of M relative to this Riemann metric?
4.2 Consider the Riemann metric
induced on S^ X S^ by the immersion (p into
E^by
X o ip(u, v) = (a — cos u) cos v,
y o ip(u, v) = (a — cos u) sin v,
Z o <p(Wj y) = sin Uj
where u and v are angular coordinates and
a > 2. What is the total surface area of
<S^ X aS^ under this metric?
4.3 Let ip map a region U of the a;2/-plane
into E^ by the formula
ip{x,y) = {x,y,F(x,y)),
Fig. 10.3
so that (piU) is the surface z = F(x, y). (See Fig. 10.3.) Show that the area of this
surface is given by
fM£r-(sr-
4.4 Find the area of the paraboloid
z = x^ -\- y^ for x^ -\- y^ < 1.
4.5 Let U C U^y and let <p: ^ -> E^ be given by
(p(u,v) = {x(u,v),yiu,v),z(u,v)),
where Xj y, z are rectangular coordinates on E^. Show the area of the surface ip{U)
is given by
i I \(dx dy dx dy\ idy dz dy dz\ Idx dz dx dy\
Ju A/ \du dv dv duj \du dv dv duf \^u dv dv duj
4.6 Compute the surface area of the unit sphere in E^.
4.7 Let Ml and If2 be differentiable manifolds, and let cr be a density on M2
which is nowhere zero. For each density p on Mi X M2, each product chart (Ui X U'2,
ai X a2)j and each X2 E U2, define the function piaii'y ^2) by
PlaiiVlj X2)(ra{a2{X2)) = PaiXa2 (^b <^2(^2) )
for all 1^1 GaiiUi).
a) Show that piaii^ij ^2) is independent of the chart ([/2, 0C2)'
b) Show that for each fixed X2 G M2 the functions piaii', ^2) define a density on Mi.
We shall call this density pi(x2).

416 THE INTEGRAL CALCULUS ON MANIFOLDS
10.5
c) Show that if p is a smooth density of compact support on Mi X M2 and a is
smooth, then pi{x2) is a smooth density of compact support on Mi.
d) Let p be as in (c). Define the function Fp on Tlf2 by
Fp{X2) = f plfe).
Sketch how you would prove the fact that Fp is a smooth function of compact
support on M2 and that
JM1XM2 JM2
5. FULLBACK AND LIE DERIVATIVES OF DENSITIES
Let if'. Ml —> M2 be a diffeomorphism, and let p be a density on M2. Define
the density <^*p on Mi by
<^*P(?l, • • • , ?n) == P(<^*?1, • • • , <^*?n)
(5.1)
for ^i G Tx(Mi) and <^^ = <^^x- To show that <^*p is actually a density, we must
check that (4.2) holds for any linear transformation A of Tx{Mi). But
<^*p(A?i, . . . , A?n) = p(<^*A?i, . . . , <^*A?n)
^ p(<^*A<^* <^*?i, . . . , <^*A<^* <^*?n)
= |det (p-^Aif^ I p(<^*?i, . . . , <^*?n)
= |det A|<^*p(?i, . . . , in),
which is the desired identity.
Let ([/, a) and {W, (3) be compatible charts on Mi and M2 with coordinates
u^, . . . , u'^ and i(;\ . . . , if^, respectively. Then for all points of U we have,
by (4.3),
(^*p).(.(.)) = P (^*^ . • • . . ^*^) = |det (g)| P (^- • -^)
(^
det
P^(^o<^(.)).
In other words, we have
(<p''p)a = |det J^o<poa-i|p(^ o <p o q:-1(-)). (5.2)
The density <^*p is called the pullback of p by <^*. It is clear that
<^*(P1 + P2) = <^*(pl) + <^*(P2)
and that
<f*{fp) = <P*U)'P*{P)
for any function /.
It follows directly from the definition that
supp <^*p = <^~^[supp p].

10.5
FULLBACK AND LIE DERIVATIVES OF DENSITIES
417
Proposition 5.1. Let ip\ M^ —> M2 be a diffeomorphism, and let p be a
locally absoluteh^ integrable density with compact support on Af 2. Then
/"*"
/"
(5.3)
Proof. It suffices to prove (5.3) for the case
suppp C <p{U)
for some chart {U, a) of Mi with (p{U) C W, where (W, ^) is a chart of M2.
In fact, the set of all such (p(U) is an open covering of M2, and we can therefore
choose a partition of unity {qj} subordinate to it. If we write p = 1] QjPj then
the sum is finite and each gjp has the observed property. Since both sides of (5.3)
are linear, we conclude that it suffices to prove (5.3) for each term.
Now if supp p C <p{U), then
fp= f P^= f
J J3(W) JBo
and
f>i.U)
P^
f<P*P = f (<P*p)a
J Ja{U)
Ja(U)
J^o(p(U)
P^y
thus establishing (5.3). D
Now let (ft be a one-parameter group on M with infinitesimal generator X.
Let p be a density on M, let ([/, a) be a chart, and let W be an open subset of U
such that cpt{W) C U for all |^| < e. Then
{^tp)a{v) = Pa(^a{v, t))
det
for V G oi{W)y
where ^a{^, t) = a o cp^ o a~^{v) and {d^a/^^){v,t) is the Jacobian of y h-> ^^{v, t).
We would like to compute the derivative of this expression with respect to t at
t = 0. Now ^ai^^j 0) = ^y and so
Consequently, we can conclude that
det
1.
0)
\^^ /iv,t
> 0
for t close to zero. We can therefore omit the absolute-value sign and write
d{(Ptp)a
dt
dpai^a)
dt
..^-«l(<'-t)l,..

418 THE INTEGRAL CALCULUS ON MANIFOLDS 10.5
We simply evaluate the first derivative on the right by the chain rule, and get
dpa f-^j = dpa{Xa{v)).
In terms of coordinates x^, . . . , x^, we can write
dpai^ajV, t)) _ y^ dPa j
dt ~ ^ dxi ^"
if Xa =^ "K X^, . . . J X^ /^ .
To evaluate the second term on the right, we need to make a preliminary
observation. Let A{t) = (dijit)) be a differentiable matrix-valued function of t
with 1(0) = id = {8i). Then
^^^^l = limi(detA(0-l).
Now aii(0) = 1 and aij(0) = 0 (i ^ j). To say that A is differentiable means
that each of the functions aij(t) is differentiable. We can therefore find a constant
K such that \aij{t)\ < K\t\ (^ 9^ j) and {aait) — 1| < K\t\. In the expansion of
det A (t), the only term which will not vanish at least as ^^ is the diagonal product
aii(0 • • • (inn(t). In fact, any other term in ^ i aii^(t) • • • ani^(t) involves at
least two off-diagonal terms and thus vanishes at least as ^^. Thus
^ (det A{t)) = li^i (aii(0 • • • a^n(0 — 0
= aii(0) + ...+a;,(0)
= tr A'(0).
If we take A = d^^c/^^j "^ve conclude that
It V dv J-^'^ dv - ^ dx^
d{iptP)a X^ dPa -^i I ^ dXa V^ ^ / vi\
Thus
*
We repeat:
Proposition 5.2. Let (pt be a one-parameter group of diffeomorphisms of M
with infinitesimal generator X, and let p be a differentiable density on M.
Then
DxP^hm^^^
exists and is given locally by
{DxP)a - 2. ^--
if Z = <Xl, . . . ,Xl> on the chart {U, a).

10.6 THE DIVERGENCE THEOREM 419
The density DxP is sometimes called the divergence of <X,p> and is
denoted by div <X, p>. Thus div <X, p> ^ DxP is the density given by
(div<X,p>)« = E^,(^aP«) on (t/,a).
Now let p be a differentiable density,
and let A be a compact contented set.
Then
I P = I e^tU)P
= I <pt{e<ptU)p)
JM
= \{^te<ptiA)){<ptp)
JiPtiA)
Jm
[eA<pt(p)
I
*
<PtP-
A
Thus
7(L/-/.^)=/j<-^^"-
Using a partition of unity, we can easily see that the limit under the integral
sign is uniform, and we thus have the formula
|(f p)\ = f Dxp= [div<X,p>.
Ctt\J^^(A) / U=0 J A J A
6. THE DIVERGENCE THEOREM
Let <p be a flow on a differentiable manifold M with infinitesimal generator X.
Let p be a density belonging to P, and let A be a contented subset of M. Then
for small values of t, we would expect the difference J^^(A) P — Ja P to depend
only on what is happening near the boundary of A (Fig. 10,4). In the limit,
we would expect the derivative of J^^(^) p at ^ = 0 (which is given by
Ja div <X, p>) to be given by some integral over dA. In order to formulate
such a result, we must first single out a class of sets whose boundaries are
sufficiently nice to allow us to integrate over them. We therefore make the following
definition:
Definition. Let M be a differentiable manifold, and let D be a subset of M,
We say that Z) is a domain with regular boundary if for every x G M there is a
chart (f/, a) about x, with coordinates x^, . . . , x^, such that one of the
following three possibilities holds:
i) UnD=0;
ii) UCD;
iii) aiU n D) = a{U) n {v = <v\ .,,,v^> eW'-.v^ > 0}.

420
THE INTEGRAL CALCULUS ON MANIFOLDS
10.6
Note that if x g D, we can always find a, (U, a) about x such that (i) holds.
If X G int D, we can always find a chart (L^, a) about x such that (ii) holds.
This imposes no restrictions on D. The crucial condition is imposed when
X G dD. Then we cannot find charts about x satisfying (i) or (ii). In this case,
(iii) implies that a(U D dD) is an open subset of U'^~'^ (Fig. 10.5). In fact,
a(U n dD) = {ve a{U) '.v"" = 0} = a(U) D [I^^~\ where we regard R^-^ as
the subspace of W^ consisting of those vectors with last component zero.
Fig. 10.5
Let a be an atlas of M such that each chart of G, satisfies either (i), (ii), or
(iii). For each (U, a) e a consider the map a \ dD:U ndD -^ W~^ C H^"".
[Of course, the maps a \ dD will have a nonempty domain of definition only for
charts of type (iii).] We claim that {([/ n dD, a \ dD)} is an atlas on dD. In
fact, let ([/, a) and (W, (3) be two charts in a such that U D W n dD 7^ 0.
Let x\ . . . , x"" be the coordinates of {U, a), and let ij^, . . . , ij^ be those of
{W, p). The map jS o a~^ is given by
<x\ . . . ,x^> K-> <y\x\ . . . ,x^), . . . ,?/^(x\ . . . ,.r'0>.
On a{U nW n dD), we have x"" =^ 0 and y"^ = 0. In particular,
2/-(x^...,x^-^o)^o,
and the functions y^{x^j . . . , x'^~^, 0), . . . ,
y''~'^(x\ . . . , x'^~\ 0) are differentiable. This
shows that (^ \ dD) o (« \ dD)'' is
differentiable on a{U n dD). We thus get a manifold
structure on dD.
It is easy to see that this manifold
structure is independent of the particular atlas of M
that was chosen. We shall denote by i the map
of dD —> M which sends each x G dD, regarded
as an element of M, into itself. It is clear that
t is a differentiable map. (In fact, {U O dD, a
charts in terms of which a o t o (« f dD)~^ is just the map of W^~'^ —> W^.)
Let jc be a point of dD regarded as a point of M, and let ^ be an element
of TxiM). We say that ^ points into D if for every curve C with C'(0) = ^,
we have C(t) G D for sufficiently small positive t (Fig. 10.6). In
Fig. 10.6
dD) and (U, a) are compatible
n —1

10.6
THE DIVERGENCE THEOREM
421
terms of a chart {U, a) of type (iii), let ^^ = < ?^ • • • , ^""^ • Then it is clear
that ^ points into D if and only if ^^ > 0. Similarly, a tangent vector ^ points
out of D (obvious definition) if and only if ^^ < 0. If ^^ = 0, then ^ is tangent
to the boundary—it lies in i^Tx{dD).
Let p be a density on M and X a vector field on M. Define the density px
on dD by
px(h, ..., ?n-i) = p(t*h, ...,t*?,_i,X(x)) for ^,eT,{dD). (6.1)
It is easy to check that (6.1) defines a density. (This is left as an exercise for
the reader.) If {U, a) is a chart of type (iii) about x and X« = <X\ . . . , X'^>,
then applying (4.3) to the chart (U n dD^ a f dD) and the density px, we see
that
(px)o,haB = p(^---.^'XJ-
Let A be the linear transformation of Tx{M) given by
The matrix of A is
d
axi
ax^-i ax^-1
Ai^ = X.
1
0
0
0
0
1 0
x^
x^
0
and therefore Idet A\ = {X"^]. Thus we have
(Px)« hai) = \X'^\pa at all points of a{U n ^2)).
We can now state our results.
(6.2)
Theorem 6.1 {The divergence theorem)."f Let D be a domain with regular
boundary, let p G P, and let X be a smooth vector field on M. Define the
function ex on dD by
(1 if X{x) points out of D,
0 if X(x) is tangent to dD,
— 1 if X(x) points into D.
Then
f div <X,p> = f €xpx. (6.3)
Jd JdD
Remark. In terms of a chart of type (iii), the function ex is given by
ex = -sgn X^. (6.4)
t This formulation and proof of the divergence theorem was suggested to us by
Richard Rasala.

422
THE INTEGRAL CALCULUS ON MANIFOLDS
10.6
Fig. 10.7
Fig. 10.8
supp p
Fig. 10.9
Fig. 10.10
Proof. Let d be an atlas of M each of whose charts is one of the three types.
Let {(ji\ be a partition of unity subordinate to d. Write p = Y. QiP- This is a
finite sum. Since both sides of (6.3) are linear functions of p it suffices to verify
(6.3) for each of the summands gip. Changing our notation (replacing Qip by p),
we reduce the problem to proving (6.3) under the additional assumption
supp p C U, where (U, a) is a chart of type (i), (ii), or (iii). There are therefore
three cases to consider.
CASE L supp pCUsindUnD=0. (See Fig. 10.7.) Then both sides of
(6.3) vanish, and so (6.3) is correct.
CASE XL supp pCU with U C int D. (See Fig. 10.8.) Then the right-hand
side of (6.3) vanishes. We must show that the left-hand side does also. But
Jd ' Ju Ja(C/) dx'' "^ JaiU) dx''
Now each of the functions X^Pa has its support lying inside a{U). Choose some
large R so that a{U) C O—r. We can then replace jaiU) by /nS-R. We extend
its domain of definition to all of U^ by setting it equal to zero outside a{U).
(See Fig. 10.9.) Writing the integral as an iterated integral and integrating with
respect to x* first, we see that
JaiU) dx^
I
= / X'pa( ...,R,...)- XWi ...,-R,...)dxUx'' dx'-^ dx
dx'" = 0.
This last integral vanishes, because the function X'p^ vanishes outside a{U).

10.6
THE DIVERGENCE THEOREM
423
CASE III. supp p is contained in a chart of type (iii). (See Fig. 10.10.) Then
I div <x,p> = f div <x,p> = j: f ^^'^«
div <X,p> = I div <X,p>
J D
Now
a(U f\D) = a{U) n {y:v'' > 0}.
We can therefore replace the domain
of integration by the rectangle
UT-r:''.^-r,^>- (See Fig. 10.11.)
For 1 < i < n all the integrals in "<-^;ii
the sum vanish as before. For
i =1 n we obtain
x{Df\U)
dx^
R>
!;^ = 0
supp Pa
/^div<X,p> = -/^„_,XV.
Fig. 10.11
If we compare this with (6.2) and (6.4), we see that this is exactly the assertion of
(6.3). D
If the manifold M is given a Riemann metric, then we can give an alternative
version of the divergence theorem. Let dV be the volume density of the Riemann
metric, so that
dViiu ...,?„) = Idet {{U, ii))\^l\ ii e T,{M),
is the volume of the parallelepiped spanned by the ^i in the tangent space (with
respect to the Euclidean metric given by the scalar product on the tangent space).
Now the map t is an immersion, and therefore we get an induced Riemann
metric on dD. Let dS be the corresponding volume density on dD. Thus, if
{ii\i=\ n-i are n — 1 vectors in T^idD), dS{^i, . . . , ?n-i) is the (n — 1)-
dimensional volume of the parallelepiped spanned by t*Ji, . . . , t*?n-i in
i^T^{dD) C T^{M). For any xGdDletnG T^{M) be the vector of unit length
which is orthogonal to uTx{dD) and which points out of D (Fig. 10.12). We

424 THE INTEGRAL CALCULUS ON MANIFOLDS 10.7
clearly have
dS{^i, . . . , ^n-i) = dV(L:^^i, . . . , t*?n-i, n).
For any vector X{x) G Tx{M) (Fig. 10.13) the volume of the parallelepiped
spanned by Ji, . . . , J^-i, X(x) is \{X(x), ii)\dS{^i, . . . , ?n-i)- [In fact, write
X{x) = {X{x),n)ii + m,
where m G ^^^^(^Z)).] If we compare this with (6.1), we see that
dVx= \{X,n)\dS. (6.5)
Furthermore, it is clear that
e{x) = sgn (X{x)j-n.).
Let p be any density on M. Then we can write
P = fdV,
where / is a function. Furthermore, we clearly have px = f dVx and
div <X,p> = div <XJdV.>.
We can then rewrite (6.3) as
[ div <X,fdV> - [ f-{X,n)dS. (6.6)
Jd JdD
7. MORE COMPLICATED DOMAINS
For many purposes. Theorem 6.1 is not quite sufficiently broad. The trouble is
that we would like to apply (6.3) to domains whose boundaries are not
completely smooth. For instance, we would like to apply it to a rectangle in U^,
Now the boundary of a rectangle is regular at all points except those lying on an
edge (i.e., the intersection of two faces). Since the edges form a set "of dimension
n — 2", we would expect that their presence does not invalidate (6.3). This is
in fact the case.
Let M be a differentiable manifold, and let D be a subset of M. We say
that D is a domain with almost regular boundary if to every x G M there is a
chart {U, a) about x, with coordinates x^, . . . , x^, such that one of the following
four possibilities holds:
i) UnD=0;
ii) UcD;
iii) a{U nD) = a{U) n {y= <v\ . . . , z;^> G IR^ : y^ > 0} ;
iv) a(U nD) = a(U) n {y= <v\ . . . , z;^> G K^ : y^ > 0, . . . , z;^ > 0}.
The novel point is that we are now allowing for possibility (iv) where k < n.
This, of course, is a new possibility only if n > 1. Let us assume n > 1 and sec
what (iv) allows. We can write a{U Ci dD) as the union of certain open subsets
lying in (n — 1)-dimensional subspaces of 1R^~\ together with a union of
portions lying in subspaces of dimension n — 2.

10.7
MORE COMPLICATED DOMAINS
425
Fig. 10.14
In fact, fork<p<n let
jy^ = {v : / > 0, . . . , z;^ = 0, z;^+' > 0, . . . , z;" > 0}.
Thus Hp is an open subset of the (n — 1)-dimensional subspace given by
v^ = 0. (See Fig. 10.14.) We can write
a{U n dD) C a{U) n {{Hi U Hl^, U---uHt)US},
where S is the union of the subspaces (of dimension n — 2) where at least two
of the v^ vanish.
Fig. 10.15
Observe that if x gU ndD is such that a(x) G H^ for some p, then there is a
chart about x of type (iii). In fact, simply renumber the coordinates so that v^
becomes z;^ that is, map IR'' -^ IR'' by sending <v^, . . . ,v''> -^ <w^,. . . , w''>,
where
y* for i < p,
w"
.,^+1
for p < i < n,
Then in a sufficiently small neighborhood U^ of x the chart (U^, <p o a) is of
type (iii). (See Fig. 10.15.)

426
THE INTEGRAL CALCULUS ON MANIFOLDS
10.7
We next observe the set of :r G dD having a neighborhood of type (iii) forms
a differentiable manifold. The argument is just as before. The only difference
is that this time these points do not exhaust all of dD. We shall denote this
manifold by dD. Thus dD is a manifold which, as a set, is not dD but only the
"regular" points of dD^ that is, those having charts of type (iii).
Theorem 7.1 (The divergence theorem). Let M be an n-dimensional
manifold, and let D cM be a domain with almost regular boundary.
Let dD be as above, and let i be the injection of dD -^ M. Then for any
p G P we have
I div <X,p> = L €xpx. (7.1)
JD JdD
Proof. The proof proceeds as before. We choose a connecting atlas of charts
of types (i) through (iv) and a partition of unity {qj} subordinate to the atlas.
We write p = 1] gjP and now have four cases to consider. The first three cases
have already been handled.
The new case arises when p has its support in U, where {U, a) is a chart of
type (iv). We must evaluate
5r^ dX^Pa
L
aiUnD)
dx^
The terms in the sum corresponding to i < k make no contribution to the
integral, as before. Let us extend X^pa to be defined on all of IR^ by setting it
equal to zero outside a{U), just as before. Then, for k < i < n we have
/ dX^Pa ^ / dX^Pa
JaiUnD) dX^ Jb dX^
where B = {y : v^ > 0, . . . , v"" > 0}. Writing
this as an iterated integral and integrating first
wdth respect to x\ we obtain
f dX%_ f
Jb dx^ J A,
X^Pa,
IB OX" JAi
where the set Ai C W^~~^ is given by
Ai= {<v\..
v^-^,v'-^\
Fig. 10.16
,z;^> :z;^ > 0, ...,z;^ > 0}.
Note that Ai differs from H^ by a set of content zero in 1R^~^ (namely, where at
least one of the i;^ = 0 for /c = ? < n). Thus we can replace the Ai by the H^
in the integral. Summing over k < i < n, we get
JaiDnU)"^ dx' a, Jh-
which is exactly the assertion of Theorem 7.1 for case (iv). D

10.7
MORE COMPLICATED DOMAINS
427
Fig. 10.17
Fig. 10.18
We should point out that even Theorem 7.1 does not cover all cases for which
it is useful to have a divergence theorem. For instance, in the plane, Theorem 7.1
does apply to the case where D is a triangle. (See Fig. 10.16.) This is because
we can "stretch " each angle to a right angle (in fact, we can do this by a linear
change of variables of IR^). (See Fig. 10.17.)
However Theorem 7.1 does not apply to a quadrilateral such as the one in
Fig. 10.18, since there is no C^-transformation that will convert an angle greater
than TT into one smaller than tt (since its Jacobian at the corner must carry
lines into lines). Thus Theorem 7.1 doesn't apply directly. However, we can
write the quadrilateral as the union of two triangles, apply Theorem 7.1 to each
triangle, and note that the contributions of each triangle coming from the
common boundary cancel each other out. Thus the divergence theorem does
apply to our quadrilateral.
This procedure works in a quite general context. In fact, it works for all
cases where we shall need the divergence theorem in this book, whether Theorem
7.1 applies directly or we can reduce to it by a finite subdivision of our domain,
followed by a limiting argument. We shall not, however, formulate a general
theorem covering all such cases; it is clear in each instance how to proceed.
EXERCISES
In Euclidean space we shall write div X instead of div <X, p> when p is taken to
be the Euclidean volume density.
7.1 Let X, y, z be rectangular coordinates on E^. Let the vector field X be given by
where r^ = x^ + 2/^ + 2;^. Show directly that
{ (X, n)dA= f div X
Js Jb
by integrating both sides. Here 5 is a ball centered at the origin and S is its boundary.

428 THE INTEGRAL CALCULUS ON MANIFOLDS 10.7
7.2 Let the vector field Y be given by
Y = YrHf + YqRq + Y^n^
in terms of polar "coordinates" r,d,,^ on E^, where rir, rift and n,^ are the unit vectors in
the directions d/dr, d/dd and d/dif respectively. Show that
div Y = ^4— \l (r' sin^ F,) + ^ (rYe) + ^ {rsm<p yA .
r2 sin ip [or ad dip ]
7.3 Compute the divergence of a vector field in terms of polar coodrinates in the
plane.
7.4 Compute the divergence of a vector field in terms of cylindrical coordinates
in E3.
7.5 Let (J be the volume (area) density on the unit sphere iS^. Compute div aX
in terms of the coordinates 6, (p (polar coordinates) on the sphere.

CHAPTER 11
EXTERIOR CALCULUS
Let M be a differentiable manifold and let co be a linear differential form in J\I.
For any differentiable curve C\ [a, h] -^ M we can consider the integral
ia (^'(Oj ^C(o) ^^^- Let [c, d] -^ [a, h] be a differentiable map given by s -^ t{s).
The curve B: [c, d] -^ M given by B{s) = C{t{s)) satisfies
B'{s) = t'{s)C'{t{s)).
Thus if t'{s) > 0 for all s,
/"' (B'is), a^Bw) ds= f' (C'it), coc(o) dt.
J c J a
Thus a linear differential form is something we can integrate over "oriented"
curves of M and is independent of the parametrization. In this chapter we shall
introduce objects which can be integrated over "oriented A:-dimensional surfaces"
of M and study their properties.
1. EXTERIOR DIEEERENTIAL EORMS
We defined a linear differential form to be a rule which assigns an element of
T*(M) to each x^M. We can regard r*(M) d.^ a^{T:,{M)). In view of this,
we make the following generalization of this definition. By an exterior
differential form of degree q on M we mean a rule which assigns an element of
Gf^i^TxiM)) to each x E M. If co is an exterior form of degree q and (11, a) is a
chart, then, since a identifies each Tx{M) with V for x ^U, we obtain an
(i^(F)-valued function, a;«, on a{U) defined by
co.W(?i, . . . , ?2) = a;(a;)(?\ • • • , ?'^) li v = a{x) and ^\ . . . ^ ^^ ^ Tx{M).
It is easy to write down the transition laws. In fact, if {W, fi) is a second
chart, we have
or, since ^^ = /^oa-i(a:(x))(^a) for ^ E T^i^l), we see that
cOaW(?i, . . . , ?2) = co^C^ ° «~'W)(/^o«-K^)?L . . . , /^o«-i(.)^^). (1.1)
In order to write (1.1) in a less cumbersome form, we introduce the following
notation. Let Vi and V2 be vector spaces, and let l\ Vi —> F2 be a linear map.
429

430 EXTERIOR CALCULUS 11.1
We define a^(/) to be the linear map of a^(T"2) -^ Gi^iVi) given by
for all IV E Gi^{V2) and vi, . . . , Vp E Fi. Note that under the identification of
anF)withF* the map a^(/) coincides with the map /*: Vt -^ f! . Note also
that if ici E a^(F2) and 102 E a^(F2), then
a^(0^ri A a^(/)?r2 = a^+^(/)06^i a 102)^ (1.2)
This follows directly from the definitions. Also, if h'. Vi -^ V2 and I2: V2 -^ F3,
then
ci^(/2 ° h) = a'\h) ° ^^(y. (1.3)
It is clear that if / depends differentiably on some parameters, then so does
a^{l) for any p.
We can now write (1.1) as
o:M - Gi'(J^oa-^(v))c^^(^ o a-\v)). (1.10
It is clear from (l.F) that it is consistent to require that co^ be a smooth
function. We therefore say that co is a smooth differential form if all the functions
coa are C"^ on a(U) for all charts (U, a). As usual, it suffices to verify this for all
charts in an atlas. We let /\^{M) denote the space of all smooth exterior forms
of degree q.
Let wi E A^(^) and a;2 E A'^(^)- We define the exterior (p + g)-form
coi A cjJ2 by
(coi A o)2)(x) = a;i(x) A co2Cr) for all x E M.
It is easy to check that coi A 002 is a smooth {p -f g)-form. We thus get a
multiplication on exterior forms. To make the formalism complete, it is convenient
to denote the space of differentiable functions on M by /\^(il/) and to denote
the product of a function / and a p-form co by /co or / A co. This product is
given by
(/ A a;)(x) = (/a;)(x) -= f(x)o:(x) for all xgM.
We have thus defined, for all 0 < p < ?i and 0 < g < ??, a multiplication
sending coi e A^'W and a;2 E A^(^) ii^to coi A a;2 E A'''^^(^) (where
coi A a;2 = 0 if p + g > n = dim M). The rules for the A-product on
antisymmetric tensors carry over and thus, for instance,
coi A a;2 = (-l)^^a;2 A coi if coi E A''(^) and a;2 E A^(^),
coi A (002 A a;2) = (coi A a;2) A coa,
^1 A (002 + a;3) = coi A a;2 + coi A cx^s,
and so on.
Let Ml and ikf2 be different iable manifolds, and let (p: Mi -^ M2 be a
differentiable map. For each co E A'^(^2) we define the form (^*a; E A^(^i) by
cpMx) = a^(<^*.)(co(<^(x))). (1.4)

11.1 EXTERIOR DIFFERENTIAL FORMS 431
It is easy to check that <p*cu is indeed an element of /\^(Mi), that is, it is a
smooth g-form. Note also that (7.5) of Chapter 9 is a special case of (1.4)—the
case g = 1. (If we make the convention that 6i^(l) = id, then the case g = 0
of (1.4) is the rule for pullback of functions.)
It follows from (1.4) that <p* is linear, that is,
<p*(a;i + C02) = <P*(coi) + <P*(co2), (1.5)
and from (1.2) that
<p*(a;i A 002) = <P*(coi) A <P*(co2). (1.6)
If <p is a one-parameter group on a manifold M with infinitesimal generator
X, then we can show that the
exists for any co E /\^(M). The proof of the existence of this limit is
straightforward and will be omitted. We shall derive a useful formula allowing a simple
calculation of Dx^ in Section 3.
Let us now see how to compute with the /\^{M) in terms of local coordinates.
Let {U, a) be a chart of M with coordinates x^, . . . , x'^. Then dx'' G /\^{U)
(where by /YiU) we mean the set of differentiable g-forms defined on U).
For any ii, . . . , iq the form dx''^ A - - - A dx""^ belongs to /\^{U), and for every
X G U the forms
{(dx'' A • • • A dx''^)(x)}i^<_...^i^
form a basis for (l^(Tx(M)). From this it follows that every exterior form co
of degree g which is defined on U can be written as
^ = 12 «^•l ^, ^^'' A • • • A dx'\ (1.7)
*1 < • • • < iq
where the a's are functions; that is,
^W = Z) «^l zM)(dx'' A • • • A dx''){x)
t'l < • • • < iq
for all X G U. It is easy to see that co G A\^{U) if and only if all the
functions a^j i are (7°°-functions on U.
If (W, 13) is a second chart with coordinates y^, . , . , y'^ and
cu= Zbj, j^dy^'^ A ••• A dy^'% (1.8)
then it is easy to compute the transition law relating the b's to the a's on U nW.
In fact, on U n W we have
dy' = Y.%dx\ (1.9)
where y^ = y\x^, . . . , :r^). Then all we have to do is to substitute (1.9) into
(1.8) and collect the coefficients of dx''^ A • • • A dx''^. For instance, if g = 2,

432
EXTERIOR CALCULUS
11.1
then we have
h<J2
ii<y2 V"-^
dx
;;d.")A(£d.^+...+£d.").
If we collect the coefficients of c/x"^ A dx''^ (remember the A-multiplication in
anticommutative), we get
CO
z
V h (^_yl^_y!l_^_y!l^_yl
dx'' A dx
,.^2
Thus
Z ^iiy2det
*'^1^2 i^^ ^JlJ2 '
J1<J2
^11
dx^i
dx^2
dx^i
dy^2
(1.10)
Although (1.10) looks a little formidable, the point is that all one has to
remember is (1.9) and the law for A-multiplication. For general q the same argumenl
gives
«^•l..■.,^•, = Z ^J\ y, det
dx'^i
dxH
dx^i
dy^
dx^Q.
(l-ll)
The formula for pullback takes exactly the same form. Let ip\ Mi —> 71/-.
be a differentiable map, and suppose that {11, a) and (TF, fi) are compatible
charts, where X , . . . , j^ ai e the coordinates of (U, a) and ?/\ . . . , ^y"" are those
of (TF, fi). Then we get that 2/' ° </^ are functions on U and can thus be written as
y^ o ^ = y^\x\ . . . , x''').
Since (^* dy^ = (:/(?/-^ o (^)^ we have
^ aa:» '^•^
(1.12)
If
Jl<---<Jq
then, by (1.5) and (1.6),
/M = Z (hu...jq ° <^)(<^*^^'0 A • • • A (<pUy''^). (i.i:;)
^l<-"<^g
The expression for (1.13) in terms of the dx's can be computed by substitutini;
(1.12) into (1.13) and collecting coefficients. The answer, of course, will look

11.2 THE INTEGRATION OF EXTERIOR DIFFERENTIAL FORMS
just like it did before. If
<p*(cu) =
then the a's are given by
433
A dx^
o^H H = S (^;i ;, ° ^) det
Jl<-'-<Jq
dxH
.dx\
dx^i
dxH,
(1.14)
Again, we emphasize that there is no need to remember a complicated
looking formula like (1.14); Eqs. (1.5), (1.6), and (1.12) (and of course the rules
for A -multiplication) are sufficient. In many cases, it is much more convenient
to do the substitutions directly than to use (1.14).
2. ORIENTED MANIFOLDS AND
THE INTEGRATION OF EXTERIOR DIFFERENTIAL FORMS
Let M be an n-dimensional manifold. Let {U, a) and (W, ^) be two charts on M
with coordinates x^, . . . , x'^ and 2/\ . . . , 2/^. Let cu be an exterior differential
form of degree n. Then we can write
and
0) = a dx^ A • • • A dx^ on U
oj = b dy^ A • • • A dy^ on W,
where the functions a and b are related on U D W hy (1.11), which, in this case
(q = n), becomes
'V ... dy^
dx^ dx^
a = b det
dy^
dx""
dx"".
or
aa{ci{x)) = bfi{l3 o a ^{a{x))) det Jfioa-^(a{x))
or, finally,
aa(v) = hifi o a-\v)) det J^oa-^{v) for v G a{U D W).
If p is a density on M, then the transition laws for p^ are given by
Pa{v) = Pfi{l3 o a:-^(z;))|det Jfioa-'{v)\'
(2.1)
(2.2)
Note that (2.2) and (2.1) look almost the same; the difference is the absolute-
value sign that occurs in (2.2) but not in (2.1). In particular, if ([/, a) and
(W, 13) were such that det J^oa-^ > 0, then (2.2) and (2.1) would agree for this
pair of charts.

434 EXTERIOR CALCULUS 11.2
This leads us to the following definition: An atlas a of ikf is said to be
oriented if for any pair of charts (f/, a) and {W, 13) of d we have
det J^oa-' (oc(x)) > 0 for all x eU n W.
There is no guarantee that there exists an oriented atlas on a given manifold M.
In fact, it is not difficult to show that there does not exist an oriented atlas on
certain manifolds. (An example of a manifold possessing no oriented atlas is
the Mobius strip.)
We say that a manifold M is orientahle if it has an oriented atlas.
Let M be an orientable manifold, and let di and (^2 be two oriented atlases.
We say that Gii and (I2 have the same orientation, and write Gii '^ (^2, if di U (^2
is again an oriented atlas. To say that di '^ (^2 meatis that for any (U, a) G Gii
and any (W, (3) E (^2 we have
detJ^.a-'(v) > 0 on a(U n W).
It is clear that '^ is an equivalence relation. An equivalence class of oriented
atlases is called an orientation of M. An orientable manifold, together with a
choice of orientation, will be called an oriented manifold. We shall denote an
oriented manifold by M. That is, M is a manifold M together with a choice
of orientation. Thus an oriented one-dimensional manifold has a preferred
direction at each point (Fig. 11.1); an oriented two-dimensional manifold has a
notion of clockwise versus counterclockwise direction (Fig. 11.2); and at any
point of an oriented three-dimensional manifold we can distinguish between
right- and left-handedness.
Fig. 11.1 Fig. 11.2
In general, let M be an oriented manifold, and let {U, a) be a chart of M
with coordinates x^, . . . , x'^. We say that (U, a) is a positive chart if J^oa~'^ > 0
for any chart {W, 13) belonging to any oriented atlas defining (i.e., belonging to)
the orientation. (It suffices to check this, of course, for all {W, ^) belonging to
one fixed atlas defining the orientation.) Note that if U is connected, then if
(U, a) is not positive, then the chart (U, a^), where
a\x) = <-v\v^,. .. ,v''> if a{x) = <v\v^,. . . .v'^y,
is a positive chart.
We shall say that (U, a) is a negative chart if det J^oa~^ < 0 for all (IF, fi)
belonging to an atlas defining the orientation. (Thus, if U is connected, then
(f/, a) must be either positive or negative.)

11.2 THE INTEGRATION OF EXTERIOR DIFFERENTIAL FORMS 435
We now return to our initial observation comparing (2.1) with (2.2).
Proposition 2.1. Let M be an oriented n-dimensional manifold. We can
identify exterior forms of degree n with densities by sending the form co
into the density p^j where for any positive chart {U, a) with coordinates
:r\ . . . , :r^, the function p^ is determined by
CO = Pa(«(•)) dx^ A - - - A dx" on U. (2.3)
Another way of writing (2.3) is
a;(a/axS ..., a/ax^) = p(a/ax\ ..., a/ax^). (2.3')
In other words, \i o^ = a dx^ A • • • A dx"^ on U, then Pa{v) = a«. That p"^
is really a density follows from the fact that (2.2) reduces to (2.1) for all pairs
of charts belonging to a positive atlas.
It is clear that this identification is additive,
P^l+^2 = p^l +p^2^ (2.4)
and that for any function,
P^" = fp""- (2.5)
Furthermore, if a;(x) = 0, then p'^ = 0 at x. By the support of a differential
form we mean, as usual, the closure of the set of x for which a;(x) 9^ 0. We say
that an n-form co is locally absolutely integrable if the density p"^ is locally
absolutely integrable. Note that to say that co is locally absolutely integrable
means that for any chart {U, a), with coordinates x\ . . . , x^ of some atlas d, if
bi = a dx^ A • • • A dx'^ on [/,
then the function a^ = a o a~^ is an absolutely integrable function on a{U).
Let T(M) denote the space of absolutely integrable ?i-forms of compact support.
It is clear that r(M) is a vector space and that /co E T(M) if / is a (bounded)
contented function and co E r(M). As a consequence of Proposition 2.1 and
Theorem 3.1 of Chapter 10, we can state:
Theorem 2.1. Let M be an oriented manifold. There exists a unique linear
function J on T(M) satisfying the following condition: If supp co C ^^
where {Uj a) is a positive chart with coordinates x\ . . . , x^, and if co =
adx^ A • • • A dx^^y then
l-L
a„. (2.6)
Observe that we can write
for all CO e T{M). (2.7)
h = 1^'
The recipe for computing Jco is now very simple. We break co up into small
pieces such that each piece lies in some U. (We can ignore sets of content zero

436
EXTERIOR CALCULUS
11.2
in the process.) If supp o: C U^ and if (C/, a) is a positive charts we express co as
o: = adx^ A • • • A dx"^.
And if a is given as a = aa(:r\ ... , x^), we integrate the function a^ over W^.
The computations are automatic. Thus one point that has to be checked is that
the chart (U, a) is positive. If it is negative, then Jco is given by —jaa-
Let Ml be an oriented manifold of dimension q, let <^:Mi -^ M2 be a
differentiate map, and let co E /\^(M2). Then for any contented compact set
A C Ml the form e^<^*(a;) belongs to T{Mi), so we can consider its integral.
This integral is sometimes denoted by J^(^A) co; that is, we make the definition
/ CO ^ / eA(p CO.
J(p{A) JMi
(2.8)
If we regard (p{A) as an "oriented g-dimensional surface" in M2, then we see
that the elements of /\^(^2) are objects that we can integrate over such
"surfaces". (Of course, if g = 1, we say "curves".)
C{c)
C{[a, h]) ~
Fig. 11.3
Fig. 11.4
Let us illustrate by some examples. Suppose that M2 = ^'^, and let A C 11^^
be the interval a < t < h. Let x^, . . . , x'^ he the coordinates of W^, and let
CO = a^dx^ + • • • + a^dx^. We regard R^ as an oriented manifold on which
the identity chart is positive (and its coordinate is ^). If C: R^ —> R^ is a
differentiate curve (Fig. 11.3), then
/ CO = / e[a,6]C*(co)
yC([a,6]) J
= / {C'{t),c^)dt. (2.9)
J a
From this last expression we see that C does not have to be differentiable
everywhere in order for /c([a,6]) co to make sense. In fact, if C is differentiable
everywhere on R except at a finite number of points, and if C'{t) is always
bounded (when regarded as an element of R^), then the function (C'(-), co) is
defined everywhere except for a set of content zero and is bounded. Thus

11.2
THE INTEGRATION OF EXTERIOR DIFFERENTIAL FORMS
437
C*(a;) is a contented density and (2.9) still makes sense. Now the curve can
have corners. (See Fig. 11.4.)
It should be observed that U o: = df (and if C is continuous), then
f df= f d{foC)= f' ifocy
•/C([a,6]) •/C([a,6]) Ja
= /(C(6)) -/(C(a)). (2.10)
In this case the integral depends not on the particular curve C but on the end-
points. In general, jc ^ depends on the curve C. We will obtain conditions for
it to be independent of C in Section 5.
In the next example let M2 = U^ and Mi = U CU^ where (u, v) are
Euclidean coordinates on U.^ and x, y, z are Euclidean coordinates on R^. Let
oi = P dx A dy -\- Q dx A dz -\- R dy A dz
be an element of /\^([R^). If (p: U -^ U^ is given by the functions x{u, y),
y{u, v), and z(u, v), then for A C U,
I CO = / e^<^*a; = / eA(p*(P dx A dy + Q dx A dz + R dy A dz)
= IA l^^ ° ^^ W a^ " a^ W + ^^ ° ^^ \TuVv -Q^Vv)
We conclude this section with another look at the volume density of Riemann
metrics, this time for an oriented manifold. If M is an oriented manifold with a
Riemann metric, then the volume density a corresponds to an n-form 12. By
our rule for this correspondence, if ([/, a) is a positive chart with coordinates
:r^ . . . , :r^, then
^= adx^ A " ' A dx"",
where, by (4.1) of Chapter 10, a{x) = |det {gij)\^'^ is the volume in Tx{M) of
the parallelepiped spanned by
dx
t(^)---^(^)-
dx""
Let ei{x)j . . . J en{x) be an orthonormal basis of Tx{M) (relative to the scalar
product given by the Riemann metric). Then
|det {gij)\
1/2
det
\h'V.
|det A|,
where A = (d/dx\ ey) is the matrix of the linear transformation carrying
ey -^ d/dx^. If o:^(x), . . . , a;^(:r) is the dual basis of the e^s, then
c^\x) A • • • A a;^(:r) = det A dx\x) A • • • A g?x^(x).

438 EXTERIOR CALCULUS ll..^
Now o:^{x), . . . , a;''(x) can be any orthonormal basis of T*(M). [T*(M) has a
scalar product, since it is the dual space of the scalar product space Tx(M).]
We thus get the following result: If a;\ . . . , co^ are linear differential forms such
that for each x G M, cx:^(x), . . . , o:'^(x) is an orthonormal basis of T*{M), then
n = zbco^ A • • • A a;^
We can write
12 = co^ A • • • A co^ (2.11)
if we know that co^ A • • • A co^ is a positive multiple of cZ:r^ A • • • A dx'\
Can we always find such forms a;\ . . . , co^ on [/? The answer is "y^s": we can
do it by applying the orthonormalization procedure to clx^, ... , dx'^. That is,
we set
1 _ dx^ where \\dx^\\{x) = \\dx\x)\\ > 0
\\dx^\\ is a C°°-function on U,
2
CO
dx^ — (dx'^, o:^) 0)^
px2 - (d:r2^a;i)a;i|
The matrix which relates the dx's to the co's is composed of C°°-functions, so that
the CO* G /\^{U). Furthermore, it is a triangular matrix with positive entries
on the diagonal, so its determinant is positive. We have thus constructed tlu^
desired forms co^, . . . , co^, so (2.11) holds. For instance, it follows from
Eq. (9.10), Chapter 9, that dd, sin 6 d<p form an orthonormal basis for Tx{S^) at,
all X G S'^ (except the north and south poles). If we choose the orientation on S'^
so that 6, (p form a positive chart, then the volume form is given by
12 = sin ^ dd A d<p.
3. THE OPERATOR d
With every function / we have associated a linear differential form df. We can
thus regard d as a map from /\^(M) to /\^(M). As such, it is linear and satisfies
c/(/l/2)=/2^/l+/l^/2.
We now seek to define a d: A^W -> A^"^^(^^) for /c > 0 as well. We shall
require that d be linear and satisfy some identity with regard to multiplication,
generalizing the above formula for g?(/i/2). The condition we will impose is that
cZ(coi A C02) = do^i A C02 + ( —l)^coi A c/co2
if coi is a form of degree p. The factor (—1)^ accounts for the anticommutativity
of A. The reader should check that d is consistent with this law, at least to the
extent that g?(coi A C02) = (—1)^^ g?(co2 A coi) if coi is of degree p and C02 is of
degree q.
We are going to impose one further condition on d which will uniquely
determine it. This condition (which lies at the heart of the matter) requires

11.3 THE OPERATOR d 439
Fig. 11.5
some introduction. Let / be a differentiable function, and let C: / —> ikf be a
differentiable curve. For any points a,hGl, the fundamental theorem of
the calculus implies that
fidb)) -/(C(a)) = I ^^dt= j^ C*df.
We can regard b and a (with i signs attached) as the "oriented boundary"
of the interval [a, h]. Let us make the convention that "integrating" an element
of A^(p) is just evaluating the function at the point p. As such, the equation
above says that the integral of the "pullback" of / over the "boundary", that is,
f(b) — f{a), equals the integral of the "pullback" of df over [a, h]. In some sense,
we would like to be able to say that if co is a form of degree /c, then the integral
of the "pullback" of co over the /c-dimensional boundary" of a (/c + l)-dimen-
sional region is equal to the integral of the pullback of do: over the (/c + 1)-
dimensional region. Without trying to make this requirement precise, let us see
what it says for the case where k = 1 and the region is a triangle in the plane.
Let <^ be a smooth map of some neighborhood of the triangle A C 11^^ into M,
and let the vertices of A be mapped by cp into x, y, and z (see Fig. 11.5). The
boundary of A consists of three curves (segments) Ci, C2, and C3 (with the
proper orientations). Let co be a linear differential form on M. We would then
expect that
I cp do) = I Cicp CjJ -{- I C2<P OJ -]- I C^cp CjJ.
If CO = dfj then the three integrals on the right become (by the fundamental
theorem of the calculus) /(y) - /(x) + /(z) - /(y) + /(x) - /(z) = 0. Thus
jcp* d(df) = 0. Since the triangle was arbitrary, we expect that
didf) = 0.
We now assert:
Theorem 3.1. There exists a unique linear map d: /\^(M) -^ /\^'^^(M)
such that on /\^ it coincides with the old d and such that
d{c^i A coa) = dc^i A coa + (-l)^a;i A do:2 if coi G A^'W (3-1)
and
d(df) = 0 if /gA^^). (3.2)

440 EXTERIOR CALCULUS ll.;')
Proof. We first establish the uniqueness of d. To do this we observe that (3.1)
implies that d is local, in the sense that if co = co' on some open set U, then
do: = (ico' on U. In fact, let W be an open set with W C U, and let <^ be a C^-
function such that (p(x) ^ 1 for x gW and supp ^ <ZU. Then <^a; = <^a;'
everywhere on M, and thus d(<^a;) = d(<^a;'). But, by (3.1), d(<^a;) = <^ c^co + d<^ A
CO = c/co on TF, since <^ ^ 1 and d<^ = 0 there. Thus dw = do:' on W. Since 14'
can be arbitrar}^, we conclude that c/co = do:' on U.
Let (r/, a) be a chart with coordinates a;\ . . . ,x'^. Every co G A^(^-^)
can be written as
CO = X) ^*i, .,u<^^^'^ A • • • A di:'^ on U.
Now [by induction on /c, using (3.1) and (3.2)] d{dx^^ A • • • A dx^^) = 0.
Thus (3.1) implies that
dco = L ^«n. .t, A dx'' A • • • A c^x''^" on U. (3.3)
Equation (3.3) gives a local formula for d. It also shows that d is unique. In
fact, we have shown that there is at most one operator d on any open subset
OcM mapping A^(0) -^ A^'^^iO) and satisfying the hypotheses of the
theorem (for 0). On the set 0 n U it must be given by (3.3).
We now claim that in order to establish the existence of d, it suffices to show
that the d given by (3.3) [in any chart ([/, a)] satisfies the requirement of the
theorem on /\^{U). In fact, suppose we have shown this to be so. Let d be
an atlas of M, and for each chart (U, a) G Gi define the operator da'- A\^(U) -^
^k+i^fj^ by (3.3). We would like to set do: = dao: on U. For this to be
consistent, we must show that daO: = d^o: on U n W if {W, P) is some other chart.
But both da and dp satisfy the hypotheses of the theorem on U CiW, and they
must therefore coincide there.
Thus to prove the theorem, it suffices to check that the operator d, defined
by (3.3), fulfills our requirements as a map of /\^{U) —> /\^'^^{U). It is
obviously linear. To check (3.2), we observe that
so
d(df) = Z d (II) A d.^ = Z {S^d.^ A d.^
= 0
by the equality of mixed partials.
Now we turn to (3.1). Since both sides of (3.1) are linear in coi and a;2
separately, it suffices to check (3.1) for coi = a dx'^^ A • • • A dx'^^ and a;2 =

11.3 THE OPERATOR d 441
b dx^^ A • • • A dx^''. Now coi A 002 = ah dx^^ A • • • A dx^^ A dx^'^ A • • • A c?:^-^''
and d(ah) = h da -\- a dh; therefore,
d(o:i A 0:2) = h da A dx''^ A • • • A dx'^p A dx^'^ A • - • A dx^"
+ adh A dx'' A • • • A dx'^ A dx^'' A • • • A dx^'%
while
G^coi A a;2 == (c^a A dx'"^ A • • • A G?:r'^) A (b dx^'"- A • • • A c^:r-^«)
=. hda A dx'"^ A • • • A G?:r'^ A G?:r-^'^ A • • • A dx^'''
and
coi A do:2 = {a dx'' A • • • A dx'^) A {dh A dx^'' A • • • A dx^''')
= (-l)^a G?6 A dx'' A • • • A g?x^^ A dx^'' A • • • A c^x^'^
so we see that (3.1) holds. This proves the theorem. D
We can draw a number of important corollaries from Eq. (3.3).
First of all, it follows immediately that for co G /\^(M), for any k, we have
d(dc^) = 0. (3.4)
(Remember we merely assumed it for /c = 0.)
Secondly, let <^: Mi —> M2 be a differentiable map. Then for co G /\^(M2)
we have
d(p*o: = <^* G^co. (3.5)
To check (3.5), it suffices to verify it for any pair of compatible charts.
But if X , . . . , ^ are coordinates on M2 and, locally,
^ = E «ti h dx'' A • • • A dx'^,
we have
d<p*o) = d{Y. (P*(ai^ if)(p* dx'' A • • • A <^* dx'')
= E ^<^*(azi ij) A c^<^V^ A • • • A dcp^'x''
= Y. <P* dai^ ik A G?<^V^ A • • • A G?(^V^
= <^*(E ^«M 4 A dx'' A • • • A G?:r^"^)
In particular, if X is a vector field on M, we conclude that
Dx dc^ = d(Dxc^). (3.6)
EXERCISES
3.1 Compute d of the following differential forms.
a) y = Hi (—iy~''^Xidxi A • • • A t^a;i_i A dxi^i A - • • A dxn
b) r~"7, where 7 is as in (a) and r = {a;? + • • • + o;^} ^^^
c) E Pt- ^5*'
d) sin (x^ + 2/^ + ^^) {x dx-\- y dy -\- z dz)

(2 2 '
442 EXTERIOR CALCULUS 11 1
Let F be a vector space equipped with a nonsingular bilinear form and an orientation
Then we can define the ^-operator as in Chapter 7. Since we identify the tangent spac*
Tx(V) with V for any x G V, we can consider the ^-operator as mapping /\^{V)
/\"~^(F). For instance, in IR^, with the rectangular coordinates (x, y), we have
*c?a: = d?/, *d?/ = —dx,
and so on.
3.2 Show that
dyy
for any function /on IR^.
3.3 Obtain a similar exi)ression for d ^ d m [R" with its usual scalar jiroduct. (Rec.'ili
that
*dx^ A ' ' • A dx^ = dx^+^ A • • • A dx""
and, more generally,
*dx^i A • • • A dx'^ = ± dx^' A • • • A dx^^-^,
where (^l, . . . , ik,ji, • • • ,jn-k) is a permutation of (1, ... , n) and the ± is the si,<;ii
of the permutation.)
3.4 Let X, y, z, t be coordinates on IR^. Introduce a scalar product on the tangent
space at each point so that
(dx, dx) = (dy, dy) = {dz, dz) = 1,
{dx, dy) = (dx, dz) = (dx, dt) = (dy, dz) = (dy, dt) = (dz, dt) = 0,
and
(c dt, c dt) = —1,
where c is a positive constant. Let the two-form co be given by
CO = c(Ei dx A dt+ E2 dy A dt+ E3 dz A dt)
-\- Bi dy A dz+ Bo dz A dx ^ B3 dx A dy.
Let the three-form 7 be given by
T = p dx A dy A dz — (Ji dy A dz^ J2 dz A dx + /s dx A dy) A dt.
Write the equations
dco = 0, d * CO = 47rT
as equations involving the various coefficients and their partial derivatives.
4. STOKES' THEOREM
In this section we shall prove a theorem which will be a far-reaching generalize
tion of the fundamental theorem of the calculus of one variable. It should,
perhaps, be called the fundamental theorem of the calculus of several variables
We first make some definitions.
Let D be a domain with regular boundary in a manifold M. We recall
(page 419) that each point of M lies in a chart ([/, a) which is one of three types

11.4
STOKES THEOREM
443
Let ([/, a) and (W, fi) be two charts of M of type (iii). Then, as on page 420,
the matrix of J^oa~^ is given by
dy^ dy^~
dy
-1
a*/"-
3x1
0
0 0
and so
det J;3o
£--
dy
0
^ n
dx'^
dy'
ax'^-i
T-^ X det /(^rai))o(arai))-i.
ax
(4.1)
Furthermore, 2j''(x\ . . . , x"") > 0 if x"" > 0, since a{U nW) D {wv'' > 0} =
a{U nW n int D). Thus dy^'/dx'' > 0 at a boundary point where Xn = 0.
Now suppose that M is an oriented manifold, and let D C M be a domain
with regular boundary. We shall make dD into an oriented manifold. We say
that an atlas Gi is adjusted if each ([/, a) G a is of type (i), (ii), or (iii) and, in
addition, if each chart of Gi is positive.
If dim M > 1, we can always find an adjusted atlas. In fact, by choosing
the U connected, we find that every {Uj a) is either positive or negative. If
([/, a) is negative, we replace it by ([/, a), where Xa^ = —x^.
If dim M = 1, then dD consists of a discrete set of points (which we can
regard as a "zero-dimensional manifold")- Each x G dD lies in a chart of type
(iii) which is either positive or negative. We assign a plus sign to x if any chart
(and hence all constricted charts) of type (iii) is negative. We assign a minus sign
to X if its charts of type (iii) are positive. In this way we "orient" dDj as shown
in Fig. 11.6.
_ D
+
Fig. 11.6
If dim M > 1, we choose an adjusted (oriented) atlas on M. It then follows
from (4.1) and the fact that dy^'/dx'' > 0 that
detJ0^dD)oia\-dD)~^ > 0.
This shows that (L' \ dD, a \ dD) is an oriented atlas on dD. We thus get an
orientation on dD. This is not quite the orientation we want on dD. For reasons
that will soon become apparent, we choose the orientation on dD so that

444 EXTERIOR CALCULUS 11.4
{U \ dD,a\ dD) has the same sign as ( — 1)''. That is, (f/ \ dD,a\ dD) is a
positive chart if n is even, and we take the orientation opposite to that
determined by (^ f dDj a \ dD) if n is odd. We can now state our main theorem.
Theorem 4.1 {Stokes^ theorem). Let M be an n-dimensional oriented
manifold, and let D C M be a domain with regular boundary. Let ^D denote
the boundary of D regarded as an oriented manifold. Then for any
CO G /\^~^(M) with compact support we have
[ L*o)= f doj, (4.2)
where, as usual^ i is the injection of boundary D into M.
Proof. For n = 1 this is just the fundamental theorem of the calculus.
For n > 1 our proof is almost exactly the same as the proof of Theorem 6.1
of Chapter 10. Choose an adjusted atlas Gi and a partition of unity {qj}
subordinate to d. Since co has compact support, we can write
where the sum is finite. Since both sides of (4.2) are linear, it suffices to verify
(4.2) for each of the summands QjO). Since supp gjO) C U, where ([/, a), we
must check the three possibilities: (U, a) satisfies (i), (ii), or (iii).
If {U, a) satisfies (i), t*co = 0, since supp co f) dD = 0, and
dco = cd do) = 0,
Jd Jm
since D f) supp co = 0. Thus both sides of (4.2) vanish.
If ([/, a) satisfies (ii), the left-hand side of (4.2) vanishes. We must show
that the same holds for the right-hand side. Let x^, . . . ^ x^ he the coordinates
on ([/, a), and write
gjO) = ai dx^ A • • • A dx"^ + ^2 dx^ A dx^ A • • • A dx"^ -f . . .
+ cin dx^ A ' ' ' A dx'^~^.
Then
dgjo) = E (-1)'-' ^^,dx' A • • • A dx"",
and thus
f dgjc. = Z i-ir~' I
daj
Since gjco has compact support, the functions ai have compact support, and
we can replace the integral over IR^ by the integral over D-R, where R =
<Ry . . , J R> and R is chosen so large that supp ai C (ZJ-r. But writing the
multiple integral as an integral, we get
□^-rI^ = L-i ^'^- • •' ^' • ■ •) - «'<• ■■,-R,---) = o,
since a^(. . . , R, . . .) = ai{. . . , —R, . . .) = 0.

11.4
STOKES THEOREM
445
Fig. 11.7
We now examine J dgjO) in case (iii). The argument proceeds exactly as
before^ except that we must compute la{Uf\D) Bai/dxi instead of ^aiUy (See
Fig. 11.7.)
We can now replace the region of integration by a rectangle of the form
□ <-¥.!??.-s.o> ^or large R. If i < n^ j dai/dxi = 0 as before. If i = n, we get
so that
Now since ^^ == 0 on f/ n dD^ we see that t* dx'^ == 0. Thus
i*co = (£*an)(t*G?x^) A • • • A {L*dx''-^),
or if (by abuse of notation) we regard x\ . . . ^ x^~^ as the coordinates of
{U ldD,a\ dD), we get
t*co = a(-, •, . . . , •, 0) G?x^ A • • • A dx''~\
In view of the choice we made for the orientation of dDj we conclude that
f t*«=(-ir/" «„(•,-,...,-, 0).
iaD Jun — l
This completes the proof of the theorem. D
Theorem 4.1, like the divergence theorem^ is not sufficiently broad for us to
apply to more general domams. For this purpose, we will again use the notion
of a domain with almost regular boundary.
We have already seen that the set of a: G dD having a neighborhood of type
(iii) forms a differentiable manifold. (Recall that these points need not exhaust
all of dD). Similarly, if M is an oriented manifold, then this collection of points
becomes an oriented manifold (with (—1)^ times the induced orientation, as
before). By abuse of language we shall denote this oriented manifold by ^D.
Thus dJ} is an oriented manifold which, as a set, is not dD but only the "regular"
points of dDj that is, the points of dD.

446 EXTERIOR CALCULUS 11.4
Theorem 4.2 (Stokes^ theorem). Let M be an n-dimensional oriented
manifold, and let D C M be a domain with almost regular boundary.
Let ^D be as above, and let l be the injection of ^D —> M. Then for any
CO G /\^~^{M) with compact support we have
f t*co = f do). (4.2)
Proof. The proof proceeds as before. We choose an adjusted atlas and a
partition of unity {gj} subordinate to the atlas. We write co = J2 dj^ and now
have four cases to consider. The first three cases have been handled already.
The new case is where
QjO) = ^ aj dx^ A • • • A dx^ f\ • • - f\ dx^,
3
where the "^ indicates that dx^ is to be omitted, has its support contained in U,
where ([/, a) is a chart of type (iv). By linearity, it suffices to verify (4.2) for
each summand on the right, i.e., for
aj dx^ A • • • A dx^ A • • • A dx'^.
Now L'^{ajdx^ A • • • A & A • • • A dx'^) = 0 unless j > /c, since dx^
vanishes on the piece of ^D n U whose image under a lies in Zfp. If j < p,
then all these dx^ occur, and thus t*(ay g?x^ A • • • A dx^ A • • • A dx^) = 0.
If j > p, then i'^iaj dx^ A • • • A dx^' A dx'^) vanish everyivhere except on the
portion of ^D which maps under a onto H^.
On the other hand,
d{aj dx^ A'" Adx' A '" A dx"") - (-1)'"^ ^:^.dx^ A - - A dx"".
We can evaluate the integral //> by integrating over the rectangle
I—j<ii!, . . . , R R>
I—l<--ii!,...,—ii!,0,...,0,0>
(where the —iJ's extend through the (/c — l)th position). Integrating first
with regard to x\ we obtain
( diajdx^ A ' ' ' Adx^ A ' ' ' A dx"") = {—lyf .aj.
Jd Jh^
On the other hand, the orientation on Hj is such that this integral has the
sign necessary to make (4.2) hold. This proves Theorem 4.2. D
As before, we can apply Theorems 4.1 and 4.2 to still more general domains
by using a limit argument. For instance. Theorem 4.2, as stated, does not apply
to the domain D in Fig. 11.8, because the curves Ci and C2 are tangent at P.
It does apply, however, to the approximating domain obtained by "breaking
off a little piece" (Fig. 11.9), and it is clear that the values of both sides of (4.2)
for D' are close to those for D. We thus obtain (4.2) for D by passing to the
limit. As before, we will not state a more general theorem covering these cases.
It will be clear in each instance how to apply a limit argument.

11.4
STOKES THEOREM
447
Fig. 11.8
Fig. 11.9
Since the statement and proof of Stokes^ theorem are so close to those of the
divergence theorem, the reader might suspect that one implies the other. On
an oriented manifold, the divergence theorem is, indeed, a corollary of Stokes^
theorem. To see this, let Q be an element of /\^{M) corresponding to the
density p. If X is a vector field, then the n-form Dx^ clearly corresponds to the
density Dxp = div <X, p>. Anticipating some notation that we shall
introduce in Section 6, let X J 12 be the (n — l)-form defined by
X J i2(f \..., r-') = (~-i)"-^i2(f \..., e~\ X).
In terms of coordinates, if Q = adx^ A - - - A dx'^, then
X JQ= a[X^ dx^ A'" Adx"" - X^ dx^ A dx^ A - " A dx""
+ h (~1)"~'X^ dx' A "' A dx""-^].
Note that
d{xjQ) = (e^-)^^' a • • • a t^x^
which is exactly the n~form Dx^, since it corresponds to the density DxP =
div <Xy p> . Thus, by Stokes^ theorem,
f L*(XJQ)== f d(XjQ)=- f div<X,p>.
JdB JB JD
We must compare X JQ with the density px on dD, By (2.2) they agree on
everything up to sign. To check that the signs agree, it suffices to compare
Px
with
(toJ aJ=f) - ^ (tei 5?^' ^)
at any x G dD. Now
t*(Z JQ)= (-l)"-iZ" dx' A '" A dx""-'
and, according to our convention, x^, . . . , x^~^ is a positive or negative
coordinate system according to the sign of (—1)^. Thus the two coincide if and only
if X^ is negative, that is.
f t*(Z jn) = f €XPX.
JdB JdD

448 EXTERIOR CALCULUS 11.4
EXERCISES
4.1 Compute the following surface integrals both directly and by using Stokes'
theorem. Let O denote the unit cube, and let B be the unit ball in R^.
a) Sdn X dij A dz+ y dz /\ dx -\- z dx /\ dy
b) SdB x^ dy A dz
c) Jan cos z dx A dy
d) Sdux dy A dz, where
U = [(x,y,z):x> 0, y > 0, z > 0, x^ + y^ + z^ < 1}
4.2 Let a; = yz dx + x dy -f dz. Let 7 be the unit circle in the i)lane oriented in the
counterclockwise direction. Compute fy cj. Let
Ai - {{x,y,z):z = 0, x^ + y^ < 1],
A2 - {(x, y,z):z == 1- x^ - y\ x^ + y^ < 1].
Orient the surfaces A i and .I2 so that^Ai = 8X2 — T. Verify that Jai doo = Jx^ do: —
Jo; b}' computing the integrals.
4.3 Let S^ be the circle and define a; = (l/27r) dS, w^here 6 is the angular coordinate.
a) Let <^: iS^ —> iS^ be a differentiable map. Show that J<^*a; is an integer. This
integer is called the degree of <p and is denoted by deg <p.
b) Let <pt be a collection of maps (one for each t) which depends differcntiably
on t. Show that deg cpo = deg <pi.
c) Let us regard S^ as the unit circle in the complex numbers. Let / be some
function on the complex numbers, and suppose that f{z) 9^ 0 for \z\ = r. Define
ifrj by setting cprjie'^) = f{re'^)/\f{re'^)\' Suppose f{z) = z'\ Compute deg <^r,/
for r ?^ 0.
d) Let/be a polynomial of degree n > 1. Thus
J{z) = anz"" + a„_iz"-i H h ao,
where a„ ?^ 0. Show that there is at least one complex number zo at which/(^o) — 0.
[Hint: Suppose the contrary. Then <Pr,a/an)f is defined for all 0 < r < 00 and deg
(Pr,o/an)f "= const, by (b). Evaluate lim7.=o and lim7.=oo of this expression.]
Let X be a vector field defined in some neighborhood U of the origin in E^, and suppose
that A^(0) = 0 and that X{x) 9^ 0 for x ^ 0. Thus A^ vanishes only at the origin.
Define the map <pr: S^ —^S^ by
ie. _ X(re' )
This map is defined for sufficiently small r. By Exercise 4.3(b) the degree of this map
does not depend on r. This degree is called the index of the vector field X at the origin.
4.4 Compute the index of
a) x-T-+y-T-> b) x- y-r-y c) y- x — •
dx dx dx dy dx dy
d) Construct a vector field with index 2.
e) Show that the index of —A'' is the same as the index of X for any vector field A^.

11.5
SOME ILLUSTRATIONS OF STOKES THEOREM
449
4.5 Let X be a vector field on an oriented two-dimensional manifold, and suppose
that X(p) = 0 for some p G M and that X does not vanish at any other point in a
small neighborhood of p. By choosing an oriented chart mapping p into zero, we get a
vector field on E^ vanishing at the origin. Show that the index of this vector field does
not depend on the choice of charts. We can thus define the index of X at p.
4.6 a) On the sphere S^ let Z be a vector field which is tangent to the meridian
circles everywhere and vanishes only at the north and south poles. What is its
index at each pole?
b) Let Y be a vector field which is tangent to the circles of latitude everywhere
and vanishes only at the north and south poles. What is its index at each pole?
5. SOME ILLUSTRATIONS OF STOKES' THEOREM
As a simple but important corollary of Theorem 4.2, we state:
Theorem 5.1. Let <p: Mi —> M2 be a differentiable map of the oriented
/c-dimensional manifold Mi into the n-dimensional manifold M2. Let co
be a form of degree k — 1 on M2, and let
D C Ml be a domain with almost regular / yq
boundary on Mi. Then we have
/ L*(p*co = / (p*(dco). (5.1)
JdB Jb
Fig. 11.10
Equation (5.1) follows directly from (4.2)
and from the fact that <^*g? = g?<^*.
We can regard the right-hand side of (5.1) as the integral of do) over the
"oriented /c-dimensional hypersurfaces" <^(D). Equation (5.1) says that this
integral is equal to the integral of co over the (/c — 1)-dimensional hypersur-
face (p{dD).
Fig. 11.11
We now give a simple application of Theorem 5.1. Let Cq: [0, 1] —> M and
Ci: [0, 1] -^ M be two differentiable curves with Co(0) = C'i(O) = p and
Co(l) = Ci(l) = q. (See Fig. 11.10.) We say that Co and Ci are (differentiably)
homotopic if there exists a differentiable map <^ of a neighborhood of the unit
square [0, 1] X [0, 1] C U^ into M such that cp{t, 0) = Co(t), cp(t, 1) = Ci{t),
(p{0, s) = p, and <^(1, s) = q. (See Fig. 11.11.) For each value of s we get the
curve Cs given by Cs(t) = <p{ty s). We think of <p as providing a differentiable
"deformation" of the curve Co into the curve Ci.

450
EXTERIOR CALCULUS
11.5
Proposition 5.1. Let Cq and Ci be differentiably homo topic curves, and
let CO be a linear differential form on M with do) = 0. Then
Jci Jc.
(5.2)
Proof. In fact,
/.
^LJ<o.o>
CO =
/n<i.i>
•/LJ<o.o>
do) = 0.
But/a □ is the sum of the four terms corresponding to the four sides of the square.
The two vertical sides (^ = 0 and ^ = 1) contribute nothing, since (p maps
these curves into points. The top gives —jci (because of the counterclockwise
orientation), and the bottom gives Jcq. Thus fco ^ — jci^ = Oj proving the
proposition. D
It is easy to see that the proposition extends without difficulty to piecewise
differentiable curves and piecewise differentiable homotopies. Let us say that
two piecewise differentiable curves, Co and Ci, are (piecewise differentiably)
homo topic if there is continuous map <^ of [0, 1] X [0, 1] —> M such that
i) <^(0, s) = p, (p(l, s) = q;
ii) cpit,0) = Co{t),<p(t,l) = Ci(t);
iii) there are a finite number of points to < ti < - - - < tm such that (p
coincides with the restriction of a differentiable map defined in some
neighborhood of each rectangle [ti, U^i] X [0, 1]. (See Fig. 11.12.)
To verify that Proposition 5.1 holds for the case of piecewise differentiable
homotopies, we apply Stokes' theorem to each rectangle and observe that the
contribution of the interior vertical lines cancel one another.
We say that a manifold M is connected if every pair of points can be joined
by a (piecewise differentiable) curve. Thus IR", for example, is connected. We
say that M is simply connected if all (piecewise differentiable) curves joining the
same two points are (piecewise differentiably) homotopic. (Note that the circle,
S^ is not simply connected.) Let us verify that IR" is simply connected. If Co
and Ci are two curves, let <^: [0, 1] X [0, 1] —> W^ be given by
cp(t, s) = sCo(t) + (1 - s)C,(t).
It is clear that (p has all the desired properties.
()
()
{)
()
Fig. 11.12
Proposition 5.2. Let M be a connected and simply connected manifold,
and let 0 gM. Let co g A\M) satisfy do) = 0. For any xgM let
J(x) = jc CO, where C is some piecewise differentiable curve joining 0 to x.
The function / is well defined and differentiable, and df = o).

11.5 SOME ILLUSTRATIONS OF STOKES' THEOREM 451
Proof. It follows from Proposition 5.1 that / is well defined. If Co and Ci are
two curves joining 0 to x, then they are homotopic, and so Jcq co = Jci co.
It is clear that / is continuous, since
fix) - M = f CO,
JD
where D is any curve joining y to x (Fig. 11.13).
To check that/ is differentiable, let ([/, a) be a chart about x with
coordinates <x\ . . . , x">. Then
S{x\ ...,x' + K...,x^)- f{x\ . . . , :r") = r CO,
Jc
where C is any curve joining p to q, where a{p) = {x^, . . . , x\ . . . , x^), and
where a(q) = (x^, . . . , x^ -{- h, . . . , x^). We can take C to be the curve given
by
aoC(t) = (x\ . . . , x' + ht, . . . , x").
If CO = ai dx^ + ' ' ' + dn dx^, then
I 0) = I hai dt = I ai{x^, . . . , x^ -{- s, . . . , x^) ds.
Jc Jo Jo
(See Fig. 11.14.) Thus
lim i [f{x\ ...,x' + h\..., x^) - f(x\ . . . , x^)] = a\
h-^o ^
that is, df/dx^ = a\ This shows that/is differentiable and that df = co, proving
the proposition. D
Fig. 11.13 Fig. 11.14
We have thus estabhshed that every co E AH^f^") with c/co = 0 is of the
form df. More generally, it can be estabhshed that if 12 G /\^(1R'') satisfies
dQ = 0, then Q = do) for some co G A^~^('I^'')-
*This is not true for an arbitrary manifold. For instance, every co G AH^^)
satisfies dco = 0. Yet the element of angle form (which is, unfortunately,
denoted by dd) is not the d of any function. The fact that d^ = 0 shows that if
Q = do), then dQ = 0. Thus the space d[A^-\M)] C A^(^) is a subspace of
the space ker^- d of elements in /\^(M) satisfying d9. = 0. The quotient space
kerA; d/d[/\!^~^] is denoted by H^{M) and is called the /cth cohomology group of
M. If M is compact, it can be shown that H^ is finite-dimensional. It measures
(roughly speaking) "how many" /c-dimensional holes there are in M.*

452 EXTERIOR CALCULUS 11.6
6. THE LIE DERIVATIVE OF A DIFFERENTIAL FORM
Let Af be a differentiable manifold, and let <^ be a flow on M with infinitesimal
generator X. For any co G /\^(M) we can consider the expression
<^f CO — CO
t
It is not difficult (using local expressions) to verify that the limit as ^ —> 0 exists
and is again an element of /\^(M), which we denote by D^co. The purpose of
this section is to provide an effective formula for computing D^co. For this
purpose, we first collect some properties of Dx- First of all, we have that it is
linear:
i)x(coi + C02) -= Dxo)i + Dxco2. (6.1)
Secondly, we have
(ft {o)i A C02) — coi A C02 = (<^«coi) A (<^«co2) — coi A C02
= (<^*coi) A (<^fco2) — ((^fcoi) A C02
+ i^to^i) A C02 — coi A C02.
Dividing by t and passing to the limit, we see that
Dx{o)i A C02) = (Dxcoi) A C02 + coi A Dxco2. (6.2)
Finally, since <^f d = dcp*, we have
Dxdo) = d{Dxo)). (6.3)
Actually, these three formulas suffice for the computation. If
CO = E (^H,...,ikdx''^ A • • • A dx''\
then
Dxco = E Dx{ai^,...,i,dx'' A • • • A dx'') by (6.1)
= L [{I>xa^,_,i,) dx'' A • • • A dx'' + a,^,...,,,(Z)x dx'') A • • • A dx''
+ • • • + (^ii,...,ik dx" A • • • A {Dx dx")] by repeated use of (6.2)
= L [(^xavi,...,t,) dx'' A • • • A dx'' + «ti,...,4 d(Dxx^) A - - ■ A dx^
+ "■ + a^^_,i^ dx'' A • • • A c/(Z)xx'^)] by (6.3).
Since this expression is rather cumbersome (the d{Dxx') have to be expanded and
the terms collected), we shall derive a simpler and more convenient expression
for Dxo). In order to do this, we make an algebraic detour.
Recall that the operator d: /\^{M) —> /\^+^(M) is linear and satisfies the
identity
(i(coi A C02) = dcoi A C02 + ( —l)^coi A do)2 (6.4)
if coi G /\^{M). More generally, any (sequence of linear) maps 6 of
AHm)-^ A^'^HM)

11.6 THE LIE DERIVATIVE OF A DIFFERENTIAL FORM 453
satisfying the identity
(9(coi A C02) = e'coi A C02 + (-l)^coi A 6*602 (6.4')
and
supp dco C supp cot (5.5)
will be called an antiderivation of the algebra /\(M).
It follows from (6.5) that if coi = CO2 on an open set U, then ^(coi) = 6(0)2)
on U. Now about every x G M we can find a neighborhood U and functions
x^, . . . , x^, so that CO G /\^(M) can be written as
CO = L ai^_,i^ dx'' A • • • A dx'^ on U. (6.6)
Then by repeated use of (6.4') we have
^(co) = L [^(a^, ij A dx'' A • • • A g?x^'^ + a^, ,,^(g?x^O A • • • A g?x^^
+ h (-l)'^~'a,, i^dx'^ A • • • A ^(g?x^^)]. (6.7)
We thus arrive at the important conclusion:
Proposition 6.1. Any antiderivation 6: /\^{M) -> A^"^H^^). /c = 0, . . . , n,
is uniquely determined by its action on /\^(M) and /\^{M). That is, if
6ii(co) = 6'2(co) for all co G A^(^^) and /\\M), then (9i(12) = 6'2(1^) for
^ E A^(-^) for any k.
Now suppose we are given maps
d: /\\M) -> A'(^^) and 6: /WM) -> /WM)
which satisfy (6.5) and (6.4') where it makes sense, that is,
e{hf2) = e{h)U + /i^(/2) and ^(/co) = ^(/) Aico + P(co). (6.
Then any chart {U,a) defines ^: A^(^)-> A^'^'(^) by (6.7). This
gives an antiderivation du on [7, as can easily be checked by the use of the
argument on pp. 440-441. By the uniqueness argument, if (TF, fi) is a second chart,
the antiderivations du and dw coincide on U dW. Therefore, Eq. (6.7) is
consistent and yields a well-defined antiderivation on /\{M). (Observe that we
have just repeated about two-thirds of the proof of Theorem 3.1 for the more
general context of any antiderivation.)
t This condition is actually a consequence of (6.4). In fact, let U be an open set
containing supp a;. Since {U, ilf-supp a;} is an open covering of 71/, we can find a
partition of unity subordinate to it. In particular, we can find a C°°-function ^ which
is identically one on supp a; and vanishes outside U. Then a; — <^a;, so that
(9(a;) - e{ipoi) - (9(<^)Aa;+<^(9(a;).
Thus supp B{oi) C supp a; U supp <p C U. Since U is an arbitrary neighborhood of
supp cj, we conclude that supp d{c^) C supp cj.

454 EXTERIOR CALCULUS 11.6
Also observe that in the above arguments, nothing changes if instead of
d:A\M)-^A^'^\M) we have 6: A\M)-^ A^"\M). [We take this to
mean d{f) = 0 for / G /\^(M).] In fact, the same argument works for
for any odd integer r. We can thus state:
Proposition 6.2. Let d: AHM) -> A^(M) and 6: AHM) -> A^-^HM) be
linear maps satisfying (6.5) and (6.8), where r is odd. Then there exists one
and only one way of extending 6 to an antiderivation 6: A^(M) -> A^^^(M)
satisfying (6.4).
As an application of this proposition, we will attach an antiderivation
d{X): A^iM) -^ A^~'\M) to every smooth vector field X on M. Since r= -1,
forfeA^iM) we set
d{X)f = 0.
For CO G A^(^) we set
diX)o) = {X, co). (6.9)
To verify (6.8) means to check that
e{X){fco) = fe{X)w,
that is, that
(Z,/co)=/(X,a>),
which is obvious.
If / is a function and 6 is an antiderivation, we denote by fd the map which
sends co -^/^(co). It is easy to check that this is again an antiderivation.
We can assert the following as a consequence of the uniqueness theorem:
Let X and Y be smooth vector fields, and let / and g be smooth functions. Then
difX + cjY) = fd{X) + gd(Y). (6.10)
By the proposition, it suffices to check (6.10) on all co G /\^(7I/). By (6.9), this
is just
(/x + ^y,co)-/(z,co) + ^(y,co),
which is obvious.
In particular, in a chart (U, h), if
then
To evaluate d{d/dx''), we use (6.8) and the fact that
<S.)/ = o. »(s)-' = |? in

11.6 THE LIE DERIVATIVE OF A DIFFERENTIAL FORM 455
Thus, for example, 6(6/dx^) dx^ A dx^ = 0 if neither p = i nor q = i, while
d{d/dx'){dx' A dx^') = dx^\ d{d/dx')idx^' A dx') = —dx\ etc.
Let us call a (sequence of) map(s) D: /\^{M) -^ /\^~^\M), where s is even,
a derivation if it satisfies (6.5) and
Z)(coi A C02) = Dcoi A C02 + coi A Dco2. (6.11)
Since s is even, this is consistent. The most important example is Dx
where s = 0. Then (6.11) is just (6.2).
All the previous arguments about existence and uniqueness of extensions
apply unchanged to derivations, as can easily be checked. We can therefore
assert:
Proposition 6.3. Let D: A^(^^) "^ A^"^'W and D: /\\M) -^ A^"^'(^^).
where s is even, be maps satisfying (6.5) and (6.8) (with d replaced by D).
Then there exists one and only one way of extending D to a derivation of
A(^/).
We need one further algebraic fact.
Proposition 6.4. Let di'. /\^ -^ A^^'^ ^^^^^ ^2- A^ -^ A^'^'' be antideriva-
tions. Then 6162 + 6261: A^ -^ /\^+^i+'2 is a derivation.
Proof. Since ri and r2 are both odd, Vi + /"^ is even. Equation (6.5) obviously
holds. To verify (6.40, let coi G A^^^)- Then
6162(0:1 A a;2) == 6i[62o:i A a;2 + ( —l)^"a;i A 620:2]
= 61620:1 A a;2+ (-l)^"+'2(92a;i A 610:2
+ { —1)^610:1 A 620:2 + coi A 61620:2.
Similarly,
6261(0:1 A a;2) -= 62610:1 A a;2+ ( — if'^''^610:1 A 620:2
+ ( — 1)^620:1 A 610:2 + coi A 62610:2.
Since ri and r2 are both odd, the middle terms cancel when we add. Hence we get
(6162 + 626i)(o:i A a;2) = (6162 + 6261)0:1 A a;2 + coi A (6162 + 6261)0:2. □
As a first application of Proposition 6.3, we observe that
6(X) o d(Y) = -6(Y) o d(X). (6.12)
In fact, by Proposition 6.4, 6(X)6(Y) + 6(Y)6(X) is a derivation of degree
—2, that is, it vanishes on A^ ^^^ A^- I^ must therefore vanish identically.
We could, of course, directly verify (6.12) from the local description of 6(X)
and 6(Y),
As a more serious use of Proposition 6.4, consider 6(X) o d -\- d o 6(X),
where Z is a smooth vector field. Since d: A^ -^ A^~^^ ^^^ ^ W • A^ -^ A^~S
we conclude that 6(X) o d + d o 6(X): A^ ~^ A^- We now assert the main
formula of this section:
Dx = 6{X) od + do d(X). (6.13)

456 EXTERIOR CALCULUS ll.G
Since both sides of (G.13) are derivations, it suffices to check (6.13) for
functions and linear differential forms. If / G /\^{M), then d{X)f = 0. Thus,
by (6.9), Eq. (6.13) becomes
Dxf = {X, df),
which we know holds. Next we must verify (6.13) for co G /\^{M). By (6.5), it
suffices to verify (6.13) locally. If we write co = ai c/x^ + • • • + a^ dx^, it
suffices, by linearity, to verify (6.13) for each term a^ dx\ Since both sides of
(6.13) are derivations, we have
Dx(a, dx') = (Dxai) dx' + a^(Dx dx')
and
[d{X) d + d^{X)]{a^ dx') = [e{X) d + de{X)]{a,) dx' + a,[^(X) d + de{X)] dx\
Since we have verified (6.13) for functions, we only have to check (6.13) for dx\
Now
Dx dx' = dDxx'
and
[d{X) d + ddiX)] dx' = dd{X) dx' = d{X, dx') = dDxx\
This completes the proof of (6.13).
In many circumstances it will be convenient to free the letter 6 for other uses.
We shall therefore occasionally adopt the notation
A^ J CO == d(X)o).
The symbol J is called the interior product. X J co is the interior product of
the form co with the vector field X. If co G /\^j then X J co G A^~^-
Equation (6.13) can then be rewritten as
Dxo: = X J dco + d{X J co). (6.14)
Let us see what (6.14) says in some special cases in terms of local coordinates.
If CO = ai dx^ -\- - • ' ~\- an dx'^ and
then
Hence
while
so
dx^ ax^
c/co = Y. d^i A dx\
X Jdw = Z) [(X J da^) A dx' — dai A X J dx']
X Jco = Y. o,jX',
d{x J w) = E ^
'(B^'^^-m^-

App. I. ^Vector analysis" 457
Thus
X'."-£(i:-Y'|;: + «,||)^.',
which agrees with Eq. (7.12') of Chapter 9.
As a second illustration, let Q = adx^ A - • - A dx'^, where n = dim M.
Then dQ = 0, so (6.14) reduces to
Dx^ = d{X J Q).
If Z = £ X\d/dx^), then
= E aX' U^ J {dx^ A • • • A dx"")
= E i-iy-'aX'dx' A • • • A dx'-' A dx'-^' A • • • A dx",
which is merely the formula introduced at the end of Section 4. Then
i^'if)'-'
Dx^ = d(X jn) = [X -f^rj dx' A '" A dx^.
Since we can always locally identify a density with an 7?-form by identifying p
with Pa dx^ A • • • A dx'^ on (U, a), we obtain another proof of Proposition 5.2
of Chapter 10.
Appendix I. "VECTOR ANALYSIS"
We list here the relationships between notions introduced in this chapter and
various concepts found in books on "vector analysis", although we shall have
no occasion to use them.
In oriented Euclidean three-space E'^, there are a number of identifications
we can make which give a special form to some of the operations we have
introduced in this chapter.
First of all, in E'^, as in any Riemann space, we can (and shall) identify
vector fields with linear differential forms. Thus for any function / we can
regard df as a vector field. As such, it is called grad/. Thus, in E^, in terms of
rectangular coordinates x, y, z,
-«-^:4'i'i^'
where we have also identified vector fields on E^ with E'^-valued functions.
Secondly, since E'^ is oriented, we can, via the ^-operator (acting on each T*),
identify A^(E^) with A^(E^). Recall that * is given by
^{dx A dy) = dZy *(c/x A dz) = —dy, ^{dy A dz) = dx. (LI)

458 EXTERIOR CALCULUS
In particular, if coi = <P,Qj R> = P dx-\-Q dy-\-R dz and 002 =
<L, Mj N> = L dx -\- M dy -\- N dZj we can introduce the so-called "vector
product" of coi with a;2. It is defined by
coi X 002 = *(a;i A a;2)
and is given [in view of (I.l)] by
<P,Q,R> X <L,M,N> = <QN - RM, RL - PN.PM - QL>.
Also we introduce the operator
curl CO = *G?a;.
Thus, if CO = <P,Q, R>, we have
_JdR,_dQd]P_dR dQ_dR,\^
\dy dz ^ dz dx ^ dx dy (
Consider an oriented surface in E^; i.e., let <^: aS -^ E^. Let Q be the volume
form on S associated with the Riemann metric induced by cp. By definition, if
?i, ?2 G T,(S), then
where dV is the volume element of E'^ and n is the unit normal vector. Another
way of writing this is to say that
where (7 = *n when we regard n as a differential form. Now let li; be a form in
E^, and suppose that <^*co = fQ for some function /. Then
/(x) = (co, *n)(<^(x)).
Thus
f <^*(a5) = f fQ = I (cj, *n)12 = f (*a5, n)Q.
J S J S J S J
Applying this to co = g?co, where cx: = P dx + Q dy + R dz, we can rewrite
Stokes' theorem as
/co= I P dx + Q dy + R dz = f (curl co, n)12,
Jc Jc Js
where S is some surface spanning the closed curve C.
If we apply the remark to the case co = *co and S = dD, we obtain, since
** = id (for n = 3),
f (co, n)Q = j d ^ 0)
Note that
, /dP . dQ . dR\. ^ , ^ .
which we write as div co; that is,
div CO = G? * CO.

APP. II. DIFFERENTIAL GEOMETRY OF SURFACES IN E^ 459
(It is in fact div {co, dV}, where dV is the volume element and we regard co as a
vector field.) Thus we get the divergence theorem again. Note that
and
since d^ = 0.
curl (grad f) = ^d df — 0
div (curl co) = G? ** G^co = c^^co =
Appendix II. ELEMENTARY DIFFERENTIAL
GEOMETRY OF SURFACES IN E^
For purposes of computation, it is convenient to introduce the notion of a
vector-valued differential form. Let £' be a vector space, and let M be a differen-
tiable manifold. By an E'-valued exterior differential form Q of degree p we
shall mean a rule which assigns an element Qx to each x G M, where Q^ is
an antisymmetric E'-valued multilinear function of degree p on Tx(M). For
instance, if p = 0, then an E'-valued zero-form is just a function on M with
values in E, An E'-valued one-form is a rule which assigns an element of E to
each tangent vector ? at any point of M, and so on.
Suppose that E is finite-dimensional and that {^i, . . . , e^} is a basis for E.
Let 12i, . . . , 12iv be (real-valued) p-forms. We can then consider the E'-valued
p-form Q = l^i^i + • • • + ^N^N, where, for any p vectors ?i, . . . , ?p in Tx{M)j
we have
^x(?b ' ' ' , iv) = ^1x(?1j • • • j iv)^l + • • • + ^Nx{il, . . . , iv)^N'
Conversely, if 12 is an E'-valued form, then real-valued forms 12i, . . . , 12iv can
be defined by the above equation. In short, once a basis for an A^-dimensional
vector space E has been chosen, giving an E'-valued differential form 12 is the
same as giving N real-valued forms, and we can write
N
12 = ^ ^iCi or 12 = <12i, . . . , 12iv>.
1
The rules for local description of ^E^-valued forms, as well as the transition
laws, are similar to those of real-valued forms, so we won't describe them in
detail. For the sake of simplicity, we shall restrict our attention to the case
where E is finite-dimensional, although for the most part this assumption is
unnecessary.
If CO is a real-valued differential form of degree p, and if 12 is an E'-valued
form of degree g, then we can define the form co A 12 in the obvious way. In
terms of a basis, if 12 = <12i, . . . , 12iv>, then co A 12 = <a; A 12i, . . . , co A 12iv>.
More generally, let E and F be (finite-dimensional) vector spaces, and let #
be a bilinear map of E X F -^ G^ where G is a third vector space. Let
{^1, . . . , ^iv}

460 EXTERIOR CALCULUS
be a basis for E, let {/i, . . . , Jm} be a basis for F^ and let {t/i, . . . , (/l} be a
basis for G. Suppose that the map # is given by
k
Then if co = XI ^t^t is an £'-valued form and 0 = X ^jjj is an i^-valued form,
we define the (j-valued form co A 0 by
CO A 0 = X! (X) ^ii^t A ^j\
k \ i,j /
gk.
It is easy to check that this does not depend on the particular bases chosen.
We shall want to appl}^ this notion primarily in two contexts. First of all^
we will be interested in the case where E = F and G = U, so that # is a bilinear
form on E. Suppose # is a scalar product and ei, . . . j e^ is^n orthonormal basis.
Then we shall write (co A ^) to remind us of the scalar product. If
CO = <coi, . . . , coa-> and U= <^i, . . . , 12,v^ ;
then
(co A fi) == L ^i A ^i.
Note that in this case if co is a j>-form and ^ is a g-form, then
(co A ^) = (-1)^^(12 A co),
as in the case of real-valued forms.
The second case we shall be interested in is where F = G and E = Hom(i^),
and # is just the evaluation map evaluating a linear transformation on a vector
of F to give another element of F. This time, choosing a basis for F determines a
basis for Hom(i^), so we can regard co as a matrix of real-valued differential forms.
If CO = (co^-y) and U = <^i, . . . , ^m>, then
CO A ^ == <2Z coiy A ^y, . . . , S o^Mj A Qj>.
The operator d makes sense for vector-valued forms just as it did for real-
valued forms, and it satisfies the same rules. Thus, if ^ = <^i, . . . , ^^y>,
then dU = <dUi, . . . , d^M ^ and
d{o: A ^) = do: A ^ + ( —l)^co A d^
if CO is an E'-valued form of degree p and 9, is an i^-valued form.
We shall apply the notion of vector-valued forms to develop (mostly in
exercise form) some elementary facts about the geometry of oriented surfaces
in E'^ Let M be an oriented two-dimensional manifold, and let (^ be a differ-
entiable map of M into E^. We shall assume that (p:^ is not singular at any point
of M, i.e., that (p is an immersion. Thus at each point p G M the space (^^ {Tp{M))
is a tw^o-dimensional subspace of 2'<p(p)(E^). Since we can identify T'<p(p)(E^)
with E^, we can regard (p^{Tp{M)) as a two-dimensional subspace of E^.
(See Fig. 11.15.) Since M is oriented, so is the tangent plane ip^{Tp(M)).
Therefore, there is a unique unit vector orthogonal to the tangent plane which,

APP. II. DIFFERENTIAL GEOMETRY OF SURFACES IN E^ 461
Fig. 11.15
together with an oriented basis of the tangent plane^ gives an oriented basis
of E^. This vector is called the normal vector and will be denoted by n(p). We
can consider n an E^-valued function on M. Since ||n|| = 1, we can regard n as
a mapping from M to the unit sphere. Note that ip(M) lies in a fixed plane of E^
if and only if n = const (n = the normal vector to the plane). We therefore can
expect the variation of n to be useful in describing how the surface (p(M) is
"bending".
Let Q be the (oriented) area form on M corresponding to the Riemann metric
induced by <p. Let Qs be the (oriented) area form on the unit sphere. Then
n*(Qs) is a two-form on M^ and therefore we can write
n*Qs = KQ.
The function K is called the Gaussian curvature of the surface (p{M). Note that
K = 0 if (p{M) lies in a plane. Also, K = 0 if <^(M) is a cylinder (see the
exercises).
For any oriented two-dimensional manifold with a Riemann metric we
let ^ denote the set of all oriented bases in all tangent spaces of M, Thus an
element of ^ is given by <fi,f2>, where </i,/2> is an orthonormal basis of
T^{M) for some x ^M. Note that /2 is determined by/i, because of the
orientation and the fact that J2 J- /i- Thus we can consider ^ the space of all tangent
vectors of unit length. For each x ^M the set of all unit vectors is just a circle.
We leave it to the reader to verify that ^ is, in fact^ a three-dimensional manifold.
We denote by tt the map that assigns to each </i,/2> the point x when
</i,/2^ is an orthonormal basis at x. Again, the reader should verify that tt
is a differentiable map of ^ onto M,
In the case at hand, where the metric comes from an immersion ^, we define
several vector-valued functions X^ ei, ^2? and 63 on 3^ as follows:
X = <^ o TT,
^l(</l,/2>) = ^Ju
^2(</lj/2>) == <^*/2,
63 = n o TT.

462 EXTERIOR CALCULUS
(In the middle two equations we regard (p^ ji as elements of E^ via the
identification of T'<p(x)E^ with E^.) Thus at any point z of 3^, the vectors ^1(2;), ^2(2;), e^{z)
form an orthonormal basis of E^, where ^1(2;) and 62(2:) are tangent to the surface
at (p{'k{z)) = X{z) and es{z) is orthogonal to this surface. We can therefore
wTite
(dX A cs) = (dX, es) = 0 and (ei, Bj) = 8i
By the first equation we can write
dX = COi^i + ^02^2, (II-l)
where
coi = (dX, ei) and 002 = (dX, 62)
are (real-valued) linear differential forms defined on ^.
Similarly, let us define the forms co^y by setting
c^ij = (dei, ej).
Applying d to the equation (e^-, Cj) = bik shows that
CO^y = —coy,-. (II.2)
If we apply d to (II.1), we get
0 = d dX = do^iei — coi A dci + do^2e2 — 0^2 /\ ^^2-
Taking the scalar product of this equation with ei and ^2, respectively, shows
(since con = 0 and 0022 = 0) that
cZcoi = 002 A 0021 and G?a;2 = coi A a;i2. (II-3)
If we apply d to the equation
dCi = ^ o^ijCj,
i
we get
0 = Y^idcx^ijej — cx^ij A dej)
and if we take the scalar product with ej, we get
k
If we apply d to the equation (dX, e^) = 0, we get
0 = d(dX, es) = (dX, de^) = (coi^i + ^02^2, coai^i + a;32e2),
which implies that
^1 A coai + ^2 A 0032 = 0. (II-5)
We will now interpret these equations. Let z = </i,/2^ be a point of ^.
For any J G Tz(^) we have

APP. II. DIFFERENTIAL GEOMETRY OF SURFACES IN E^ 463
Therefore,
= (^*?,/i), (11.6)
since the metric was defined to make <^^ an isometry. In other words, (?, coi)
and (?, 002) are the components of tt^J with respect to the basis </i,/2>.
If 7} is another tangent vector at z, then coi A a;2(?, i?) is the (oriented) area of
the parallelogram spanned by tt^ J and tt^t;. In other words,
coi A a;2 = 7r*12, (II.7)
where Q is the oriented area form on M.
Similarly,
{^,des) = n^TT^?, (II.8)
and we have
^*7r^? = (J, coai)^! + (?, ^032)^2
= (?j ^3l)<^*/l + (?, C032)<^*/2. (II.9)
Since we can regard ei and 62 as an orthonormal basis of the tangent space to the
unit sphere J we conclude that cosi A co32(^, ^7) is the oriented area on the unit
sphere of the parallelogram spanned by n^7r^ J and n^ir^r]. Thus
^31 A 0032 = 7r*n*12^
Let
be the matrix of the linear transformation n^: Tx{M) -^ Tn{x)(S^) in terms of
the basis </i,/2> of T^(M) and <eu e2> of Tnix){S^)> Then comparing (II.6)
with (II.9) shows that
^31 = ^^1 + ^^co2 and 0032 ^ ?)coi + ca;2. (11.10)
If we substitute this into (II.5), we conclude that h = 6\ i.e., that the
matrix of n^ is symmetric. This suggests that it corresponds to a symmetric
bilinear form of some geometrical significance. In other words, we want to
consider the quadratic form
acoi + 26a;ia;2 + ccoi
[where it is understood that this is the quadratic form on T2{^) which assigns
the number
a(?, a;i)2 + 26(?, coiX?, 0^2) + c{^, 0^2?
to any f G ^^(3^)].
a
b'
= TT*
ii:fi
= Kwi A W2
6'
c

464 EXTERIOR CALCULUS
EXERCISES
11.1 Show that
11.2 The quadratic form which assigns to each f G Tx{M) the number (<^*f j ^*f)
is called the second fundamental form of the surface. We shall denote it by 11(f).
(What is usually called the first fundamental form is just ||f||^ in our terminology.)
Let C be any smooth curve with C'(0) = f. Show that
11(f) = - I ^4;^ (0). n{x)
)■
Thus 11(f) measures how much the curve cp o C \^ bending in the n-direction. Suppose
we choose C to be such that (p <=> C lies in the plane spanned by cp^^ and n{x).
[Geometrically, this amounts to considering the curve obtained on the surface by intersecting
the surface with the plane spanned by cp^^ and n{x).] Show that 11(f) is the curvature
of this plane curve.
In this sense, the second fundamental form 11(f) tells us how much the surface is
bending in the direction of f.
Note that
Let Xi and X2 be the eigenvalues of the matrix
a b
J) c
Thus
Xi= max 11(f) and X2 = min 11(f) for ||f|| = 1.
If Xi 5^ X2, there are two orthogonal eigenvectors which are called the directions of
principal curvature of the surface. (Note that they must be orthogonal, since they are
eigenvectors of a symmetric matrix.)
If A is a Euclidean motion of E^, then ^ = ^1 o <^ is another immersion of
M and it is easy to check that both the Riemann metric induced by ^ and the second
fundamental form associated with ^ coincide with those attached to <p. What is not
so obvious is the converse: If ^ and <p induce the same metric and the sa7ne second
fundamental for 7n, then 4^ = A o <pfor some Euclidean motion A. We will not prove this fact,
although it is a fairly easy consequence of what we have already established.
We have seen the meaning of coi, 002, 0031, and 0032 in geometric terms. Let us
now interpret the one remaining form, a;i2.
Let 7 be a differentiable curve on M. A differentiable family of unit vectors
/i(-) along 7 [where/i(s) E Ty^s){M)] is the same as a curve C in 3^ with w oC = J.
[Here C{s) = <fiis)j f2(s)>.] Let us call the family /i(s) parallel if the unit
vectors are all changing normally to the surface in three-space. In other words,
/i(s) is parallel if the vector
ds

APP. II. DIFFERENTIAL GEOMETRY OF SURFACES IN E^ 465
is normal to (p(M) for all s. Let us see how to express this condition. Let ^s
be the tangent vector to the curve C at C{s). Then, by the definition of dei^
^<^*T(s)(/i(s)) = {^s,dei).
Note that ((?«, dei), ei(C(s))) = 0 and ((?«, dei), e2{C(s))) = (?«, CO12). Now
ei and 62 span the tangent space to (p(M), so saying that /i(-) is parallel is the
same as saying that (Jg, a;i2) = 0.
Thus /i(s) is parallel along 7 if and only if (Js, CO12) = 0.
Let M and M be two-dimensional manifolds with Riemann metrics.
Let u: M -^ M be a differentiable map which is an isometry. Let ^ be the
manifold of orthonormal bases of M, and let ^ be the manifold of orthornormal
bases of M. Then u induces a map u oi ^ —^ ^ by
u{<fuf2>) = <uJuuj2>-
Let coi be the differential form on ^ given, as in (IL6), by
{i.o^i)= (7r*?,/i) for ?gT,(3^),
where z= </i,/2^, with the corresponding definition for 002, coi, and C02.
Then for any J G Tz{^) we have, since t ^ u ^ u o w,
<?, ^t^coi) = (?2^J, coi) = (tt^?2^?, 1/Ji) = (i^*7r^J, 7^*/i) = (tt^J, /i) = (?, coi).
In other words,
u*cibi = coi and ^^*co2 = a;2.
Now suppose that the metrics on M and M come from immersions <p and ^.
Then we get forms co^-y and co^j. Now by (II.3) we have
Thus
or
iIt*(coi A CO21) = 'W*G?coi = d{u'^G)i) = do^i = coi A a;2i.
coi A 'W*co2i = coi A a;2i and 002 A 'W*coi2 = 002 A ^12?
(?2*coi2 — a;i2) A coi = 0' and (i;t*coi2 — a;i2) A 002 = 0.
Since the differential forms coi and 002 are linearly independent, this can only
happen if
^*^12 = ^12-
In other words, if the two surfaces (p(M) and (p(M) are isometric, they have
the "same" a;i2, that is, the same notion of "parallel vector fields". Observe
that a piece of a cylinder and a piece of the plane are isometric, even though they
are not congruent by a Euclidean motion. In different terms, while the forms coi 3
and 0023 depend on how the surface is immersed in E^, the form 0J12 depends
only on the Riemann metric induced by the immersion.
Now we have (II.4):
G?a;i2 = CU13 A CU23 = —■ 7r*Qs = — K^-

466 EXTERIOR CALCULUS
From this we conclude that the Gaussian curvature K also does not depend only
on the immersion, but only on the Riemann metric coming from the immersion.
Since a;i2 does not depend on <^, we should be able to define it for an arbitrary
two-dimensional manifold with a Riemann metric. Note that the preceding
argument shows that a;i2 is uniquely determined by Eq. (II.4). It therefore
suffices to construct an co 12 on a coordinate neighborhood so as to satisfy (II.4).
It will then follow from the uniqueness that any two such coincide to give a
well-defined form. Let f/ be a coordinate neighborhood of M, and let ^: U -^ ^
be a differentiable map such that tt o ^ = id. Thus ^ assigns a basis </i, /2>
to each x G U, in a differentiable manner. (One possible way to construct ^ is
to apply the orthonormalization procedure to the vector fields < d/dx^, d/dx^ >.)
Once we have chosen ij/, any basis of T^ differs from ^(x) by a rotation.
If we let r denote the (angular) coordinate giving this rotation (so that r is
only defined mod 27r), then we can use the local coordinates on U together
with r as coordinates on 7r~^{U). More precisely, if x^ and x^ are local
coordinates on Uj we define y^, y^yThy
?/^ = X^ o T, 2/^ = X^ o TT,
and t(z) is given for z = <ei, e2> by
ei = cos T{z)fi + sin T{z)f2, ^2 = —sin T{z)fi + cos r(2)/2, (II.H)
where <fij2> = H^) when <ei, e2> e T^{M).
Now let
^1 = ^*(coi) and 62 = ^*(co2),
so that ^1 and 62 are forms defined on U and are, in fact, the dual basis for ^(x)
at each x G M. If we set
0:1 = 7r*^i and 0:2 = 7r*^2j
then (11.11) gives
coi = cos Tai + sin Ta2 and 002 — —sin roii + cos 70:2.
Note that
coi A a;2 = oil A 0:2.
Define the functions li and I2 on M by
ddi = lidi A 62 and de2 = h^i A 62.
Let hi — li o TT and k2 — h ° ^r, so that
dai — kiai A 0:2 and 6^0:2 = k2(xi A 0:2.
Now
dcx^i — —sin T dr A ai + cos t dr A (X2 + (/ci cos r + /c2 sin 7)0:1 A 0:2,
G?a;2 — —cos T dr A ai — sin r G?r A 0:2 + (+/c2 cos t — ki sin 7)0:1 A 0:2.

APP. II. DIFFERENTIAL GEOMETRY OF SURFACES IN E^ 467
Since coi A 002 = ai A 0:2, we can rewrite these equations as
do^i = (dr + (/ci cos r + /c2 sin r)a;i) A 002,
G?a;2 — —{dr — (+/c2 cos r — ki sin 7)0^2) A coi.
We thus see that the form
a;i2 = dr -\- (/ci cos r + /c2 sin r)a;i + (—/ci sin r + /c2 cos r)a;2
= dr -\- kiai + /c2a:2
satisfies the desired equations.
As before, on any two-dimensional Riemann manifold we will call a family
of unit vectors parallel along a curve T if (^s, CO12) — 0. With this definition of
parallel translation we can state the following:
Theorem. Let 7 be any differentiable curve on M. Given the unit vector
Qi G T'7(0)(M), there is a unique parallel family of unit vectors ^i(s) along 7,
with ^i(O) == ^1. If g[{Qi) is another unit vector of Ty(^Q){M) differing from gi
by an angle c, then ^'i(s) differs from ^i(s) by the same angle a for all s.
Proof. It is clearly sufficient (by breaking 7 up into small pieces if necessary) to
prove the theorem for curves 7 lying entirely in a coordinate chart. Then we
can use the local expression for a;i2.
Let us rewrite the condition for parallel translation along 7(s). In terms of
local coordinates, the unit vector ^i(s) is given by a function r(s), where
Then
gi{s) = cosr(s)/i(7(s)) - sin r(s)/2(7(s)).
<?s, CO12) -= {is dr) + <?s, /ciai + k2a2)
-^^^'^ + {is,7r'^(k,d, + k2d2))
— <7r*?s, kidi + /C2(92).
ds
dr{s)
ds
But TT^Js ^ fs is the tangent vector to 7 at 7(s). Thus
where Fy(s) = (fs, kidi + /c2^2) is a function depending only on s. In
particular, ^i(s) is parallel if and only if
From this we see that given ^i(O) there is a unique parallel family ^i(s), starting
with ^i(O). Furthermore, if g[{0) is a second unit vector at 7(0), the angle

468 EXTERIOR CALCULUS
between gi{s) and g[(s) is equal to the angle between gi(0) and g[{0). Thus
parallel translation preserves angles, which proves the theorem. D
Note that if M is (locally isometric to) Euclidean space, then we can choose
fi = -^ and /2 = 73
so that ^1 = dx^ and 62 = dx^. In this case, ki = k2 = 0 and r is just the
angle that gi makes with d/dx^, that is, with the Xi-axis. Thus 0012 = —dr
in this case. Then the condition for parallel translation becomes dr/ds = 0,
which coincides with the usual notion of parallelism in Euclidean geometry.
Note that in Euclidean space the parallelism does not depend on the curve 7.
This is not true in general.
Exercise 11.3. Let 7i and 72 be two arcs of great circles joining the north and south
poles on S^. Suppose that 7i and 72 are orthogonal at the poles. Let f be a tangent
vector of the north pole. Compare its translates to the south ])ole via 7i and 72.
Let M be any two-dimensional Riemann manifold. For any curve 7 on M
there is an obvious way of choosing unit vectors along 7: just let gi(s) be the
unit tangent vector to 7 at 7(s). Thus for every curve 7 on M we get a curve,
which we shall call y, on 3". [Here y = (y{s), gi{s), g2{s)) and gi{s) is the
tangent to 7(s).]
We call the form 7*(a;i2) the geodesic curvature form of 7. [In the Euclidean
case this is just the ordinary curvature (see the exercises).]
Let us consider those curves whose geodesic curvatures vanish, i.e., those
curves whose tangent vectors are parallel. We shall call such a curve a geodesic
with respect to the given Riemann metric. Note that the condition that a curve
be geodesic is given, in local coordinates, by a second-order differential
equation. Therefore, a geodesic C{-) is uniquely specified by giving C(t) and C'(t)
at any fixed value of t. In Chapter 13 we use the term "geodesic" to mean
a curve which locally minimizes length. It is the purpose of the next few exercises
to show that geodesies in our present sense have this property.
EXERCISES
II.4 Let X, y be local coordinates on U C M. Through each point of the curve
y = 0 (that is, the x-axis in the local coordinates), construct the unique geodesic
orthogonal to this curve. (See Fig. 11.16.) Let s be the arc-length parameter along
the geodesic, so that the geodesic passing through {u, 0) is given by
{y(u, s), x{u, s)).
Show that the map {u, s) ^-^ {y{u, s), x{u, s)) has nonzero Jacobian at (0,0) and
therefore defines a coordinate system in some open subset U' C U.

APP. II. DIFFERENTIAL GEOMETRY OF SURFACES IN E^ 469
Fig. 11.16
Fig. 11.17
II.5 We are going to make a further change of coordinates,
field on U' defined by the properties
Let Y be the vector
iJ^ii = 1,
(^.|)
0, {Y,du)>Q.
Thus Y is orthogonal to the geodesies u = const and points in the increasing tZ-direc-
tion. Let us consider the solution curves of this vector field parametrized by the initial
position along the geodesic u = 0. That is, let v be the arc-length parameter along the
geodesic u = {), and consider the map
{u^ v) H-> {u^ s{u, v)),
where s(u, v) is the s-coordinate of the intersection of the solution curve of Y passing
through (0, v) with the geodesic given by u. (See Fig. 11.17.) Again the existence
theorem and smooth dependence on parameters, together with the fact that the curves
u = 0 and s = 0 are already orthogonal, guarantees that we can find some
neighborhood W so that (u, v) are coordinates on W. We have thus constructed coordinates
such that the curves u = const are geodesies and the curves u = const and v = const
are orthogonal. Such a system of coordinates is called a geodesic parallel coordinate
system.
II.6 Let {u, v) be a coordinate system on U C M for which (d/du, d/dv) = 0, so
that the metric takes the form
ds^ = p du^ + q dv^.
Define the choice of frame ^ by normalizing d/du, d/dv so that ^(x) = <fi,f2^,
where /i = {d/du)/\\id/du)\\ and /2 = id/dv)/\\id/dv)\\. Show that the forms ^i
and 02 are given by
di = p du, 62 = Q dv,
and
and
C012
K
(idp ^ ^ I dq \
= dr — irn- ^ du-\- - —- dv]
\q dv p du /
pq \_du \p du/ dv \q dv J

470 EXTERIOR CALCULUS
11.7 Let (u, v) be a geodesic parallel coordinate system, as in Exercise II.5. The
curve Cu given by Cuiv) = (u, v) is a geodesic. Thus {C'\v), a;i2) = 0. But in terms
of our local coordinates, {C"(v), dr) = 0, since C {v) is always parallel to one of the
base vectors /2, and
{C\v),f^du) = {CHv),du) = 0,
since u = const along C. Thus we conclude that dq/du = 0, or q = q{v). Let us
replace the parameter vhy w = Jq q{t) dt. Then {u, v) is a geodesic parallel coordinate
system for which we have
ds^ = p du^ + dw^,
and now the arc length along any curve u = const is / dw.
11.8 Show that for \w\ sufficiently small, any curve joining (0, 0) to (0, w) must have
arc length at least \w\. Conclude that (since the choice of our original curve x = 0
was arbitrary) the geodesies locally minimize length.
11.9 Let <w,z>- be local coordinates on an open set f/ of a Riemann manifold
with the property that the curves Cz given by Cziw) = {w, z) are geodesies
parametrized according to arc length. Thus z = const is a geodesic and ||^/^ti;|| = L Let
\dw dzj
Show that da/dw = 0. [Hint: Show that by orthonormalizing {d/dw, d/dz), we obtain
a map ^ whose associated forms ^i and 62 are given by ^1 = dw ^ a dz, 62 = b dz,
where b = \\d/dz — a{d/dw)\\. Then compute h and h and use the fact that z = 0
is a geodesic]
11.10 Construct geodesic polar coordinates. That is, for a fixed p E M let 6 be an
angular coordinate on the unit circle in Tp{M). For each 6 let Ce{-) be the unique
geodesic, parametrized by arc length, such that C(0) = p and C'(0) corresponds to
angle 6 in Tp(M). Show that the map (r, 6) 1—> Ce(r) gives "coordinates" on U — {p},
where U is some neighborhood of p. By taking <r, 6> to be the <w, z> of Exercise
II.9 and passing to the limit r = 0, conclude that
\dr ddj
dr ' do) ^'
Thus in terms of the "coordinates" <r, 6> on U — [p], the Riemann metric takes
the form
ds'^ = dr^+ A dd'^.
These coordinates are the analogue, for a general two-dimensional Riemann manifold,
of the polar coordinates introduced on the plane and on the sphere at the end of
Chapter 9. The argument given there applies generally to give another i)roof of the
fact that geodesies locally minimize arc length.
We now continue to study the consequences of the equation
Let D be a domain with regular boundary on M, and let ^ be a map of some
neighborhood of D -^ 5 satisfying tt o ^ = identity. Then, by Stokes'

APP. II. DIFFERENTIAL GEOMETRY OF SURFACES IN E^ 471
theorem,
/ ^*(coi2) = - I ^*7r*K^ = - I Kn, (11.12)
JdD Jd Jd
where ^ is the area form on M.
Let us apply this to the map ^ constructed as follows: Let 7 be a vector field
on a compact manifold iff which vanishes at only a finite number of points
pi, . . . , pr. At any x E M where 7^ ^ 0, put fi(x) = Y:c/\\Yx\\, so ^(x) =
<x, fi{x), /2(^) > • About each point p^ choose a chart (f/^, /i^) such that f/^
contains no other point pj and /^t(pt) == (0, 0) in the plane. Let 7^-,^ be h~^{Cr),
where Cr is the circle of radius r in the plane. [Here r is chosen so small that Cr
lies in h^(U^).] Now let D = M — Ui (interiors of 7^-,^). We must compute
Jy. ^ ^*(a;i2), where 7^,^ is oriented clockwise rather than counterclockwise.
For this purpose, we introduce about p^ the orthonormal frames coming from
the coordinates x^, x^. Thus we have coordinates x\ x^, t, where T{x,fi,f2)
measures the angle that/i(x) makes with
d I
dx^lx
Thus (taking yt,r,T clockwise)
- I dr(h{s)) = 27r
(index of Y at p^). But ^*a;i2 = ^*G?r + ^*(/ciq:^ + ^2«^), so
— f ^*coi2 == 27r (index of 7 at p^) + f kiO^ + /c2^^.
Now as r -^ 0, the second term on the right vanishes and Jd -^ Jm on the right
of (IL12). Thus we have proved the following:
Let y be a vector field which vanishes at a finite number of points pi, . . . , p^.
Then
X; index (7) = -^- / K dA. (IL13)
i Pi ^TT Jm
Li particular, the sum of the indices of a vector field is independent of the
vector field, and the total integral of the curvature is independent of the
Riemann metric.
Thus, for instance, if if/ is S^, the vector field tangent to the meridian circles
has zeros only at the north and south poles, and the index at each of these
points is +1. Thus (IL13) says that the sum of indices of any vector field on S^
must equal 2. (In particular, it is impossible to construct a vector field on S^
which does not vanish anywhere.) Furthermore, (IL13) says that no matter
what Riemann metric we put on S^, we have
/.
KdA = 47r.
Similarly, if M is the torus, we can put it on a vector field which does not
vanish anywhere, so the integer in (IL13) is 0.

472 EXTERIOR CALCULUS
EXERCISES
Let M denote the torus with angular coordinates <pi, <p2 (defined up to 2t). The map F
will denote the immersion of M —^ Euclidean three-space given by
xi = (2+ cos <^i) cos <^2,
0:2 = (2+ cos <^i) sin <^2,
0:3 = sin <pi.
11.11 What is the Riemaiiii metric on M induced by F? That is, compute (d/dcpi,
d/d(pi)p, (d/d<pi, d/d(p2)p and (d/d<p2, d/d<p2)p for each p G M. (These can be given as
functions of cpi, ip2.) Let/i,/2 denote the vector fields obtained by orthonormalizing
d/difi, d/d(p2. What is the explicit expression for/1,/2?
11.12 What is the total area of the torus relative to this Riemann metric?
11.13 In terms of the vector fields /i,/2 given in Exercise 11.11 we can introduce
(angular) coordinates (pi, ip2, r on 3^. In terms of these coordinates, what are the
explicit expressions for the vector-valued functions X, f^,f^,f^, on 3^? (Each of these
vector-valued functions should be given in terms of the fixed standard basis of Euclidean
three-space, i.e., a triplet of functions of (pi, (p2, r.) Thus
X(<^i, <p2, r) = < (2 + cos (pi) cos (p2j (2 + cos (pi) sin <^2, sin (pi > .
What are the corresponding expressions for/1,/2, and/a?
11.14 What are the explicit expressions for the forms cji, 0^2, and a;i2 on 3^ given in
terms of d(pi, d(p2, and dr?
11.15 What is the Gaussian curvature K oi Ml (Again, K can be given as a function
of (pi,(p2') For which points of M is the Gaussian curvature positive, and for which
points is it negative?
11.16 On any Riemann manifold there is an obvious way of identifying a linear
differential form with a vector field [via the identification of T^iM) with T^iM) given
by the scalar product in Tx{M)]. If g' is a function, the vector field associated to the
form dg is denoted by grad g. Let M be the torus with Riemann metric as above, and
let xi be the function given by xi{(pi, (p2) = (2 + cos (pi) cos <^2, as before. Show that
the vector field X = grad xi vanishes at exactly four points, and compute its index
at each of these points.
Let x^ x^ be a local system of coordinates on U C M, and let ^: U -^ ^ he given by
orthonormalizing <d/dx^, d/dx^>. Thus coordinates on t~^(U) are x^ o tt^ x^ o tt, r,
where t{z) is the angle that /i makes with d/dx^ U z = </i,/2>". Then
a;i2 = dr-\-kiai-\-k20C2 = dr + 77*^*0; 12
on T~'^{U). Let C be a curve in U with C\t) > 0, and let C be the curve in tt{U)
assigning to each t the unit tangent vector C\t)/\\C\t)\\. Then
/ a;i2 = /C*coi2 = / \^*a;i2+ / dr,
Jc J Jc Jc
Jc
where r is the angle that C\t) makes with the x-axis.

APP. II. DIFFERENTIAL GEOMETRY OF SURFACES IN E^ 473
From this formula we can deduce a number of interesting consequences which we
flJiilo in exercise form.
11.17 Show that the sum of the exterior angles of a geodesic triangle D is given by
'*" Jd KQ.. (A geodesic triangle is a domain whose boundary consists of three
K«MHiosics.)
11. Itt Suppose that D is a domain which in the local coordinates is
Kivcii by a simple polygon. Thus dB = Ci U • • • U Cjt, where each
i\ is a curve which is a straight line segment in the local coordinate
*NHt(»m. Let ai, . . . , ak be the interior angles. (See Fig. 11.18.)
Hliow that
kTr+ Z^ = 27r — 2 L ^12- Fig. 11.18
11.19 This gives us another way of computing the integer in (11.13) if M is a compact
R>ii face.
Definition. A cellulation of a compact differentiable two-dimensional manifold M
is a finite collection of closed subsets (called the cells) Fi, . . . , Fm such that
m
M = [j Fi
satisfy the following:
1) For each Fi there is a one-to-one bidifferentiable map fi of a neighborhood of
Fi onto a polygon with rn edges {m > 3).
2) For I ^ j\ either Fi n Fj is empty or fiiFiOFj) is a fixed edge or vertex of
the corresponding polygon.
Let/ be the number of faces, e the number of edges, and v the number of vertices
ui (he cellulation of a two-dimensional Riemann manifold. Show that
f-e+v = f
Jm
m.
Tims/ —■ e-^ V does not depend on the cellulation. We have thus given three distinct
N\ iiys of computing an integer attached to the manifold. This integer is called the Euler
I h(tracteristic and is denoted by X{M). Thus
X(M) = f K^ = f - e+v =J^ index Y.
Jm ^i
11.20 By a regular cellulation we mean a cellulation such that each face has the same
number of edges and each vertex is the union of the same number of edges. Show that
(luu-e are at most five possibihties for the number of faces in a regular cellulation of the
Hphcre. Conclude that there are at most five "regular solids".

CHAPTER 12
POTENTIAL THEORY IN E^
1. SOLID ANGLE
In what follows x^,...,x'^ will denote Euclidean coordinates on E'\ Let
r^ = (^x^y + ' ' ' + (x"")^; then dr^ = 2r dr, so that we have
r dr = J2 ^^ dx\
*r dr = 2: i-iy-^x' dx^ A " ' d?" ' A dx^,
d * r dr = ndx^ A ' ' ' A dx^.
Let i denote the injection of S'^~^ —> E^ as the unit sphere. Let Vn denote the
volume of the unit ball and An-i the volume of the unit {n — l)-sphere. The
volume of the sphere of radius r is thus r^~^A^_i, and so
Vn= / r''-'An-idr = -An-i.
Jo n
Since z*(*7' dr) is an (n — l)-form invariant under rotations, it is some multiple of
the volume form on S'^~^. By Stokes' theorem,
/ i*{^r dr) = f d^rdr = n\ dx^ A • • • A dx^.
Js^-'^ Jb^ J b^
Comparing this with the above, we conclude that z*(*r dr) is the volume form on
the unit sphere.
Let p denote the projection of E^ — {0} onto the unit sphere. Set
r = p*i*{^r dr).
Then r is called the element of solid angle. Integrated over any {n — 1)-surface,
it gives the volume (counting sign of orientation and multiplicity) of its
projection on the unit (n — l)-sphere.
We have
dr = 0,
since
dr = dp*i*{*r dr)
= p*(df{*rdr)) = 0,
because z*(*r dr) is an (n — l)-form on the (n — l)-dimensional manifold aS^~\
and d of any (n — l)-form on an (n — l)-dimensional manifold must vanish.
474

12.1
SOLID ANGLE
475
Let Ir denote the injection of the S^ ^ into E"" as the sphere of radius R
(so that ii = i). It then follows directly from the definitions that
irt
dSR
where dSR is the induced volume form. (See Fig. 12.1.)
Fig. 12.1
Now the volume of the ball of radius R is R'^Vn and the surface volume
(area) of the sphere of radius R is R'^~^An-i = R~^(nR'^Vn)' Now iR{^dr) is
an (n — l)-form and thus i*R(*dr) = f dSR for some function /. Since ^dr and
dSR are invariant under rotation, we conclude that / is a constant. But
f i%(*r dr) = nf dx^ A " - A dx^ = nR'^Vn,
form which we conclude that
dSR = i%(^) = i%{^dr).
Thus
.* /*r dr\
Now let X be any point in E^ — {0}, and let ?i,
at X. Then
^n-i be tangent vectors
[r - :^y Uu • • •, ?n-i
will vanish if all the J's are tangent to the sphere centered at the origin and
passing through x, by the above equation. If one of the J's, say ?i, is a multiple
of {d/dr)x, then again the expression will vanish, because t(Ji, . . . , ?n-i) =
z*(*r dr){p^^i, . . . , p*?n-i) and p*?i = 0, and because *dr{^i, ...,•)= 0- By
multilinearity, we conclude that the expression vanishes identically and
therefore that
_ j^ _ ^rdr _ E j—iy-'^x' dx^ A " ' A dx' A " ' A dx""
where the ^^ indicates that the corresponding term is omitted.

476 POTENTIAL THEORY IN E""
12.2
Fig. 12.2
2. GREEN'S FORMULAS
Let u and v be smooth functions on E"". Then
du
^b"-'
so
:,du = L (-1)'-' ^.dx' A "'d^'"' A dx"",
and therefore
d ^ du =
where the operator
dx^ A • • • A dx'^ = Au dx^ A • • • A dx'^,
is called the Laplacian.
Let [/ be an open subset with compact closure and almost regular boundary.
(See Fig. 12.2.) Set
Du[Uj v] = j du A *G?f = dv A ^du = / ( 2 '^~i J~i) ^^^ A • • • A dx'^.
Now jdu u ^ dv = ju d(u * dv). But d{u * G?y) = du A "^dv -\- u A d * dVj so
[ u* dv = Du[u, v]+ f (uAv) dx^ A ' " A dx"". (2.1)
JdU Ju
Since Du is symmetric in u and v, we have
/ u ^ dv ~ V * du = I (uAv — vAu) dx^ A - - - A dx'^. (2.2)
J du Ju
Suppose U contains the origin, and let 11^= U — Be, where Be is the ball of
radius e centered at the origin. Let v = r^~''.t Then dv = {2 — n)r^~'^ dr and
*G?y = _(^ _ 2)t, so that d * dv = 0. Substituting into (2.2) and taking e
tFor n = 2, set v = log r.

12.3 THE MAXIMUM PRINCIPLE 477
small enough so that dUe = dU — dB^, we get
-in-2) I iuT + ^^^^-1^) + (n-2) I ur + e""" [ ^du
J BU \ n ~ I J J BBe JdBe
= - / r^-^'Audx^ A ••• A dx"".
JUe
Let us examine what happens as e -^ 0. The first term on the left does not
depend on e. For the second term, let us write u = u{0) + e{e). Then the
second term becomes (n — 2)An-iu{0) + 0(e). For the third term on the left,
we write
/ *du = d * du = All dx^ A • • • A dx^.
JdBe JBe J Be
Thus the third term on the left vanishes to order e^~^. Since r^~^ is an integrable
function on E^, the right-hand side tends to — ^^ r'^~'^ Au dx^ A • • • A dx^.
Therefore, if we let e -^ 0, we get
An-Mid) = [ (ur + '''~"*^-) ^ f r^-^ Audx' A'" A dx\
J du \ n — Z / n — z Ju
(2.3)
In particular, if i/ is a harmonic function, that is, Au = 0, then
^(0) = -IT— \ ur-\- -, r ^ [ ^ ' (2.4)
^ ^ An-i Jeu An-i(n - 2) Jeu r^-^ ^ ^
In the special case U = Ba, the second term on the right-hand side of (2.4)
becomes
*du = 7 zrr—. / d * du ^ 0.
(n - 2)An-ia^-^ jbb, {n - 2)An-ia^-^ Jb
Thus (by the definition of A^_i and r)
«(0) = ^, (2.5)
where Sa is the sphere of radius e and dS is its volume element. In other words,
if 1/ is a function that is harmonic in some domain, then the value of u at any
point is equal to its average value on any sphere centered at that point whose
interior is completely contained in the domain.
3. THE MAXIMUM PRINCIPLE
Equation (2.5) has a number of startling consequences which we shall now
develop. Let i/ be a function that is harmonic in a domain U. Let xq be some
point of [/, and suppose that there is a neighborhood W of xq such that W GU
and
u{x) < u{xo) for all x gW.

478
POTENTIAL THEORY IN E^
12.3
Fig. 12.3
Fig. 12.4
(See Fig. 12.3.) Let Sa be a sphere of radius a centered at xq^ where a is so small
that Ba C W. Then u(x) < u{xo) for all x G Sa. Therefore,
/.
udS <
i(xo) f
Js.
dS,
If there were some point x G Sa with u(x) < u(xo), then u(y) would be less than
u(xo) for all y sufficiently close to x, since u is continuous. But then the above
inequality would be a strict inequality, i.e.,
[ udS < u(xo) [ dS,
which contradicts (2.5). We must therefore have
u(x) = u(xq) for all x G Sa-
Now suppose that W is an open set that is connected; that is, suppose that any
two points of W can be joined by a continuous curve lying entirely in W. Let y
be any point of TF, and let C be a curve joining xq to y. About each x on the curve
we can find a sufficiently small ball with center x lying entirely in W. By the
compactness of C we can choose a finite number of these balls which cover C.
We can therefore formulate the following: There are a finite number of spheres
Sa^, . . . , Saj^ such that each sphere and its interior lie entirely in W, Sa^^ has
center xq, the center Xi of Sa.^^ lies on Sa., and y G Sa^. (See Fig. 12.4.) But this
implies that u(xo) = u{xi) =^ - - - :=z u{xk) == u{y). In other words, we have
established:
Proposition 3.1. Let u be harmonic in a connected open set W, and suppose
that u achieves its maximum value at some xq G TF. Then u is constant
onTF.
An immediate corollary of this result is:
Proposition 3.2. Let C/ be a connected open set with U compact. Then if
1/ is a function that is continuous in U and harmonic in [/,
unless 1/ is a constant.
u{x) < max u(y)
yGdU
(3.1)

12.4 green's functions 479
Proof. In fact, since U is compact and u is continuous, u must achieve its
maximum at some point xq of U. If we could actually choose xq E U, then u
would have to be a constant by Proposition 3.1. If u is not a constant, then
xo ^ dUj and we have (3.1). D
From this we deduce:
Proposition 3.3. Let C/ be a connected open set with U compact. Let u
and V be functions that are continuous in U and harmonic in U. Suppose
that
M{y) = v{y) for all y G dU,
Then
u(x) == v{x) for all x G 17,
Proof. In fact, u — v satisfies the hypotheses of Proposition 3.2 and vanishes
on dU. Thus u{x) — v{x) < 0 for x G C/. Similarly v{x) — u{x) < 0, which
implies the proposition. D
An alternative way of formulating Proposition 3.3 is to say that on a domain
U a harmonic function is completely determined by its boundary values. This
is a uniqueness theorem: there is at most one harmonic function with given
boundary values. It suggests the problem of deciding whether the corresponding
existence theorem is true. This problem is known as Dirichlet's problem.
Dirichlet's problem. Given a continuous function / defined on dU, does
there exist a function u that is continuous in U and harmonic in U and such
that u{y) = f{y) for all y G dU?
We shall show in Section 9 that we can always solve Dirichlet's problem for
domains with almost regular boundary.
4. GREEN'S FUNCTIONS
Suppose that C/ is a domain for which the Dirichlet problem can be solved.
We shall show in this section that the solution can be given "explicitly" in terms
of a certain integral over the boundary.
We first introduce some notation. For each x G E^ set"!"
^»(^. 2/) = (^_ 2)4^^11:.-2/||n-2 f«r 2/eE"-{x} (n>3),
SO that (for fixed x)
*dKn(x, •) = -z Tx on E"" — {x},
where r^ denotes the solid angle about the point x. Then for x G C/, Eq. (2.3)
tThis is for n > 3. For w = 2, set K2(x, y) = (l/27r) In \x — y\.

480 POTENTIAL THEORY IX E^ 12.4
can be rewritten as
u(x) - j [w * dKnix, •) + Knix,«) * diij - J^ Kn(x, ^) Au dV^ (41)
Now for fixed x G U the function Knix^ ij) is a cliffereiitiable function of y as y
varies on dV\ By assumption^ we enn therefore find a function hiXy •) that k
harmonic in U and continuous in U and such that
h{x, y) - -Kn(x, y) for all y G dl\ (1,2)
Furtlierniore, for fixed y, /i(x, y) is a continuous function of x. In fact, by the
maximum principle,
I^C^i. y) ^ h(j^2, y)\ < max \Kn(x2, z) — Kn(xi, z)l
star
and Kn(x, z) is clearly uniformly contmuous in x for all z E ()U so long as i
stays a fixed distance away from dL\ We have thus constructed a function in
such that
i) hr(x, y) is a contmuous function of x and y for x, y E U] and is differeii-
tiable in y for // G L';
ii) for each fixed x, Ayhu = 0; i.e., AhuiXy -) ^ 0;
in) Gu(x, y) = ii„(x, y) -r ^ir^(^, I/) "^ 0 for (/ e dU.
The function (rV i^^ called the Green function of the domain l\ Let us suppoM*
for a moment that Gu exists, and let us derive some of its properties. We write
Q = Gf^r when there m no po^sibiiity of confusion. We fixst show that for x F V
and 1/ G r we have
G(.x,y)= (7(1/, x). (4J^
Let B^^^ and Bj^,, be small balls about x and y. Let ti == (?(x, •) and v —
G{yj •) in (2.2), where U is replaced by C -— B^^t ^ ^y,i^ Since both function-
arc harmonic in the domain and vanish on dU^ we obtain
( G{:c, •) * dG{y, ■) + [ (?(x, •) * dG{y, ■)
- ( G{y, ■) * dG{x, •) ^ f G^y, ■) * dG{x, ■
We will show that the left-hand side approaches G{x, y) and the right-hand sidt*
approaches Giy^ x) as e --^ 0. By symmetry, it suffices to look at the left-han<l
aide.
Now
f G{x, 0 * dGiy, 0 - f Gix, •) * dKniy, 0 + f Gix, ^) * dh.
JdBy^t JdBy^^ JdBy^^
The first term on the right-hand side approaches Gix, y) as in the proof of (2J .
since A^^i * dK is the solid angle about y. The second term tends to zero

12.4 green's functions 481
since G(.r, •) and h are smooth functions on By^^, On the other hand,
f Gix, ■) * dG{y, ■) = [ K{x,-)*dG{!j,-) \ f h{x, ■) * da{y, ■).
JiB^^t JBB^,^ JdBj^^
The second term tends to zero, as above. The first term can be written, (for
n > 3)
since fi(ij^ •) is harmonic in B^^^, A similar argument works for n = 2 with
^n-2 r^pig^cg(j |3y iQg £^ This proves (4.3).
Lei M be any smooth function on U, Apply (2.2) to u and v = G(x, •)
on U — Bj^f. We get (since G(x, -) = 0 on dU)
f M * fl(?(x, •) - [ u* dG{x, ') + [ G(x, 0 * clii
JdU ' JdBr,i ^ ia5x.€
= [Gixr)^udx^ A " - A Ac''.
The tMrd nitegral on the left-!iand side can be written as
I lCn(Xj •) * du + I /i(x, •) * du
JdB.^t J ' if f
= I /„/^ 9t^V"^2 I *^'^' i I ^'('^'>")
::; a(€"-^) + 0(€^'-^^)
A(n - 2)e^-^
and so tend^ to zeni. (The usual niodificaiion wnrks for n ^ 2.) We get
nix) = [ G{i\ •) An dx^ A • • • A dx" + f ii * iiG(:r, •). (^^4)
if Jar/
We observe that (4.4) shows that ij we know that there exists a solution to
AF = /with the boundary conditions F = 0^ then it is given by
Fix) = JGix, y)f{jj) dy.
Similarly^ if we know that there exists a smooth sohition to the problem
Am = 0, ti{x) = /(:r) for x G BU, (4.5)
then it is given by
u{x) ^ M * dG{xj *)• (4-^)
It is important to observe that these formulas are consequences of the existence
of Green's function for IL Thus they are valid whenever we can find the
function h such that properties (ii) and (in) hold.

482 POTENTIAL THEORY IN E^ 12.5
5. THE POISSON INTEGRAL FORMULA
In this section we shall explicitly construct the Green function for a ball of
radius R. Let Br be the ball of radius R centered at the origin.
For any .r E E"* — {0} let x' be the image of x under inversion with respect
to the sphere of radius R:
Define the function Gr by
1 R''-'-
An-i{n-2)GR{x,y)
\y-4n-2 ||,,||.-.||^,^_^,.||.-,
1 1
l||/y||^^-2 Rn-2
if X 9^ 0,
(5.1)
if X = O.t
If X G Br, then x' ^ Br, and so the second terms on the right-hand side of
(5.1) are continuous and harmonic on Br. We nmst merely check that property
(iii) holds. Now for \\y\\ = R we have, by similar triangles (or direct
computation),
R \\u - A\ ,,^,
ll'^il h - 4
so that
Gr{x, y) = 0 for \\y\\ = R.
(See Fig. 12.5.) This is (iii), so we have verified that Gr is the Green function
for the ball of radius R.
To apply (4.4) we must compute ^cIGr on the sphere of radius R. Now by
(5.1) we liave (for x 9^ 0)
j^n-2
An-\ * dGR{x, ') = T^ — ippn-r^-'
But, by (5.2),
R""
v\\n—2\\i, —
\\,j-.r\\" ||,-||'-2||^_.,'||4*^f/-
if ||/y|| = R.
R'A\y - xt
tif n = 2, sot
(?«(.r, y) = log \\y - x\\ - log M \\y _ ^'|| if :c 9^ Q,
= log |h/|| - log R if X = 0.

12.5
THE POISSON INTEGRAL FORMULA
483
Fig. 12.5
We thus see that for \\y\\ = R,
i*R{An-i * cIGr) = i*R
IR
IR
\y — X
R'-
\\y - x^R
R^ —
"^22y *dy
R^ y wxr^'ji
* dy^
■r dr\ .
But z^(*r dr) = R dSu, where dSn is the volume element on the sphere of
radius R. If we substitute into (4.6), we obtain
u(x)
R'
Wxf
RAr.
u{y)
SrWV
dSu (the Poisson integral formula). (5.3)
In the proof of (5.3) we used the assumption that the function u is differ-
entiable in some neighborhood of the ball Br and is harmonic for ||x|| < R.
Actually, all that we need to assume is that u is differentiable and harmonic for
||x|| < R and continuous on the closed ball ||x|| < R. In fact, for any ||x|| < R,
Eq. (5.3) will be valid with R replaced by Ra, where ||x|| < Ra < R- If we then
let Ra approach R, we recover (5.3) by virtue of the assumed continuity of u.
Equation (5.3) gives the solution of Dirichlet's problem for a ball provided
we know thai the solution exists. That is, if u is any function that is harmonic on
the open ball and continuous on the closed ball, it satisfies (5.3). Now let us
show that (5.3) is actually a solution of Dirichlet's problem for prescribed
boundary values. Thus suppose we are given a continuous function u defined on
the sphere Sr. Then we are given u{y) for all y G Sr. Define u{x) for ||x|| < R
by (5.3). We must show that
a) u is harmonic for ||x|| < R, and
b) u{x) -> uiy^) if X -> 1/0 and ||i/o|| = R.
To prove (a) we observe that Gr{Xj y) is a differentiable function of x and y
in the range ||x|| < Ri < R, Ri < \\y\\ < R^/Ri, and is, by construction, a
harmonic function of y. For ||x|| < R and \\y\\ < R, we know, by (4.4), that
Gr{x, y) = GRiVy x)'

484 POTENTIAL THEORY IN E"" 12.5
Thus for fixed y with \\y\\ < R, Gr{', y) is a harmonic function on Br — {y}.
Letting \\y\\ -^ R, we see that Gr{x, y) is a harmonic function of x for
||x|| < Ri < R for each fixed y G Sr. Thus
is a harmonic function of x for each 1/ G Sr. In other words, all the coefficients
of *dGR{', y) are harmonic functions of x for each y G ^Si^, and therefore so is
each coefficient of u{y) ^ dGr{', y). It follows that the function u{x) =
Jsr u * dGR(Xj •) is a harmonic function of x, since the integral converges
uniformly (as do the integrals of the various derivatives with respect to x) for
||x|| < Ri < R. This proves (a).
To prove (b) we first remark that the constant one is a harmonic function
everywhere, so Eq. (5.3) applies to it. We thus have
R^-Jxf f dSR , „ 11 11 / r> rKA^
—-A n- / Ti ru = 1 for any \\x\\ < R. (5.4)
An-lR Jsr \\y - Xt J W W
Now let yo be some point of Sr, and let u he a continuous function on Sr.
For any e > 0 we can find a 5 > 0 such that
\u{y) — u{yo)\ < e for \\y — yo\\ < 28 {y gSr).
Let Zi= {yeSR: \\y - yo\\ > 28} and Z2= {yGSRi \\y - yo\\ < 28}.
Then by (5.3) and (5.4) we have, for ||x|| < R,
R' - \\xf f u{y) - u{y^)
{An-l)R JSr
nix) - uiyo) = ^f^r-f^ r: zr^dsu,
so
where
and
\u{x) — u{yo)\ < I1 + 12,
_ R^ - ||xf f \u{y) - 1/(7/0)1
'' An-lR Jz, Wlj-Xt
J ^ R^ — \\xf f \u(y) — u{yo)\
^ An-iR Jz2 \\y - ^Ih
dS,
dSi
Now if \\yo — x\\ < 8, then for all y gZi we have \\y — x|| > \\y — yo\\ —
\\x — yo\\, so that \\y — x\\ > 8. Thus for all x such that ||x — yo\\ < 8 the
integral occurring in /i is uniformly bounded. Since ||x|| -^ i^ as x -^ 1/0, we
conclude that /i —> 0 as x -^ 1/0.
With respect to 12, we know that \u{y) — u{yQ)\ < e for all y G Z2, so that
h <
R' - llxf
An — lR
by (5.4). This proves (b).
[ edSR ^ (r' - ||xf f dSR \
Jz2\\y-xt V An-lR JsR\\y-x\\/

12.6 CONSEQUENCES OF THE POISSON INTEGRAL FORMULA 485
We have thus proved:
Theorem 5.1. Let u he Si continuous function defined on the sphere Sr.
There is a unique continuous function defined for \\x\\ < R which coincides
with the given function on the sphere Sr and is harmonic for ||x|| < R.
This function is given by (5.3) for all ||x|| < R.
Fig. 12.6
6. CONSEQUENCES OF THE POISSON INTEGRAL FORMULA
In the proof of Theorem 5.1 we assumed that u is continuous in the closed ball,
possesses two continuous derivatives in the open ball, and is harmonic in the
open ball. However, it is clear from (5.3) that u possesses derivatives of all
orders when ||x|| < R. In fact, if ||x|| < Ri < Rj we can differentiate (5.3)
under the integral sign any number of times, since all the integrals we obtain
will be uniformly integrable in ||x||. Now if u is harmonic in some open set U
and X G Uy we can choose R sufficiently small so that the ball of radius R
centered at x will be contained in U, (See Fig. 12.6.) We can then apply (5.3).
We thus conclude:
Proposition 6.1. Let u he Si function defined on an open set U, possessing
two continuous derivatives, and satisfying Au = 0. Then u has continuous
derivatives of all orders.
We can improve Proposition 6.1 by examining (5.3) a little more closely.
Let \\y\\ = R, and let Ri < R. Therefore, all ||x|| < Ri. The multinomial
theorem allows us to expand
= L ^ii in(y)^i'''' ^n"", (6.1)
\\y - x\\
where the coefficients depend on y but the series converges uniformly in x for
all \\y\\ = R, Furthermore, if D is any operator of partial differentiation with
respect to x, then we obtain an analogous expansion
D{\\y - x\\-"} = z 5„. aSy)^V ■ ■ ■ 4",
where this series is obtained from (5.4) by term-by-term differentiation and
converges uniformly in the same region. If we now substitute (6.1) into (5.3),
we can integrate the series term by term because of the uniform convergence in

486 POTENTIAL THEORY IN E^
y, and we conclude that
12.6
U{x) — zL ^ai,...,a„^l^ * * * ^n"j
where the series converges uniformly for ||x|| < Ri. Furthermore, we can
differentiate the series term by term. Doing this and evaluating at x = 0,
we see that
whereas <q:i, . . . , q:^> .
We thus have proved:
Proposition 6.2. Let u be harmonic in the ball ||x|| < R. Then
^W = Z A ^^"^(O)^", (6.2)
where the Taylor series (6.2) converges for all ||x|| < R and converges
uniformly for all ||x|| < Ri < R. In particular, u is determined throughout
the ball by its value and the value of all its derivatives at the origin.
Fig. 12.7
Let U be some connected open subset of E^, and let u and v be two harmonic
functions on U. Suppose that u and v have the same value and the same
derivatives of all orders at some point x G U. Let y be some other point of U. We
can then connect x to i/ by a series of balls lying in U, where the center of each
ball lies in the interior of the preceding ball, and such that x is the center of the
first ball and y lies in the last ball. (See Fig. 12.7.) We thus conclude that
y^iy) = v{y).
Thus we have:
Proposition 6.3. Let u and v be harmonic functions defined on an open
connected set U, Suppose that u and v, together with all their derivatives,
coincide at some point of U. Then u ^ v onU, In particular, if u{x) = v{x)
for all X G IF, where W is some open subset of [7, then u{x) = v{x) for
all X G [/.

12.7 harnack's theorem 487
We continue to examine the consequences of (5.3). Let u be given by (5.3).
Let D denote any operator of partial differentiation with respect to x. Thus
^ai+...+a„
Then D^u{x) is given by an integral over Sr of a function involving, at worst,
inverse powers of \\y — x\\ for y on Sr and the function u on Sr. In particular,
if ||x|| < Ri < R, then we can estimate the maximum absolute value of D^u
in terms of the values of u on Sr and the difference R — Ri. In short,
iD^'uix)] < c(D'', R, Ri) max \u(y)\ for ||x|| < R^ (6.3)
\\y\\=R
where c depends only on a, R, and Ri. Now suppose that u is harmonic in some
open set U, and let Ki and K2 be compact subsets of U with
Ki C int K2CK2C U.
About each x G Ki we can draw an open ball Br^ such that Br^ C int K2,
so that
Sr^ C K2.
We can also draw a ball Br^^ of slightly smaller radius about x. Now the open
balls Br^^ cover Ki. Since Ki is compact, we can select a finite number of these
balls which cover Ki, By applying (6.3) to each of these balls, replacing \u{ij)\
by the larger max^e^a W{^)\-, and using the smallest of the finitely many
constants c, we conclude that
max \D''u{x)\ < c(a, Ki, K2) max \u(z)\. (6.4)
xGXi 2GX2
In particular, we have:
Proposition 6.4. Let {uk} be a sequence of harmonic functions defined on
an open set U that converges uniformly on any compact subset of U. Then
the sequence of partial derivatives {D^u} also converges uniformly on any
compact subset. In particular, the limit of the sequence is again a harmonic
function.
In fact, for any compact subset Ki we can always find a compact subset K2
such that Ki C int K2. Applying (6.4) to the harmonic functions u^ — Uj
establishes the uniform convergence of {D^uj^} on Ki. Since the sequence of
partial derivatives converges uniformly, Au = lim Auk = 0.
7. HARNACK'S THEOREM
We continue to reap the consequences of Eq. (5.3). In addition to what we
assumed in the preceding section, let us suppose that u{y) > 0 for all y G Sr.

488
POTENTIAL THEORY IN E""
12.7
y--^\\
Now for ||x|| = Ri < R we have (see Fig. 12.8)
R - Ri < \\y - x\\ < R + Ri for all
Then, by (5.3),
Fig. 12.8
11^11 = R'
R'
An-lR(R +
R\ f
+ Rir Js
udSR < U{x) < -r-
R'
R
.iR(R
udSi
Now according to (2.5), the integrals occurring on the right and left of this
inequality are equal to An-iR'^~^u{0). Thus
/ r>2 r>2\ r>n—2 / r>2 r>2\ r>n—2
[K — III) ti (f\\ ^ ( \ ^ \'^ ~ t(j\)K ,^.
-1/(0) < U{X) < -r^ ^-^^ 1/(0).
{R+ Rl^
(R - Rir
(7.1)
Inequality (7.1) is known as Harnack^s inequality. It has the following
consequence. Suppose that {uj^} is a sequence of functions satisfying (5.3) and
such that
ui(y) < U2{y) < ■ ' ' < Uk{y) < • • • for all ye Sr.
If we apply (7.1) to the functions Uj — Ui (j > i), we conclude that
\ujix) - Ui{x)\ < d{R, Ri)\ujiO) - u^{0)\ for all ||:r|| < Ri,
where d{R, Ri) depends only on R and Ri.
In particular, if the sequence converges at the origin, it converges uniformly
for ||:r|| < Ri < R, By applying our usual device of joining two points by a
sequence of balls, we deduce:
Proposition 7.1 {Harnack^s theorem). Let {uk} be a sequence of harmonic
functions defined on a connected open set U and such that
Ui{x) < U2{x) <
for all X gU.
(7.2)
Suppose that the sequence {uk{p)} converges for some p gU. Then the
sequence of functions {uk} converges uniformly on any compact subset of U
and, by Proposition 6.4, the limit function is again harmonic.

12.8 SUBHARMONIC FUNCTIONS 489
A useful variation of Harnack's theorem is:
Proposition 7.2. Let {uk} be a sequence of harmonic functions defined on
an open set U and satisfying (7.2). Suppose that there is a constant M such
that Uk(x) < M for all k and all x gU. Then the sequence of functions
{uk} converges uniformly on any compact subset of [/ to a harmonic
function.
To prove Proposition 7.2 we remark that this time the convergence of
{uk(x)} at each x G U is automatic because it is a monotone (nondecreasing)
sequence of bounded real numbers. Proposition 7.1 guarantees that the
convergence is uniform on compact subsets to a harmonic limit function.
8. SUBHARMONIC FUNCTIONS
In this section and the next we shall show that Dirichlet's problem can be solved
for bounded open sets U whose boundaries satisfy certain regularity conditions.
The proof that we shall present (there are many others) is due to Perron and
makes essential use of the concept of a subharmonic function. Let i/ be a function
defined on the open set U. We say that u is subharmonic if
a) u is continuous; and
b) for any connected open subset W of U and any harmonic function, v
defined on W, the function u — v satisfies the maximum principle on W.
In other words, for any such W and v, if there is some Xq gW with
u{x) — v{x) < u(xo) — v(xo) for all x GWy
then u{x) — v{x) = u{x(y} — v{xo) in W.
In order to understand condition (b) a little better, we study some of its
consequences. First of all, we can let v be the harmonic function that is
identically zero. Then (b) says that u satisfies the maximum principle on every open
subset of U,
Next we let Br be some ball with center z whose closure is contained in U.
In particular, its boundary, Sr, is contained in U, and the function u is
continuous on Sr. We can therefore find a harmonic function v defined on the open
ball and taking on the values u(y) for y G Sr. We take W to be the open ball
and V to be the function we have just constructed. Then u — v vanishes on Sr,
so (b) implies that u{x) < v{x) for all x gW. In particular, u{z) < v{z). But
We have thus shown that if t^ is a subharmonic function defined on U and Sr
is a sphere with center z lying in [/, then
*(^)^A„_'ii;"-i/.,"^'^«-

490 POTENTIAL THEORY IN E^ 12.8
In other words, a subharmonic function is always less than or equal to its
average value over a sphere about a point. This property is frequently taken as the
definition of a function's being subharmonic, and the hypothesis of continuity is
somewhat relaxed. However, the definition we gave above is more suitable for
our present purposes.
Let Wi and t(;2 be two subharmonic functions defined on an open set U.
Define the function Wi V W2hy setting
(wi V W2)(x) = max [wi(x)j W2(x)],
The function Wi V W2 is again subharmonic. The fact that Wi V W2 is
continuous is established as follows: Let x be any point of [/. Then Wi V W2{x) is
either Wi(x) or W2{x), say wi{x). Since Wi is continuous, for any € > 0, we can
find a 5 > 0 such that \wi{y) — Wi{x)\ < e for \y — x\ < 8, Similarly, we
can arrange to have W2{y) < W2{x) + € for that same range of y, so that
"^2(1/) < Wi(x) -\- €, since ii;i(:r) = max [wi{x), W2{x)]. For these values of y
we thus have
wi{x) — e < wi(y) < wi y W2{y) < Wi{x) + e,
so that Wi y W2\s continuous. Now let T7 be a connected open subset of U
and let y be a harmonic function on W. Suppose that Wi \J W2 — v takes on
its maximum value at some point :ro of W, Suppose that Wi V t(^2(^o) =
{xq). Then for all :r G T7
Wi{x) — v{x) < Wi y W2{x) — v{x) < Wi{xo) — v{xo).
Since Wi is subharmonic, the right and left sides of this inequality must be equal.
Thus Wi V W2 — V satisfies the maximum principle on W, which shows that
Wi V W2is subharmonic.
Let whe Si subharmonic function defined on an open set U, and let B be a
ball whose closure is contained in U. Let u be the solution of Dirichlet's problem
for the ball B with boundary values w. Thus u is the unique function continuous
in the closed ball, harmonic in the open ball, and coinciding with w on the
boundary S of B. As we have already observed, w{x) < u{x) for x G B. Now
define the function ivb by setting
/ N lw(x) for X G U —~ B,
^^^^)=U(:.) for xeB.
We claim that the function wb is again subharmonic. The proof is just as before.
The continuity is obvious, as before. If W is subset of U and z; is a harmonic
function on W, then wb — v cannot achieve a strict maximum at any interior
point: suppose wb{x) — v{x) < wb{xo) — v{xo) for all x gW, (See Fig. 12.9.)
If xq G U — B, then we h8ivew(x) — v(x) < wb{x) — v{x) < wb{xo) — v(xo) =
^(^0) ~~ ^(^o)- Since w is subharmonic, all these inequalities must be equalities.
On the other hand, ifxoGB, then since wb is harmonic in the open ball, we must
have wb{x) — v(x) = wb{xo) — v{xo) for all x gW. By continuity, this imphes

12.9
DIRICHLET S PROBLEM
491
Fig. 12.9
that wsiy) ~ v{y) = wb{xo) — v{xo) ior y G W 0 S. If W oS p^ 0^ we can
take 2/ as a new xq and the problem is reduced to the previous case.
In shorty to every subharmonic function w defined on U we have attached
another subharmonic function wb such that wb is actually harmonic in the
interior of B and coincides with winU — Bj and such that w < Wb throughout
17. It is clear from the method of construction that if Wi and W2 are two
subharmonic fimctions defined on 17 with Wi < W2, then
9. DIRICHLET'S PROBLEM
Let U be an open subset of E^ with U compact. We say that U has a touchable
boundary if for every p E dU there is a ball B such that B oU = {p}. Thus
Fig. 12.10(a) represents a domain with a touchable boundary, while Fig. 12.10(b)
and 12.10(c) represent domains that have untouchable points on their boundaries.
(a)
(b)
(c)
Fig. 12.10
Let 17 be an open subset of E^ with U compact and with touchable boundary.
Let/be a bounded function defined on dUj and suppose that i/(p)i < M for all
p E dU. Let Wf denote the class of all functions defined on 17 which satisfy the
following two conditions:
a) each w G Wf is subharmonic; and
b) for each p GdU,
lim sup w{x) < /(p).
xeu

492 POTENTIAL THEORY IN E"* 12.9
We can rephrase condition (b) as follows: For each p E dU and each e > 0
there is a § > 0 (which depends on p, w, and e) such that
w{x) < f{p) + € for \\x — p\\ < 8.
Note that the family of functions Wf is nonempty, since the constant function
which is identically —M clearly belongs to Wf. Also note that condition (b)
implies that lim sup w{x) < M sls x —^ dU. Since w gW/ is subharmonic, we
conclude from the maximum principle that \w\ < M for all w G Wf.
Now define the function u/ by setting
Uf(x) = lub w(x).
wGWf
In view of the preceding remarks, the function Uf is well defined and, in fact,
\u{x)\ < M for all x G U. We shall show that if / is continuous on dU, the
function Uf is the solution of the Dirichlet problem for U. We must thus show
that Uf is harmonic and takes on the boundary values /.
We first show, without any continuity restrictions on /, that Uf is harmonic.
To do this it is sufficient to show that Uf is harmonic in any open ball B, where
B CU. Let X be some point of B. Let wi, W2, - - - , Wk, . . . , be a sequence of
functions belonging to Wf such that lim Wi(x) — Uf(x). (Such a sequence
exists by the definition of Uf.) Now define
w\ = Wi,
W2 = Wi V W2j
w^i = Wi V ' ' ' V Wi.
Then w[ < W2 < - - - < wj^ < • • • is a monotone increasing sequence of
functions belonging to Wf with lim Wi(x) — Uf(x). Now replace each Wi by Wis-
The function Wis belongs to T7/, since it is subharmonic and coincides with wl
near dU. Furthermore, since w < wb for any subharmonic function, we can
conclude that lim Wisix) — Uf(x). Inside the ball B, each of the functions wIb
is harmonic, so that the sequence {wis} is a monotone nondecreasing sequence
of harmonic functions in B. Since all the functions of Wj are bounded by M,
we conclude from Proposition 7.2 that the sequence {wis} converges uniformly
on any compact subset of J5 to a function u which is harmonic in jB, and we have
Uf(x) — u(x). Furthermore, by definition, we have u(y) < Uf{y) for any y E B.
We would like to show that we actually have u{y) = Uf{y) for all such y. To this
effect, we follow the same procedure at ?/, namely, we select a sequence {vIb^
where each ViB is harmonic in J5, belongs to T7/, and is such that the sequence is
monotone nondecreasing and lim v^Biy) = '^fiv)- In this way we obtain another
function v which is harmonic in B and satisfies v < Uf throughout B, and
v{y) — Uf{y). Finally, let the functions Si be defined by
Si = W^iB V v'iB,

12.9 dirichlet's problem 493
and consider the functions Sis- On the one hand, they all belong to W/ and are
harmonic in B. Furthermore, we have w^b ^ SiB and Vis < Sis- Finally, the
sequence {s{b} is nondecreasing and therefore converges to a harmonic function
§ in B, By construction, we have
u < s and v < s
throughout B, while u{x) = s(x) = Uf(x) and M(y) = s{y) = Uf(y). Applying
the maximum principle to the harmonic functions u — s and v — s,we conclude
(since the maximimi value 0 is achieved at x and y, respectively) that s =
-M = ?; in all of B. We thus see that Uf(y) = u(y) for any y G B, which shows
that Uf is harmonic in U, Note that in proving that % is harmonic we did not
make use of any properties of the boundary of U or any continuity assumptions
of/.
Fig. 12.11
Now let p G ^17 be a touchable point of the boundary, and assume that / is
continuous at p. We shall show that Uf(x) -^ f{p) as x -^ p. Let the ball Br
of radius R with center z touch Tl at p, that is, Br nU = {p}, (See Fig, 12.11,)
Since / is continuous at p, for any € > 0 we can find an Ri > R such that
l/(g) - f(p)\ < e for all qedUn Br,. (9.1)
Define the function b by setting b(x) = \\x ■— z\\^'^'^ — R^^^J Note that b is
defined and harmonic on E^ — {z} and is negative for \\x — z\\ > R.
Furthermore, b{x) < i?^ < 0 if ||x — z\\ > Ri. In particular, for all qGdU we have
m <^Hq) + e+f{p). (9.2)
In fact, this is just (9.1) if ||g — z\\ < Ri, and it follows from |/(g)| < M if
l\q — z\\ > Ri. By the maximum principle for subharmonic functions we
conclude that w{x) < {2M/K)b{x) + e + /(p) for any w gW/ and any a: e 17.
We thus have
Uf{x) < -^ b(x) +/(p) + € for any x G U.
On t^e other hand, we have, for the same reason as before,
2M
f{p) - € " -^ b(q) < fiq) for any q E BU,
t h{x) = In iR/\\x - z\\) if n = 2.

494 POTENTIAL THEORY IN E^ 12.9
SO that the harmonic function f{p) — e — {2M/K)h actually belongs to Wf.
In particular, we have
/(p) — € 7^ h{x) < Uf{x) for any x E.U.
K
Putting the two inequalities together, we conclude that
\M^) - Kp)\ < ^ Hx) + € for all xeU. (9.3)
Now the function h is continuous and vanishes at p. Thus by choosing x
sufficiently close to p we can arrange that the right-hand side of (9.3) is less than
2€. Thus Uf{x) -^ f(p) SiS X -^ p.
In particular, if all points of dll are touchable, and if/ is continuous at all
points of the boundary, we have proved:
Theorem 9.1. Let U be an open subset of E^ with compact closure and
touchable boundary. Let / be a continuous function defined on dU. Then
there exists a unique function Uf which is continuous on U and harmonic
in U, and which coincides with f on dU.
Some remarks concerning Theorem 9.1 and its proof are in order.
a) The only time we used the assumption that dll is touchable was when
we constructed a function h which was subharmonic in U, vanished at p, and took
on values less than some negative constant on all points oi dll outside some
neighborhood of p. Now a more careful analysis will show that we can always
construct such a function so long as we can touch p from the outside by a cone.
Thus we can solve the Dirichlet problem even at points like the p in Fig. 12.12.
Fig. 12.12
b) On the other hand, some condition on the boundary is necessary. For
instance, if dll contains an isolated point p, then we cannot assign the value at p
arbitrarily. In fact, if u is continuous in a neighborhood W of p and harmonic in
W — {p}, then the Poisson integral formula implies that the various derivatives
of u will also be bounded mW— {p}. Therefore, the proof of the mean-value
theorem applies to the point p itself, and so u{p) is determined and cannot be
assigned arbitrarily. More generally, a more delicate argument shows that
the Dirichlet problem cannot be solved for domains whose boundaries
contain spikes pointing inward or (in dim >3) sufficiently sharp cusps (as in

12.10 BEHAVIOR NEAR THE BOUNDARY 495
Fig. 12.13
Fig. 12.13). This is the mathematical analogue of the physical fact that a very
sharp conductor cannot hold a charge, but will spark. (The relation with
electrostatics will be discussed in Section 12.)
c) For the purpose of applying the results of Section 4, i.e., the construction
of the Green function of the domain and the various identities. Theorem 9.1
is still not quite enough. We need to know not only that the Green function
exists, but also that it has continuous derivatives up to the boundary. This fact
requires further argument and additional assumptions about the nature of the
boundary; it will be discussed in the next section.
Fig. 12.14
10. BEHAVIOR NEAR THE BOUNDARY
The purpose of this section is to discuss the behavior of the solution of the
Dirichlet problem near the boundary. In fact, we shall prove:
Theorem 10.1. Let U he Si domain with regular boundary and compact
closure in E^. Let / be a continuously differentiable function defined on dU,
and let u be the solution of the Dirichlet problem with boundary values /.
Then the partial derivatives du/dx^ can be extended to continuous functions
on U. Furthermore, if p ^ dU and J = < ?\ • • • , ?^> is tangent to
the boundary at p, then
{^,df) = {^,du) = j:(du/dx')e-
The following proof was suggested to us by Professor Ahlfors and is
reproduced here with his kind permission.
Proof. Let p be some point of dll, and let us arrange, by a Euclidean motion,
that the tangent plane to dU 2it p is the plane x'^ = 0. (See Fig. 12.14.) Then
nearpthepointsof^t/are all points of the form (x\ . . . , x^~\ <p{x^,. • •, x'^~^)),

496 POTENTIAL THEORY IN E^ 12.10
where <^ is a function defined near the origin of IR^~^ and vanishing at the origin,
together with all its first derivatives. In particular, there is a constant C > 1
such that
\Ax\..., x^-')\ < c{{x'f + ■■■ + ix^-'n
in some neighborhood of the origin of W^~^. We can therefore choose a
sufficiently small R < 1/2C so that the (open) ball of radius R with center
(0, . . . , 0, R) lies entirely within U and the ball of radius R with center
(0, . . . ,Qj —R) lies inside the complement of U. (See Fig. 12.15.)
Let z = (0, . . . ,0, —R). Then for all q E: dll sufficiently close to p we have
\\q - zf = {x'r + . . ■ + (a;—^)2 + R^ - 2E<p{q) + /(g)
> (1 " 2RC){{x^Y + . . . + {x'^-^f} + fi2 ^ /(g),
so that
\\q - z\\- R> k{(x'y H h (x^'-Y},
where k is some positive constant. Let (p(r) = r^"^ (=ln r if n = 2). Then
(p'(R) 9^ 0, and therefore |(p(||g — ^Ij) — ip{R)\ > C\\\q — z\\ — R\ for a
suitable C > 0. In particular, we have the inequality
1
'|g — z|h-2 R^-2
or
iig — z
In-
>K{(xy + --- + (x^-Y} (10.1)
> KixY if n - 2
R
for some positive constant K.
Now let us replace the function / by / - [/(p) + Y.V^ W/dx'){p)x%
Then u — [/(p) + YJi"^ {df/dx'){p)x''\ is the solution of the corresponding
Dirichlet problem. In this way we may assume (changing our notation
accordingly) that /, together with its first derivatives, vanishes at p. Since / is assumed
to have continuous first derivatives, we may apply Taylor's theorem to conclude
that there exists a c > 0 such that for all qon dU sufficiently close to p we have
|/(g)| <c{{x'Y + --- + {x--'Y}. (10.2)
As in the last section, let
h{x) - \\x - ^f-^ - i^'-"
— In -M \r if n — 2.
h - ^11
If we compare (10.2) with (10.1), we see that for all q E dU sufficiently close to
p we have
1/(3)1 < I l&(3)l-
On the other hand, since b is strictly negative outside some neighborhood of p

12.10
BEHAVIOR NEAR THE BOUNDARY
497
ie«-^0
«(0,...,0, "20
Fig. 12.15
Fig. 12.16
(on dU)f and since / is bounded on dUj we can (using the above inequality for
all q near p) find a constant A such that
|/(^)i < A\biq)\ for all q E dU. (10.3)
Since the function b is harmonic in C7, the maximum principle applied to m — Ab
and Ab —■ u allows us to conclude that
\u{x)\ < A\bix)\ for all xeU, (10.4)
Now the function I? is a differentiable fimction of the distance from z which
vanishes when this distance is R, In particular, there is a constant a > 0
such that
\b{x)\ < ad{x),
where d(x) denotes the distance from x to the sphere of radius R with center z.
Now let B be the ball of radius R with center —z = (0^. . . ^ 0, U), so that B
lies in U^ and let S = dB. Note that S is tangent to the sphere about z at the
point p. (See Fig. 12.16.) Thus for points y on Sj d{y) < Ci\\y — p\\^. If we
substitute this into (10.4) using the previous inequalityj we conclude that
\u{y)\ < L\\y - p\\
for all y &S,
(10.5)
for some constant L. We apply the Poisson integral formula to the ball B and
the fimction u and differentiate with respect to x* to obtain
/u
(y)
■dS +
R'
\\x + z\\
An~lR
/w
II
iy)ix' - y')
2/11"
+2
dS.
(10.6)

498 POTENTIAL THEORY IN E^ 12.10
Now let X be on the normal to the boundary through p, so that x = (0,. . . , 0, x'^).
If |x^| < R, then
\\y -p\\ < \\y -x\\ + \x^\ < 2\\y ~x\\.
Thus
[ r^^in ds\ <2^f II '""^^^L dS < 2^2) f\\y - pf "^ dS.
J \\x — 2/11^ I J \\p — 2/lh J
Since the sphere is {n — 1)-dimensional, this last integral converges absolutely,
and we thus see that the first integral in (10.6) is uniformly absolutely
convergent. (Note that the term containing this integral vanishes if i < n.) We now
show that the second term in (10.6) tends to zero as x'^ tends to zero. In fact,
R^ — \\x + ^11^ = x^{2R — x^), so we can write the second term of (10.6) as
An_iR J \\x-yt+^
This time, since x" < \\x — y\\ for all y & S, we can assert that the integrand
is smaller in absolute value than
\u{y)\W--y'\ \u{y)\ k(^)|2-+^^^ ^ m^+i/^ii,, ^ ^||3/2-n
\\y _^ a:||^+3/2 - \\y _^ :r||n+l/2 - \\y _ p||n+l/2 ^ ^^ 11^ PW
which is again absolutely integrable. Therefore, the integral occurring in the
last expression is uniformly bounded for all values of x'^ > 0. Since (x^)^^^ -^ 0,
we conclude that the second term of (10.6) approaches zero.
If we now rephrase the result we have obtained independently of the
special choice of coordinates, we see that we have proved the following: Let p
be any point of dUj and let x be a point on the normal line to dU through p.
If ^ is any vector parallel to a tangent vector at p, then (^, du) -^ (^, df) as
X -^ p along the normal line. If d/dn denotes the unit vector in the normal
direction (pointing into U), then
) _, _z^ [ <y) - y ds, (10.7)
/ An-i J y - pt
E~ '^^ / A I II
dn / An-i J \\y — p
where the integral is taken over the sphere of radius R tangent to dU dhi p and
lying inside U.
So far, we have proved convergence only \ix -^ p along the normal direction.
If we go back and examine the argument, we see that the radius R and the
constant D in (10.5) can be chosen uniformly for all p G dU. (In fact, since dU
is compact, we can cover ^[/ by a finite number of neighborhoods, of type (iii),
such that a ball of radius R about each p E: dll lies in one of these neighborhoods,
where R is sufficiently small. In such a situation it is clear that the choice of R
(still smaller, if necessary) depends only on the second derivatives of the change
of charts from Euclidean coordinates to the adjusted charts. Also the choice
of D depends only on the constants involved in the Taylor expansion of / and
on R. Thus the choice of R and D can be made uniformly.) Since, by assump-

12.11 dirichlet's principle 499
tion, the partial derivatives of/are continuous and the right-hand side of (10.7)
is continuous in p, we conclude that the limits of the partial derivatives of u
exist as we approach the boundary in any direction, and that in fact we have
proved Theorem 10.1. D
We can now construct a Green function for any domain with regular
boundary by the solution of Dirichlet's problem, as in Section 4. Theorem 10.1
tells us that it is differentiable in the closed set U in the sense that the partial
derivatives are continuous up to the boundary. If we want to derive the various
formulas of Section 4, we can do so by applying a limit argument: We simply
replace [/ by a smaller domain U^ such that U^ -^ U and dU^ -^ dU SiS a -^ 0.
(Actually, a more careful and delicate argument will show that if / has two
continuous derivatives, then the du/dx^ we obtain as limits on the boundary are
in fact continuously differentiable. Since du/dx^ is the solution of the Dirichlet
problem with these boundary values, we conclude that the second derivatives
of u can be extended so as to be continuous on U. In this way one can prove that
if / is C°°, all the derivatives of u can be extended so as to be continuous on U.
In particular, all the derivatives of the Green function G{x, •) can be extended
to continuous functions at the boundary for any x E: U.)
11. DIRICHLET'S PRINCIPLE
The solution of Dirichlet's problem has an interesting and useful
characterization as the solution of the problem of minimizing the Dirichlet integral
D[Uj u] = JYl {du/dx'^Y with given boundary values. More precisely.
Theorem II.I (Dirichlet's principle). Let U he a, domain with compact
closure and regular boundary, and let / be a continuously differentiable
function on dU. Let u be the solution of the Dirichlet problem with
boundary values /, and let lo be any function which is differentiable in U
and takes on the values of / on the boundary, and whose derivatives can
be extended to be continuous in U. Then
D[u,u] < D[w,wl (11.1)
and equality holds if and only if u = w.
Proof. Let us write w = u ^ v, where now v is continuously differentiable on
U, continuous on [/, and vanishing on dU. Then
D[w, w] = D[u + v,u + v] = D[u, u] + D[v, v] + 2D[u, v], (11.2)
since D is bilinear and symmetric in its arguments.
Now suppose for the moment that u possesses two continuous derivatives
and V possesses one continuous derivative in a neighborhood of U, Then
D[u, v] = Du[u, v] = dv * du = I v * du — vAu = 0. (11.3)
Ju Jdu Ju

500 POTENTIAL THEORY IN E"" 12.12
Now D[v, v] = JX! (dv/dx^)^ > 0 and vanishes if and only if all the derivatives
of V vanish, so that z; is a constant on each component of U. Since v vanishes on
dU, we conclude that D[v, z;] = 0 if and only if z; = 0.
In case u and v are not necessarily differentiable in a neighborhood of dU,
we establish that D[u, z;] = 0 by writing
Du[Uj v] = lim Du^[Uj v]j
where U^ is a sequence of domains which approach [/ as a -^ 0 and dll^ -^ dll.
Applying the previous argument to each of the domains U^j we conclude that
D[u, z;] = 0 and D[v, z;] = 0 if and only if z; = 0. This proves the theorem. D
12. PHYSICAL APPLICATIONS
The study of Laplace's equation and its solutions plays an important role in
many theories of classical physics. This is essentially due to the intimate
connection between the Laplace operator and Euclidean geometry. Since this
is not the place to elaborate on the physical applications, we refer the reader to
any physics text for the details. (See, for instance, Feynman, Leighton, and
Sands, The Feynman Lectures on Physics, Addison-Wesley, Reading, JNlass.,
1964, vol. II, especially Chapter 12.) We shall briefly mention some of the
relevant physics in this section.
In electrostatics it is assumed that the electric field E satisfies the equations
d* E = p, dE = 0,
where p is the density of charge and we identify the vector field E with a linear
differential form via the Euclidean metric on E^. Here p is assumed, for the
moment, to be a smooth density. (It is also convenient to consider the limiting
cases of "surface distributions" and "point distributions".) By the second of
these equations, we know that E can be written as
E = —d(p,
where <^ is a smooth function known as the potential of the electric field. The
first equation then implies that
A(p = d * d(p = —p.
If p is given, then (p is determined by the formula
^^^^ 4:7r J \\y — x\\
(since ^2 = 47r). As limiting cases, we obtain, for charge distributed along a
surface S with density a dA, the formulas
^^^ iTTjs \\y - x\\

12.12 PHYSICAL APPLICATIONS 501
where dA is the area density of .the surface, and
where Ids is a, linear distribution along a curve C,
For a point charge located at Xj we obtain
where e is the magnitude of the charge.
There can be no electric field along a conductor (since this would result in a
motion of the charge, which contradicts the assumption of a static field). Thus
d(p = 0 and (p = const.
In most problems that arise the distribution of charge is not known in
advance, but must be determined. For example, suppose that a unit charge is
placed at a point x inside a cavity whose boundary is a conductor which is
grounded, i.e., kept at zero potential. (See Fig. 12.17.) We want to find the
electric field inside the cavity. There will be a charge distribution induced on
the boundary surface that we also wish to determine. Since there is no charge
distribution anywhere inside the cavity, except at x, the potential (p must be
harmonic everywhere except at x, and, in fact, differs from
1
47r|| • — a^ll
by a harmonic function. Since we want «^ == 0 on the boundary, we see that the
desired potential <p is exactly the Green fimction for the domain. The surface
density can then be determined from d(p along the boundary. (This is another
reason why it is important to know that the solution of Dirichlet's problem is
differentiable at the boimdary.)
More frequently, we are interested in the electric field outside conducting
surfaces. Here, strictly speaking, the theory we have developed of Dirichlet's
problem does not apply, since we considered only bounded domains. We can
handle this problem in one of two ways. First, we can reduce to the Dirichlet
problem by considering everything contained inside a very large conducting
sphere (at potential zero). Then we let the radius go to infinity to get the
desired potential as a limit. Second, we can modify our arguments in Sections 3
through 10 to include the case of unboimded domains, but restrict all our
functions to vanishing as c/r at infinity. The details are left to the reader.
In the theory of diffusion one assumes that material of some nature is
flowing according to a vector field X. If the density of the material is p dV,

502 POTENTIAL THEORY IN E"* 12.12
then the amount of material flowing per unit time through an oriented piece of
surface S is given by
f p(X, Ti)dA=f *pX,
Js Js
where n is the unit normal vector to the surface, dA is the element of area, and,
we are regarding pX on the right-hand side as a linear differential form via the
Euclidean metric. Thus the net amount of material flowing out of any region is
d * pX = div <X, p>. Now "material" may be produced in the region at a rate
s dV (where s is the function describing the rate of productivity of the sources
in the region). Thus the total change of density is given by
^dV = div <X,p> -» sdV.
If, as we shall assume, the situation is stationary, i.e., the density does not
change, we obtain the equations
div <X,p> = sdV.
On the other hand, it frequently happens that the flow is given by the gradient
of some function, i.e.,
PidX = dN
for some function N. Combining these equations, we obtain
AN = s.
In the theory of heat N is the temperature, pi^X is the rate of flow of heat, and
s is the density of the sources of heat. The equation pi^X = dN says that the
rate of flow of heat is proportional to the gradient of temperature. In the theory
of diffusion of particles it is assumed that the rate of flow of the material is
proportional to the change (i.e., gradient) of the density of the particles. (This
says that the particles tend to flow from a region of higher density to a region
of lower density.) Then N represents the density of the particles and s represents
the rate of production of particles.
Suppose, for example, that the boundary of a region D is kept at some fixed
temperature /, where / is a given function on dD and there are no sources of heat
inside D. Then the distribution of temperature in D is given by the function N
which satisfies the differential equation AN = 0 and the boundary condition
N = f on dD. In other words, the temperature is determined by the solution
of Dirichlet's problem.
In the theory of steadyj incompressiblej irrotational flow of a fluid it is assumed
that a vector field X is given which represents the flow of the liquid. By the
incompressibility it follows that c? * X — 0. It is assumed that there is no
circulation, that is, dX = 0. Thus X ^ d(p for some function cp which is a
solution of Laplace's equations. The natural boundary-value problems that arise
in this case are different from those we have been discussing, so we simply refer
the reader to standard works on fluid mechanics.

12.13 PROBLEM set: the calculus of residues 503
SUPPLEMENTARY EXERCISES
1. Let Vi and V2 be vector spaces with scalar products. A linear map I: Vi —> V2
is said to be conformal (or a similarity) if
(Iv, Iw) = civ, w),
where c is a positive constant. (Note that if c = 1, then I is an isometry.)
Let Ml and M2 be manifolds each with a Riemann metric. A differentiable (C°°j map
(f is said to be conformal if <^*^ is conformal for each x G Mi. Thus, to say that (p is
conformal means that
i^*.^i, <P*.h) = c{x){ii, b) if h, h G T,{Mi)
for any x G Mi. (Note that this time the c can depend on x.)
a) Let ip: [7 —> R^ be a conformal map, where [7 C U^^ and we use the Euclidean
metric on U and on R^. Suppose (p is given by
ip{x,y) = (u,v),
where u = u{x, y) and v = v{x, y), and where {x, y) and {u, v) are rectangular
coordinates in the plane. Show that either the equations
hold or the equations
hold,
b) Conclude that if u and v are as in (a), then they are both harmonic functions.
2. a) Let w be a harmonic function defined on all of W'. Show that if u is bounded
then it must be a constant,
b) Using (a) and Exercise 1, show that there is no conformal map of R^ onto a
bounded subset [7 C R^.
13. PROBLEM SET: THE CALCULUS OF RESIDUES
On a manifold M we can consider complex-valued smooth functions,
differential forms, etc. For instance, we say that the function
f =: u -\- iv
is a complex-valued C°°-function if each of the real-valued functions u and v are
real-valued C°°-functions. Similarly, we consider'the complex-valued linear
differential form
CO = (7 + iT,
where (7t G /\^{M) are linear (real-valued) differential forms. We can then
perform the usual operators, remembering that z^ = — L Thus
du
dx
du
dx
dv du
dy' dy
dv du
dy' dy
dv
dx
dv
dx
fo) = (ua — vt) + i{v(T + IJ-t)'

504 POTENTIAL THEORY IN E^ 12.13
Similarly, if
0)1 = (Ti + iTi and C02 = 0'2 + ^^2,
then
coi A C02 = (cFi A 0-2 — Ti A T2) + i{cri A T2 + Ti A 0-2)
is a complex-valued exterior two-form.
We similarly have the operator d given by
df = du + i dv, do) = da + i dr, etc.
If Xi and X2 are vector fields, we define the "complex vector field" Xi + ^'^2
as that differential operator on complex-valued functions which is given by
(Xi + iX2)/-Xi/+iX2/
- {X^u - X2V) + i{X^v + X2U).
From now on, let M be an open subset of R^ with rectangular coordinates
{x, y). We set 2 = a^ + iy, so
dz = dx-\- i dy.
We let
2 =: X -— iy, dz = dx — i dy,
and then define the complex vector fields d/dz and d/dz by
d_
dz
\(d .d\ . d l/d . .d\
We then set
dz
for any complex function /.
EXERCISES
13.1 Show that for any complex-valued smooth function / we have
df = %dz + %dz.
dz dz
13.2 Show that Leibnitz's rule holds for d/dz and d/dz; that is,
A function/is called/lo^omorp/iic (or complex analytic) if ^//^2 = 0. For a holomorphic
function / we write
^ dz

12.13 PROBLEM set: the calculus of residues 505
13.3 Show that dz and dz are independent in the sense that
if adz-^^h dz = Oj then a = 0 and 6 = 0.
13.4 Conclude that / is holomorphic if and only ii df = h dz for some h.
13.5 Show that
diz"") = nz""-^ dz
(on R^ — {0} if n < 0) for all integers n, so that z"^ is holomorphic.
13.6 Conclude that every polynomial p given by p{z) = J^q ^kZ^ ^^ holomorphic and
that
n
p'(z) = X kakZ^~^.
1
13.7 Define the function e^ by setting e^ = e^(cos y -j- i sin y). Show that
de^ = e^ dz.
13.8 Show that the product of two holomorphic functions is holomorphic.
13.9 Let / = w + ^^ be a holomorphic function, and consider the map sending
(^j y) -^ (wj v) as a map oi M d^^ into R^. Show that this map is conformal and that
its Jacobian determinant at the point {x, y) is |/'(2;)P \i z = x^ iy.
13.10 Let gr be a holomorphic function defined on the image of M under the map of
M C K^ corresponding to /. Define g ° fhy
9 ° fix, y) = 9(u(x, y), v(x, y))
[which we can write, for short, as g ^ f(z) = g(f(z))]. Show that g ^ f is holomorphic
and that
ig°f)'iz) = 9'{fiz))f'(z).
13.11 Let U he Si domain with almost regular boundary in the plane. By Stokes'
theorem we have
[ V = [ dv
JdU Ju
for any complex-valued linear differential form v. (This is to be interpreted, as usual,
as the equality of the real and imaginary parts of both sides.) Conclude that
/ g dz = I dg A dz = I —dz A dz.
JdU Ju Juoz
13.12 Show that dz A dz = 2i dx A dy, so that
/ gdz = 2i /dx A dy = 2i /'
JdU Ju dz Ju oz
13.13 Conclude that if / is a smooth function defined in a neighborhood of U, and
if / is holomorphic in U, then
f fdz = 0.
JdU

506 POTENTIAL THEORY IN E^
12.13
Fig. 12.18
Fig. 12.19
13.14 Let J = ^ -\- ir}, where (f, ??) E t/ and g is a smooth function defined in a
neighborhood of U. (See Fig. 12.18.) Show that
9(0 =
1
2Tn
JdU ^ — k Ju
dU/dz
dz A dz
[Hint: Apply Exercise 13.11 to the function U{z)/{z — J) and the domain U — Bg,
where B^ is a disk of radius e about J. Then let e —> 0.]
13.15 Conclude that if / is holomorphic in [7, then for any J E f/,
M)
2« Jev
m
dz.
IdU Z — ^
13.16 Let / be holomorphic in U — {ai} U • • • U {ak} (and let / be smooth in
IF - {ai} U--- U {ak},
where IF 3 C/). Suppose that in a neighborhood of each ai
f(z) = BhU^ - a,)"'^' + • • • + Bi,i{z - a)-' + <pi{z),
where cpi is holomorphic in the neighborhood. (See Fig. 12.19.) Show that
if
27ri Jbu
J{z)dz = B11 + B12-] \-Bik.
[Hint: Apply Exercise 13.13 to U — U A.e where Di,g are small disks around the ai,
and let e —> 0.]
The result of Exercise 13.16 can frequently be used to evaluate definite integrals. For
example, suppose we wish to evaluate
/•2ir
/ R{cos e, sin 6) dS,
Jo
where R is some rational function of two variables. If we set z = e^^, then this integral
becomes
•i, 4
^{z+z ').^(^-^'
■')]!■

12.13 PROBLEM set: the calculus of residues 507
where Di is the unit diak. To apply Exercise 13.15, we must merely find the points %
and the coefficients Bij to evaluate the integral. For instance, if a > 0,
Jo a + COS ^ 2 Jo a + cos ^ JdDi
dz
z^+2az+ 1
Now 2^+ 2a2+ 1 = (2 — ai)(2 — 02), where ai = —a+ (a^ — 1)^''^ and a2
—a — (a2 — 1)1/2^ Thus only ai E Di, and
1
^2 + 2a5J + 1 ai
so the integral is evaluated as
Jo a-\- cos B
/2
- 02 V — ai ^ — 02/
13.17 Evaluate
13,18 Evaluate
L
dS
0 a + sin2 6
2jr
(a > 0).
/
Jo
™'^ :de.
/o a + 5 cos ^
Let P and Q be polynomials such that Q(x) 9^ 0 for any real x. Suppose that
P(z) = a„„22;"~2 + a„^32:*--3 H \-aiz+ao
and
Q(2) = ft^^'^ + hn^iz^-^ H h 6i2 + &0,
where hn 9^ 0. In other words, the degree of Q is at least two more than the degree of
P. Then P/Q is absolutely integrable and
l^dx= lim / ^.
Fig. 12.20
Now consider §bu [P{z)/Q{z)] dz, where 17 is a semidisk of radius R. (See Fig. 12.20.)
The integral over d U splits into two parts:
r p{x) f p(z)

508
POTENTIAL THEORY IN E^
12.13
where Tr is half the circle of radius R. For large values of R we have, for ||2;|| = R,
\P(z)\
Q(z)
^ \an\ + \an-i\'-^-] i-ko|-^;r=^ ^
- ^7 ] Tx ^'r^'
so
(n-|6„-.|-| N-^)
f 1^,11 - \-7?rEi^
KQiz)n - rJo Q{Re«>.
n --R
and thus lim/2_,oo Str = 0. We can then apply Exercise 13.16, since there will be only a
finite number of zeros of Q in the upper half-plane.
13.19 Evaluate
J —c
dx
(^2 _|_ ^2)2
13.20 Show that
/■
J —i
dx
(a > 0).
(2n-2)!
(a;2+ 1)*^ 22('^-i) [(n — 1)!]2

CHAPTER 13
CLASSICAL MECHANICS
In this chapter we shall present a brief introduction to the study of classical
theoretical mechanics. We emphasize that what we are studying is a branch
of mathematics—idealized from the mathematical considerations common to
many problems arising in the physics of mechanical systems. We will thus
formulate, on an axiomatic basis, a mathematical model that describes features
that arise in the study of the equations of mechanics. The model will apply to
most situations arising in the mechanical applications. In order to simplify the
mathematics, we have sacrificed a certain amount of generality, and therefore
there will be some situations in classical mechanics where our model is inadequate.
Mechanics is devoted to the study of how a physical system evolves in time.
The first fundamental assumption is that the system can be described (locally)
by a collection of continuous parameters. More precisely, we assume that the
set of all possible "positions" or configurations of the physical system is a
differentiable manifold M which is called the configuration space.
For example, if the physical system consists of three particles free to move
in space, we can describe the "position" of the system at any given instant by
giving the positions of each of the three particles. Thus, in this case, the
configuration space is
E^ X E^ X E^.
If we insist that no two particles be able to occupy the same position of space at
the same time, then the configuration space M is given by
M - E^ X E^ X E^ - >S,
where
S = {<Vi, V2, vs> : Vi G E^ and vi == V2 or vi = vs or V2 == Vs}-
If the physical system is a rigid body (say a top) spinning about some point
held fixed in space, then the position of the system is completely described by
giving the position in space of three orthonormal vectors on the body (drawn
through the fixed point, say). Since these vectors can be placed in any position
but must remain orthonormal and cannot change orientation (from right- to
left-handedness), we see that M is the set of all oriented orthonormal bases of E^.
In this case M is a three-dimensional manifold. If we fix some arbitrary initial
orthonormal basis <ei, 62, e^^, then any possible position of the system is
509

510 CLASSICAL MECHANICS
given by <Vi, V2, vs>, where Vi — Aei and A is a rotation, that is, A is an
orthogonal linear transformation with positive determinant. Thus M is diffeo-
morphic to 0"^(3), the space of all orthogonal three-by-three matrices with
positive determinant.
The basic problem of mechanics is to describe how the configuration of the
system changes in time. As the system evolves, the configuration at any instant
t will be given by some point C{t) G M. Our problem is to give a reasonably
simple description of the possible curves C which can arise as actual changes
in the configuration of the mechanical system.
The second fundamental assumption of classical mechanics is that the
curve C{t) can be determined from a knowledge of the "state" of the system
at any given time. That is, we may (and, in general, will) have to know more
about the system at a given time than its configuration in order to be able to
predict its future configuration. However, if we do have enough such
instantaneous information, we can determine the curve C{t). The total amount of
relevant information is called the state of the system. It is assumed that the set
of all states is itself a differentiable manifold, S. Since the state of a system
contains more information than the configuration, we can assign to every state s the
configuration 7r(s) G M of the system in the state s. In other words, we have a
map TT: S —> M. We assume that this map tt is differentiable.
It is assumed that if we know the state s of the system at time to, we can
predict its state at any future time t. Thus we are given a map
^t,to' S —> S
such that if the system is in the state s at time to, it will be in the state (pt,to{s)
at time t. Now there is nothing special about the time ^o- If ^ > ^1 > ^0 and s
is the state at time ^0, then (Pti,to{s) is the state at time ^1, and therefore
is the state at time t.
In other words.
Now it turns out that in most (basic) mechanical systems, if the time is suitably
parametrized, the function (ptj^ really depends only on t — to- (This fails to
hold in the so-called nonconservative systems. A typical nonconservative
system is one involving friction. This is usually a consequence of not studying
a sufficiently complete system. In the case of friction, for example, the heat loss
must be taken into account.) Let us assume that cptjo depends only on t — to.
Then, if we write
Si == ^1 — ^0 aiid S2 = t — ti,
then the previous equation can be written as

13.1 THE TANGENT AND COTANGENT BUNDLES 511
This looks like the defining relation for a one-parameter group except that so
far we have been restricting ourselves to nonnegative values of s. In point of
fact, it is assumed that this restriction is unnecessary and, in fact, we are given
a flow (f on S.
To repeat, we are given a difTerentiable manifold M representing the set of
all possible configurations of our system, a differentiable manifold S representing
all possible states of the system, a differentiable map tt : S —> M, and a flow (p on
S. Then the curves C{t) are all assumed to be of the form
C{t) = TTiptix) for some x G S.
We must therefore describe S, tt, and cp for any given configuration space M.
Classical mechanics makes a very definite assumption about the nature of the
space S (and the map tt). It asserts that the state of a system is completely
determined by its (configuration and its) "momentum". What is the momentum
of our abstract setup? As usual, the momentum should be something that resists
change in velocity. It turns out that an appropriate object representing
"infinitesimal resistance to change in istantaneous velocity" is a cotangent vector. A
heuristic motivation for this, which the reader may choose to ignore, is the
following: At any given configuration x E M, the set of all possible velocity vectors
is just Tx{M). At any given v G Tx{M), a "resistance to change in v" would be
some function defined near v and vanishing at v. To first order we could replace
such a function by its differential. Thus "infinitesimal resistance" is a linear
function on Tv{Tx{M)). Since Tx{M) is a vector space, we may identify
Ty{Tx{M)) with Tx{M) for each v. Thus all possible "momenta" become
identified with all elements of r*(M).
The set S is thus taken to be the set of all momenta, i.e., the set of all
cotangent vectors at all points of M. We must first show how to make this
space into a differentiable manifold.
1. THE TANGENT AND COTANGENT BUNDLES
Let M be a differentiable manifold, and let us consider the set T{M) of all
tangent vectors to all points of M. Thus
T{M) - U Tx{M).
xGM
Let TT denote the map of T{M) onto M which assigns to each tangent vector the
point of M at which it is defined; that is, if J G Tx{M), then 7r(J) == x. We
claim that T(M) can be made into a differentiable manifold in a natural way,
so that TT is a differentiable map. In fact, let d be an atlas on M. For any
(f/, a) G a, define T{a): Tf-^U) -^ V ^ V hy setting
T(a)^ = <a(x), ^«> if ^ G Tx(M), xeU. (1.1)
We claim that the collection (7r~^C/, T{a)) is an atlas on T{M). To check that

512 CLASSICAL MECHANICS 13.1
T{a) is one-to-one, we observe that if J G Tx(M) and rj G Ty(M) with x 9^ y,
then a{x) 9^ 0L{y); while if ^ 9^ rj and both lie in Tx{M), we have ^a 7^ Va-
To check that the transition law is satisfied, we note that Eq. (4.3), Chapter 9
implies that if {U, a) and (W, 0) are two charts of d,
T(fi) o T(a)-\v, U - <p o a-\v), /^o«-iW?«> (1.2)
for <v, ^cc> eT(a)7r-\U nW), that is, vea(UnW) and J« arbitrary
in 7.
The fact that the structure on T(M) does not depend on the choice of Gi
is obvious. That tt is differentiable is also clear; in fact, (7r"~^(C/), T(a)) and
(U, a) are compatible charts in terms of which a o tt o T(a)~^(v, J^) ^ v. We
call T(M), together with its structure as a differentiable manifold, the tangent
bundle of M.
If M is finite-dimensional, let {U, a) be a chart of M with coordinates
x\ . . . , x^, so that
a(x) = <x\x),...,x''{x)> GR^
We will denote the coordinates associated to the chart T(a) on Tr~^(U) by
<g\ . . . , g"", g\ . . . , g''>. Hence if J G T^^iM), where a: G f/, we have
T{a)(0 - <cc{x), ^«>
- <g^?),...,g^(l),g'(?),...,r(?)>,
so that
g^ = x^7r and ? = E s'^) (^); (1-3)
In other words, the q^ are just the coordinates x^ regarded as functions on T{M)
via TT, and the q^(^) are the components of J relative to the basis
{[dxijj * * * ' Ua^V J *
We can follow a similar procedure for
r*(M)^ U ^*(M),
xGM
which we call the cotangent bundle. If (U, a) is a chart of an atlas Gi of M,
define the map
T*{a):7r-\U) -^7 0 7*
by setting
T*{a){l)= <a{x),l.> (1.4)
for I G T*{M) and x G U. As before, this defines an atlas on T*{M) which,
in turn, defines a differentiable structure on T*{M) which is independent of the
choice of atlas of M, (Note that we have used the same letter, tt, to denote the
two projections: that of T{M) -^ M and that of T*{M) -> M. Whenever there
is any confusion, we shall denote these maps by ttt and ttt*-)

13.2 EQUATIONS OF VARIATION 513
If M is finite-dimensional and (U, a) is a chart on the coordinates x^,..., x^,
we shall denote the coordinates of (7r"~^(C/), T*{a)) by
Thus, if I G T*(M), where x G f/, we have
so that
g^- = q^oTT and Z - L v\l){dx^)x' (1-5)
2. EQUATIONS OF VARIATION
Let Ml and M2 be two differentiable manifolds, and let <^: Mi —> M2 be a differ-
entiable map. Then (p induces a map T{ip) of T{Mi) —> T{M2) when we set
n^)?--<^*x(?) if ?Gr,(Mi). (2.1)
To check that T{(p) is differentiable, let us choose compatible charts (f/, a) on
Ml and (T7, iS) on Mg. Then it follows from the definition of T{ip) that
(r(C/), T{a)) and (r(T7), r(^))
are compatible charts. Furthermore,
T{&)oT{^)oT{a)-\vi,V2) - <{&-<P-cr^)vi,J^.^.a-^{vi)v2>. (2.2)
This establishes that T{(p) is differentiable. Observe that if \p: M2 —> M3 is a
second differentiable map, then it follows from Eq. (4.4) of Chapter 9 that
T{^p o ^) = T{yp) o T{^). (2.3)
Furthermore, it is clear from the definition that
TT o T{ip) = cpo TT. (2.4)
In other words, the diagram
T{Mi) '—^ T{M2)
commutes in the sense that it doesn't matter which path one takes to get from
T{Mi) to M2.
In particular, if {(ft} is a one-parameter group of diffeomorphisms of M, we
get a one-parameter group T{(pt) on T{M)^ where, by (2.4),
Tv-T{^t)i= <^^(7r(^)). (2.5)

514 CLASSICAL MECHANICS 13.2
If X is the infinitesimal generator of {cpt}, let us denote by T{X) the
infinitesimal generator of {T{(pt)}. It follows from (2.5) that if J G Tx(M), then
7r*Kr(X)^) ^ X,. (2.6)
Let us obtain the expression for T{X) in terms of a chart: Let {U, a) be a
chart, and choose an open set W such that W C U and e > 0 such that (pt {W)cU
for 1^1 < €. Then
T{a) o T{cpt) o T(a)-^<v, w> = <{ao cpto a-^)v, J(ao^^oa-i)(v)w> ,
SO that, differentiating with respect to ^ at ^ == 0, we see by Eq. (4.9) of Chapter 9
that
T(X)Tia)<v,w> = <XM,dX^^,,(w)>. (2.7)
If M is finite-dimensional, X J ... J X are the coordinates of (U, a), and
Xa = <X , . . . , X^ > ,
then we can rewrite (2.7) as
TiX)T,^,<v, w> = -(x'iv), . . .,X"iv), J:^-^w\..., Y^^-^w^y ,
(2.7')
where if — <w^, . . . , w'^>. In other words, in terms of the local coordinates
^^\ • • • J ^^j ?\ • • • J ?^^ the differential equation corresponding to T(X)
take the form
^^ = XV, •••,«") (2.8)
and
dq' dX' .1 dX' .„
_=__g +...+__g. (2.9)
Note that Eqs. (2.8) are just the local equations corresponding to the vector field
X, Since q^ = x^ o tt, this is simply another way of writing (2.6). Suppose that
cpt(x)= <x\t),...,X^(t)>
is a solution curve of X lying in U. Then we can regard the coefficients
as functions of t alone. Thus (2.9) takes the form
■dt = ^^')^
of a linear differential equation for w. This linear differential equation is called
the equation of variation of X along the curve (p(x). Roughly speaking, it repre-

13.3 THE FUNDAMENTAL LINEAR DIFFERENTIAL FORM ON T*{M) 515
sents, to linear approximation, how solution curves of X near (p{x) are deviating
from (p{x).
We now go through a similar construction for the cotangent bundle. If
cp: Ml -^ M2 is a differentiable map, then <^*: !r*(x)(M2) —> T^{Mi) is going in
the wrong direction. We therefore restrict our attention to maps which are
locally diffeomorphisms, and we define T*{(p) by setting
r*(<^)z ^ (cp-yi = (cp^,.,ru if IG tUmi). (2.10)
Since <^ is a diffeomorphism,
is well defined, and we have
TT o T*(<p) = <poTr. (2.11)
If ([/, a) and (W, jS) are compatible charts on Mi and M2, then
T*(fi) o T*(cp) o r*(a)-i<2;i,2;2> - <(^o .^o «"%, [/^o^o«-i(2^i)]*-'2^2>,
(2.12)
and so T*{(p) is differentiable.
If <^: Ml —> M2 and i^: M2 -^ Ms are diffeomorphisms, then
SO that
r*(^ o ^) = T^iyp) o r*(<^). (2.13)
In particular, if {ipt} is a flow on a manifold M, we obtain a flow T*((pt)
on r*(M). It satisfies
7roT*{cpt)l^ cptiiril)), (2.14)
If X is the infinitesimal generator of {(ft}, we denote by T*(X) the infinitesimal
generator of {T*((pt)}. It satisfies
7r*r*(Z)z-X, for leTt(M). (2.15)
3. THE FUNDAMENTAL LINEAR DIFFERENTIAL FORM ON T*(M)
Before returning to our study of mechanics, we study in some detail the geometry
of the cotangent bundle. Let M be a differentiable manifold, and let 2 be a point
of T*(M). Let J be a tangent vector to T*(M) at the point z, so that
jGr,(r*(M)).
Then tt* J is an element of !r^(2)(M). Since z G T*(^z){M) is a linear function on
TTr(z){M)j we can consider the expression
which depends linearly on J. We denote this linear function of J by dz- We have

516 CLASSICAL MECHANICS 13.3
thus defined a linear differential form 6 on T*{M) by setting
(?, 6,) = (7r*,^, z) for J G T,{T*{M)). (3.1)
The form ^ is called the fundamental linear form of T*{M). Let us obtain the
expression for d in terms of a chart (7r~^(C/), !r*(a)). Since T*{a) maps 7r~^(C/)
onto an open set, 0 of F © V*, the expression dT*(a) should be a function from
0 to (F © F*)* which can be identified with V* © V. Let us evaluate this
function. In terms of the chart {U, a) on M and (tt-^U), T*{a)) on T*{M),
we have
?r*(«) = <v,w*> eV ^ V* and (7r*J)« -= t; g F,
so that
(tt*?, 2> -= (z;, 2«>.
Thus {< V, w*^, (Pz)T*{a)) — (^j 2!a); that is,
(^^)r*(a) - <Za, 0> G F* © F - (F © F*)*.
In other words, the local expression 6^ for the differential form d in terms of
the chart !r*(a) is given by
ea<ui,ut> - <ut 0>. (3.2)
If M is finite-dimensional and we use the local coordinates <5\ • • • , 5^,
p\ . . . , p'*>, then
while
Thus
while
so that
or
TT* (t^) = 0 and 2 = ^ P^(2) rfa;^.
e^ Zv'dq\ (3.3)
Of course, (3.3) is just a way of writing (3.2) in terms of a basis of F == U^.
Let (p: Ml —> M2 be a diffeomorphism, let ^1 be the fundamental linear form
on T*{Mi), and let 62 be the fundamental linear form on T*{M2). Let
^ G T,{T*{Mi)) and 7r(2) == a: g Mi.

13.4 THE FUNDAMENTAL EXTERIOR TWO-FORM ON T*(M) 517
and so
In other words,
(r*(^))*^2- ^1. (3.4)
In particular, if {(pt} is a flow on M, then
(r*(^,))*^- ^.
If X is a vector field on M, then the infinitesimal version of the last equation is
DT*iX)e = 0. (3.5)
Note that any vector field X on M defines a function fx on T*{M) by
fx{z) = (Z,, z) if 7r{z) = X. (3.6)
We also have
fx = {T*{X), d\ (3.7)
since (r*(X)„ ^,) = (7r*r*(X)„ ^> ^ (X,, z) by (2.15).
Finally, in view of (3.5) and Eq. (6.14), Chapter 11, we have
0 - DT*,x,d = d{T%X), 6) + r*(X) J de,
so that
dfx-= -T*{X)Jdd. (3.8)
For reasons which will become clear later on, the function fx is sometimes called
the momentum function associated to the vector field X.
4. THE FUNDAMENTAL EXTERIOR TWO-FORM ON T*{M)
It turns out that the exterior two-form
n= de (4.1)
plays a fundamental role in mechanics, and we therefore study some of its
properties. First of all, since d^ — 0, we have
dn == 0. (4.2)
We claim that ^z is a nonsingular bilinear form on Tz{T*(M)) for each
z G r*(M). That is,
if ^ G Tz(T*{M)) is such that |J 12;, -- 0, then ^ = 0. (4.3)

518 CLASSICAL MECHANICS 13.4
For this purpose we compute the local expression for 12 in terms of a chart
(7r~^([/), T*{a)) [where z G Tr~^{U)]. The map T*{a) gives a diffeomorphism
of Tr~^{U) with a subset of F © V*, and by (3.4) the form 6 on M carries over
to the corresponding form da on F © F* == T*{V). Also, da can be regarded as
a (F* © F)-valued function which is given by (3.2). Let us denote Jr*(a) by Ja,
and let
^a= <XuX2> ev ev*
be considered a constant vector field on F © F*. Then
and so <:/(?«, ^a) is the linear differential form given by
{Va, diU Oa)} = (Xi, F2) if Va= <Yi,Y2> &V® V*.
On the other hand, since J^ is a constant vector field, the Lie derivative D^^Sa
reduces to the ordinary derivative of the Hnear function <U2,U2> »-> <U2,0>.
This derivative is just the constant <X2, 0>. Thus the Lie derivative D^^da
is given by the constant linear form
D^Ja = <X2,0> eV*^ V;
that is.
Now ^a -\^a == ?a J dda = D^Ja — d{^a, ^a), SO we See that ^a J ^a IS the
constant form
^aJcld= <X2,-X,> eV*ev.
To recapitulate, if J G Tz(T*{M)) is such that Jr*(a) is given by Jr*(a) ==
<Xi, X2> G F© F*, then
(? J^.)r*(«) - <X2,-Xi> G F*© F. (4.4)
Equation (4.3) clearly follows from this.
Since (4.3) is of fundamental importance, we present an alternative
derivation for the case of a finite-dimensional manifold. If we use the coordinates
<5^...,5^p^...,p^>,then
d=T.p'dq\
so that
n=Zdp' A dq\ (4.5)
If
x = T.{a<±, + b'^),
then
X A^^Y.{B^dq^ - A^ dp^. (4.6)
Thus (X AQ),^ 0 if and only if X^ = 0. This shows that on T*{M) we
have a one-to-one correspondence, X »-> X J12, between vector fields and

13.4 THE FUNDAMENTAL EXTERIOR TWO-FORM ON T*{M) 519
linear differential forms. Let us denote by cox the form corresponding to X,
so that
c^x^ X A 12;
and let us denote by X^ the vector field corresponding to the linear differential
form CO. Thus
CO == X^ J fl == cox„.
Observe that
c?co -= 0 if and only if Dxjl = 0. (4.7)
In fact, by (4.2),
DxSl ^ d{X^ J 12) -= c?co.
In particular, there is a distinguished class of vector fields on T*{M)—those
corresponding to functions, i.e., the vector fields of the form Xar, where i^ is a
function on T*(M). These vector fields are called Hamiltonian vector fields.
*In view of (4.6), if Dx^ = 0, then locally, at least, X is of the form XdF,
since any form co with do) = 0 can be written locally as co — dF. If we make the
topological assumption that T*{M) has vanishing first cohomology group, then
Dx^ — 0 is equivalent to X being Hamiltonian. This assumption is really a
restriction on the nature of the configuration space M. Since we do not wish to
restrict M in this manner, we will not take Dx^ = 0 as the definition of a
Hamiltonian vector field.*
Note that if X and Y are Hamiltonian vector fields, so is aX -^bY, where a
and b are constants. In fact, li X = XdF and Y — Xdo, then
aX -\- bY = Xd(aF+hG)'
Furthermore, [X, Y] is also a Hamiltonian vector field. In fact,
Dx{Y jn) = DxY J 12 + F J Dx^
= DxY J 12,
since Dx^ = 0. Since DxY = [X, Y], we see that
[X, Y]JQ=DxdG= dDxG.
In other words, we have
[XdF, Xdo] = XdiX^pG)' (4.8)
We thus see that we have a binary operation on functions corresponding to the
Lie bracket on Hamiltonian vector fields. It is called the Poisson bracket and is
denoted by {F^ G}. In other words, we define
{F, G} = XdFG, (4.9)
so that we can rewrite (4.8) as
[XdF, Xdo] — Xd[F,G\' (4.8')

520 CLASSICAL MECHANICS 13.5
Note that
XarG ^ {XaF, dG) - {Xar, X^g J^} = {Xao A X,^, 12), (4.10)
so that, in particular,
{F,G} = -{G,F}. (4.11)
In terms of local coordinates <5^ . . . , 5^, p^ . . . , p^>, we have
so that by (4.4),
^"^ = ^ (^ 5^ ~ w w)' ^*-^^^
and therefore
from which the antisymmetry (4.11) is apparent. A consequence of (4.11) is
the following:
Proposition 4.1. If F and G are functions on T*(M) such that
XdpG == 0,
then
XaoF - 0.
In other words, if G is constant along the solution curves oi XdFj then F is
constant along the solution curves of Xao-
In fact,
XapG ^ {F, G} = -{G,F} = -X^gF.
It will turn out that Proposition 4.1 is the prototype of all the "conservation
laws" of mechanics.
We close this section with the following observation. Let F be a vector
field on M. Then the momentum function of F is a function fy on T*{M).
Equation (3.8) asserts that
-T*(F) ^ X,f^. (4.14)
5. HAMILTONIAN MECHANICS
As we indicated in the introduction, the first fundamental assumption of
mechanics is that the evolution of the system is determined by a flow on T*{M)j
where M is the configuration space of the system. The second fundamental
assumption concerns the character of the flow. It is assumed that the
infinitesimal generator of the flow is a Hamiltonian vector field. That is, it is assumed
that there is a function H (called the energy) on T*{M) such that the vector
field X—dH is the infinitesimal generator of the flow on T*{M) describing the

13.5 HAMILTONIAN MECHANICS 521
evolution of the system. (The minus sign is a consequence of certain standard
conventions.) In order to see what this means, let us express the equations of
motion in terms of q- and p-coordinates for a finite-dimensional system. Thus
X_,
dH
- Y (— - — —^
Thus, if <q^('), . . . , q^'i'), P^{'), • • • , p''(')> is an integral curve of the flow,
it must satisfy the differential equations
dq' dH , dp' dH
dt dp^ dt dq^ ^ ^
A trivial consequence of (4.11) is that
X_dHH = 0.
In other words, the function H is constant along trajectories of the system.
This principle is known as the law of conservation of energy.
More generally, we can formulate Proposition 4.1 as:
Proposition 5.1. Let X_dH be the infinitesimal generator of a mechanical
system with energy H. Let /^ be a function such that
XdpH - 0.
Then F is constant on the trajectories, i.e., solution curves, of the flow
generated by X_dH'
Proposition 5.1 is the prototype of all the "momentum conservation" laws
we shall derive later; see, for example, the discussion at the beginning of Section 6.
In order to specify the mechanical system, one must give the function H.
It turns out (in many but not all cases) that the energy is the sum of two terms,
H = K + U, where K is called the kinetic energy and U is called the potential
energy. They each have a very special form which we now describe.
The kinetic energy is a function on T*(M) which is associated with a
Riemann metric on M. Let ( , ) be a Riemann metric on M. It gives a scalar
product ( , )x on each Tx{M), and therefore induces an isomorphism of Tx{M)
with T*{M), and thus gives a scalar product on T*{M) which we will continue
to denote by ( , ). The function K is then defined by
K{1) = iil, I)- (5.2)
To understand the relevance of a Riemann metric to mechanics, let us consider the
most elementary case—the study of a single particle of mass m in E^. The usual relation
between velocity and momentum,
p — mq
can be formulated as follows: Consider the Riemann metric on E^ which is
m X the Euclidean metric. That is, if <x, y, z> are rectangular coordinates on

522 CLASSICAL MECHANICS 13.5
E^ and <qx, Qy, Qz, qx, iy, qz^ are the corresponding coordinates on T(E^),
then
II(^x, Qy, qz)\\^ = ml + mql + mql.
Then the map of r^(E^) -^ T*(E^) sends
<qx, qy, qz, qx, iy, qz> "-^ <qx, qy, qz, Px, Py, Pz>,
where
Px = ^gx, Py = mqy, pz = mq^, (5.3)
and
K{qx, qy, qz, Px, Py, Pz) = 2"- ipl + pI + P^z)- (5-4)
Thus the passage from velocity to momentum depends on the choice of a
Riemann metric, which determines a map from T{M) -^ T*{M). This can be
regarded as a generalized "choice of mass" for the configuration.
The function U is assumed to be of the form U = U o tt, where f/ is a
function on M. The form F = — df/ is called th^ force field whose potential is U.
It can be regarded as a vector field on M in view of the Riemann metric on M:
(?, F,) = -(?, dU) for any ? G T,{M).
In the special case of a single particle in E^ with mass m, substituting (5.3)
and (5.4) into Eq. (5.1) gives, when we write H = K + U,
or, since m is constant.
mg = n (5.6)
which is the usual rule stating that force = mass X acceleration.
Returning to the general theory, we now formulate a useful corollary of
Proposition 5.1.
Proposition 5.2. Let H = K + U be the energy associated with the
Riemann metric ( , ) and the function U on M. Let F be a vector field on
M which is an infinitesimal isometry of ( , ) and is such that YU = 0.
Then the momentum function fy is constant under the flow generated
by X_dH'
Let {(ft} be the flow generated by Y. Since (pt is a local isometry, (<^*/, (pfl) =
(/, /) wherever (pfl is defined, so that
K{T*i<pt)l) = Kil),
and thus
T*(Y)K = 0.

13.6 THE CENTRAL-FORCE PROBLEM 523
Also,
T*(Y)U = {T*(Y),dU)
- (r*(F), 7r*dU) = (7r*7^*(F), dU) = (Y, dU) - 0,
so that T*{Y)H = 0 and Proposition 5.1 applies [see Eq. (4.14)].
6. THE CENTRAL-FORCE PROBLEM
Before proceeding with the general theory, we illustrate the previous results in
a simple but important case. We will study the motion of a particle in E^ acted
on by a "force centered at the origin". That is, our configuration space M will
be taken to be E^ — {0}, and the Riemann metric is m X the Euclidean metric,
where m is the "mass" of the particle. We also assume that the function U
depends only on the distance to the origin. Thus U{x, y, z) = P{r), where
r^ — ^2 _j_ ^2 _j_ ^2 Under these circumstances it is clear that any rotation
about the origin is an isometry of the Riemann metric and preserves U. We
can therefore apply Proposition 5.2 to the infinitesimal rotations x{d/dy) —
y{d/dx) to conclude that the corresponding momentum function xpy — ypx is
constant. (This momentum function is known as the angular momentum about
the 2;-axis.) Similarly, the functions xpz — zpx and ypz — zpy must be constant
on any trajectory of the flow. If we write x == <x,y,z> and p == <px, Py, Pz >,
then these three conservation laws can be combined to read
X A p == const, (6.1)
which is known as the law of conservation of angular momentum. Here x and p
are considered vector-valued functions on T*{M). In order to study the
implications of (6.1), let us first distinguish two cases: where the constant 9^ 0 and
where the constant vanishes. If the constant occurring in (6.1) does not vanish,
then (6.1) implies that the plane spanned by x and p does not change. In
particular, the motion is such that x lies in a fixed plane. If x A p — 0, we
argue as follows:
Since p — mx, we have x A x — 0. Since x 9^ 0, this implies that
X — Xx
for some function X of time. Now ||x|| — (x, x) ^^^, so
^llx|| = ^(x,x) = (^,x)=M|x||, (6.2)
and therefore
Xx Xllxllx
d / X \ _ Xx
dt\MJ ||x|| llxp ^'
SO that x/||x|| == const; that is, x lies on a fixed ray through the origin.

524 CLASSICAL MECHANICS 13.6
If we differentiate (6.2) and make use of the fact that x/||x|| is constant, we
get, by (5.6),
(~ ,m--i)=- (j^r^, P'(||x||) j^^ ,
\M dt) viixii ^" "^iixiiy
that is,
m~=-P'(r), (6.3)
where r == ||x||.
In any event, in all cases the particle x moves in a plane. We may therefore
restrict our attention to the plane. That is, we can consider a "new" mechanical
system where M = E^ — {0} and its Riemann metric, the function U = P{r),
etc. Let us introduce polar "coordinates" in the plane. Then ^/^^ preserves the
metric and the potential. If < r, 6, pr, pe > are the corresponding coordinates
on T*(M), then Proposition 5.2 implies that
Pe = const. (6.4)
(Note that this is really just that part of (6.1) that we haven't yet used.) Now
in terms of polar coordinates, the Euclidean metric has the form of Eq. (8.7)
of Chapter 9, so that the Riemann metric associated to the mass m which is
m X the Euclidean metric is given by
In particular, the associated map from T(M) to T'^{M) is given by
Pr — mf and pe — mr'^B. (6.5)
Thus (6.4) says that
r'^e ^ const. (6.40
To understand the significance of (6.4'), consider a curve x(-) in the plane.
Consider the region %, bounded by the portion of the ray from 0 to x(0), the
portion of the ray from 0 to x(t), and the curve x(-) from 0 to t. (See Fig. 13.1.)
Then
r 9 f ^^^)
i/ r^ do = rdr A dd
is the area of 11. On the other hand, since dd — 0
on the rays, we see that
t
f r^ dd = f r^d dt.
Jd'n. Jo
Thus (6.4') is exactly the content of Kepler's second law: The particle sweeps
out area at a constant rate.
Thus we have seen that the spherical symmetry of the system implies that
the motion is in a plane and that Kepler's second law holds.

13.6
THE CENTRAL-FORCE PROBLEM
525
We have not yet made use of the conservation of energy, which in the
present context reads
^mf + imr 6 + P{r) = const =
1
1
2m^' + 2^^^^ + ''^^^- ^'-'^
Let us examine a particular solution curve for which p^ = A = const. Then
differentiating (6.6) gives
dpr
dt
d'r A' ^,. .
(6.7)
We can interpret (6.7) as the equations of motion of a one-dimensional
mechanical system. The kinetic energy of this system is ^mf^, and the potential
energy Q is given by
Q{r) = P{t) +
2mr2
(6.8)
The second term in (6.8) is known as the centrifugal potential and the
corresponding term, A^/mr^, occurring in (6.7) is called the centrifugal force. Note that if
A = 0, then (6.7) reduces; to (6.3), which is what we would expect.
E,
Fig. 13.2
We can now use the following procedure for solving the equations of motio^:
First, for a given angular momentum find the various solution curves r(-) of (6.7).
For a solution ?^(-), determine d{') by integrating 6 = A/mr^ to get
e{t)
J to
mr^{s)
ds + const.
(6.9)
We can obtain a good bit of information about the nature of the solutions
of (6.7) by using the law of conservation of energy. We draw a graph of the
function, as shown in Fig. 13.2. Suppose that the trajectory r(-) has the
constant energy Ei = ^f^ + Q{r). Then if the set {r : Q(r) < Ei} is bounded,
r(t) must always be in this set, since imf^ > 0. In fact, suppose that the
interval a < r < h is such that Q(a) == Q{h) = Ei and Q(h + e) > Ei and
Q{a — e) > El for small €. Then if a < r{to) < h for some value of to,it follows

526 CLASSICAL MECHANICS 13.6
that a < r{t) < h for all t. Furthermore, if Q(r) < Eiior a < r < h, then for
(^ < ^'(^o) < ^, ^(0 cannot vanish. The particle is thus moving to one of the
limits r = a and r = h. If we use the law of conservation of energy, we see that
lmf^ + Q(r) = El,
so that
-=w^
dr_,_^ IE,- Q(r)
dt
For a < r < h we see that r is a monotone function of t, and we can solve to
obtain t as a function of r by the formula
.1/2 / dx
t{r) = Mirny" , + ^0 if r{to) = To. (6.10)
'to VEi - Q{x)
In particular, if Q'{a) 9^ 0 and Q'(6) 9^ 0, the integral in (6.10) converges, so
that it takes a finite amount of time for r to get to a or to h. Thus (since
dfldi — Q'(f) 7^ 0), the function r oscillates between a and h taking the time
(Jm)
1/2 / dx
/« VE'i - Q{x)
to get from one side to the other.
If {r : Q(f) < E2} is not bounded, then the motion with energy E2 need
not be bounded. For instance, suppose that Q{r) < E2 for r > a. Then if
r(^o) > ct and f{to) < 0, r will decrease until it hits a at which time it will turn
around and then go off to infinity; if f{to) > 0, then r will simply go to infinity.
The trajectory in this case is that of r coming in from infinity, turning around
at a, and then going back to infinity.
We recall that this is a descriptive analysis of the function r(-). The curve
d(-) is then to be determined by (6.9).
Frequently we are interested in the tajectories as curves in the r^-plane
without reference to the time dependence. Suppose that the energy is E and the
angular momentum is A. For the range where r f^ 0 we can substitute
dd _ A
dt mr^
into (6.10) to obtain [since dd/dr = (dd/dt){dt/dr)]
A dx
'roxWE - Q{x)
^«=±(27^y''/ ^^T^^^^^+^M-
In the important case where P{r) — —a/r corresponding to the inverse-square
law of gravitational attraction, we have
Q{x)
A^
2mx^ X

13.6
THE CENTRAL-FORCE PROBLEM
527
so that the integmnd is
1 A
(2m)
1/2
v^-
A^ a
2mx^ X
2 2"|l/2
h'-ii-^y^'^]
dx
arccos■
A
X
ma
A
Thus
4
2mE +
m^a^
d{r) — d(ro) = arccos
A
r
ma
~A
4
2mE +
2ry2
Let us choose Tq to be the minimum value of ?% and let us choose ^(?^o) = 0.
Let us set
A'
ma
and
= ^1 +
2^:42
ma^
We see that the equations for the orbit are
- = 1 + e cos 0.
(6.11)
This is the equation of a conic section (Kepler's first law), li E < 0, then
e < \ and (6.11) represents an ellipse, whose major and minor semiaxes are
p _ a
a ==
and
1 - ^2
V
2\E\
A
Vl - e^ \/2m\E\
We leave to the readers the details of working out the hyperbolic and parabolic
orbits.
For the elliptic orbit, Kepler's second law implies that the total area of the
interior of the ellipse is swept out at the uniform rate A/2m. Thus the area,
Tvab, of the ellipse is given by {A/2m)T. Thus
2m
T = Tvah — Tva -
V2m\E\
and hence
T == 27ra^'Wm/a.
In other words, the square of the period of motion is proportional to the cube
of the linear dimension of the orbit (Kepler's third law).

528 CLASSICAL MECHANICS 13.7
7. THE TWO-BODY PROBLEM
Let us consider the mechanical system consisting of n particles each having
mass rrii, so that the configuration space is E^ X • • • X E^ (n times), and if
a ^ <a\ . . . , a''> G M, where a^ = <x^, y^, z^> E E^, then
MdV == im,\\dT + ' ' ' + hrnn\\d^r
= imAixY + (yy + (zY] + • • • + imn[(x-)' + {rY + (^")']
is the kinetic energy of the system. As we indicated in Section 1, we may wish
to restrict the manifold M to be that subset of E^ X • • • X E^ for which no
ai == aj for any i 9^ j. Let us further assume that the potential energy depends
only on the mutual distances between the particles. That is, let us assume that
[7(a^ . . . ^a"") ^ P{\\a2 — aJ, ||ay — aJ, . . . , \\an — an_i||).
Then if A is a Euclidean motion of E^ and we apply A simultaneously to all the
particles, the kinetic and potential energy are conserved. That is, the
transformation
<a\ . . . , a^> ^ <Aa^, . . . , Aa^>
is an isometry of the Riemann metric on M and preserves TJ.
Let <x^ 2/^ ^\ • • • J ^^j y^j ^^^ be the Cartesian coordinates on M,
and let
^^xl, ^2/1, Q.zl, Qx2, . . . , Qyn, Qzn, Pxl, • • • , Pzn^
be the (Corresponding coordinates on T*{M). Now d/dx is the vector field
representing the infinitesimal translation in the x-direction in E^. Therefore,
dx^ ^ dx^^ dx^
is the infinitesimal generator of the one-parameter group
<x\y\z\x^...,x'',y'',z''>
^ <x' + t, y\ z\ ^2 + t, y^, z^...,x'' + t, 2/^ ^">.
We can therefore apply Proposition 5.2 to conclude that the function
Pxi+Px2-\ h Pxn
is constant in any trajectory. This function is known as the total linear
momentum in the x-direction. Similarly, the total linear momentum in the ^/-direction,
Pyl-\ h Pynj
and the total linear momentum in the ^-direction,
Pzl + ' ' ' + Pzn,
must be conserved. If we define p* G E^ by setting p* == <Pxij Pyu Pzt^ , then

13.7 THE TWO-BODY PROBLEM 529
we can say that the E^-valued function
must be conserved. This is the law of conservation of total linear momentum.
(For two particles the assertion that pi + P2 is conserved is just Newton's law
of "equality of action and reaction". In our setup we see that this law is a
reflection of the invariance of the physical situation under translations.)
The vector field x(d/dy) — y(d/dx) represents infinitesimal rotation about
the 2;-axis in E^. Therefore,
1 d 1 d . 2 ^ 2 ^ , I n ^ n ^
^ dy^^ ^ dx^'^'' dy^ y dx^'^""^'' ar ^ Qxn
represents simultaneous infinitesimal rotation of all the particles about the 2;-axis.
Therefore, the function
QxiPyi Qy^Pxl -f- • • • -f- QxnPyn QynPxn
is conserved. This is the law of conservation of total angular momentum about
the 2;-axis. Similarly, we obtain the law of conservation of total angular
momentum about the x- and y-Sixes, If we set
q' = <qxijqyi,qzi> G E^,
we can combine the three equations by saying that the function
gi A p^H hg"" A p""
with values in /\^(E^) is constant on the trajectories of the motion.
Let us examine more dlosely the law of conservation of total linear
momentum. In view of the fact that
dx^
PxV=m,^.
etc., we have p* == mi(da^/dt), and thus
-r, (mia^ + ni2a^ -f . . . -f ntnO^) =
In other words, the center of mass
^ _ niia^ _j- . . . _]- rrtnO^
const.
mi-\ + rrin
moves in a straight line with constant velocity.
Suppose there are only two particles. Then it is reasonable to introduce the
center of mass
mi + ^2
and the relative position vector
d = a^ — a^

530 CLASSICAL MECHANICS 13.8
as new coordinates. If we solve for a^ and a^, we get
mi + m2 mi + 1712
and therefore the kinetic energy is given by
while f7(C, d) = P{\\d\\).
Thus the motions of the system have the following description: The center
of mass has constant linear motion, and the relative position vector d = a^ — a^
satisfies the equations of motion of a single particle with mass
mim2
nil + ^2
in the central-force field with potential U — P(||d||). We can thus apply the
results of the preceding section to determine the motions of the particle. In
particular, if P(||c/||) — q:/||c?|| is the inverse-square potential, the corresponding
two-body problems can be completely solved.
Note that if m2 is very large compared to mi, then C is very close to a^.
In this case c/(-) is a good approximation to the motion of the smaller particle
relative to the larger one.
This is the situation that arises, for example, in the study of the motion of
of the planets relative to the sun.
8. LAGRANGE'S EQUATIONS
Our discussion of mechanics has led us to a certain kind of vector field X on
!r*(M), where M is the configuration manifold. In the case where H = K + U,
the Riemann metric giving K induces an isomorphism <£ [that is, a diffeo-
morphism which is linear on each Tx(M)] from T{M) to T*(M). We therefore
obtain a vector field Y on T(M) such that <£*F^ == ^£(1;) at any v G T{M),
We can therefore inquire about the form of this vector field Y in terms of local
coordinates <q^, . . . , q^, q^, . . . , q^> on T{M), associated with coordinates
<x^, . , . ,x'''> on M. Suppose that the Riemann metric is given by
If <5S . . . , 5^, pS . . . , p^> are corresponding local coordinates on !r*(M),
then the diffeomorphism <£ is given by
q'-q\ p'= i:giM\'".(fW' (8.1)
We could proceed to use (8.1) to obtain the local expression for <£ and thereby
the local expression for F. However, it is more convenient to argue a little
differently. Let L be the function on T{M) defined by
L{q\ . . . , g^ gS . . . , r) - iZ gm'q' - U(q\ . . . , g^), (8.2)

13.8 Lagrange's equations 531
that is,
L= K - U,
where K{v) = i\\v\\^ and U(v) = U{Tr{v)) are the kinetic and potential energy
expressed as functions on T(M), We can rewrite (8.1) as
^' = «'' ^' = W ^^-^^
Then for any I e T*(M), we have
H{1) = {£-\l), I) - L(£-i(0). (8.4)
In fact, by definition,
{£-\l),l)= ||Zf = \\£-\l)r,
so that
(£-'(Z), I) - L(£-\l)) = \\ir - Mir + U(7r{l))
= i\\iV+u(i)
= H{1).
In terms of local coordinates we can write (8.4) as
H{q\ . . . , 5^ pS . . . , p^) - L p¥ - L(q\ . . . , 5^ gS . . . , g^), (8.5)
where the g* are regarded as functions of the q's and p's via £~^ We can write
the map <£"~^ as
«' =«'' ^' = ff' (^-6)
since H{1) = M^V + U(-k{1)). Furthermore, by (8.5),
dL o £-1
dq'
by (8.4).
Now let v{-) = <q^{-), ■ ■ ■, g"(-)> ?'(")> • • • > ^''C')'^ be a solution curve
to the vector field Y, so that £ » v{-) is a solution curve of X on r*(ilf). Then
by (5.1),
dq^^dH^^ .i
dt dpi ^
and
dp' d{dL/dq') ^ _ 5F^aL
In other words, Eqs. (5.1) are equivalent to the equations
dq' .i d{dLM) dL_

532 CLASSICAL MECHANICS 13.9
on T(M), The first of Eqs. (8.7) says that we are dealing with a system of second-
order differential equations which is given implicitly by the second of Eqs. (8.7).
Equations (8.7) are known as Lagrange's equations.
For certain purposes Lagrange's equations are very convenient. We
illustrate by establishing the "principle of mechanical similarity". Note that
Eqs. (8.7) are unchanged if we replace L by cLj where c is any nonzero constant.
Suppose that M is an open subset of a vector space with linear coordinates
x^, . . . , x^ and that the Riemann metric is given by ^ Oiji^q^j where the Qij are
constant. Let m^ denote the linear map consisting of multiplication by a > 0,
that is, ma(x \ . . . , x^) = < ax \ . . . , ax^ >. Suppose that C/ is a homogeneous
function of degree p, so that
U(ax\ . . . , ax"") = a^U{x\ . . . , x^).
Now let us change our time scale by a factor /3, replacing ^ by s — ^t. Then
tT '
daq^ _ a dq^
~ds' ~ '^~dt
and
1 ^ daq^ daq^ _ ^^ 1 \^ ^5* ^^^
2^ ^''~di ~di ~ J^2^ ^'' ~dt '~dl'
Let us choose /3 so that a^/^^ = oP^ that is,
Then
T I \ n oidq^ adq\ vt ( \ n dq^ dq''\
In other words, replacing q^ by aq^ and t by 0t carries solutions into solution:
if we change the linear scale by a and change the time scale by a^~^^i^^p^ we
obtain an isomorphic situation. For instance, if U is homogeneous of degree —1
(as in the case of an inverse-square law of attraction), then /3 — a^'^. In
particular, the period of any periodic orbit is proportional to the f-power of its
linear dimension, which is just Kepler's third law. We thus see that Kepler's
third law is an exact consequence of the inverse-square law of attraction.
Returning to the study of the general Lagrangian system, we observe again
that it is a system of second-order differential equations. We can therefore
apply the fundamental existence and uniqueness theorem to it to conclude that
for every x G M and for every ^ G Tx{M) there is a unique curve C(-) which is a
trajectory of the system and for which C(0) = x and C'(0) == ?.
9. VARIATIONAL PRINCIPLES
The function L plays a crucial role in the study of variational principles of
mechanics. Consider the following problem: Let p and q be two points of M,
and let ^i < ^2 be two real numbers. For any differentiable curve C defined on

13.9 VARIATIONAL PRINCIPLES 533
the interval [^i, ^2] we set
I[C]^ C" L{C'(t))dt, (9.1)
Note that C'{t) G T{M) and L is a function on T{M), so the integrand makes
sense. Among all curves joining p to q, find that curve for which I[C] takes on
its minimum value. We shall see that a necessary condition for I[C] to be a
minimum is that it be a trajectory of a mechanical system. (In fact, if a suitable
notion of neighborhood is introduced on the space of curves, it is also a necessary
condition for C to be even a local minimum.)
Before establishing this result, it is convenient to have another expression
for I[C], As before, let <£ be the map of T{M) -> T*(M) given by the Riemann
metric. Let V be the curve in T*(M) given by
^(0 - £{C\t)),
Then
I[C] = f_e - C"H{V{t)) dt, (9.2)
Jc Jti
where 6 is the fundamental linear form on T*(M).
In fact, in terms of local coordinates,
since the curve C by definition is such that
q\t) = % (0.
But, by (8.5),
z / f| m dt = Y: f(p' ° £)(tmt) dt = jw»£(c'(o) + Lic'm at,
so (9.2) holds.
Let Z be a vector field on M which generates a flow ip. For all sufficiently
small s the curve ips o C(-) will be defined and
(v» ° cy{t) = <p,*c'{t) = TMic'it)),
so that
IWs ° C] = p LiTMC'it)) dt
= [ e- f'"H{£oTi<p,)(C'{t)))dt. (9.3)
j£o(pgoC' Jti
Since I[C] is to be a minimum, we must have
ds
We will now compute this derivative so as to derive the consequences of the above
equation.

534 CLASSICAL MECHANICS 13.9
Now £ o T[(ps] o <£~^ is a flow on T*(M) which satisfies
TT o £ o T(cps) o <£~^ -= cps^
Let Z be the infinitesimal generator of this flow, so that at all points of T'^(M)
we have
7r*Z -- Z, (9.4)
If we differentiate (9.3) with respect to s at s — 0, we obtain
ds
I[cps-C]= IdzO- / DzH{V{t))dL
Jc Jti
Now
Dzd = d(Z, d) + Z J dd and (Z, d)i = (tt^Z, /> - (Z, /> - fz(l),
so we get
^ ly. oC] = f_Z Jde- f^ DzH{V{t)) dt+fzi^{t2)) - fzi^i)).
(9.5)
Now suppose that the vector field Z vanishes at p and q. Then all the curves
<Ps ° C join p to q. If C is to minimize the integral /, then the derivative
d(I[(ps o C])/ds must vanish at s == 0. Note that in this case the two last
terms of (9.5) vanish and we must have
fZjdd- C' DzH{V(t)) = 0 (9.6)
Jc Jti
for all vector fields vanishing at p and q. In particular, let us take
Z^yp—. for xGf/,
- 0 for X g 17,
where i^ is a C°°-function whose support lies in some coordinate neighborhood U
of M with coordinates <x^, . . . , x^>. Suppose further that T^(p) == yp{q) == 0
if p G f/ or 5 G f/. Then by (2.7')
a^ ^dqod^o
so that
where jB^ are some functions on T*{M) which depend linearly on Z. [This, of
course, is just a restatement of (9.4).] Then
Z Ade^'Z AY.dp' A dq' ^ ^ B'dq' - rP dp\

13.9
VARIATIONAL PRINCIPLES
535
while
d-zh = >^§+i:b
dqi
Thus if the curve "C is given by C(t)
Eq. (9.6) becomes
<q\t),,,,,q-{t),pHt),,,,,p-(t)>,
/[2-'(f^g)-*(f+!!)]--»•
dpj/ ^ \dt ' dq\
Now by construction, ^(t) = £> o C'{t), so on C we have
dq^
dt
T =
dpj
(9.7)
by (8.6). Thus the first sum occurring in the above integral vanishes and we
must have
A(¥+i)-=»-
dqi
This must hold for all functions \p whose support lies in a coordinate
neighborhood and vanishes at p and q. Clearly, this can happen only if
dp^
dt
m
dq^
(9.8)
This must hold for all i. Since (9.6) and (9.8) are exactly (5.2), we can assert:
Proposition 9.1. A necessary condition for C to minimize the functional
I[C] is that C is a trajectory of the corresponding dynamical system.
The question of when this necessary condition is also sufficient is a more
complicated one. We shall not discuss it in any generality here, but refer the
reader to any standard source on the calculus of variations.
Fig. 13.3
By the way, we can derive a bonus from (9.5). Suppose that we consider
the following problem: Let Ni and N2 be two submanifolds (Fig. 13.3) of M,
and suppose that we require that C minimize / among all curves joining Ni to N2
and not merely among those joining p to q. In this case (9.5) will have to vanish
for all vector fields Z which are tangent to iVi at p and to N2 at q. Now observe

536 CLASSICAL MECHANICS 13.9
that if C is a solution to this minimum problem, it certainly is a solution to the
problem of minimizing / among all curves joining p to q. In particular, C must
be a trajectory of the mechanical system. As the reader can easily check, this
implies that
j_Z AS- j{DzH){V{t)) dt^O
for all vector fields Z, Thus, if C solves the more difficult minimization problem,
we must have
Now
fzi^ih)) = (ZpMi)) and /z(^fe)) = (2«,^(<2)>.
Since ii p ^ q we can choose Zp arbitrarily in Tp{Ni) and Zq arbitrarily in
Tq{N2), we conclude in this case that C must also satisfy
{^,^iti)) = 0 for all ^GTpiNi),
iv, ^(<2)) = 0 for all V e TqiN2).
Since ^(<i) = £C'{ti), we have
so we can write (9.9) as
(^,C'«i)) = 0 for all ^eTp{N^),
{v,C'{t2)) = 0 for all veTq{N2).
(9.9)
(9.10)
In other words, the curve C must be orthogonal to the submanifolds A^i and A^2-
Although our statement of Proposition 9.1 was couched in the framework of
dynamical systems, it actually can be formulated in a more general context.
Let L be any function on T(M)—not necessarily of the type K — U. We can
then define the integral / as in (9.1) and again pose the minimization problem.
This is the typical problem in the calculus of variations. We have already
discussed this problem from a different point of view in Section 3.15. We leave
to the reader the task of showing that our arguments carry over in this more
general case if the matrix
is nowhere singular. (Here the map £ of T(M) —> T'*'(M) is given by
I n . 1 . n,. >^ 1 n ^^ ^^ V
<x J . . . J X J X J , . , J X >»->-<^x,...,x, -^ri J ' ' ' J -^^ >,
and the nonsingularity guarantees that this map is locally a diffeomorphism.)

13.10 GEODESIC COORDINATES 537
10. GEODESIC COORDINATES
In this section we depart momentarily from the study of mechanics in order to
exhibit some applications of the results of the preceding paragraphs to the
study of Riemann manifolds. Note that a Riemann manifold always defines a
mechanical system by its kinetic energy if we set the potential energy equal to
zero. It is this special kind of mechanical system that we wish to study in this
section.
Let M be a finite-dimensional manifold with a Riemann metric and, as
above, define L on T{M) by setting
L(v) = i\\v\\^.
This then determines a vector field Y on T{M) which corresponds to the system
of differential equations (8.7) in terms of local coordinates. Let p be a point of M,
For every ^ G Tp{M) there is a unique trajectory Q(-) such that
Q(o) - p, cm == ?.
In terms of local coordinates, if ? == < ?^ • • • , ?^^, then
where q\ are the unique solutions of the differential equations
dqi .i d{dL/dct) _dL^
dt ^^' dt dqi
with the initial conditions
5i(o) = x\p), 4(0) = r.
By the fundamental existence and uniqueness theorem the functions q\ are
defined for sufficiently small t. We can regard C^(t) as dependent on both ? and t.
In other words, we have a map C.(-) assigning to each ^ G Tp(M) and each
sufficiently small t a point of M, The map C.(-) is, in fact, defiijed in some
neighborhood of Tp(M) X {0} C Tp{M) X U.
The above is true for any Lagrangian function. In the case of no potential
energy we can say a lot more. Let s be a real number. Consider the curve
Q(s •); that is,
Q(sO - <ql(st),,,,,q'^(st)>.
Then (suppressing the subscript ? which is to be understood in the following
computations)
^ = ^^'■(«^)'
II (q\st), ..., q'^ist), sq\st), ..., sq'^{st)) = 1 s^ ^ ^ q''{st)q\st),

538
CLASSICAL MECHANICS
13.10
and
d_
dt
i(^'(^^)'-
q\st),sq\st),...,sq''{st))
d
^^sJ2 9ik(q\st), . . . ,q\st))q\st)
dtdqi
?"(■))
In other words,
ddL^
dt dq^
d dL / I. .
dL
?>-))-|^(5^(«-),-..,.-.,^g>-))
II (.H-),...,^"^)-I
I iq\-), ..-, ?"(•))
= 0.
Thus the curve C^{s ■) is again a trajectory of the system, and we clearly have
CKs-)lo = sCK-)lo = s?.
By the uniqueness theorem for differential equations we thus must have
Ciist) = C,i(t). (10.1)
We are therefore led to define a map, which we shall call exp, from
Tp(M) -^ Mhy setting
exp (0 = Q(l). (10.2)
Note that the map exp is defined and differentiable in some neighborhood of
the origin in Tp(M), In fact, by (10.1),
exp(0 = C^jU\\(U\\),
where now J/|| J|| lies on the unit sphere in Tp(M) [the unit sphere with respect
to the Euclidean metric given by the scalar product on Tp{M)]. Since the unit
sphere is compact, there is some e > 0 such that C^(0 is defined for all rj on the
unit sphere and all |^| < e. Thus exp will be defined for all || J|| < e.
The map exp is a differentiable map from some neighborhood of the origin
in the vector space Tp(M) into the manifold. Let us compute
exp^o--To[Tp(M)]^Tp(M).
Let J G Tp(M), and let us consider the straight line through 0, /^, in Tp{M)
given by /^(O — t^. Since Tp{M) is a vector space, we can identify Tq[Tp{M)\
with Tp{M) via the identity chart, in which case we identify
But
1[{Q) with ^
exp [im = exp {to = C,K1) = Q(0,
(10.3)

13.10 GEODESIC COORDINATES 539
and the tangent to this curve at 0 is just ^. In other words,
exp*o [um - ?.
Thus, if we identify To[Tp{M)] with Tp{M), we can assert that
exp*o: To[Tp{M)] -^ Tp(M)
is given by
exp*o (0= ^' (10.4)
In particular, the map exp:j:o is nonsingular, so that by the implicit-function
theorem the map exp is a diffeomorphism in some neighborhood of the origin.
We have thus constructed a diffeomorphism exp from some neighborhood
of the origin in Tp(M) into M, which by (10.3) carries straight lines through the
origin into trajectories through p. In the case of no potential energy the
trajectories are called geodesies for reasons which will soon become apparent. If we
identify Tp(M) with V by some chart {U, a), we can then use exp to introduce
a new chart (U\ n^) by setting
n~\^a) = exp (?).
The chart n^ has the property that rtaip) — 0 and n^ carries geodesies through p
into straight lines through the origin. The chart (U\ n^) is called a geodesic
normal chart, and corresponding coordinates are called geodesic normal coordinates
on M.
Let us consider the curve Q(-) == exp (• ^) which is defined for 0 < ^ < 1
so long as ||?|| < e. We have
mi-)] = C L(cm) dt = if' \\cmf dt.
Jo Jo
But, by the conservation of energy, for any trajectory of our system we have
H o £(C'(0) -= const. In this case, since U = 0, H o £(C'(t)) = L(C'{t)) ^
i||C^(^)||2 3^ const. Since C^(0) - ?, we have ||C^(0|| == ||?||, and therefore
/[Q(-)]-i/^ ll?ir^^-^- (10.5)
Now let {jSs} be a one-parameter group of rotations in Tp{M). Then
(Ps = exp o^^o exp~^
defines a one-parameter group on the open set U = exp {«^: ||«^|| < e} C ^.
If ll^ll < e, we have by (10.5)
so, by (9.10), we get
f/i'Ps ° QO] = 0 = /z,(C(l)) = {€'{!), Zca)),

540 CLASSICAL MECHANICS 13JO
exi) Sjjiii
Tp{M) -^^ M Fig. 13.4
where Z is the infinitesimal generator of ip. But ^ccd = exp#o Y^, where F is the
vector field on Tp{M) which is the infinitesimal generator of the one-parameter
group of rotations. Now we can choose fi arbitrarily. Therefore, F| can be any
vector tangent to the sphere of radius || J|| in Tp(M). We thus conclude that
{ri, C|(l)) == 0 for any rj wjiich is tangent to exp ;S||||l, where S\\^\\ is the sphere
of radius || J|| in Tp(M). (See Fig. 13.4.) In other words, not only are the rays
through the origin orthogonal to the spheres in the Euclidean metric of Tp{M),
but also the image of a ray through the origin under exp is orthogonal to the
image of a sphere about p in the Riemann metric of M,
In particular, we can transfer "polar coordinates" on Tp(M) to M so as to
get "geodesic polar coordinates" on M. This has the following effect: Let r be
the "radial coordinate", that is, r is the function defined in U by
r(x) = ||exp'~^ x||.
Then for any x e f/, x 5^ p, and any f G T^{M) we have
llrll >\{^,dr)l (10.6)
with equality holding only if f is tangent to a geodesic through p. In fact,
suppose that f G Tx{M), where x = exp J for some J G Tp(M), Then we can
write f = f 1 + f 2; where f 1 is some multiple of C|(l) and f 2 is tangent to
exp ;S!i|!i. By the above result, (f 1, f 2) = 0, so
llff =||fl||^+llf2f
and we obtain (10.6), with equality holding only if ||f2|| = 0.
We are now in the same fortunate position we were in Section 9 of Chapter 9.
Let D be any curve joining p to x, where x = exp J. Let ^1 be the first time that
D{t) G exp ;S|||l!. Then the length of
D = f \\D\t)\\ dt > C' \\D'{t)\\ dt > C' \{D'{t),dr)\
Jo Jo Jo
> C'{D\t),dr)^T[D{h)]^ Ul
Jo
Furtherniiore, equality holds only if D'{t) is a nonnegative multiple of a tangent

13.11 euler's equations 541
to a fixed geodesic through p. Then Z) must be the geodesic C|(-). In short,
we have proved:
Theorem 10.1. Let € > 0 be so small that the map exp is a diffeomorphism
on B, = {J G Tp(M) : || ||| < e}. Let x = exp J be a point of C/ = exp B,.
Then the geodesic Q joining p to x has length |||||, and any other curve D
joining p to x is strictly longer unless D differs from C| only in a (monotone)
change of parameter. In other words, loosely speaking, geodesies are locally
the shortest curves joining two points.
Since we have come this far, let us show in addition that geodesies also
locally minimize the energy. Let D be any curve from [0, 1] to M. Then by
Schwarz^s inequality we have
(I' IID'COII dty < p \\D\t)fdtf' 1 dt
= 2/[D],
with equality holding only if ||D'(0|| is constant. If C|(-) is the geodesic joining p
to X = exp J, we thus have
iiD] > i (f^ wD'rn dty > i (/^' iic'coii dtf = iiiif.
Now equality holds in the second inequality only if D\t) is proportional to C'(0,
while equality holds in the first only if ||D'(0|| = const. We thus conclude that
11^'(Oil = II III, that is, D\t) = C\t). In short, we have proved:
Proposition 10.1. Under the hypotheses of Theorem 10.1 the curve C|(-)
is a strict absolute minimum for I[C] among all curves C: [0, 1] —» M such
that C(0) = p and C(l) = x.
11. EULER'S EQUATIONS
Under certain circumstances, which we shall presently describe, the equations of
motion take a particularly elegant form. The first special assumption that we
shall make is that there is an isomorphism of T(M) with M X F. More precisely,
we assume that there is a diffeomorphism, t, of T(M) with M X V such that
t(j) = <m, t;>, where m — 7r(J), and for each x G M the map J h-> t; of
Tx(M) —» F is a linear isomorphism of vector spaces. For example, if M is an
open subset of F, then the identity chart defines such an isomorphism
t(J)= <7r(J), Jid>.
A slightly less trivial example is furnished by the n-dimensional torus T^ =
S^ X ' ' ' X S^. Then we can introduce "angular variables" ^\ . • • > ^^i where
^* is the angular variable on the iih circle. We thus obtain n vector fields
d/dd^j . . . , d/dd^ which are linearly independent at each point of M. We have a
basis of Tx(M) and therefore an isomorphism of Tx{M) with U^ = V which

542 CLASSICAL MECHANICS 13.11
defines the desired map t. We shall encounter a more complicated example in
the next section. We should point out that only very special kinds of manifolds
admit such an isomorphism l of T(M) with M X V.
For the rest of this section we shall identify T{M) with M X F and T*(M)
with M X V* via the corresponding (adjoint) isomorphism.
The rule which identifies each Tx(M) with V can be regarded as a F-valued
linear differential form on M. Let us denote this form by co. In other words,
the identification l is given by t(J) == (J, ojx) if I E Tx(M). We can therefore
study the F-valued exterior two-form doj. For each pair of tangent. vectors
^j V ^ Tx{M) we obtain {rj, J J dcj) as an element of V. Now we can identify J
and 7} with vectors of V. We thus obtain a F-valued antisymmetric bilinear form
on F; if we call it Gix, we have
ax{v, w) = {t], ^J do))j
where (J, co^) — v, {rj, co^) — tf, and J, r; G Tx{M). Note that, in general, the
bilinear form Gix depends on x. Our second fundamental assumption about the
identification l is that Gix is independent of x. That is, we assume that there is a
F-valued bilinear form d such that
(F, Xjdc^)= a{{X, CO), (F, CO)) (11.1)
for all vector fields X and Y on M.
In the examples given above (the open subset of F on the torus) dcj = 0,
so that (11.1) holds trivially. In the next section we shall come across a case
where dco ^ 0.
To understand (11.1) a little better, let us introduce the following notation:
For any z; G F let ?) be the vector field on M given by (23,-co)^ == v for all x G M.
Then for any Vj w G Vj we have
{w, V Jdo)) = D%{w, co) — D^{v, co) — {[v, w], co).
Now {Wj 0)) = w and {v, co) — z; are constants, so the first two terms on the right
disappear. Thus we can rewrite (11.1) as
a{v,w) = -{[v.wic^). (11.1')
For the kinetic energy of a mechanical system we need a Riemann metric
on M. A Riemann metric on M gives a scalar product on each Tx(M). This
means giving a scalar product ( , )a; on F for each x G M. Our third special
assumption is that ( , )x does not depend on x. Thus we are given a scalar
product on F which gives the Riemann metric on M via the identification of
T(M) with M XV,
We wish to describe the vector field on T*(M) as X-dH, where H = K + Uj
K being the kinetic energy'of the Riemann metric and U being some potential
energy. Since T*(M) == M X F*, a vector field X on T*(M) can be uniquely
written as X — Xi + X2, where Xi is tangent to M and X2 is tangent to F*.
Furthermore, we can regard Xi as a F-valued function and X2 as a F*-valued

13.11 euler's equations 543
function (identifying the tangent space to vector space V* with V*). Then
(X, e)<.,i> = (X„ I)
Sit any <Xj l> G M X V*. We should really write this as
(•, 6} = {{; W*C0), V).
where co is the form defined above and the F*-valued function p:M X V* -^ V*
is the projection onto the second factor, p(m, l) — /. Then
<•, XAde) = <(•, 7r*w), (X, dp)) - «X, 7r*«), (•, dp» - ((•, Xtt* J d«>, p>.
Substituting Y = Y\-\- Y2 and using (11.1), we obtain
(F, XAde)= (Fi, X2) - (Xi, F2) - (a(Xi, Fi), 0-
Now H = K+U, where K(Z) = i(Z, I) and [/(a;, 0 = U{x). Thus
(F, rfF) = (F2, 0 + Fi[7,
and the equation
{Y,X Jdd)= -{Y,dH)
becomes
(Fi, X2> - (Xi, F2) - (a(X„ Fi), Z) = -(F2, Z) - (Fi, rfiJ), (11.2)
which must hold for all choices of Y.
Setting Yi = 0, we get
<Xi,-)= (-.Z). (11.3)
In fact, the scalar product occurring in the last equation is on V*. Transferring
it to an equation on F, we get
(Xi, •) = (•, Z). (11.4)
Let t »-> <C(t), v{t) > be a solution curve of our system transferred to T{M) ==
M X y by the Riemann metric. Then this last equation says C'{t) — v{t).
If we set F2 == 0 in (11.2) and use (11.4), we get
(•,X2)- (a(X„-),Xi) = -{■,dU).
Now for any solution of (C, v) we have
<■■« = (■. a)'
SO that these last two equations can be rewritten as
C'{t) = v{t),
(•' j) = («(^' •)' ^^ - <^' '^^>' ^^^'^^
which are known as Euler's equations.

544 CLASSICAL MECHANICS 13.12
12. RIGID-BODY MOTION
We shall apply the results of the previous section to the study of the motion of
a rigid body. For simplicity, we shall confine ourselves to the study of the
motion of a rigid body with one point fixed. The more general case where the
body as a whole is allowed to move can be handled by similar methods.
(Frequently, by considering first the motion of the center of gravity, the more
general case can be split into two parts: the motion of the center of gravity and
the motion of the body relative to the center of gravity. This then reduces the
problem to the one we are studying.)
In order to exhibit the generality of our method, we shall consider the
equations of motion of a rigid body in E^. Only at the end will we make use of
the fact that n — 3. Let us fix some positive orthonormal system "drawn through
the fixed point of the body". In other words, we fix some initial position Xq —
<h^, . . . ,h^> of the body. Any other position, Xi, of the body can be obtained
from Xq by a rotation: Xi — RiXq. Let R{t) — e^^ be a one-parameter group
of rotations. Then
RiR{t)xo = RiR(t)R^^Xi
is a curve of possible positions of the body. The tangent to this curve at Xi will
be denoted by A^^ Thus A^^ G T^^iM), If X2 = R2^o = R2R7^Ri^o ^
R2R1 ^:^i, then Ax2 is the tangent to the curve R2R{t)xo = R2R1 ^RiR(t)xoj
so that Ax2 = (R2RT^)*^xi' It is clear from its definition that Ax^ == 0 if and
only if A = 0.
We can regard the map A »-> A^i as a map from the space of skew adjoint
linear transformations to Tx^{M), Let V denote the space.of skew adjoint linear
transformations. Then since
dim V = dim M = ''^'' ~ ^^
and the map A »-> Ax^ is an injection, we conclude that it is an isomorphism.
We thus have a trivialization T(M) »-> M X V. Consequently, we get a F-
valued linear differential form co, as in Section 11.
Let us describe once more the meaning of co^: Tx(M) -^ V. If ? G Tx{M)
represents an infinitesimal motion of the body, then since the body is rigid, we
can regard ? as an infinitesimal rotation of the body relative to an observer
situated outside the body (fixed in space). Thus ? — J5 for some J5 G F, say.
Then (^, co) is the corresponding infinitesimal rotation expressed in terms of
the basis attached to the body; that is, (?, co) — RJ^BRi if x — RiXq,
We denote by A the vector field x^-^ Ax corresponding to A E.V, Let (ft
be the one-parameter group generated by 5 for some B E.V. Note that at any
X2 — R2^o we have «^ + ^2 — R2^^^^0' Then at any xi — RiXq we have
^As D —1_ r) „As^ D AsBt^
*-• - — (ptKie Xq = Kie e
r, Bt/—BtAsBt\^
— Kie [e e e jxq,

13.12 RIGID-BODY MOTION 545
SO that ^—^^^
If we differentiate this equation with respect to ^ at ^ — 0, we conclude that
[i, S] = (AB - BA) = -[A, Bl
so that, according to (H.l), we have
a{A, B) = [A, B] = BA - AB, (12.1)
We now show how a mass distribution on the body determines a Riemann metric
on M. Let p be a particle on the body with mass m. We will assume that the
particle p has coordinates -<p^, • • • , p^>" relative to the axes drawn on the
body. Suppose that the body is at position x = <hi, . . . ,hn^- Then the
particle p will be situated at the point p^hi + - - - + p'^hn E E"*. If R(t) == e^^
is a one-parameter group of rotations, then when the body is at RiR(t)x the
particle p will be situated at
Ri[p'R{t)hi + '"+ p^R(t)hn] = RiR{t)p,
Thus the velocity of the particle p as the body undergoes the motion generated
by A is RiApj and the kinetic energy of the particle p is im||i2iAp||^ —
Jm||Ap||^, since i2i is an orthogonal linear transformation. We define the kinetic
energy of Ax to be the total kinetic energy of all the particles of the body. Thus
illi.f = i| m\\Apf. (12.2)
body
Note that \\Ax\\ depends only on A, so that our third requirement of the last
section is satisfied, provided (12.2) does indeed define a norm on V. (Note: That
the mass distribution could be such that Ap = 0 for all p in {p : m(p) > 0}
does not imply that A = 0. For example, suppose that all the mass were
concentrated along a line l. If A represents infinitesimal rotation about I, so that
Ap = 0 for p E Ij then \\A\\ — 0. However, it is clear that if the set of p for
which m > 0 spans E^, then (12.2) defines a Riemann metric. In fact,
\\A\\ = 0 =^ Ap== 0
for all p belonging to a spanning set, and thus A = 0.)
Let us examine the scalar product given in (12.2) a little more closely. Let/
denote the linear function on E^ 0 E^ corresponding to the scalar product on
E^ (which is a bilinear form); in other words,
/(ai ® 6i H [-ak®hk) = (ai, hi)-\ h (ak, hk).
Let s be any element of F ® F. Then s defines a bilinear form on Hom(E^)
given by
s{A,B) = f{{A0B)s).
Note that if the tensor s is symmetric, so is the corresponding bilinear form. In

546 CLASSICAL MECHANICS 13.12
our present context the scalar product (12.2) comes from the tensor / G F 0 F,
where
/ — / mp (X) p
body-
is called the inertia tensor of the body.* Thus (12.2) can be written as
llill^.- I(A,A),
Euler's equations in this case become
/ dA\
V' dt)
C\t) ^ A{t) and \^, -^J = 1(1, A], A) - {Ac,t„ dU), (12.3)
Now the tensor / is symmetric. We can therefore find an orthonormal basis
e^, . . . , e^ oi E^ which diagonalizes /, so that
/ - he^ ®e' + he^ ® e^ + h /n^" ® 6".
Let Eij (i < j) he the antisymmetric matrix defined by
Eije' = e'\ Eije' - -e\ Eije^ - 0 (I 9^ ij).
Then
I(Eij,Eki) = 0 if i,j ^ kl
and
I{Eijj Eij) = li + Ij-
Now let us see what Eqs. (12.3) say for the case n == 3, where we have
[^12j ~^13] — ^23j [^12,^23] — ~^13j [^13j^23] — ^12-
Suppose A(t) == ai(t)E23 — ^2(0^13 + ^3(0^12, and let
B = hiE23 — &2^i3 + &3^i2 == const.
Then substituting B into (12.3) and comparing the coefficients of 61, 62? ^i^d
63, we get
(I2 + h)^= (h - I2)a2a3 + a,{E^2, dU),
(/i + ^3) ^ - (/i - /3)aia3 + a2(-^i2, dU), (12.4)
(/i + ^2)^= (I2 - Ii)a,a2 + asiE^s, dU).
A simple case of these equations arises when there is no potential, that is,
17—0. First of all, suppose that the body has a spherically symmetric distri-
* This differs slightly from that which is usually called the inertia tensor in physics
texts. In terms of the coefficients li introduced below, what is usually called lij is
li + Ij in our setup.

13.12 RIGID-BODY MOTION 547
bution of mass. Then Ii = I2 = 13 and Eqs. (12.4) become
dai da2 da^ ^
dt dt dt
In other words, A — const. The motion of the body consists of a steady rotation
about some axis fixed on the body. Of course, this means that for an observer
in space also, the body undergoes a steady rotation about a fixed axis.
Next, let us consider the case of an axially symmetric rigid body moving
freely; that is, /i — 12 and [7 — 0. The equations of motion then become
dai
~dt
da2
"dt
das
dt
where
K-
= Ka2asj
= —Kaiasj
= 0,
/3+ I2
The solutions of these equations can be written down immediately: as =
s — const and
ai — Ci cos Kst + C2 sin Kstj
a2 = —Ci sin Kst + C2 cos Kst.
Thus for an observer situated on the body the instantaneous axis of rotation
describes a circle around the axis of symmetry of the body. This motion is
known as regular precession. (This motion should not be confused with the
astronomical precession of the earth's axis, which is due to the gravitational action
of the sun and moon.)
If no two of 11, 12, and Is are equal, then the equations of motion can still
be solved in terms of integrals, although the expression is rather complicated.
We refer the reader to any standard book on mechanics for the details.
So far we have been considering the motion with no potential. If [/ f^ 0,
then Eqs. (12.4) are usually not so easy to solve. Let us treat the case of a
symmetrical top; that is, /i == 12 and 17 is given by the gravitational potential.
In order to solve this problem it is convenient to use the Euler angles of
astronomy as the local coordinates on M. In order to avoid confusion, we reproduce
the definitions here: We let 8x, 8y, dz be the basis vectors of E^ corresponding to
the rectangular coordinates x, y, z, where we take the origin of E^ to be the fixed
point of the body. We may assume that the center of mass of the body is distinct
from the fixed point—otherwise there will be no gravitational force acting on
the body. It is easy to see that the center of mass c lies on the axis of symmetry.
We shall assume that the vectors drawn on the body are such that 63 points in
the direction from the fixed point to the center of mass, i.e., from 0 to c. Then

548
CLASSICAL MECHANICS
13.12
0 < ^ < TT is defined to be the angle between &3 and 5^:
cos 6 = (&3, dz).
The line of nodes is the intersection of the plane spanned by &i and 62 with the
xy-plsme. (Thus, in order for it to be defined, we must restrict to the open set
b 9^ :hdz.) We define the unit vector n along the line of nodes to be the one that
makes §«, 63, n a positive (right-handed) basis of E^.
"^xLine of nodes
Fig. 13.5
The angle 0 < ^ < 27r is defined to be the angle that n makes with 8x and
0 < «^ < 27r is defined to be the angle that n makes with hi. We now wish to find
the transformation relating the basis of Tx{M) given by d/dd^ d/dip^ d/dtp with
the orthogonal basis jE/12, -~jEi3, JE23 introduced earlier. Suppose that x has
the coordinate <6ftp, <p>. (See Fig. 13.5.) Now {d/dd)x represents an
infinitesimal rotation about the line of nodes and n = (cos (p)hi — (sin <p)h2i so that
\ddjx
{cosip)E23 — (sin (p){—Eis).
The vector (d/d(p)x represents infinitesimal rotation about 63, so that
Finally, {d/d^)x represents infinitesimal rotation about 5^. Now
{K ^3) = cos 6,
Furthermore, since (5^, n) == 0, the projection of bz onto the plane spanned by
§1 and ^2 must still be orthogonal to n, since n lies in that plane. It is therefore
easy to check that this projection is given by (sin (p)hi + (cos «^)62- Thus
Bz = sin d [(sin (p)hi + (cos (p)h2] + (cos d)hsy
and therefore
W/x
(sin d sin (p)E23 + (sin d cos «^)(—jE/13) + (cos d)Ei2.

13.12 RIGID-BODY MOTION 549
If ? e Ta:{M) is given by
we have
2m) ^ur
= d^cos e)Hl2 + Is) + (sin ^)Hh + Is) + i>\li + 12)
+ 2(i4{li +12) cos e
+ xl.'iim' e)[{h + Is) sin^ ^ + (/i + U) cos^ ^] + (/i + /2) cos^ 6).
Now J by assumptioa Ii = I2, Let us set
Af 1 = /i + /3 ^ ^2 + /3 and Mg = /i + /2.
Then the above expression simplifies to
K{e, ^, <p; 6, i, ^) = JMi(d2 _^ ^2 gi^2 ^) ^ i^^(^ ^ ^ ^^g ^)2^ (12.5)
Let us now obtain the expression for the potential energy. It is proportional
to the height of the center of gravity if we assume (as we shall) a uniformly
vertical gravitational field. Thus
U(d, tp.ip) =: k cos 6, (12.6)
where k = mg\\c\\ and m is the total mass of the body, g is the force of the
gravitational field, and ||c|| is the distance of the center of gravity from the fixed
point. Thus the Lagrangian is given by
L{d, ^,ip\6, i/', ^) = 4Mi(iJ2 + ,/.2 sin^ e) + hM%{^ + ^ cos (9)2 - fc cos 6,
(12.7)
Note that 6/dip and d/d^ are both isometries of the Riemann metric which
leave V invariant. Thus the corresponding momenta are constants of the motion.
In other words, for any motion of the system we have
Pa/Bf = Ci and Pd/d(p = C^2)
where Ci and C2 are constants. But
Pdld^ = TT = Af 1 ^ sin^ e + M2 cos e{(p + ^ cos 6) = Ci
and
Pd/d^ = Af2(^ + ^ cos e) = C2.
Solving these equations gives
<p + xl^cosd=^~-r Ml ^ sin^ e=^Ci -C2 cos 6. (12.8)
M2

550 CLASSICAL MECHANICS 13.12
Let us substitute (12.8) into the expression for the energy,
E == K+U = Wi(6^ + ^^ sin^ 6) + \M2{<p + ^ cos 61) + ^ cos 6,
to get, for a given value of Ci and 'C2, the expression
^ + I M,e' + ^^^7/lT/^' + kcose^ const. (12.9)
2M2 ' 2 ' ' 2Misin2(9
Thus, just as in our treatment of the central-force problem, for a fixed
value of Pd/d(p and P^/^^|/, the motion of 6 is determined by a one-dimensional
mechanical system whose energy is given by (12.9). After solving this
mechanical system for 6, we can then obtain i^(-) and <^(-) by integrations from (12.8).
In order to obtain qualitative information about the behavior of the solutions
d{'), we can apply the method of Section 5 to our one-dimensional mechanical
system. Note that if Ci ^ C2 (as would be the case if the body were "spinning
fast")J the kinetic energy tends to infinity as 0 -^ 0 and as ^ -^ tt. We
therefore conclude that 6 oscillates between two values 0<^i<^<^2<'7r.
In other words, the axis of symmetry of the body executes a periodic up and down
motion (called nutation). As d oscillates, xj/ satisfies (12.8). Let us graph the
curve that 63 traces out on the unit sphere. If Ci > C2 cos 0 > 0 for ^1 <
6 < 62, then xp > Q although it oscillates in magnitude. The motion is thus as
given in Fig. 13.6. Another possibility is that
Ci — C2 cos 01 < 0 and Ci — cos 62 > 0,
so that i is negative near 6 = di and positive max d = 62. In this case the
average of xj/ over a period is still positive, so that the motion is as shown in
Fig. 13.7.
Fig. 13.6 Fig. 13.7 Fig. 13.8
A limiting case is where Ci — C2 cos di = 0, where the motion of 63 is as
shown in Fig. 13.8. (This is the case that arises if the axis of a spinning top is
held fixed at some position 0, ^ and then allowed to fall.)
The motion of the axis of body around the 2;-axis in all these cases is called
precession. It should be remembered that all this time the top is spinning about
its axis with constant angular momentum 62-

13.13 SMALL OSCILLATIONS 551
13. SMALL OSCILLATIONS
Suppose we are given a mechanical system on a manifold M with energy
H = K + U. Suppose that the "force field" dU vanishes at some xq G M.
Then the constant curve C{t) = a^o is a trajectory of the system. In fact, let us
choose a chart (Wj a) with coordinates x^, . . . , x^ such that
«(^o) = <0, . ..,0>.
Then in the corresponding coordinates <q^y. . . , q^> on 7r~^(W) we have at
the point <0, . . . , 0, . . . , 0>
dH dU ^ ^ dH dK ^
It is therefore natural to expect that for small initial values of q and p and
for small intervals of time, the solutions of the system should be well
approximated by the following linear system:
Replace the potential energy,
U(x\...,x'') = ZaijxV+ Us,
[where Us = 0(\\x\\^)', that is. Us vanishes to third order at x = 0] by the
quadratic potential energy,
U2{x) = JE aijxVj
and replace the kinetic energy corresponding to the given Riemann metric,
K(q,q) = hY.giM)i"i\
by the one corresponding to the Euclidean metric at xq,
K2(q,q) = hEgij{0,..,,0)q'q\
We thus obtain a mechanical system H2 whose corresponding equations
(5.1) are actually linear. [The reader should check, as an exercise, that these
equations are exactly the equations of variation (introduced in Section 2) of the
vector field X_dH along the curve C(t) = <0, ....... 0> in T*{M).]
Of course, as time increases, the values of q^ and p* might become quite
large and the linear approximations useless. However, under certain
circumstances we can guarantee that q^ and p* remain small for all time. In fact,
suppose that the quadratic form X! ciijxV is positive definite. Then U has a
strict minimum at xq—say U(xo) = 0. In particular, if we start at Xq with a
kinetic energy K = E, where E is sufficiently small, then x will be restricted to
the neighborhood of xq defined by U(x) < E, and the momenta will be restricted
by the condition K < Ey since, by the conservation of energy, K + U = E.
(See Fig. 13.9.) Thus the q^ and the p* will remain small. This does not mean
that the solutions to the original mechanical system with Hamiltonian H will
remain close to one fixed solution curve of the linearized system. It does mean

552
CLASSICAL MECHANICS
13.13
Fig. 13.9
that for any short interval of time (that is, a short interval near any time in the
future), the trajectory will be close to some trajectory of the linearized system.
It is therefore important to study the behavior of such mechanical systems.
We are thus interested in the following kind of mechanical system: The
configuration space M is a vector space. The Riemann metric is given by a Euclidean
metric on the manifold. The potential energy is a positive definite quadratic
form on this vector space. Let us choose rectangular coordinates x^, . . . , x^
with respect to the Euclidean metric. Thus
and the map £ is given by
Thus
iqy + --- + (ry
V = 9 •
L{q, q) = K{q) - U{q)
and Lagrange's equations become
-dt = '^'
dt
= — E aaq'-
(13.1)
(Of course, these are just the Euler equations (11.15) for the case at hand where
a = o.)
Equations (13.1) can be written more suggestively as follows: Let A be the
linear transformation whose matrix is (a^y). Stated invariantly, A is the unique
self-adjoint linear transformation such that
U(x) = {Ax, x).
(13.2)
Then the trajectories v(') of the system are the solutions of the second-order
differential equations
d^v .
df^ = -^^•
(13.3)
To find the actual solutions of (13.1) and (13.3) we apply Theorem 3.1 of
Chapter 5. According to that theorem, if M is finite-dimensional, we can find an
orthonormal basis e^, . . . , e^ of eigenvalues of A. In other words, if 2* are the

13.14 SMALL OSCILLATIONS (CONTINUED) 553
rectangular coordinates corresponding to this basis,
U{z\...,z^) = Xi(^i)2 + ... + x,(^-)2
and Eqs. (13.1) and (13.3) become
Thus the general solution of (13.3) is given by
v{t) = (ai cos Xi( + &i sin \it)e^ + • • • + (^n cos X„< + 6„ sin X„0«",
(13.4)
where the constants a ^ and h ^ are determined by
v{{)) - aie^-\ \-ane''
and
jj(0)- \,h^e^ + '" + \nhne\
Thus the general motion is a superposition of independent oscillations, the
frequency of each oscillation being determined by the eigenvalues {X^}. That
is why the mechanical system (13.1) is called a system of "small oscillations".
14. SMALL OSCILLATIONS (Continued)
So far, we have been considering the linearized equations (13.1) as an
approximation to a finite-dimensional mechanical system. The philosophy has been
that the solutions to the actual mechanical system exist, but are hard to find.
We use (13.3) as good approximating equations.
Fig. 13.10
It turns out that the method has more extensive applications, even to the
case of infinite-dimensional systems where the very existence of solutions to the
"actual" mechanical systems may be difficult to establish. Let us illustrate by
the mechanical system consisting of a stretched string which is held fixed at two
endpoints. For simplicity in illustration, we shall assume that the string is
restricted to move in the x2/-plane, although this is in no way essential to the
argument. Let us assume that the string is homogeneous and that the two
fixed points are (0, 0) and (0, 1). Then the configuration space should be all
possible smooth curves joining (0, 0) to (0, 1). Thus the curve shown in Fig.
13.10 would be a possible element of our configuration space. In some sense.

554 CLASSICAL MECHANICS 13.14
the configuration is an "infinite-dimensional manifold" and with suitable work
this idea can be made precise. However, what is of interest to us is behavior
near the "equilibrium" curve C{t) = {t, 0). For such curves we will have
dx/dt > 0, and so the curve can be described by using x as independent variable,
i.e., by giving a function u{x). In other words^ we are replacing the big
configuration space by the approximating vector space V of all functions u of one
variable, with u{0) = u{l) = 0, Thus V is regarded as the "tangent space"
to our system, in the sense that u is the "tangent vector" to the curve Cs(-),
where Cs(r) — (r, su{t)). (Remember that our configuration space is a
collection of curves, so that a curve in our configuration space is a one-parameter
family of curves.) Now we expect the "kinetic energy" of u to be the total of
the kinetic energy of all the particles on the string. The particle at r has velocity
u{t) and therefore kinetic energy ^mu(T)^. If we assume that the mass density
is constant, we thus get
K^iu) — im I u^ dx
Jo
as the expression for the kinetic energy. This makes V into a pre-Hilbert space
as in Section 6 of Chapter 6.
We expect that the potential energy depends on the stretching of the string,
i.e., that it is some function of the length:
C/(C)=F^'||C'(0||d<) = F(L),
where L is the length of the curve and F is some smooth function with F{1) = 0.
For curves parametrized by x, the length is given by
f!^!^
2
dx.
Now
Vl + ci^ — 1 + h^^ + higher-order terms.
Thus using a Taylor expansion for F,
F{L) ^ F\\){L - 1) + quadratic terms in (L - 1)
and
du
dx
L — 1 = - I \T~) ^^~^ I higher-order terms i
we see that 172, the quadratic approximation to U, is given by
^^^^) = f io \W '^^^

18.14 SMALL OSCILLATIONS (CONTINUED) 555
By analogy with (13.3) we expect that the "small oscillations" of the string are
solutions Ut{') of the equations
d^ut .
where A is the self-adjoint linear operator such that
\ m f
Now
{Au, u) = ^ I u{Au) dx.
^ Jo
Since u{0) = u{i) = 0, we have, by integration by parts,
I du du _ _ i tdSc\
Jo dx dx Jo \dx^/ '
so that
C d\
Au= -jT^ •
Equation (13.3) thus becomes
Hh^^'^^' ..(0) = «.(!) = 0. (14.1)
Note that we have derived (14.1) by reasoning by analogy. We have not
formulated the actual (nonlinear) infinite-dimensional mechanical system, nor
have we any guarantee that there is one that can be solved. Furthermore, we
don't actually know the function F. We only need to know its form and the
value of F\l), Nevertheless, Eq. (14.1) gives a good explanation of observed
physical phenomena.
The solution of (14.1) proceeds just as in the finite-dimensional case due to
the fact that the operator A with the boimdary conditions u{0) = u{l) = 0
is a Sturm-Liouville system, so the results of Sections 6 and 7 of Chapter 6
apply. In fact, we can choose the functions
Un{x) = sin (mrx)
as an orthogonal basis of eigenvectors of A, where Un has the eigenvalue
n^(c/m)7r^. Thus the general solution is given by
Ut(x) = (aI cos at + hi sin at) sin (ttx)
+ (^2 cos 2at + &2 sin 2at) sin (27rx) + * * *,
where a = (c/m)7r^. In other words, the general solution is a superposition

556
CLASSICAL MECHANICS
13.14
(linear combination) of the "harmonics"
sin nat sin mrXj cos nat sin rnrx.
These can be regarded as "standing waves", as, for example, in Fig. 13.11.
Isin Sat sin Sttx
Fig. 13.11
As another illustration of this method, let us consider the "vibrating
membrane" in n-dimensions. Here we are given a domain D with almost regular
boundary in E^. We consider a stretched membrane in E^"^^ == E^ X E^
which is fastened along dD in E^ X {0}. Again, as our linear approximation V
to the configuration manifold, we take the space of all functions u on D which
vanish on dD. To be precise, we let V be the space of functions which we
defined, are of class C^ in some neighborhood Z), and vanish on dD. Thus the
membrane is the surface in E^"^^ whose points are of the form
■KX , . . . , X^, U(X y . . . ,
where <x^j . . . , x^> G D. As before, we define the kinetic energy as
K(u) = if u\
Jd
while the potential energy is to be some function of the total volume (area)
which vanishes at ijl(D). Now the total volume (area) of the hypersurface is
given by (see Exercise 4.3, Chapter 10)
Thus, as before.
U2{U)
D[u, u],
where D is the Dirichlet integral introduced in Section 11 of Chapter 12. By
Green's formula we have (since u = 0 on dD)
D[u, u]== - f
Jd
uAu.
Thus the operator A of (13.3) is given by
(c/m)A, and Eq. (13.3) becomes
(14.2)

13.14 SMALL OSCILLATIONS (CONTINUED) 557
Note that this is exactly (14.1) if n — 1. In order to solve (14.2), we must find
a complete set of eigenvalues for (14.2). If {un} is such a basis, where X^ is the
eigenvalue associated to Uij then the general solution of (14.2) would be given by
y^tix) = X) (O^n COS XJ + hn Slli \nt)Un{x).
The problem of showing that —A has a complete set of eigenvectors is
somewhat more difficult for n > 1, and will be discussed in the exercises.
EXERCISES
In order to study the eigenvalue problem it is convenient to replace the space of
C^-functions vanishing on ^D by the space Hi (where we refer back to page 361 for the
definition of the spaces H^.) To show that this replacement is legitimate, we have:
14.1 Show that if </? is a function which is C^ in a neighborhood of D, then ip G H^
if and only if ip vanishes on dD.
14.2 Let the operator K be defined as Kf = (1 — A)/ for smooth / and extended as
in Section 14 of Chapter 8 to a map of Hg —> Hs-2' Let g G Hq. Since
|(^, v)\ < ll^llollHIo < ll^llollHli for any v G H?,
conclude that there is a bounded linear map L from Hq to H^ satisfying
(Lg,v)i = {g,v)o for all vGH^.
14.3 Let/ and g be locally integrable functions on D. By this we mean that/</? and
gcp are integrable for every test function (p E Co(D). We say that the differential
equation Kf = g holds weakly on D if (/, Kcp) = (g, cp) for all test functions (pECo(D).
Show that this generalizes the notion of being a solution by proving the following
lemma.
Lemma. If the functions / and g are respectively in the classes C^{D) and C^(D),
then the equation Kf = g holds in the weak sense if and only if it holds in the
classical sense.
In proving this lemma you may assume that if /i is a continuous function on D
such that johip = 0 for every test function </?, then h = 0.
14.4 Prove that the operator L of Exercise 14.2 is a right inverse of K in the weak
sense. That is, show that \i g E. Hq and/ = Lg, then Kf = —g weakly on D.
14.5 We now want to show that / is actually in C^{D) if g is suitably smooth.
Roughly speaking, what we want to do is to "round off"/ and g near the boundary of C
in such a way that we can consider the adjusted functions to be defined on the whole
of U^, and can then apply Exercises 14.25 and 14.30 of Chapter 8.
Our rounding off process will simply be multiplication by an arbitrary but fixed
function \l/mCo{D). Prove, to begin with, that multiplication by such a i/' is a bounded
linear mapping of Hg into itself for any s.
14.6 We know that Dj = d/dxj is a bounded linear mapping from Hg to Hg—i for
every s. Combine this fact with the result of the above exercise to show that \i Kf = g

558 CLASSICAL MECHANICS 13.15
weakly on D, and if xp is any fixed element of Co(D), then there is a differential operator
R of order 1 defined on the whole of U^ such that
1) K{\l/f) = \l/g+ Rg weakly on D;
2) h^-^ Rhis & bounded linear mapping from Hs to Hg-i for every s.
In order to consider R to be defined on IR^ we have to extend rp to R^ in an
obvious way. The proof is essentially an integration by parts.
14.7 We say that a function h defined on D is locally in Hg if (fh G Hg (when extended
to U^) for every (p E:Co{D). Use the above exercise to prove the following lemma:
Lemma. Suppose that Kf = g weakly in D, that / G Hj locally in D, and that
g ^ Hm locally in D. Then/G //min(m+2,y+i) locally in D.
[Hint: In order to prove this crucial lemma, show first that the weak differential
equation of Exercise 14.6 holds for all test functions cp on IR^. The crux of the matter
is that there is a function X EiCoiD) such that X = 1 on the support of yp. We extend X
to R^ as above, and then for each test function (p on U^ we write
cp =Xcp+ (1 -X)cp,
where Xcp E:Cq{D). Now use the fact that the test functions cp on IR^ are dense in Hg
for every s, that ypf G Hj, and that ypg G Hm-]
14.8 Suppose now that g G Hq^ that/ = Lg G Hi (Exercise 14.2), and that kf = g
weakly on D (Exercise 14.4). Apply the above lemma repeatedly to show that if
g G Hm locally in D, then / G Hm+2 locally in D. Conclude from Sobolev's lemma that
if m > n/2 + j and g E Hm locally in D, then / = Lg E 0^^{D).
14.9 Show that \\Lg\\i < \\g\\o and conclude from Exercise 14.31 of Chapter 8 that if
we regard L as an operator from Hq to Hq, it is compact, and all of its eigenvectors
belong to Hf. Use Exercise 14.8 to show that every eigenvector belongs to C°^{D).
15. CANONICAL TRANSFORMATIONS
In Sections 1 through 5 we formulated the notion of a mechanical system as a
flow of a certain type on the cotangent bundle of the configuration space. The
defining equations for the vector field X generating this flow were X J^ —
—dH. Thus the basic property of the cotangent bundle used in singling out the
class of flows is the existence of the two-form 12. It turns out that in studying
the equations of mechanics it is sometimes convenient to forget that the flow is
on the cotangent bundle and to concentrate on the form 12. For instance, we may
be able to introduce charts that don't arise from the configuration manifold but
in terms of which the vector field X takes a particulary simple form. We shall
therefore want to consider a manifold N which carries a two-form 12, subject to
certain restrictions which we shall describe below. On such manifolds we shall
study vector fields satisfying X J 12 = —dH. It will be convenient to allow H
to depend on the time t, as well as being a function on N, so that X will be a
time-dependent vector field. The reason for this is twofold. First of all, it allows

13.15 CANONICAL TRANSFORMATIONS 559
the consideration of "nonconservative" mecftanical systems. Secondly, even in
the study of the systems we have introduced so far, it is sometimes convenient
to make a time-dependent change of coordinates to simplify the equations. This
has the effect of changing a time-independent vector field into a time-dependent
one. Now to the definitions:
Definition. A manifold N is said to possess a Hamiltonian structure (or to
be a Hamiltonian manifold) if there is an exterior two-form Q. defined on N
such that
i) do. = %\ and
ii) 12 is of maximal rank in the sense that (4.3) holdJs.
Remxirks
a) As we have seen, if iV == T*(M), then iV is a Hamiltonian manifold
where Q. is given by (4.1).
b) If N is finite-dimensional, then it must be even-dimensionaii In fact,
condition (ii) says that Q. restricted to each tangent space is an antisymmetric
bilinear form which is nonsingular. This can happen on a vector only if it is
even-dimensional.
c) It can be proved that if iNT is a finite-dimensional Hamiltonian manifold,
then one can always find local coordinates ^i, . . . , qn, Pi, - - - j Pn such that
^ = Z dp>i A dqi.
[We know this to be the case if AT == T*{M).] The point of this result (which
we shall not prove here) is that locally all Hamiltonian manifolds of the same
finite dimension look alike.
We shall now single out a class of vector fields on TV X R.
Definition. The vector field X is a Wamiltonian vector field if there is a
function H = Hx on iV X IR such that
Xt= (X.dt)^ 1, (15.1)
where t is the standard coordinate on U, regarded as a function on iV X IR,
and
X J (7r*12 - dH A dt) - 0, (15.2)
where tt is the projection of iV X IR onto N\ 7r(x, t) == x. Note that H is
determined up to a function of t alone.
Let us set
CO = 7r*12,
so that (15.2) can be written as
X J{c^ -dH A dt) = 0. (15.20

560
CLASSICAL MECHANICS
13.15
t f
iw^O
(It
Fig. 13.12
If we consider the direct sum decomposition of the tangent space of iV X IR,
then condition (15.1) says that we can write
X
(-•i)'
where X is a time-dependent vector field on N; that is, X is a rule which assigns
a tangent vector X(x, t) G Tx(N) to each x and t. Since co does not involve dt,
we can write
X J CO = X(-, t) J^ at any time t. (15.3)
[Strictly speaking, this equality should be written as follows: Let it: N ^^ N X U
be the map defined by it{x) = (x, t). Then
Also,
i*(XJco) = X(., 0 J^.]
dH
(15.30
(X, 6^^> = <X(-, 0, ^iy(-, 0) + -^ at any time t.
Thus (15.2) can be split up into two equations. When we compare the terms not
involving dt, we obtain
Thus
X{',t) JQ= -dH{', t) for every fixed t.
X J CO = -dH + ^ dt
(15.2a)
(15.2b)
Note that (15.2a) is just the condition stated at the beginning of Section 5 with
the novelty that H (and therefore X) can now depend on time.
Definition. A diffeomorphism, ^, of iV X IR ^ iV X IR is called a canonical
transformation if
i) ^*(co) = CO — dW A dt, where W = W-^ is some function depending
on Jp] and

13.15 CANONICAL TRANSFORMATIONS 561
X
Fig. 13.13
ii) Tp is time-preserving, i.e., 7p has the form ^(x, () — (<p{Xj 0, 0? where
<^(-, 0 is a diffeomorphism of N for each t.
Observe that if ^ is a canonical transformation, then so is ^~^ and
TF^„i= -{ip^)*W^. (15.4)
Also observe that if cp and ip are canonical transformations, then so is i?' o ^,
where
Wj^^ = ^*W^ + Wj, (15.5)
These facts follow directly from the definitions and will be left as exercises for the
reader.
We note next that if ^ is a canonical transformation and Z is a Hamiltonian
vector field, then ^*(X) is also a Hamiltonian vector field. In fact,
<P*X J [co - d{W^ + cp*H) A dt] = cp*X J [^*co - ip*(dH A dt)]
= <P*[X Jic^-dH A dt)] = 0.
Thus we may take H^*x as
H^*x -=W^+ <p*H, (15.6)
Let Z be a Hamiltonian vector field, and let 7p be the map of iVxK^iVX U
obtained by letting the system evolve from time t == 0 according to the flow
generated by X. That is, let the map <p(-, t) be defined so that the curve
t H-» (p(x, t) is a solution curve to the (time-dependent) vector field X which
passes through x at time ^ = 0. To put it another way, the curve 11-^ (<^(x, t), t)
is the solution curve to the vector field X which passes through (x, 0) at time
zero. (See Figs. 13.12 and 13.13.)
Note that it follows from the definition of ^ that
^*(l)
^<p(X,t)'
We claim that ^ is a canonical transformation. In fact,
it {<P co) = (Tp o ^) TT Q = (p{', t)*Q^

562 CLASSICAL MECHANICS 13.15
since Tp o {^(Xj t) = ((p{x, t)j t). But
d
§- i<p(', s)*12) = <^(., s)*Dx,.,n^ = 0
ds
by (15.2a). Thus
it cp 0) = 12,
or, in other words, <^*co is of the form o) + 6 A dt To determine 6 it suffices to
take the interior product with d/dt, since co doesn't involve dt. But
(f^ J <p*7r''n\ = cp* U^ (j\ J 7r*12l =<p*(Xj 7r*12) = <p* (-dH + ^-§dt\.
so 6 = 7p* dH. Thus ^ is a canonical transformation and
T7- = -7p*Hx. (15.7)
Note that (15.7) is just what we would expect from (15.6). In fact, <^*Z = d/dt,
and we may take HdiQt = 0.
Equations (15.6) and (15.7) are used in conjunction in the following way:
Suppose that H = Hq + Hi, where we know how to solve the differential
equations corresponding to ^o- In other words, we can find the map 7p
corresponding to the vector field Xq, where Xq J (co — dHo A dt) = 0. If
X J {c^ - dH A dt) = 0,
then <^*Z is a vector field whose corresponding Hamiltonian function is ^*Hi
by (15.6)
This method was first introduced by Lagrange in the study of the n-body
problem. We can let ^o be the Hamiltonian obtained by ignoring the terms in
the potential energy coming from the interaction of the planets, and let Hi be
the rest of the Hamiltonian H. The solution for ^o is then given by having the
planets move about the sun according to Kepler's laws. For simplicity in
discussion, let us restrict our attention to that portion of phase space where the
motion is elliptical. Then the motion of the planets is specified by giving the
various parameters of each ellipse (such as the plane of the ellipse, its major
axis, its eccentricity, etc.) and telling the position of the planet on its ellipse at
time t = 0. This corresponds to the use of the map (p. One then regards the
equations of motion of the whole system as differential equations for the
parameters of each ellipse. This corresponds to studying the vector field ip*X.
This idea of introducing the parameters of the ellipses as "generalized "
coordinates was one of the key steps leading to the notion of the invariant calculus
on manifolds.
We have seen that solving the differential equations corresponding to a
Hamiltonian H is the same as looking for a map 7p satisfying (15.7). Under
certain circumstances this can be reduced to looking for the solution to a certain
partial differential equation. Suppose that we have local coordinates qi, ^ - - ^ qn^

13.15 CANONICAL TRANSFORMATIONS 563
Ply . . . yPn such that 12 == I] ^Pi A %. Let V = V(qi, . . . , ^n, Pi, • • • , Pnj 0
be a function such that the maps <^i and <^2 are diffeomorphisms, where
M^l, • • • , ^n, Pi, . . . , Pn, 0 ^ < ^1, • • * ' ^^' d^ ' * * * ' ^ ' ^/^
and
<^2(gi, . . . , gn, Pl, . . . , Pn, t) = < ^ ' * * * ' ^ ' ^1' • • • ' ^^' Y
We claim that ^ = <^i o <p~^ is a canonical transformation and that
TT^-^^'*^- (15.8)
Note that co — c?(X) Pi dq^) — —d(^ qi dpi). Thus
<^*a) — CO -= cZ(^* ^ p^ c^g^ + ^ qi dpi)
^ ^<^2"^*(<^i 23 Vi dqi + <^2 2] ^z ^Pz)
of
= _,(,-..*!) ^,,
If we substitute into (15.7) we see that (p solves our differential equations if and
only if
^-1*^ + ^*^ = 0.
But :p*H = (p^^*ip*H, so we can write this equation as
^ + ^t^ = 0 (15.9)
or
dV
dt
+ ff(3x,...,3.|^.---.£./) = 0. (15.90
Equation (15.9) is known as the Hamilton-Jacohi equation.
We therefore have a prescription for (locally) solving the equations of
motion: Find a solution V of (15.9) which has the property that (pi and <^2 ^re
diffeomorphisms. Under certain circumstances, a proper choice of coordinates
allows us to solve (15.9) by the method of "separation of variables''. We
illustrate this method in the following examples.

564 CLASSICAL MECHANICS 13.15
Example 1. Central-force motion again. Here
when we use polar coordinates in the plane. Equation (15.9) becomes
Since the variables t and 6 do not occur explicitly in this equation, we seek a
solution of the form
V= V,{t) + V2{d) + Vs{r)
and conclude that both V[(t) and ¥2(0) depend only on r, and so must be
constants. We may thus write
V[{t) = ~E, V'2(d) - Ae,
where E and Ae are constants. (This just reflects the conservation of energy and
angular momentum.) We then get the equation
'^ = Viir) = ^2m{E - t/(r)) - f
Thus
Aee +
f
Jo
2m{E - U(s))
1/2
ds — Et
is a solution of (15.9). Here we consider V a function of the variables r, 6, Ej Ae.
Then the map <^2 is given by
cp2(r, d;E,Ae) =
-t +
m ds
[ [2m{E - Uis)) - j{\"
Ae ds
l^s^[2m(E-ms))-§\^'' '
E,Ae
Now the map <^i o <^2 ^ takes the flow into the constant flow, so that we must
have
m ds
t-to =
and
^0-
\2m(E - U(s]
Ae ds
', .' [imiE - VM) - 0\
1/2

13.15 CANONICAL TRANSFORMATIONS 565
where ^o and ^o are constants. Note that the second of these equations gives the
orbit explicitly, which can then be solved to give r as a function of t.
Example 2. The simple harmonic oscillator. Here
so Eq. (15.9) becomes
2m\dqJ ^ 2 ^ dt
Again, since time doesn't enter explicitly, we can write
V = -Et + T7,
where T7 is a function of q alone which must satisfy
or
/2
W ^ Vmk I [^ — s^ ) ds.
Then
Thus
=(r/:(¥-r''-'-
Solving for q in terms of t gives
Example 3. The motion of a particle attracted by two fixed point masses. Here
where ri and ?^2 are the distances to the two points and A and B are constants
(determined by the masses of these two points).
For the purpose of solving this problem, it is convenient to introduce so-
called elliptical coordinates. Let us assume that the two fixed points lie on the
X-axis, each at a distance c from the origin. We may take c — 1 for simplicity.

566
CLASSICAL MECHANICS
13.15
(-1,0)
Fig. 13.14
(See Fig. 13.14.) In the xiy-plane define the local coordinates ? and rj by setting
? = Wi + ^^2), V = i{ri — r2).
Thus the curves ? — const represent ellipses with semimajor axis ? and foci at
the two fixed points, while the curves r) — const are hyperbolas with semimajor
axis rj and the same foci. Note that 0<|r7|<l<?<oo. The equations of
these two curves are
2 2
^__(__ y
= 1
and
y
= 1,
so that
Thus
and therefore
^2 ' ^2 _ 1 ^ ^ ^2 I - r)^
x' ^ ^'v' and y' = U' - 1)(1 - ry^).
^d^ rjdrj
dx __ d^ drj
dy
y
?2
1-772
If we now rotate about the x-axis to get the analogue of cylindrical coordinates
in space, we have the coordinates
<x,p,d> and <J, r;, ^>
in space, and the Euclidean metric is given by
dx^ + dy^ + dz^ = dx^ + dp^ + p^ dd^
- (?' - ^') {-^i + r^) + (?' - i)(i - ^') ^^'-
Also,
jj _A I B _Ar2 + Bri
1
^2_,2
(a? + M,

13.15 CANONICAL TRANSFORMATIONS 567
where a = A + B and 0 ^ A — B. Then H takes the form
H(^y rj, d, P^, P,, Pe, t) ^ 2m(?2 _ ^2)
X [(?' - l)Pf + (ry" - l)Pl + (^^^ + r^) ^'' + «^ + ^^] •
Since t and ^ do not occur explicitly in (15.9), we may write
V = —Et + AeS + W,
where now W must satisfy
(£^ - .) (f)% tf - .) (15) + (^; + ^) ^1 + <+„
= 2m{f - v^)E.
Note that if we set W = T7i(Q + ^72(^7), this equation separates into two, and
we can explicitly solve each of them by quadratures. This gives the solution to
the original equations of motion. We leave the details to the reader.

SELECTED REFERENCES
Chapters 1, 2, and 7
A more extensive treatment of linear algebra, over arbitrary coefficient fields, can be
found in
BiRKHOFF, G., and S. MacLane, A Survey of Modern Algebra, 3rd ed., Macmillan,
New York, 1965.
Hoffman, K., and R. Kunze, Linear Algebra, Prentice-Hall, Englewood Cliffs, N. J.,
1961.
For linear algebra over commutative rings that are not fields, see
Lang, S., Algebra, Addison-Wesley, Reading, Mass., 1965.
Zariski, 0., and P. Samuel, Commutative Algebra, 2 vols.. Van Nostrand, Princeton,
1959, 1960.
Chapter 3
Difficult but rewarding is
DiEUDONNE, J., Foundations of Modern Analysis, Academic Press, New York, 1960.
The differential notation in this book is Df(a) (instead of dFa) and Df(a) • ^
(instead of dF«(^)).
Chapters 4 and 5
Good books on general (point set) topology are
Kelley, J., General Topology, Van Nostrand, Princeton, 1961.
Simmons, G., Topology and Modern Analysis, McGraw-Hill, New York, 1963.
The remaining standard examples of Banach spaces and Hilbert space require the
Lebesgue integral and are therefore beyond our scope. However, the interested reader
can pursue the abstract theory of Banach and Hilbert spaces in Simmons and in such
books as
HiLLE, E., and R. S. Phillips, Functional Analysis and Semigroups, American
Mathematical Society, Providence, R. I., 1957.
Murray, F. J., An Introduction to Linear Transformations in Hilbert Space, Princeton
University Press, Princeton, 1941.
RiESz, F., and B. Sz-Nagy, Functional Analysis, Ungar, New York, 1955.
569

570 SELECTED REFERENCES
YosiDA, K., Functional Analysis, Springer, Berlin, 1965.
The books by Murray and Yosida are more advanced and harder.
Chapter 6
Standard books on ordinary differential equations are
BiRKHOFF, G., and G. C. Rota, Ordinary DijSferential Equations, Ginn, Boston, 1962.
HuREWicz, W., Lectures on Ordinary Differential Equations, M.I.T. Press, Cambridge,
Mass., 1958.
Advanced treatises are
CoDDiNGTON, E., and N. Levinson, Theory of Ordinary Differential Equations,
McGraw-Hill, New York, 1955.
Hartman, p.. Ordinary Differential Equations, Wiley, New York, 1964.
Chapter 8
This chapter has been devoted to the theory of content. For modern mathematics
the more powerful theory of Lebesgue measure and integration is needed. For one-
dimensional theory the reader can consult
RuDiN, W., Principles of Mathematical Analysis, 2nd ed., McGraw-Hill, New York,
1964.
For the general theory the reader can consult
Halmos, p.. Measure Theory, Van Nostrand, Princeton, 1961.
Another way of computing certain definite integrals, involving the "residue calculus",
is discussed in a set of exercises at the end of Chapter 12, and can be read independently
after Chapters 8 and 11.
For the relationship of integration to "generalized functions'' see
Gel'fand, I. M., and G. E. Shilov, Generalized Functions, vol. 1, Academic Press,
New York, 1964. A little complex variable theory is necessary to read this book.
Chapters 9, 10, and 11
For a more abstract treatment see
Lang, S., Introduction to Differentiable Manifolds, Interscience, New York, 1962.
For a less abstract treatment see
Flanders, H., Differential Forms with Applications to Physical Sciences, Academic
Press, New York, 1963.
Fleming, W., Functions of Several Variables, Addison-Wesley, Reading, Mass., 1965.
Spivak, M., Calculus on Manifolds, Benjamin, New York, 1965.
A more extensive treatment of differential geometry will be found in
WiLLMORE, T., An Introduction to Differential Geometry, Oxford University Press,
London, 1959.

SELECTED REFERENCES 571
Chapter 12
The classical book on potential theory is
Kellogg, 0., Foundations of Potential Theory, Springer, Berlin, 1929.
The relationship between harmonic functions on the plane and analytic functions is
studied in standard texts on the latter subject, such as
Ahlfors, L., Complex Analysis, 2nd ed., McGraw-Hill, New York, 1966.
HiLLE, E., Analytic Function Theory, Ginn, Boston, 1959.
Chapter 13
A standard modern book on mechanics is
Goldstein, H., Classical Mechanics, Addison-Wesley, Reading, Mass., 1952.
For the classical astronomy-oriented aspect of mechanics see
PoiNCARE, H., Legons de mecanique celeste, Gauthier-Villars, Paris, 1905-1910.
Whittaker, E. T., Analytical Dynamics, 4th ed., Cambridge University Press,
Cambridge, England, 1937.
A leisurely geometrical study of classical mechanics will be found in
Lanczos, C, The Variational Principles of Mechanics, University of Toronto Press,
Toronto, 1960.
For a book which treats classical mechanics in the spirit of this chapter, and which
also studies statistical mechanics and quantum mechanics, see
Mackey, G., Mathematical Foundations of Quantum Mechanics, Benjamin, New York,
1965.
An elegant treatment of the geometrical applications of the calculus of variations will
be found in
MiLNOR, J., Morse Theory, Princeton University Press, Princeton, 1965.

NOTATION INDEX
General Conventions
D end of proof
U the real number system
Z the integers
C the complex number system
E^ Euclidean n-space
boldface letters for n-tuplets: a = <ai, . . . j an>
Special Symbols
Symbol Page Symbol Page
(Vx) 1
(3x) 2
& 3
4
=» 4
^ 5
G 6
C 7
0 7
{} 7
< > 9
R-^ 9
AxB 9
\ 10
i^[A] 10
•-> 11
572
—>
5^
n
Xa
n
o
Fix, •)
u, n, u, n
A'
T
e([a, b])
2Z ^iO^i
L{A)
8'
La
Try
11,
12
12
12
14,
14
16
17
17
19,
24,
27
27
28,
32
33,
117, 202
43
53,56
121
31,74
47

Symbol Page
NOTATION
Symbol
I, ^'"*
dA
piA, B)
Br[A]
Q{A, W)
(m{A, w)
(ii)
A^
^2.^, ALB
V*®W*
(7*)@
a"
\ /\ n
*M
m(A)
SD
«*-'min
Da^
M*, M
«^con
5J
c^(x)
ffp
'J'"con
«/ ^
JA
S
/,/
•
//s
(f^z, «z)
4:
I>^
INDEX
Page
197, 198
198
199
212
219
219
38, 248
250
250
305, 306
307
310
316
320
321
322
324, 326
324
331
332
333
337
338
339
150, 342
351
356
357
359
361
364
372
373
573
N{T), 'Sl{T)
R{T), (R(r)
Hom(F, W)
dj
©
d{V)
U
V*
A°
jr*
(*
A(r)
tr(r)
Z)(t)
lub, gib
il II
II 111, II lU, II lU
BM)
(S,{A, W)
^, e, ©
AF„
dF^
SD„(F, W)
D^F
d'F„
dFi
3(2/1, . . . , 2/„) , ,
7} ;: (a)
d(Xi, . . . ,X„)
d''F„
class C*, e*([a, b])
c*
P(x, y)
Br{x)
34
34
45
47
56
78
78
82
84
85, 262
93
99, 312
99
101, 312
119
122, 128
122
123, 196
122, 123
138
141, 142
141, 142
143
147
150, 186
153
159
192
194
194
196
196

574 NOTATION
Symbol
m
T.{M)
^*x
f*x
F«
Dxf
nx]
IX,Y]
Tt{M)
<?, I) - K&
dL dKx)
d
d
dx'
supp^
P,Pa
supp p
P
cp*P
Dxp
div <X,p>
INDEX
Page
373
374
374
376
379
384
384
388
390
391, 83
391
394
157, 395
405
408
408
409
416
419
419
Symbol
a^
A'
the operator d
Dxco
X J CO
curl CO
div CO
A
Wi V W2
T{M)
T{a)
r*(a)
e
12
fx
(^x,X^
exp
GixiVj W)
V
A
00
w^
Page
429, 430
430
438
452
456
458
458
476
490
511
511
512
516
517
517
519
538
542
542.
544
559
560

INDEX
absolute convergence, of an infinite
series, 221
absolute value, 121, 242
absolutely integrable function, 351
adjoint of T, 85
alfine independence, 81
affine span, 56
affine subspace, 40, 52
affine transformation, 53
algebra, 30, 46
Banach, 223
of matrices, 92
alternating tensor, 310
amplitude, 292
analytic plane R^, 10
angular momentum^ 523
annihilator, 84
antisymmetric tensor, 310
associative law, for composition, 14
atlas, 364
ball, closed, 124
open, 123, 196
Banach algebra, 223
Banach space, 217
Banach-Tarski paradox, 321
basis, 71
dual, 82
in Hom(F, IF), 96
infinite, 75
for pre-Hilbert space, 254
basis isomorphism, 72
BesseFs inequality, 254
bijective, 12
bilinear, 67
bilinear functionals, 305ff
block decompositions, 65ff
bound variable, 2
boundary, 198
boundary-value problem, 294ff
bounded below, 128
bounded function, 122
bounded linear mapping, 127
bounded set, 123
braces, 7
calculus of variations, 182ff
canonical basis, 104
canonical transformation, 558, 560
Cartesian axes, 38
Cartesian product, of an indexed
collection, 14
of two setSy 9
Cauchy sequence, 216
cellulation, 473
center, of gravity, 349
of mass, 529
central force, 523, 564
centrifugal force, 517
centrifugal potential, 517
chain rule, 153
change of basis, 94
change of variable formula, 342, 355
characteristic function, Fourier
transform, 337
of a set, 12
characteristic polynomial, 259
classical mechanics, 509
closed set, 124, 197
closure, 198
codimension, 80
codomain, 12
cofactor, 315
column space (of a matrix), 91
575

576
INDEX
column vector, 93
commutative diagram, 86
compact transformation, 264
compactness, in general topology, 214
on a manifold, 403
sequential, 206
compatible atlases, 370
complement, 17
complementary subspaces, 57
completeness, 217
completion, 222
complex conjugation, 241
complex normed linear space, 244
complex numbers, 240ff
complex vector space, 24, 241
complexification, 244
component, 58
composite-function rule, 143
composite truth-functional forms, 5
composition, 14
configuration space, 509
conjugate space, 82
connectives, logical, 3
conservation, of angular momentum, 523
of energy, 521
laws, 521
constant coefficient equation, 278
constrained maximum, 173f
content, 332
contented function, 339
contented sets, 331f
continuity, 126, 196, 201, 203
contraction mapping, 229
contravariant vector, 95
convergence, of infinite series, 221
sequential, 202
convex set, 125
coordinate, 43, 72
coordinate correspondence, 36f
coordinate functional, 33, 72
coordinate isomorphism, 72
coordinate n-tuple, 72
coordinate projection, 43
correspondence, 9
coset, 52
cotangent bundle, 511
CO variant tensor, 307
covariant vector, 95
Cramer's rule, 101, 315
critical point, 161, 188ff
cylindrical coordinates, 345 (Ex. 11.5)
definite (form), 115
degree (of a polynomial), 64
De Morgan's law, 17
dense subset, 212
density, 408
dependent set of vectors, 73
derivative, 146
in a Banach algebra, 226
over C, 242
directional, 147
determinant, 99ff, 108, 312ff
diagonal matrix, 262
diffeomorphism, 376
differentiable manifolds, 363, 369ff
differential, 142
differential forms, 390f
differential geometry, 459
differentiation under the integral sign,
181
dimension, 78
dimensional identities, 78f, 84
direct sum, 56
directed frame, 9
directed line segment, 37
directional derivative, 147
disjoint (family of sets), 18
distance, in a metric space, 196
between vectors, 122
divergence theorem, 421
division algorithm, 64
domain, with almost regular boundary,
424
with regular boundary, 419
domain, of a relation, 9
of a variable, 8
dual basis, 82
dual space, 82
duality, 15, 68
dyad, 86
echelon matrix, 104
eigenvalue, 34, 258

INDEX
577
eigenvector, 34, 258
element (of a set), 6
elementary bilinear functionals, 306
elementary matrices, 105f
elementary row operations, 103
elliptical coordinates, 566
empty set, 7
equations of variation, 513f
equicontinuity, 215
equivalence relation, 19
equivalent directed line segments, 37
equivalent norms, 132, 204
equivalent truth-functional forms, 5
Euclidean norm, 38, 122
Euler angles, 548
Euler's equation(s), 184, 541, 543
existential quantification, 2
exponential function, 226, 228
exterior algebra, 316ff
exterior calculus, 429
exterior differential forms, 429
exterior forms, 316
fiber ing, 19
finite-dimensional, 28
first variation, 183
fixed-point theorem, 229ff
floor of a paving, 329
flow, 381
formal adjoint, 295
Fourier coefficient, 254
Fourier series, 254, 301 ff
Fourier transform, 355, 357
frame, 71
free variables, 2
frequency, 292
function, lOf
function space, 24
functional dependence, 175ff
fundamental form on T*iM), 515
fundamental solution, 277
fundamental theorem, of algebra, 243
of calculus, 238
of ordinary differential equations, 267
geodesic, 539
geodesic coordinates, 537
geodesic curvature, 468
geodesic polar coordinates, 470
germ of functions, 138
global solutions, 269
graph, 9, 162
greatest lower bound, 119
group, 15
Hamilton-Jacobi equation, 563
Hamiltonian mechanics, 520
Hamiltonian structure, 559
Hamiltonian vector fields, 519, 559
Heine-Borel property, 214
Hermitian symmetry, 249
Hilbert space, 249
homogeneity, of a norm, 121
homogeneous equation, 277, 282ff
homogeneous function, 148
homogeneous polynomial on V, 145
hyperplane, 41
idempotent, 59
if ... , then . . . , 4
image, 10
immersion, 399
implicit-function theorem, 167, 230
inclusion (set), 7
increasing norm on R", 134
increasing sequence, 207
independent set of vectors, 7If
independent subspaces, 56
indexed collection, 13
inertia tensor, 546
infinite series, 221, 224ff
infinitesimal, 137
infinitesimal generator, 379, 381
inhomogeneous equation, 277, 288ff
initial condition, 267
injection (product), 47
injective, 12
inner content, 331
inner paving, 331
integral, of a parametrized arc, 236ff
integration, 321, 336
interior, 197
interior point, 124
intermediate-value theorem, 120

578
INDEX
intersection, 17
invariant subspace, 54, 63
inverse, of a function, 15
of a relation, 9
inverse-mapping theorem, 167
isomorphism, 34
between normed linear spaces, 132
iterated integrals, 346
Jacobian determinant, 159
Jacobian matrix, 158f
Kepler's first law, 527
Kepler's second law, 524
Kepler's third law, 527, 532
kernel, 34
kinetic energy, 521
Kronecker delta function, 74
Lagrange multipliers, 174
Lagrange's equations, 530
law of cosines, 41, 250
least upper bound property, 119
left inverse, 14
lemma of Du Bois-Reymond, 184
length of a curve, Riemannian manifold,
401
Lie bracket, 389
Lie derivative, of a density, 417
of an exterior differential form, 431,
452
of a function, 383f
of a linear differential form, 392
of a vector field, 385, 388
limit, of a function, 116, 126
line (straight), 38
line of nodes, 548
linear combination, 27
linear combination mapping, 31
linear differential equations, 276ff
linear functionals, 32, 82
linear transformation, 30
Lipschitz continuity, 126
local solution, 269
locally absolutely integrable density,
408
locally absolutely integrable n-form, 435
locally uniformly Lipschitz, 267
mapping, 12
matrix, 32, 88
maximal solution, 269
maximum, relative, 161
mean-value theorem, 148f
member (of a set), 6
metric space, 196
minimal polynomial, 261
momentum, 511
momentum function of a vector field,
517 (Eq. 3.6)
monomial on F, 145
monotone sequence, 207
multi-index notation, 193
multilinear functionals, 306ff
multivector, 318
name, 7
natural isomorphism, 69, 86
negation, 6
neighborhood, 137, 201
deleted, 137
nilpotent, 50
nonnegative transformation, 257
nonsingular, 93
norm, 121
norm metric, 196
normal distributions, 355 (Ex. 13.1), 356
normed linear space, 121
nth-order equation, 270ff
null space, 34
nutation, 550
one-parameter group,
one-to-one, 11
open set, 124, 196
or (connective), 4
ordered basis, 71
ordered n-tuple, 13
ordered pair, 9
orthogonal, 250
orthogonal projection, 253ff
orthogonal transformation, 262
orthonormal, 254
orthonormal basis, 112, 254
oscillating system, 291 ff
outer content, 331
outer paving, 331

INDEX
579
pair set, 7
parallel translation, 40
parallelogram law, 251
parametric equations (of a line), 38
parametrized arc, 146
Parseval's equation, 254
partial derivative, 156f
partial differential, 153
partition, 19
partition of unity, 405
subordinate to an open covering, 406
patch manifold, 172
paved function, 338
paved set, 324, 326
paving, 324
permutations, 308
phase angle, 294
plane, 40
Poisson bracket, 519
polar coordinates, 345
polar decomposition, 263
polynomials, 62, 64
positive definite, 115, 190
precession, 547
pre-Hilbert space, 249
principle of mechanical similarity, 532
product, matrix, 91
product norm, 133
product rule (for differentials), 155
product vector space, 43
projection, 19, 58
projection-injection identities, 47
property, 7
puUback, of densities, 416
of exterior differential forms, 392
of functions, 372
of linear differential forms, 392
of vector fields, 384
Pythagorean theorem, 41, 251
quadratic form, 112
quotation marks, 7
quotient space, 54
range, of a linear transformation, 34
of a relation, 9
rank, of a linear transformation, 85
of a matrix, 91
real vector space, 23f
rectangles, 323f
regular differential equations, 282, 298
regular solid, 473
relation, 9
Rellick's lemma, 362 (Ex. 14.30)
resonance, 294
restricted variables, 8
restriction (of a relation), 10
Riemann metric, 397
right inverse, 14, 61
rigid-body motion, 544
row-reduced echelon form, 104
row space (of a matrix), 91
row vector, 93
scalar, 23
scalar product, 38, 248
Schwarz inequality, 125, 249
second conjugate space, 83
second difference, 188
second differential, 150, 186ff
self-adjoint, 257
se minor m, 125
semiscalar product, 248
separation of variables, 563f
sequential convergence, 202ff
set, 6
shear transformation, 56
sign of a permutation, 309
signature (of a quadratic form), 113
simple harmonic motion, 293
skeleton, 32
small oscillations, 551
smooth arc, 146
Sobolev's inequality, 361
solid angle, 474
span, linear, 28
spherical coordinates, 345 (Ex. 11.4)
standard basis, 74
star operator, 320
state, 510
statement, 1
statement frame, 1
stop function, 237
Sturm-Liouville system, 298
submanifold, 172, 367
subsequence, 205

580
INDEX
subset, 7
subspace (vector), 24
successive integration, 346
support, of a density, 408
of a differential form, 425
of a function, 336
surjective, 12
symmetric tensor, 310
tangent bundle, 511
tangent plane, 162
tangent space, 373
tangent vector, 146, 373
tautology, 5
Taylor's formula, 191ff
tensor product, 305f
topology, 201
total linear momentum, 528
totally bounded, 212
trace, 99
translation, 40, 53
transpose, 90
triangle inequality, for metrics, 196
for norms, 121
truth-functional forms, 5
truth table, 3
two-body problem, 528
two-norm, 250
undetermined coefficients, 289
uniform conditions, 21 Off
uniform continuity, 179, 210
uniform convergence, 211
uniform norm, 123
uniformly absolutely integrable, 353
union, 17
unit set, 7
universal quantifier, 1
upper triangular, 66
variation of parameters, 289
variational principles, 532
vector analysis, 457
vector field, 44
vector space, 23
volume density of a Riemann metric, 411
volume of an immersed hypersurface,
413, 415
weak sequential convergence, 245
wedge operation, 316
Weierstrass approximation theorem, 304