Differential Forms
with
Applications to the Physical Sciences
HARLEY FLANDERS
Purdue University
Lafayette, Indiana
1963
ACADEMIC PRESS
New York • London

Copyright © 1963, by Academic Press, Inc.
all rights reserved
no part of this book may be reproduced in any form,
by photostat, microfilm, retrieval system, or any
other means, without written permission from
the publishers.
ACADEMIC PRESS, INC.
Ill Fifth Avenue, New York, New York 10003
United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
Berkeley Square House, London W1X 6BA
Library of Congress Catalog Card Number: 63-20569
Fifth Printing, 1972
PRINTED IN THE UNITED STATES OF AMERICA

To June

Foreword
After several friendly discussions of the pros and cons of tensors versus
differential forms in the solution of engineering problems, I persuaded my
colleague Dr. Flanders to prepare a number of lectures on differential forms.
The result was an outstanding series of lectures which was presented to a
group of interested faculty members within the several schools of Engineering
at Purdue University.
It became obvious to those attending that the use of differential forms
would give them another tool for the analysis and synthesis of engineering
systems. There are certain problems, normally very difficult to solve by
using tensors only, for which results are more quickly and directly obtained
with differential forms.
The author was encouraged to formalize his notes to the extent necessary
for publication, to enable others to study this important subject. The text
is recommended highly because differential forms and related concepts which
have evolved from modern mathematics are new and powerful analytical
tools for use by the engineer and scientist.
George A. Hawkins, Dean
Schools of Engineering and Mathematical Sciences
Purdue University
November 20, 1962
vii

Preface
Last spring the author gave a series of lectures on exterior differential
forms to a group of faculty members and graduate students from the Purdue
Engineering Schools. The material that was covered in these lectures is
presented here in an expanded version. The book is aimed primarily at
engineers and physical scientists in the hope of making available to them new
tools of very great power in modern mathematics. Although none of our
applications goes very deep, it is hoped, nevertheless, that enough ground
is covered in each case to indicate the usefulness of this machinery.
A word about the organization of the book is in order. The first chapter is
introductory and sketches where we are going and why. Chapters II, III,
and V include all of the theoretical material; a knowledge of this opens the
door to the applications. Probably on first reading, one should aim more at
developing some intuition for the subject and getting a firm idea of what the
various different things which are defined look like, rather than at working
out proofs in detail. Applications to questions in differential geometry
(including many topics of considerable use in physical sciences) are mostly in
Chapters IV, VI, VIII, and IX. Applications to various topics in ordinary
and partial differential equations will be found in Chapter VII. Finally,
applications to several topics in physics are in Sections 3.5, 4.6, 6.4, and
Chapter X.
What is presupposed of the reader is first of all a certain amount of
scientific maturity, the precise direction not being too important. While the book
is not really advanced mathematics, it is not exactly ground floor
mathematics either, and a reasonable knowledge of the calculus of functions of
several real variables is necessary, as is a working knowledge of linear algebra
through the ideas of linear combination, basis, dimension, linear
transformation. Some exposure to a minimum amount of the ground rules of modern
mathematics, sets, cartesian products, functions on sets, is helpful but not
essential. This material is usually picked up by osmosis anyway, and the
Glossary of Notation at the end of the book should be helpful. The reader
should also know about the existence of solutions of ordinary differential
equations. A passing familiarity with tensor methods is useful, but not
essential.
If our audience consisted of mathematicians alone, it would be in order to
use somewhat more care in our formulations of definitions and proofs of
theorems and to discuss in considerably more depth numerous technical
points we here pass over lightly. Our goal, however, is to develop an intuition
IX

x PREFACE
and a working knowledge of the subject with as much dispatch as is possible.
This perhaps could be done in less space except for our insistence on a degree
of rigor matching that found in the better treatises on theoretical physics.
This falls short of the extremely great precision which is customary in modern
abstract mathematics and pretty much inherent in its nature. One who quite
rightly is searching recent developments in mathematics for applicable
material must find this precision a considerable barricade, overpedantic if not
downright tedious—a very real factor in the great separation between modern
mathematics and modern science. Making his craft available to science is not
a light task for the mathematician and the extent to which this book makes a
contribution therein must necessarily be its primary measure of success.
In spite of all this, we do not hesitate to recommend this material to
graduate students in mathematics as an introduction to modern differential
geometry; indeed, a well-trained advanced undergraduate should find the
book quite accessible. Considering the degree to which present day
mathematical training consists of one abstraction after another, some of the things
in this book could be a bit of an eye-opener, even to a mathematics student
who is well along. For example, one could envisage such a student meeting
here a parabolic differential equation, or a matrix group, or a contact
transformation for the very first time.
It is my pleasant duty to acknowledge the substantial help and
encouragement I have always had from my teachers, colleagues, and students. In this
respect a special vote of thanks is due George A. Hawkins, Dean of the
Schools of Engineering and Mathematical Sciences of Purdue University.
Finally, I wish to express my gratitude to Elizabeth Young, whose beautiful
typing of the manuscript was a substantial contribution.
July 1963
Habley Flanders

Contents
FOREWORD
PREFACE
I. Introduction
1.1. Exterior Differential Forms
1.2. Comparison with Tensors
II. Exterior algebra
2.1. The Space of p-vectors ..
2.2. Determinants
2.3. Exterior Products
2.4. Linear Transformations
2.5. Inner Product Spaces
2.6. Inner Products of p-vectors
2.7. The Star Operator
2.8. Problems
III. The Exterior Derivative
3.1. Differential Forms
3.2. Exterior Derivative
3.3. Mappings
3.4. Change of Coordinates
3.5. An Example from Mechanics
3.6. Converse of the Poincare Lemma
3.7. An Example
3.8. Further Remarks
3.9. Problems
IV. Applications
4.1. Moving Frames in E3
4.2. Relation between Orthogonal and Skew-symmetric Matrices
4.3. The 6-dimensional Frame Space
4.4. The Laplacian, Orthogonal Coordinates
4.5. Surfaces
4.6. Maxwell's Field Equations
4.7. Problems
XI

CONTENTS
V. Manifolds and Integration
Introduction
Manifolds
Tangent Vectors
Differential Forms
Euclidean Simplices
Chains and Boundaries
Integration of Forms
Stokes' Theorem
Periods and De Rham's Theorems
Surfaces; Some Examples
Mappings of Chains
Problems
VI. Applications in Euclidean space
Volumes in En
Winding Numbers, Degree of a Mapping
The Hopf Invariant
Linking Numbers, the Gauss Integral, Ampere's Law
VII. Applications to Differential Equations
Potential Theory
The Heat Equation
The Frobenius Integration Theorem
Applications of the Frobenius Theorem
Systems of Ordinary Equations
The Third Lie Theorem
VIII. Applications to Differential Geometry
Surfaces (Continued)
Hypersurfaces
Riemannian Geometry, Local Theory
Riemannian Geometry, Harmonic Integrals
Affine Connection
Problems
IX. Applications to Group Theory
Lie Groups
Examples of Lie Groups
Matrix Groups
Examples of Matrix Groups
Bi-invariant Forms
Problems . .

X. Applications to Physics
10.1. Phase and State Space
10.2. Hamiltonian Systems
10.3. Integral-invariants
10.4. Brackets
10.5. Contact Transformations
10.6. Fluid Mechanics
10.7. Problems
CONTENTS
xiii
Page
163
165
171
179
183
188
193
BIBLIOGRAPHY
GLOSSARY OF NOTATION
197
199
INDEX
201

I
Introduction
1.1. Exterior Differential Forms
The objects which we shall study are called exterior differential forms.
These are the things which occur under integral signs. For example, a line
integral
Llcfo + Bdy + Cdz
leads us to the one-form
co — Adx + Bdy + Cdz;
a surface integral
Pdydz + Qdzdx + Rdxdy
leads us to the two-form
a = Pdydz + Qdzdx + Rdxdy;
and a volume integral
Hdxdydz
leads us to the three-form
X — Hdxdydz.
These are all examples of differential forms which live in the space E3 of three
variables. If we work in an n-dimensional space, the quantity under the
integral sign in an r-fold integral (integral over an r-dimensional variety) is an
r-form in n variables.
In the expression a above, we notice the absence of terms in dzdy} dxdzf
dydx, which suggests symmetry or skew-symmetry. The further absence
of terms dxdx, • • • strongly suggests the latter.
We shall set up a calculus of differential forms which will have certain
inner consistency properties, one of which is the rule for changing variables
in a multiple integral. Our integrals are always oriented integrals, hence
we never take absolute values of Jacobians.
Consider
A(x, y)dxdy
1

2 I. INTRODUCTION
with the change of variable
x — x(u, v)
y = y(u, v).
We have
,y)dxdy.
which leads us to write
Ll(a;,y)etedy =
Six ?/)
A[x(u, v), y(u, v)] — dudv,
o(u, v)
dxdy — —-—dudv
d(u, v)
dx
du
dx
dv
dy dy
du
dv
dudv.
If we set y = x, the determinant has equal rows, hence vanishes. Also if
we interchange x and y, the determinant changes sign. This motivates the
rules
(dx dx = 0
dydx = —dxdy
for multiplication of differentials in our calculus.
In general, &n {exterior) r-form in n variables xl, • • • , x" will be an expression
oo — — Y Ait , dxli • • • dxlr,
where the coefficients A are smooth functions of the variables and skew-
symmetric in the indices.
We shall associate with each r-form a> an (r -f- l)-form dco called the exterior
derivative of a>. Its definition will be given in such a way that validates the
general Stokes' formula
co = dco.
J 01 J I
Here L is an (r + 1 )-dimensional oriented variety and 3£ is its boundary.
A basic relation is the Poincare Lemma:
d(dco) = 0.
In all cases this reduces to the equality of mixed second partials.
1.2. Comparison with Tensors
At the outset we can assure our readers that we shall not do away with
tensors by introducing differential forms. Tensors are here to stay; in a
great many situations, particularly those dealing with symmetries, tensor

1.2. COMPARISON WITH TENSORS
3
methods are very natural and effective. However, in many other situations
the use of the exterior calculus, often combined with the method of moving
frames of fi. Cartan, leads to decisive results in a way which is very difficult
with tensors alone. Sometimes a combination of techniques is in order. We
list several points of contrast.
(a) Tensor analysis per se seems to consist only of techniques for
calculations with indexed quantities. It lacks a body of substantial or deep
results established once and for all within the subject and then available for
application. The exterior calculus does have such a body of results.
If one takes a close look at Riemannian geometry as it is customarily
developed by tensor methods one must seriously ask whether the geometric
results cannot be obtained more cheaply by other machinery.
(b) In classical tensor analysis, one never knows what is the range of
applicability simply because one is never told what the space is. Everything
seems to work in a coordinate patch, but we know this is inadequate for most
applications. For example, if a particle is constrained to move on the
sphere S2, a single coordinate system cannot describe its position space, let
alone its phase or state spaces.
This difficulty has been overcome in modern times by the theory of
differentiate manifolds (varieties) which we discuss in Chapter V.
(c) Tensor fields do not behave themselves under mappings. For
example, given a contravariant vector field a1 on #-space and a mapping </>
on #-space to y-space, there is no naturally induced field on the t/-space.
[Try the map t —► (t2, t3) on E1 into E2.]
With exterior forms we have a really attractive situation in this regard. If
</>: M—► N
and if a> is a p-form on N, there is naturally induced a ^-form </>* co on M.
Let us illustrate this for the simplest case in which co is a O-form, or scalar,
i.e., a real-valued function on N. Here </>* co = co o (j), the composition of the
mapping </> followed by co.
(d) In tensor calculations the maze of indices often makes one lose sight
of the very great differences between various types of quantities which can

4
I. INTRODUCTION
be represented by tensors, for example, vectors tangent to a space,
mappings between such vectors, geometric structures on the tangent spaces.
(e) It is often quite difficult using tensor methods to discover the deeper
invariants in geometric and physical situations, even the local ones. Using
exterior forms, they seem to come naturally according to these principles:
(i) All local geometric relations arise one way or another from the
equality of mixed partials, i.e., Poincare's Lemma.
(ii) Local invariants themselves usually appear as the result of applying
exterior differentiation to everything in sight.
(iii) Global relations arise from integration by parts, i.e., Stokes*
theorem.
(iv) Existence problems which are not genuine partial differential
equations (boundary value or Cauchy problems) generally are of the type of
Frobenius-Cartan-Kahler system of exterior differential forms and can be
reduced thereby to systems of ordinary equations.
(f) In studying geometry by tensor methods, one is invariably restricted
to the natural frames associated with a local coordinate system. Let us
consider a Riemannian geometry as a case in point. This consists of a
manifold in which a Euclidean geometry has been imposed in each of the
tangent spaces. A natural frame leads to an oblique coordinate system in
each tangent space. Now who in his right mind would study Euclidean
geometry with oblique coordinates ? Of course the orthonormal coordinate
systems are the natural ones for Euclidean geometry, so they must be the
correct ones for the much harder Riemannian geometry. We are led to
introduce moving frames, a method which goes hand-in-glove with exterior
forms.
We conclude the case by stating our opinion, that exterior calculus is here
to stay, that it will gradually replace tensor methods in numerous situations
where it is the more natural tool, that it will find more and more applications
because of its inner simplicity, body of substantial results begging for further
use, and because it simply is there wherever integrals occur. There is
generally a time lag of some fifty years between mathematical theories and
their applications. The mathematicians H. Poincare, fi. Goursat, and fi.
Cartan developed the exterior calculus in the early part of this century; in
the last twenty years it has greatly contributed to the rebirth of differential
geometry, now part of the mathematical main stream. Physicists are
beginning to realize its usefulness; perhaps it will soon make its way into
engineering.

II
Exterior Algebra
2.1. The Space of p-Vectors
Notation:
R = field of real numbers a, b, c, • • • .
L = an n-dimensional vector space over R with elements a, jS, • • • .
For each p = 0,1,2, • • • , n we shall construct a new vector space
over R, called the space of p-vectors on L. We begin with
A°l = r, A1 ■"-■••
Next we shall work out ^2 L in some detail. This space consists of all sums
Z «i(«£ A Pt)
subject only to these constraints, or reduction rules, and no others:
((«!(*! + a2<x2) a p - a^a! a /?) - a2(<x2 a /?) = 0,
a a (6^ + 62jS2) - Ma a ft) - 62(a a j82) = 0,
a a a = 0,
aAj3 + j3Aa = 0.
Here a, j3, etc., are vectors in L and a, b, etc., are real numbers; a A jS is called
the exterior product of the vectors a and jS. If a and jS are dependent, say
/? = ca, then
a a ]5 = a a (ca) = c(a a a) = c-0 = 0
according to our reductions. Otherwise a A j3 ^ 0.
Suppose or1, • • • , <Jn is a basis of L. Then
a = £ af(7f, j5 = X fyr*,
a a j8 = (X af<7<) a (X bjo^) = £ a,6,(<r' a ^).
We rearrange this as follows. Each term a1 A a1 — 0 and each <7J A a1 —
— a1 a <rJ for i < j. Hence
a a jS = Z (a*&/ "" apiW a (T7*.
5

6 II. EXTERIOR ALGEBRA
The typical element of ^2 L is a linear combination of such exterior
products, hence the 2-vectors
a1 A <rJ, 1 ^ i < j ^ n,
form a basis of A 2 L. We conclude
a 2 . n(n— 1) (n\
In general, we form A p L (2 ^ p ^ n) by the same idea. It consists of
all formal sums (p-vectors, or vectors of degree p)
£ a^ a • • • a ap)
subject only to these constraints:
(i) (a<x + b/}) a a2 a • • • a ap
= a(a a a2 a • • • a ap) + 6(j5 a a2 a • • • a ap),
and the same if any oct is replaced by a linear combination.
(ii) ol1 A • • • A ap = 0 if for some pair of indices i ^ j, af = a,-,
(iii) at a • • • A ocp changes sign if any two af are interchanged.
It follows easily from (i) that <xt a • • • a <xp is linear in each variable; we may
replace any variable by a linear combination of any number (not just two)
of other vectors and compute the value by distributing, for example
a a {bjt +M2+M3) a y aS
= bt(oc a Pi a y a d) + &2(a a jS2 a y a (5) + 63(a a jS3 a y a 5).
It follows from (iii) that if n is any permutation of the {1, 2, • • • , p}, then
an(1) a • • • a an(p) = (sgnjc)^ a • • • a ap.
Exactly as in the case p — 2, we can show that if
a1, •••,*"
is a basis of L, then a basis of A p L is made up as follows: for each set of
indices
H = {hti h2 , • • • , hp}, 1 ^ hi < h2 < • • • < hp ^ n,
we set
<jH = (t*i A • • • A (J**.
Then the totality of aH is a basis of A p L. We conclude that
dim A" >- = (*)>
the number of combinations of n things taken p at a time. In particular
dimy\wL = l.

2.2. DETERMINANTS 7
If AisinyY'L, then
* = Z ^^
summed over all of these ordered sets H. One can also sum over all p-tuples
of indices by introducing skew-symmetric coefficients:
1
r; I ^i.-'-.fcp0*1 A •"• A(ji
.^
*'*!, — .*
where the bhi.. ,h is a skew-symmetric tensor and
&fcl.. -ftp = aH for ^ = {*!,-•■ , hp}, hl<h2< ••• <h.
p'
This skew-symmetric representation is often quite useful.
Let us note why we do not define /\p L for p > n. (Sometimes it is
convenient to simply set /\p L = 0 for p > n.) We express each a in a
product <xt A • • • A ap as a linear combination of the basis vectors a1, • • • , <?
and completely distribute according to Rule (i). This leads to
<*! A • • • A <Xp = X «V • .h, J*' A • • • A (J*'.
Each term or*1 A • • • A <7ftp is a product of p > n vectors taken from the set
a1, • • • , an so there must be a repetition; by Rule (ii) it vanishes. We are
left with (x1 A • • • A <xp = 0 as the only possibility.
We close with a very important property of the spaces /\p L.
In order to define a linear mapping / on A p L it suffices to present a
function g of p variables on L such that (i) g is linear in each variable
separately, (ii) g is alternating in the sense that g vanishes when two of its
variables are equal and g changes sign when two of its variables are
interchanged. Then
/((*! a ■•• Aap)=gr(a1, • • • , ap)
defines / on the generators of /\p L.
It can be shown that this property provides an axiomatic characterization
of A p L. In the next section we apply this property to define the
determinant of a linear transformation.
2.2. Determinants
As above L is a fixed linear space of dimension n. Let A be a linear
transformation on L into itself. We define a function g = gA of n variables
on L as follows:
9a(*i > '' * > 0 = A(Xi A ''' A A*n>

8 II. EXTERIOR ALGEBRA
where X" ^ denotes the cartesian product. Since g is multilinear and
alternating, there is a linear functional f = fA,
satisfying
fA(*i a • • • a aj = gjctj,, • • • , a„) = A<xt a • • • a Accn.
But f\n L is one-dimensional so the only linear transformation on this space
is multiplication by a scalar. We denote the particular one here by \A\
and have
Aol1 a • • • a A<xn = \A\(ol1 a • • • a a„).
This serves to define the determinant \A\ of A. We must not fail to note
that this definition is completely independent of a matrix representation
of A.
We observe next
\AB\fa a • • • a a„) = (AB<xt) a • • • a (AB<xn)
= \A\(B*, a ••• AtfaJ
= |4|-|J8|(a1 a ••• a a.),
hence
M5| = |^|-|5|.
We can relate this to the determinant of a matrix as follows. Let
a1. - • • , g" be a basis of L and ||afj|| an n x n matrix. Set
a* = Z au<7"/-
Then
a! a • • • A a„ = Kj|crl a • • • a <7w.
In particular, if one obtains the matrix representation of A with respect to
the basis (<jl) by
Aal = $>';<7;,
then Aal a • • • a Ac" = la'^cr1 a • • • a <7n, |4| = |aly|.
2.3. Exterior Products
We now observe that our spaces /\p L have a built-in multiplication process
called exterior multiplication and denoted by a for obvious reasons. We
multiply a ^-vector \i by a #-vector v to obtain a (p + q)-vector \i a v (which
is 0 by definition if p + q > n):
*■■ (APL)X(A,L)^AP+*1-
A on generators and use the basic prir
1 it to all p- and #-vectors:
((*! A • • • A (Xp) A (/?! A • • • A /3q) = <Xt A • • • A 0Cp A fit A • • • A /3q .
It suffices to define a on generators and use the basic principle at the end of
Section 1 to extend it to all p- and #-vectors:

((*! A a2 A a3) A (ftx A p2)
2.3. EXTERIOR PRODUCTS 9
The basic properties of this exterior product are
(1) X A \i is distributive,
(2) X a (fi a v) = {X a fi) a v, the associative law
(3) flAA = (-l)pqAAIH.
Property (3) simply says that any two vectors of odd degrees anticommute,
otherwise vectors commute. The following will illustrate why this is the
case:
((*! a a2 a a3) a P = —(ax a a2 a j? a a3)
(-l)2(a1 a p a a2 a a3)
(-l)3j3 a (o^ a a2 a a3),
-1)3/?! A ((*! A a2 A a3) A jS2
1)3(-I)3(j5i AjS2)A(ai Aa2Aa3)
= (-l)3'2(j5i a p2) a (a! a a2 a a3).
Examples. We take for L the linear space based on the differentials
dx, dyt • • • and, as is customary, omit the exterior multiplication sign a
between dx's. Thus dxdy denotes dx a dy.
1. {A dx + Bdy + Cdz) A(Edx+ Fdy + Gdz)
= (£G - CF)dydz + (OS - 4G)efeete + (4.F - BE)dxdy,
illustrating the vector-, or cross-product of two ordinary vectors.
2. (Adx + Bdy + Cefe) a (Pdydz + Qdzdx + Rdxdy)
= (AP + BQ + CR)dxdydz,
illustrating the dot-, or inner-product of vector algebra.
3. Let a be any form of odd degree. Then
a2 = a a a = 0.
For if a and /? are of odd degree p, then
P a a = —a a jS.
We set jS = a to have
a a a = —a a a, 2(a a a) = 0, a a a = 0.
4. Here we take
CD = dpi dq1 + • • • + dpn dqn,
a form arising in mechanics. The two-forms dptdql all commute, hence
of = (n\)dpi dq1 dp2 dq2 • • • dpn d(f
= (_i)»<»-!>/*(„])^ '^dpndql • • • dg".

10
II. EXTERIOR ALGEBRA
The product dpt • • • dqn is called phase-density. We shall discuss this
further in Chapter X.
We apply the exterior product to obtain the Laplace expansion of a
determinant by complementary minors.
Let ||a£.|| be an n x n matrix. For H — [ht, • * • , hp), set
1pM
*P,hP
Set p + q = n. For K — {kl, • • • , kq), set
cv =
ap+l,ki ' ' ' ap+l,kq
ln,ki
*ntka
Thus if K = H', the complementary set of indices to H (always arranged in
natural order), then bH and cK are complementary minors of ||a^-||.
Now set
«£ = Z aij°j
where (aJ) is a basis of L. We easily see that
hence
But
and
ai A ••• Aap = £&H<7H,
ap+1 a ••• a a„ = Zcx^
ax a • • • a a„ = (o^ a • • • a ap) a (ap+1 a • • • a a J
<7H A <7*
at a • • • a a„ = \au\{Gl a • • • a an)
(0 if ^ # #'
eH'H' (a1 a ••• A(TW) ifif = #',
K-\ = Z eH,H' W •
If il = {&!,-•• , &p}, H' = {kl, • • • , JfeJ, then
Ui K ••• V*
hence
2.4. Linear Transformations
In this section we deal with two linear spaces M and N with
dim M = m, dim N = n.

2.4. LINEAR TRANSFORMATIONS 11
Let us agree that when we need bases, a1, • • • , am will denote a basis of M and
t1, • • • , xn a basis of N.
Let ibea linear transformation,
A. M—>N.
The mapping
(ai, ''' > ocP) —> Aax a • • • a Aap
sends
It is alternating multilinear, hence defines a linear transformation, denoted
hp A on J\f M to j\f N. This exterior p\h power of A is defined on
generators by
(Ap A)(at a • • • a (xp) = ^4ax a • • • a ^4ap.
Suppose .4 is represented by the m x n matrix ||al -|| according to
Act1 = JV^tA
The (TH and tk form bases of ^p M and /\p N, respectively, where H and K
are ordered sets of p indices. We have
(ApA)(TH = ^(T*1 A ••• A Aoh*
= £ At • • • Ap Tfcl A • • • A T**
= Za»KT*.
Hence Ap A is represented by the matrix
ii«hkii
of all p x p minors of \\alj\\. This is sometimes called the pth compound of
iK-ii-
Suppose one has three spaces L, M, N and this situation:
L * >^M
A
N
a a,) = (ABaJ a • • • a (AB^)
= (S'A)[(Bol1)k ••• A(Bap)]
= (A» A)[(A» B)^ a ••• Aap)]
= [(A'W £)](<*! a ••• Aap),
We compute Ap (AB):
A" (AB)(<xt A • • •

12 II. EXTERIOR ALGEBRA
hence
Ap (AB) = (ApA)(Ap B).
It follows that the pth compound of the product of two matrices is the
product of their pth compounds, a nontrivial result.
We must consider one other matter. Again let A : M —> N. Suppose
a) is in /\p M and rj is in /\q M. Then
(Ap+q A)((o Af|) = (Ap A)(co) a (Aq A)(rj).
For if we take monomials, a> = ax a • • • a ap, rj = jSx a • • • a fiq, then
(Ap+q A)(co Arj) = (Ap+q A)^ a • • • a ap a fit a • • • a fiq)
= A(Xt A • • • A ^4jS^
= (^4a1 a • • • a A<xp) a {Apt a • • • a ^4/y
= (A^)(a>) a (AqA)(rj).
2.5. Inner Product Spaces
In the remainder of the chapter we shall study a space L which has an
inner product (a, jS). This is a real-valued function on L Y L which is
(i) Linear in each variable,
(ii) Symmetric: (a, jS) = (jS, a),
(iii) Nondegenerate: if for fixed a, (a, jS) = 0 for all jS, then a = 0.
Example 1. The Euclidean inner product on E" is given by
a = («i, • • • , an), £ = (&!,..., &„),
(a,]8) = o161+ ••■ +«A-
Example 2. The Lorentz inner product in four-space:
a = K , • • • , a4), jS = (&!, • • • , 64),
(a, jS) =«!&! + a262 + a363 — c2a464
where c is the speed of light.
Condition (iii) is equivalent to the following. If a1, • • • , an is a basis of L,
then
I(ff',<r0l*0.
(The left-hand member is the Gram determinant, or Grammian.) For this
determinant vanishes if and only if there is a nontrivial solution (04 , • • • , aj
of the homogeneous system

2.5. INNER PRODUCT SPACES
13
But this is the same as having the vector
a = Z "P1
satisfy the relation (a, jS) = 0 for all p.
An orthonormal basis of L consists of a basis a1, • • • , an such that
If there are r plus signs and s minus signs, then r + s = n, and t = r — s is
the signature of the inner product. It does not depend on the choice of basis.
It is a basic fact that each inner product space L has an orthonormal basis.
This is proved in several steps.
1. If dim L > 0, there is a vector a in L such that (<r, 6) #= 0.
For if (a, a) = 0 for all a, then
0 = (a + fi, a + fi) = (a, a) + 2(a, /?) + (0, /?)
= 2(a, j3), (a, j8) = 0 for all a, j3,
a contradiction to nondegeneracy.
2. Pick a maximal sequence a1, • • • , ar of vectors satisfying
(cr£, (rO = ± 5".
Let M be the subspace of L these vectors span. Then dim M = r. [The
a1 are independent since £ ap1 = 0 implies £ ^((T1, oJ') = 0, + a • = 0.] We
suppose r < w.
3. Let N be the orthogonal complement of M, i.e., N is the space of all
vectors P such that (a, jS) = 0 for all a in M. Since N is determined by the
r relations (a1, jS) = 0, dim N ^ n — r. But obviously MnN=0 (i.e.,
the only vector common to M and N is 0), hence dim N = n — r, M and N
together span L, M + N = L.
4. N itself is an inner product space relative to the inner product of L.
Only the property (iii) of nondegeneracy must be checked. Suppose jS is in
N and (y, p) = 0 for all y in N. But (a, /?) = 0 for all a in M, hence (a, /?) = 0
for all a in L since M and N together span L. Hence p — 0.
5. By (1), there is a vector a in N such that (a, a) ^ 0. We set
<rr+1=a/|(a,a)|1/2
and see that we have constructed a sequence a1, • • • , <rr + 1 longer than a
maximal one. Since this is impossible we conclude that we must have had
r = n in the first place, which completes the proof.
There is another basic property of inner product spaces which we shall
need below.
Let f be a linear functional on L. Then there is a unique vector p in L such
that
/(a) = (a, /*)■

14 II. EXTERIOR ALGEBRA
This is easily established by taking an orthonormal basis a1, • • • , a". We
set bt =zf(al) and for /? simply take
0 = I ± &,v = s>7'> <rJHV-
For then
2.6. Inner Products of p-Vectors
Again we start with an w-dimensional vector space L with an inner product
(a, jS). We shall define an induced inner product on each of the spaces
/\p L. We set
(A, ii) = |(a„ fij)\
for X = ocx A • • • A olp, \x — Pi A • • • A Pp. This definition works because
the determinant on the right is an alternating multilinear function of the
a's, ditto the /Ts. This means the formula defines a scalar-valued function
on (A p L) X (f\P L) which is linear in each variable. Next (//, X) = (2, jn)
because interchanging the rows and columns of a matrix (transposing) does
not change its determinant.
The nondegeneracy of this inner product is most easily seen by
computing with respect to an orthonormal basis a1, • • • , Gn of L. As usual the
gh, H — {hx < h2 < • • • < hp], form a basis of A p L. We have
If H ^ K, this is zero since the determinant has a row (also a column) of
zeros. If H = K, all but the diagonal elements vanish and these are +1,
hence
(<7H, <7K) = + 8H-K
In other words, the gh form an orthonormal basis of A p L, nondegeneracy
follows free of charge.
In particular g — g1 A • • • A Gn is an orthonormal basis of /\n L and
(<r, a) = (a1, a1) ■ ■ ■ (<j», a") = (-l)<-<>/2
where t is the signature of L.
For another example, set
af = (T1 A ••' A G1'1 A Gi + 1 A ••• A(7W,
forming a basis of f^"1 L. Clearly
{a\ a<) = (<r, g)/(g\ g[) = (a, g){g\ g%
hence

2.7. THE STAR OPERATOR
15
2.7. The Star Operator
Again let L have inner product (a, jS). We shall take a definite orientation
of L which will remain fixed. (This simply means we take one basis for L
and only consider other bases which are expressed in terms of this one by a
matrix with positive determinant. The space L has two orientations and
we take one of them.) We only use bases coherent to the orientation.
We shall define an operation *, called the (Hodge) star operator. This will
be a linear transformation on A p L onto f\"~p L. This operator depends,
of course, on the inner product and also depends on the orientation.
Reversing orientation will change its sign.
We note that the orientation of L determines a definite orthonormal basis
,ofA"L.
Now fix X in /\p L. The mapping
\i —► X A \x
is a linear transformation on f\"~p L into the one-dimensional space A n L.
We may write
* A/i=/A(jK)(7
where fx is a linear functional on An~p L. By our result at the end of
Section 2.5, there is a unique (n — p)-vector, which we denote *A to indicate
its dependence on X, such that
X a \x = (*A, jj)g.
This equation defines the * map which is evidently linear on Ap L into
A""'L- ...
In order to compute *X for generators of A p L, in view of the linearity, it
is enough to compute *X where X = a1 a • • • a ap and where a1, • • • , an is
an orthonormal basis. Let K run over sets ofq — n — p indices. Then
X a aK = (*A, aK)a.
The left-hand side vanishes unless K — [p + l,p + 2, • • • , w}, hence
*X = CGP + * A • • • A Gn
and the constant c is determined taking K = [p + 1, • • • , n]:
a = X a aK = c(gk, aK)a,
c = {oK,oK)= ±1,
*X = (<7*, <7*)<7*
For definiteness, set H = {l, • • •, p), K = [p + 1, • • • , n}. We have proved
*<jH = (<7*, GK)(JK.

16 II. EXTERIOR ALGEBRA
Since aK a aH — ( — l)p{n~p)aH a aK, we deduce, taking orientation into
account,
*aK = (-l)«"-'XoH9oH)oH,
hence
* (*<T") = (-1)*»-*>(<X» <7*)(<7*, GK)aH,
*(*GH) = (-l)*n-PX(T, (T)(TH
= (_l)P(»-P) + ("-0/2(TH#
w&ere £ is the signature.
It follows that if a is any p-vector, then
**a = (_i)p(»-p) + (»-«/2a.
Another consequence of these formulas is this result.
If a, fi are p-vector s, then
a a *£ = p a *a = (-l)(w-f)/2(a, j3)<r.
For when jS = gh as above, the only generator a = <7J for which both sides
do not vanish is a = <7H, and then
a A *j3 = GH A (<7X, <X*)(7X = (<7*, <7*)<7
= ((7H,(7H)(-l)(w-^2(7
= (-l)(w-r)/2(a,jS)(7.
Example 1. We take 4-space with coordinates so normalized that dx1, dx2,
dx3, dt is an orthonormal basis with (dx1, dx1) =1, (dt, dt) = — 1. We have
n = 4, t — 2, ( —l)(w~r)/2 = —1. We shall study certain two-forms. For
p = 2, p(n — p) — 4. Thus
* (eta* d£) = dx* dxk
where (i, j, k) is cyclic order,
* (dx* dx*) — — dxldt.
Let Et be the components of electric field strength, H( the components of
magnetic field strength (all in free space) and consider the form
a) = (E1 dx1 + E2 dx2 + #3 dx3) dt + (Ht dx2 dx3 + H2 dx3 dx1 + H3 dx1 dx2).
Then
*o> = ~(H1dxl + H2dx2 + H3dx3)dt
+ (#! efo2 dx3 + #2 efo3 efo1 + E3 dx1 dx2).
We shall see the use of these forms in Maxwell's equations later.

2.8. PROBLEMS
17
Example 2. E3 with the ordinary metric. If/ and g are functions,
df = d/dx+8Jdy+8fdz
ox oy oz
and we have
C$ -P 3 -P ^5 -P
*df = — dydz H dzdx + —-dxdy
ox oy oz
df a *dg= i —— + — — + —— \dxdydz.
\ox ox oy oy oz dz)
2.8. Problems
1. Let L be an w-olimensional space. For each p-vector a ^0 we let Ma be
the subspace of L consisting of all vectors a satisfying a a g = 0. Prove
that dim(Ma)^#. Prove also that dim(Ma)=£> if and only if a =
a1 a • • • a Gp where at, • • • , ap are vectors in L.
2. (Continuation) Let a be any (n — 1)-vector. Prove that a =
Gt A ••• A (Tn-i-
3. Let L be an ^-dimensional space and a a 2-vector. Show that there
is a basis g^ , • • • , an of L such that
a = ax a a2 + G3 a <t4 + • • • + a2r-1 a a2r.
The number 2r, which depends only on a, is called the rawJfc of the 2-vector a.
Show that ar # 0, ar+1 = 0.
4. (Continuation) Let o^ , • • • , an be a basis of L and let A = ||a<7.|| be a
skew-symmetric matrix. Show that the rank of the matrix A coincides
with the rank of the 2-vector a = £ £ #o <rj a <7y.
5. We are given a linear transformation
A: L—>L,
where dim L = r&. Find the value of the determinant
|AP^|.
6. Let A be an m x n matrix and B an n x m matrix, where m <n.
Prove
\AB\=ZaHbH
where # runs over ordered sets,
H = {hl9h29--, hm), 1 ^ ^ < &2 < • • • < hm ^ n,
and where
*i/»i
*l/»rr
*»»,/»*,
hm,m

18 II. EXTERIOR ALGEBRA
Note that the special case
lal a2 a3\
A = , B = 'A
yields a well-known formula of vector algebra,
|axb| = |a|2|b|2-(a-b)2.
7. Let A be an n x n matrix and denote by cof A the matrix of cofactors
of A [so that ^4(cof^4) = (cof A)A = \A\ •/]. Let bH K be a typical element
of Ap (cof A). Show that
bH.K= ±\A\p-laH,tK.
where H' is the set of indices complementary to H, ditto K', K, and an, K,
is the corresponding element of
Nn'p A.
8. Express in terms of exterior algebra the formula from vector algebra
ax (/?xy) = (a-y)/?-(a-/?)y.

Ill
The Exterior Derivative
3.1. Differential Forms
Let P be a point in E". The one-forms at P are the expressions
n
^didx1, at constants
1
These form an w-dimensional linear space L = Lp. The p-forms at P are
the elements of
i.e., expressions
£ aHdxhx • • • dx^p, aH constants.
Note that we are dropping the notation " a " so that differentials dx[
juxtaposed will always be multiplied by exterior multiplication.
Now let U denote an (open) domain in E". A p-form on U is obtained by
choosing at each point P of U a p-form at that point, and doing this smoothly.
Thus a p-form co has the representation
a> — Y, aii(xl> ''' » %")dxH,
where the functions aH(x) are smooth functions on U, differ entiable as often
as we please.
The exterior algebra applies at each point of U and so may be interpreted
on the differential forms on U itself. Thus if co is a p-form and rj is a
g-form on U, then co A t] is a (p + g)-form on U. (Of course co A rj = 0 if
p + q > n.) If
00 == X aHd%H, y\ — X bKdxK,
then
co a t] — X aH°K dxH d>xK
so that the coefficients of co a y\ are again smooth functions, being
polynomials in the coefficients of co and r\.
For example a one-form
co = Pdx + Qdy + Rdz
19

20 III. THE EXTERIOR DERIVATIVE
may be identified with an ordinary vector field (P,Q, R) in E3, a two-form
a = A dydz + Bdzdx + Cdxdy
may be identified with a polar vector field in E3.
3.2. Exterior Derivatives
We denote by
P(U)
the totality of p-forms on U. In particular F°(U) is simply the set of all
smooth functions on U.
We shall now set up an operation d which takes each p-form co to a
(p + l)-form dco. In E3 it will work this way. For a 0-form/,
df = —dx + — dy + — dz.
ox oy oz
For the one-form co above,
doo = (- I dydz + (- -Adzdx + I- —\dxdy,
\oy oz) \oz ox) \ox oy)
da = (^ + ^ h ~-1 dxdydz.
while for the two-form a above,
/dA dJB dC\
\dx dy dz)
Thus the operator d subsumes the ordinary gradient, curl or rotation, and
divergence.
It will turn out that d is completely independent of coordinate systems.
This will be more or less clear when we axiomatize d.
We shall establish the existence and uniqueness of an operator
d: R(U)'—*P + 1(U)
such that
(i) d(co + r\) = dco + drj
(ii) i(AA//)=«A|M+(-l)(de|A)iAi/I
(iii) For each co, d(dco) = 0
(iv) . For each function /,
Let us note the consistency of (iv) as it applies to the coordinate functions.
For example xl is a function on U and d(x1) the effect of d on this function
x1 is the symbol dxl. Thus from (iii), d(dx1) = 0 once we have d.

3.2. EXTERIOR DERIVATIVES 21
First we prove there is only one such operation d. Suppose we are given
such a d. We first show that
d(dx*1 • • • da*') = 0
by induction on p. We have just noted this for p =1. If it is true for
J9 — 1, then by (ii),
did11 (do?2 • • • dx**)] = dx*11 • • • dx**,
d(efe*« • • • dx**) = d{d(xhldxh> • • • dx**)} = 0
by (iii). Now if a> is a p-form,
da> = J] d(aHdxH)
which shows that the recipe (i-iv) completely determines doo. To prove that
there exists such an operator d, we simply set
da> = Yd-p!.dxUxH
^ dxJ
for a> = £ aH dxH and check that the properties are satisfied. Properties
(i) and (iv) are fairly clear; let us look at (ii) and (iii). Evidently if we can
establish these for monomials, by summation they will follow generally.
Suppose
X — adxH, \x — bdxK.
Then
d(X A /i) = d(abdxHdxK)
r d(ab)
dx1
■dxldxHdxK
= Y —: bdxldxHdxK +y a —: dx[dxHdxK
^ dx1 ^ dx1
= Y,(8^idxidxH\ A(bdxK)
+ ( _ l)WegA) £ (ocfeH) A /<* ^il^jA
= (dA) Afl+(-l)(degA);i Acfyl.

22 III. THE EXTERIOR DERIVATIVE
The sign results from
dx[dxH = (-l)(dc*X)dxHdx\
This proves (ii).
Again, let a> = adxH. Then
did^^dfrp-.dxidxA
= ^e^hdxJdxidxH
2^\dxldxJ dxJdxl)
= 0,
which verifies (iii).
Property (iii) is nothing more than the equality of mixed second partial
derivatives. It is the source of most "integrability conditions" in partial
differential equations and differential geometry. It is usually referred to as
the Poincare Lemma.
3.3. Mappings
We study the following situation: U is a domain in Em, V is a domain in
En and </> is a smooth mapping on U into V. We write
4>: u—>V.
Also, we denote by x1, - - - , xm the coordinates of Em and by yl> • • • , i/1 the
coordinates of E". Then we can write
to show that the point with coordinates x is transformed by </> to the point
with coordinates y. The functions yl(x) are smooth.
As before, R denotes the reals. If g is any real-valued function on V,
g: V—>R
then we may combine this with </> to obtain a function on U to R which we
write
Thus
4>*: F°(V)—>F°(U).
From the mapping </> on U to V we have constructed a new (induced)
mapping </>* on F°(V) to F°(U).

u
3.3. MAPPINGS
-*-v
23
<f>*g = go<f>
We are now going to define a map </>* taking p-forms on V to p-forms on U :
4>*: F"(V) —>P(U).
(Strictly speaking we should index </>* and write (f)p*, p = 0, 1, • • • , but we
shall skip this.) We have taken care of p = 0 already. The crucial case is
p = 1; after we do that, the algebraic considerations of Chapter II do the
rest of the work.
The basic idea is substitution of coordinate functions, replacing dyl by
^ dxJ
Thus if a> = £ a((y) dyl is a one-form on V, we set
We now have
4>*: F^V)—>F!(U).
By the method of Section 2.4, we extend this mapping to the exterior products
to obtain
<£*: F*(V) — P(U).
As an example,
^(dy'dy2) = ((j>*dyl)(<t>*dy2)
2 ^ \ ftr' dxJ dxJ dxlJ
2^ d(xl,xJ)

24 III. THE EXTERIOR DERIVATIVE
We now list the basic properties of 0*.
(i) </>* (co + rj) = (/)* co + </>* r\
(ii) ^(2A/i) = (f2)A(^/l)
(iii) If co is a p-form on V,
d{<l>* a>) = <l>* (da)
(iv) If</>: U —Vand^: V — W, then
(^ o </>)* = </>* o iA*.
The first property is evident and the second follows from the final formula
of Section 2.4.
Property (iii) is essentially the chain rule for partial derivatives. First
we take a 0-form g on V.
^ = £ ti dyj>
8yJ
We proceed inductively, supposing we have verified (iii) for (p — l)-forms.
It suffices to verify (iii) for ^-forms co which are monomial since each £>-form
is a sum of such. Suppose then that
(o = gdyH = gdri
where rj = ]/tl dyhl • • • dyhp is a (p — l)-form. Then
<j)*co = (</>* g)(<j)* dr\) = ((j)*g) a (d(j)*rj),
d((f)* co) = d((j)* g) a d(</>* r\),
and
dco — dg a drj,
</>* dco = (</>* e&7) a (</>* rffy)
= d((j)*g) a d((j)*rj) = d(j)*co;
we have pushed through the next case.
We now look at the final property (iv).
For a 0-form (function) h on W we have
[(* o *)**](X) = h[(xj> o 4>)(X)] = hty[4>(x)]}
= W*h][<t>(x)] ={$*[** h]}(x)
= [(tf>* o ^*)*](X),

3.4. CHANGE OF COORDINATES
25
hence
(^o^)**=(0*o^*)A.
An induction similar to that above establishes the property in general. All
it means is that one can substitute directly the expressions for the coordinates
U
<t>
-*-v
F'(U
(l/>o</>)*
F'(W)
zk on W in terms of the coordinates a;' on U, or indirectly by first going
through the coordinates yJ of V; the results are the same.
What has really been seen in this section is that one can carry on fearlessly
with the most obvious kind of calculations with differential forms.
Examples. Consider the map <j>\ t—► (x, y) on E1—> E2 given by
x = t2,y =t3. Ifco = xdy,a, one-form on E2,
<j>*(D = (t2)-£di = 3t*dt.
ct
Take the map \\t: (x, y) —► t = x — y.
\j/* (dt) = dx — dy.
One final remark. Suppose m < n and </> is a map on the domain U of Em
into the domain V of E". If o is a p-form on V and p > m, then necessarily
0*co = 0.
3.4. Change of Coordinates
We apply the results of the last section to the special case in which U and
V are both domains in E" and </> is a one-to-one mapping on U onto V with
both (j) and ij/ = </>"l smooth. (Note the map x —► y = x3 on E1 —► E1 is
one-to-one and smooth. But the inverse map y —► x = ylf3 is not smooth—
no derivative at y = 0.) In each figure, i is the identity map, f(x)= x. It
follows that </>* is a one-one map on FP(V) onto FP(U) and its inverse is \j/*.
If we interpret the coordinates y of V as new coordinates on U, the result
d(f)*a> = (j)*daj
means that the exterior derivative of a differential form is independent of the
coordinate system in which it is computed.

26
III. THE EXTERIOR DERIVATIVE
This inner consistency of the differential form calculus is most important.
Later we shall base the global theory (forms on manifolds) on this.
U
<t>
->-v
-*-u
We note in passing that with a proper formulation this independence of d
on the coordinate system can be obtained as a consequence of the four basic
defining properties (i-iv) of the exterior derivative in Section 3.2.
3.5. An Example from Mechanics
The following problem is taken from fi. Goursat [15, p. 85]. We work in
a region with coordinates (x, u) = (a^ ,•••,#„,%,••• , un). We are given
a function
<{> = <j)(X, U)
which is supposed homogeneous of degree 2 in the variable u. (For example,
a kinetic energy form £ aij(x)uiuJ.) Define
We assume that the mapping (x, u) —> (x, p) defines a regular change of
variables. We then write
0(x, u) = Mx,.p).
The problem is to prove the relations
# d4>
#
The proof depends on two things, the Euler formula for homogeneous
functions which in our case implies
11^ = 2*
i.e.,
X pa = 2</>>
and the fact that exterior relations are independent of how they are derived.

3.6. CONVERSE OF THE POINCARfi LEMMA
27
We have
d<b = Y —- dx: + Y —- duk = y —- dx{ + y pkduk
r ^ dxt l ^duk k ^ dxt l ^*k k
and
2d(f) = Y^pkduk + £%%,
hence by subtracting
# = - Z -fa dxi + Z Mp* •
Now everything follows from </> = i/f and
^ ^f ^ dpk
3.6. Converse of the Poincare Lemma
The Poincare Lemma, d(dco) = 0, has these interpretations in 3-space:
curl(grad /) = 0
div (curl v) = 0
according to the examples at the beginning of Section 3.2. In vector
analysis one proves that a curl-free vector field is a gradient by line integrals
and that a divergence-free vector field is a curl, usually by a brute-force
method. We are now going to prove a general result. If co is a p-form
(2>= 1) and dco = 0, then there is a (p — l)-form a such that co = da. The
result is hard if p > 1 because there are many solutions. Also the result is
valid only in domains which are not too complicated topologically.
The demonstration is based on a "cylinder construction." We begin with
a domain U in E". We denote by I = [0, 1] the unit interval on the £-axis
and consider the cylinder or product space.
■ Xu.
This consists of all pairs (t, x) where 0g<^ 1 and x runs over points of U.
We single out the two maps which identify U with the top and bottom of
the cylinder, namely,
jt: U-lXU, j1(x) = (l,x)
j0: U-lXU, jo(x) = (0,x).
Thus
jt*: P(IXU)-P(U) (i = 0,l).
For example, to form j^co where co is a form on I ^ U. simply replace
t by 1 wherever it occurs in co (and dt by 0 correspondingly).

28
III. THE EXTERIOR DERIVATIVE
We now form a new operation K,
K: Fp+1(lXU)~>Fp(U);
K is denned on monomials by the formulas
K(a(t,x)dxP) = 0
K(a(t,x)dtdxJ)
(\ a(t,x)dt\dxJ,
and on general differential forms by summing the results on the monomial
parts. Here is the basic property of K: If co is any (p + I)-form on
I X U> then
K(da>) + d{Kco) = ji*(jo-j0*oo.
It is enough to check this for monomials.
Case 1. co —a(t, x)dxH.
We have Kco = 0, dKco =0,
doo = — dtdxH + [terms free of dt],
ot
Kdco = (\ ydt\dxH = [a(l,x) - a(0, x)]efaH.
But jt* co — a(l, x)efoH, j0* a> = a(0, x)etaH, so the formula is valid.
Case 2. co = a(t,x)dtdxJ.
First j j* co = j0* cd = 0. Next,

3.6. CONVERSE OF THE POINCARE LEMMA 29
Kdco = k\- Y—.did^daA
dKco = d ( a(t, x)dt\dxJ
s rf1 i
= Zti a(^ x)eft dxldxJ
so the formula again works.
Definition. A domain U is deformable to a point P if there is a mapping
0: lXU""~>U
such that
0(1, x) =x,
0(0, x) = P.
The boundary conditions may be interpreted in terms of the jf as follows:
0oJ!=I, 0ojo = P.
For a (p + l)-form w on U we have as a consequence
j\*[0* a>] = a>, io*[0* m] — ®'
Now we can state and prove the main result.
Let U be a domain in En which can be deformed to a point P. Let co be a
(p + l)-form on U such that do = 0. Then there is a p-form a on U such
that
co = da.
We merely substitute 0* co in the formula above to have
X[tf(0* co)] + d[X (0* co)] = ©.
But d(0* co) = 0* (dco) = 0, hence co = da with a = i£(0* co).
It is interesting to see how far the solution of the equation da — co is
determined. If jS is another solution, then d/3 = co = da, d(a — /?) = 0. If
p ^ 1, we conclude by the main result again that a — (I = dk where A is a
(p — l)-form. In other words, given one solution a, the general solution is
a — dk where X is absolutely arbitrary. (When p = 0, a and jS are functions
and we conclude that a — jS is constant.)

30 III. THE EXTERIOR DERIVATIVE
3.7. An Example
We shall illustrate this whole method in the case n =3, p = 2. Thus we
take a two-form
co = A dydz + Bdzdx + Cdxdy
in E3 for which do = 0, i.e.,
dA d£ dC___
dx dy dz
The space E3 can be deformed to 0 by the map
(j)(t, x, y, z) = (tx, ty, tz).
The assertion is that co = da where
a = Kef)* oo.
First we compute </>* co :
(f)*oo = A(tx, ty, tz)d(ty)d(tz) + • • •
= A(tx9 ty, tz)(tdy + ydt){tdz + zdt) H
= A(tx, ty, tz)(ytdtdz — ztdtdy) + • • • + (terms free of dJt).
Now we have
a = K((j)*oo) = I I A(tx, ty, tz)tdt)(ydz — zdy)
( 4(te,ty,te)*<»|(
+ I B(tx, ty, tz)tdt\(zdx — xdz)
+ I 0(to, ty, tz)tdt\(xdy — yrfa;).
One verifies after some calculation that indeed da = co.
3.8. Further Remarks
For
co — A dydz -r Bdzdx + Cdxdy
the problem of finding
a = Pdx + Qdy + Rdz
so that
da = co
is that of finding three unknown functions P, Q, R of the three variables
x, y, z so that the system

3.9. PROBLEMS
31
dR
dy
dP
~dz"
8Q_
dx
dQ
" dz
dR
dx
dP
dy
of three partial differential equations is satisfied, the given functions A, B,C
being subject to the necessary condition
dA dB dC
dx dy dz
It is remarkable that this system (and the more general ones covered in
Section 3.6) can be solved by an explicit formula involving quadratures. In
general, the theory of exterior differential forms exposes many types of
systems of partial differential equations which are reducible to systems of
ordinary differential equations and often solved by quadratures.
Another point to be noted is this. If we are dealing with a (p + l)-form oo
such that dco = 0 and oo happens to depend on several parameters smoothly,
then we can find an a such that dot =oo and a depends on the same parameters
just as smoothly. This again follows from the explicit formulas of Section 3.6.
3.9. Problems
1. Let
oo = i £ <^ijdxldxj, djj + Uji = 0.
Prove that
*•-*!(£' + & + &)*"■'"
2. Consider a linear transformation </>: E" —> E", (^(x1, • • • , x") =
(y1,'" >2/")> where
yl = ]T aljXj + 6', a1 j, b( constant.
Whatis^cfc1 •• •£&")?
3. Consider the mapping
on E2 into E2. Compute </>* (dx), </>* (dy), and </>* (ydx).
4. Complete the unfinished calculation of Section 3.7.

Applications
4.1. Moving Frames in E3
We first point out that in dealing with vectors in Euclidean space, no
matter where we draw them for picturesque purposes, when we deal with
them analytically, they always start at the origin.
We attach to each point x of E3 a right-handed orthonormal frame
ei' e2> e3 and suppose that the vector fields e£ are smooth fields.
What we shall do is express everything in sight in terms of the ef, apply d
to these relations to derive further ones, and continue until we obtain no
further results.
First of all, dx is a vector with one-form coefficients, for example, dx =
(dx, dy, dz) — dx i -f dy j + dz k. We express dx in terms of the frame
el5 e2, e3 at the point x, which we certainly may do, say, by first expanding
i, j, k in terms of the e- and then collecting terms:
dx = (7lel + <r2e2 + <73e3 ,
32
IV

4.1. MOVING FRAMES IN E* 33
where the ox are one-forms. We do the same with each e,:
det = (oilel + o)i2e2 + coi3e3 (i = 1, 2, 3)
where the a>fj. are one-forms.
Since ei-ek = 8ik, we have
deleft + et-dek = 0,
that is,
<*>ik + <*>ki = 0.
In particular, a>u = 0.
It will be convenient to introduce some matrix notation. We set
<x = (^ , <r2 , <r3), Q = ||©v||
and have these structure equations:
dx = (je,
de = Qe,
Q + rQ = 0.
Here applying ^toa matrix means simply applying it to each element. In
the last equation, the left-hand superscript t denotes transpose of the matrix,
i.e., interchange of rows and columns, so this equation expresses the skew-
symmetry of Q.
From d(dx) — 0 we have
da e — a de = 0,
da e — aQ e = 0,
{da - aQ) e = 0.
Because the et are linearly independent, this means
da = aQ.
Similarly, from d(de) = 0, we have
0 = dQe - Qde = (dQ - Q2) e,
e£Q=Q2.
In summary, then we have
Structure equations Integrability conditions
e
dx = <jrx/\ f do* =aQ\
de = Qe
lQ+rQ = 0
dQ = Q:

34
IV. APPLICATIONS
Further differentiation does not lead to new results. We shall see in our
study of Riemannian geometry that the equation dQ — Q2 = 0 expresses the
lack of curvature of Euclidean space.
A point to be noticed is that the three-form at a g2 a <t3 is precisely the
element of volume in E3:
ax A a2 A <73 = dxdydz.
We shall verify this in the next section.
It will be observed that the calculations of this section work equally well
in E".
Example. Spherical coordinates. The orthonormal unit vectors el5 e2, e
are taken in the directions of increasing r, $, 9, respectively. From
x = (r sin </> cos 0, r sin </> sin 9, r cos </>)
we have
dx = (sin (f) cos 6, sin cj) sin 0, cos cj>) dr
+ (r cos (f) cos 6, r cos <f> sin 9, — r sin </>) d(f)
+ (— r sin (f> sin 0, r sin cj) cos 0, 0) d6
= (dr) e! + (r d(j>) e2 + (r sin </> d0) e3

4.2. ORTHOGONAL AND SKEW-SYMMETRIC MATRICES 35
with
(et = (sin (j) cos 9, sin</>sin0, cos$)
e2 = (cos (j) cos 0, cos 0 sin 0, — sin (f>)
e3 = ( — sin 9, cos 0, 0)
and so
al=:dr, cr2=zrd(f), a3 = rsin(f)d9.
Differentiating,
cZex = (cZ</>)e2 + (sin</>d0)e3
de2 = ( — dcf))e1 + (cos $d9)e3
hence since Q is skew-symmetric,
(0 d(f> sin <t>d9\
-d(f) 0 cos(f)d9 J .
— sin (j)d9 — cos (j)d9 0 /
The volume element is
gx A <72 a <r3 = r2 sin (j)drd(f)d9.
4.2. Relation between Orthogonal and Skew-symmetric Matrices
It is no accident that Q turns out to be skew-symmetric. This is a
consequence of the principle that the first-order approximation to an orthogonal
transformation is a skew-symmetric one. We shall look at this from several
viewpoints.
A matrix B is orthogonal if its transpose equals its inverse, fB = B~l, or
B'B= XBB — I. Suppose A is skew-symmetric, A + lA — 0. Then for
small e we set B — I + eA and have
B'B=(I + sA)(I -eA) = I + 0(e2)
so that B is orthogonal up to first-order terms.
Here is another approach. Let A be skew-symmetric. Since the
characteristic roots of A are pure imaginary, I + A and I — A are non-
singular. Set
I-A
Then
'•'-m)(iii)-'
so that B is orthogonal.

36 IV. APPLICATIONS
Next we re-examine the calculations of the last section. Let
-0
where the i • are the fixed unit vectors in the x, y, z directions, respectively
(i, j, k in usual vector notation). Then
e« = E hfo e = m
leading to a matrix B = ||6fj|| which is clearly orthogonal:
/ = e'e = B\ 'i <B = BVB = B'B.
(Now we can prove the fact dxdydz = g1 a g2 a g3 mentioned at the
end of the last section. We have
dx = (dx, dy, dz) i = (je = aB\,
(dx, dy, dz) = aB,
hence
dxdydz = l-BI^ a a2 a <t3 .
But from *BB = / we have \B\2 = 1, |jB| = ±1. Since we are supposing e
is a right-handed system, \B\ — +1,
dxdydz = al a g2 a <t3 .)
Then we have
de = dB\ = (dB)B~1e
so that
Q = (dB)B~1.
We note this general result: // A is an orthogonal matrix whose elements are
functions of any number of variables, then
(dA)A~l
is a skew-symmetric matrix of one-forms.
For we have
<AA=I,
fdA A + 'AdA=09
*A-ltdA -t-dAA-1 =0,
'(dAA'^+dAA'1 =0.
There is also a converse which is important. Suppose A is a matrix of
functions defined on a domain U. Suppose A is orthogonal at a single point
of U and that
dA = AA

4.3. THE 6-DIMENSIONAL FRAME SPACE
37
where A is a skew-symmetric matrix of one-forms. Then A is orthogonal on
all of U.
We set C — XA A and have
dC = (<dA)A + fA(dA) = (-tAA)A + 'A(AA) = 0,
hence C is a constant matrix on U. But we are assuming C = / at one point
of U, hence C = I on U, XA A — I on U, A is orthogonal.
Another point is this. If A is a variable orthogonal matrix
(transformation), each point v0 of space is sent by the general A to
v = Av0.
dv = dAv0 = (dA)A-lv
We then have
so that one passes from v to the ' 'infinitely near" vector v + dv under the
action of the general A of our family by means of
v—>v + dv = [I + (dA)A~1]v
with the skew-symmetric (dA)A~l representing this ''infinitesimal
transformation."
All of these considerations work equally well in E".
4.3. The 6-dimensional Frame Space
We consider the space of all right-handed orthonormal frames El5 E2, E3
at all points x of E3. This space is 6-dimensional because we have three
degrees of freedom in choosing x, two degrees of freedom in choosing the
unit vector Et, one degree of freedom in choosing the unit vector E2
perpendicular to Ex and then E3 is determined.
We write
E2
and have
E=Ae
where A is a variable (three parameter) orthogonal matrix and e = e(x) is a
definite moving frame.
Then
dx = (re — (tA~1E,
dE = (dA) e + Ade = [dA+ AC1] e
= [dA+An]A~iE.
We set
5 = oA~\ ft = (dA)A'1 + AQA-1.

38
IV. APPLICATIONS
These are matrices of one-forms on the 6-dimensional frame space and we
have
Structure equations Integrability conditions
dx = oE\ tdo = oQ]
dE = ClE \ [dCl = ft2.
|^ + fQ=0
To check the integrability conditions we note
0 = d(dx) = d5E — 5dE = (do — dd)Ey do = oCiy etc.
In making a penetrating study of the differential geometry of E3 one is
necessarily led to this 6-dimensional frame space and its differential forms
&l, (bij which, it will be noted, are entirely independent of the choice of the
moving frame e on E3.
4.4. The Laplacian, Orthogonal Coordinates
We continue the considerations of Sections 4.1 and 4.2. The forms
dx, dy, dz make up an orthonormal basis for the Euclidean geometry of the
space of one-forms at each point; these are related to the fixed (absolute)
frame i. From
e = B\, dx= oe = (dx, dy> dz) i
we have
oB= (dx, dy, dz)
as already noted. As B is orthogonal, we see that olf <r2, o3 is an orthonormal
basis for one-forms at each point.
Let/ be a function on E3. Then we have
df = ^dx + ^dy + ^f-dz,
ox dy dz
s$ -P £ -P %-P
*df = — dydz + — dzdx + — dxdy,
ox dy dz
[d2f d2f d2f\
d*df= 1—2 +zr-2 + -z-j\dxdydz = (Af)dxdydz.
The Laplacian A/ of/ is known as soon as the three-form d *df is known, for
this has turned out to be the Laplacian multiplied by the volume element
dxdydz.

4.4. THE LAPLACIAN, ORTHOGONAL COORDINATES 39
Now we know that the * operator can be computed equally well in any
orthonormal coordinate system. Also at a g2 a g3 = dxdydz, so our
procedure is this. We express df in terms of the at,
df — alal+ a2 o2 + a3cr3.
Then
*df = al a2 a3 + a2 a3 al + a3 al a2 ,
d*df= (Af)ala2a3.
A coordinate system u,v,w in a domain in E3 is called an orthogonal
coordinate system if the vectors
dx dx dx
du' dv' div
are mutually perpendicular. This means that for suitable functions A, /i, v,
the vectors
1 dx I dx I dx
1 X du' 2 n dv' 3 v dw
form an orthonormal, or moving frame. We shall presuppose that this is a
right-handed one. (Otherwise we merely permute w and v.) We have
, dx dx dx
dx — du — + dv — + dw —
du dv dw
= (Xdu)el + (fidv)e2 + (vdw)e3
so that
<7j = Xdu, o2 — \idv, a3—vdw
build an orthonormal frame for one-forms. Now we compute the Laplacian:
df = fudu + fvdv + fwdw
= (fJX.)°l+(fvll*)°2 + (LM03'
*df = (/«M)^2^3 + (full*) °3°1 + (/w/v)<7l<72
= dnvfJX)dvdw + (Xvfvlin)dwdu + (Xfifjv)dudv.
We compare this to
d*df= (Af)ala2a3 = Afiv(Af)dudvdw:
A/zv \du \ k du) dv\jx dv) dw \ v dw)J '

40 IV. APPLICATIONS
Let us apply this to spherical coordinates r, 0, 6:
(x = rsin^cosfl
y = r sin (j) sin 9
[z = rcos0.
The orthogonality is easily checked (it is obvious geometrically) and we have
al=dr, <72 = rd(j), a3 = rs'mcfrdQ,
A/ =
1
r sin (j)
Jr {r2s[n<t> I) + ^ ("^ |) + M fe D.
4.5. Surfaces
We study a smooth surface £ in E3. We choose a moving frame e at each
point x of L in such a way that e3 is the normal to the surface. Then et
and e2 span the tangent plane at each point. We shall see how the equations
of Section 4.1 specialize.

4.5. SURFACES
41
Since x is constrained to move in the surface, dx must lie in the tangent
plane, <r3 = 0:
dx = alel + <72e2 .
It is clear that the two-form axg2 represents the element of area of £.
unit normal
tangent plane
We exploit the skew-symmetry of Q by writing
Q
-COi
-C00
The structure and integrability conditions now reduce to
Structure equations
( dx = ale1 + <r2e2
de1 = G7e2 — a>1e3
de2 = — w^i — co2e3
^e3 = coiel + o>2e2 J
Integrability conditions
jdal = ma2 \
da 2 = —wg1
i Gitt>i + <x2a>2 = 0 |
j dm + (Di(D2 — 0
\d(o2 = —wo)1 )

42
IV. APPLICATIONS
In a certain sense, all of local surface theory is contained in these equations.
It remains to interpret them in terms of curvatures, curves on the surface,
etc. We illustrate a little of this.
As already remarked, (T1(72 is the element of area on E. As x moves over
S, e3 moves over a region on the unit sphere S2, called the normal, or
spherical, image of L. Since ex and e2 are orthogonal to e3, they lie in the
tangent plane to the spherical image and form a frame there. We see that
the equation de3 = co^ + a>2e2 pl&ys the same role for the spherical image
as dx = 0^! + <r2e2 does for L, hence a>1a>2 represents the element of area
of the spherical image.
Since there is only one linearly independent 2-form on the 2-dimensional
space L, we have
where K is a scalar called the Gaussian curvature. We shall see shortly that
it is entirely independent of the choice of et and e2 .
Similarly g1cd2 — <r2a>1 is a 2-form on £, and so
0i(D2 — (T2(o1 — 2Hala2
defines a scalar H called the mean curvature of £.
The one-forms a^, co2 are linear combinations of (Tt and g2 . Because of
the relation
GlCOi + G2(02 = 0
we have a symmetry in the coefficients:
g>i = peri + qo2
a>2 = q<Ti + ra2-
We easily have from this
2H = p + r, K = pr -q2.
The characteristic roots of the symmetric matrix
CD
are called the principal curvatures fc1}fC2ofL We consequently have
2H = Kt + k2, K =k1k2 .
From the relation dm + (1)^2 = 0 we have
efo +K<jia2 = 0.
This relation gives us K once we know al, a2 and tzj. But the relations
do1 = wg2 , <^cr2 = — WGi

4.5. SURFACES
43
suffice to determine w once ax and a2 are given. (For then dal = aala2 and
da2 = bo~lG2 are determined and we must have w — aal + ba2 •) In total
then, K is completely determined analytically from crl and a2 . This contains
the theorem of Gauss that the curvature K is an intrinsic invariant of E,
independent of how E is imbedded in E3, so long as the distance between
points of E measured along E (on geodesies, or shortest paths) is preserved
locally.
When we apply vector operations to vectors with differential form
coefficients, we must always combine the coefficients according to the rules
of exterior algebra and pay strict attention to the ordering of the factors.
With this we form vector (cross) products:
dx x dx = (a^i + <r2e2) x (a^t + <r2e2)
= ffi2(ei x ei) + ^22(e2 x e2) + <ri<r2(ei x e2) + ^2^1 (e2 x e,).
Now at2 = 0 (and et x et = 0), etc. Also
(T2(T1(e2 x ex) = (-<Ti<r2)(-e1 x e2)
= (^1^2)e35
so finally
dx x dx = 2(<71<72) e3
and we have obtained the vectorial area element.
Precisely, the vectorial area element is
(0^2) e3,
a vector directed along the normal with magnitude ata2, the element of
area of E. Since
dx x dx = (dx, dy, dz) x (dx, dy, dz)
= 2(dydz, dzdx, dxdy)
we have
(dydz, dzdx, dxdy) = (o^o^) 63 .
If v = (P, ©, R) is a vector field, then
(Pdydz + Qdzdx + Rdxdy) = v-(<71<T2e3)
= (v*e3)((T1(T2)
is the /Zwa; of v through E.
Similarly we have
dx x dx = 2(<71<72)e3
dx x de3 = 2#(<71<72)e3
de3 x de3 = 2K(ala2) e3

44
IV. APPLICATIONS
which shows the independence of H and K on the tangent vectors et, e2.
If / is a function on £ with
d/= a^! +a2<j2,
then on L,
*d/ = — a2al +al<j2f
d*df= d(—a2(7l + a^) = (A/Jcr^
defines the Laplacian of f on the surface or the second Beltrami operator A.
The same works for vectors and we have
dx = axex + <r2e2 ,
*dx = g2^1 — (T1e2 .
We notice that
dx x e3 = (at^! + <r2e2) x e3
= ^2ei -<7ie2>
hence
*cZx = cZx x e3 ,
d*dx = --dx x de3 — — 2H(g1g2)^2
and so
Ax = (Ax, Ay, Az) = —2^63 .
A minimal surface (surface of stationary area) is one for which the mean
curvature vanishes, H = 0. We have proved: The coordinate functions
x, y, z are harmonic on each minimal surface. (That is, they satisfy Ax =
Ay=Az = 0.)
In this section we have given a sample of how the exterior calculus fits
into the classical differential geometry of surfaces. Further material will be
found in Sections 8.1 and 8.2, but there is much of the subject that we cannot
cover in this text. A treatment from this point of view of exterior calculus
which is not quite completely satisfactory and which unfortunately is
embellished with historical comments often in bad taste is found in Blaschke [3].
4.6. Maxwell's Field Equations
In classical eleqtromagnetic field theory one deals with the following
quantities:
E = electric field H = magnetic field
B = magnetic induction J = electric current density
D = dielectric displacement p = charge density.

4.6. MAXWELL'S FIELD EQUATIONS 45
These are all functions of the space variables X , X , X 3 and the time t. The
basic Maxwell equations in ordinary vector language are
1 dB
(i) curl E = —- (Faraday's law of induction)
c ot
4:71 1 3D
(ii) curl H = — J H — (Ampere's law)
c c ot
(iii) div D = 4:71 p (continuity)
(iv) div B = 0 (nonexistence of true magnetism)
Here c is the speed of light. We shall put these equations into the language
of exterior forms. To this end, we set
a = (Eldxl + E2dx2 + E3dx3)(cdt)
+ (Bldx2dx3 + B2dx3dxl + B3dx1dx2)i
0= -{H^dx1 +H2dx2 + H3dx3)(cdt)
+ (Dtdx2dx3 + D2dx3dx1 + D3dxldx2),
y = (Jt dx2 dx3 + J2 dx3 dx1 + J3 dxl dx2) dt — p dx1 dx2 dx3.
Equations (i) and (iv) become
Equations (ii) and (iii) become
dp + 4ny = 0.
Applying d to this last equation yields
dy = 0,
in vector notation
dp
divJ+^ = 0.
From the equation doc = 0 one concludes, at least in any region of space-time
which can be shrunken to a point, that there is a one-form X such that
dX = a.
We introduce the vector potential A and a scalar A0 by writing
X = Aldx1 + A2dx2 + A3dx3 + A0cdt.
The equation dX — a in vector form is
curl A = B
a a l 8A c
grad^0--— = E.
c ot

46 IV. APPLICATIONS
In free space, everything simplifies according to
E = D, H = B,
J = 0, p = 0
so that the Maxwell equations become
curl E = div E = 0
c dt
curlH =-^ divH=0.
c dt
We introduce the Lorentz metric into 4-space whereby
dx1, dx2, dx3, cdt
is an orthonormal basis:
(dx1, dxj) = Sij, (dx1, cdt) = 0,
(cdt, cdt) = — 1.
The signature is 3 — 1 = 2.
According to the formulas of Section 2.7,
*(dxidx2) = —dx3(cdt), etc.,
*(dxl cdt) = dx2 dx3, etc.
We see that
a = (E1dxi + -•-)(cdt) + (Hldx2dx3 + •••),
P= -(H^dx1 + •--)(cdt) + (Eldx2dx3 + •••)
= *a.
Consequently Maxwell's equations in free space are simply
f da = 0
\d *a = 0.
We return to the general situation and refine our analysis by introducing
one-forms:
<ot = El dx1 + E2 dx2 + E3 dx3
a>2 = Bx dx2 dx3 + B2 dx3 dx1 + B3 dx1 dx2
a>3 = Hl dx1 + H2 dx2 + H3 dx3
a>4 = !>! dx2 dx3 + D2 efo3 dx1 + D3 da:1 da;2
co 5 = Jj da:2 da;3 + J2 dx3 dx1 + J3 dx1 dx2.

4.6. MAXWELL'S FIELD EQUATIONS
47
These involve space variable differentials only. Now we interpret d' to
denote the exterior derivative with respect to space variables only. We
introduce d/dt in this form
rs
— (&>i) = d)l= Et dx1 + • • • , etc.
ot
Now the Maxwell equations are
d,col = (b2
c
47r l .
a a>3= — a>5 + - co4
c c
d'a>2 — 0
eZ'a>4 = knpdx1 dx2 dx3.
The Poynting energy-flux vector S is introduced by
(s)'-H
that is
—) (Oi A a>3 = St dx2 dx3 + S2 dx3 dx1 + #3 dx1 dx2.
4:71/
Poynting's theorem,
fe)6-H + E-J + (^)E-6 + divS:=0'
follows from
df(o)i A a>3) = d'a>l A ft)3 - (Ot A d'a)3
/ 1 . \ /4tc 1 . \
= I C02 I A a>3 — a>! A I CO5+-CO4I
1 4tc 1
= (b2 a a>3 a)! a a>5 a^ A a>4.
c c c
For bodies at rest, one assumes D = kE, B = juH where the dielectric constant
k and the permeability \i are constant in time. Then Poynting's theorem
becomes
-^ = divS + E.j
where
8n

48
IV. APPLICATIONS
is the energy density of the field. The quantity E • J is called the thermo-
chemical activity.
4.7. Problems
1. Develop the formula for the Laplacian in cylindrical coordinates.
2. A complex matrix A is unitary if AA*=I, where A*—tAi the
transpose conjugate of A. We call A skew-hermitian if A* + A = 0.
Discuss the connection between unitary and skew-hermitian matrices.
3. Show that eA is orthogonal if A is skew-symmetric. Here
An
to!
-* = / + £ -
for the real matrix A.
4. Set up a frame and the structure equations for a sphere of radius R.
Compute the curvatures.
5. Find Gaussian curvature of the surface of revolution obtained by
revolving the curve
/ x = cos0 + In tan (6/2)
] y = sin 6
-<0<n
x2
about the #-axis.
6. Given a surface in the form z =f(x,y), develop formulas for H and K
in terms of/ and its partial derivatives.
7. Let L be a surface. Let ex, e2 and ei, e'2 be two moving frames of
tangent vectors to L. Determine the relation between the corresponding
w and w' and verify that dm = dm'.
8. Let ex, e2, e3 and ei, e2, e'3 be two moving frames in E3. Set up
the orthogonal matrix relating these frames and determine how the
corresponding Q and Q' are related.

V
Manifolds and Integration
5.1. Introduction
An w-dimensional manifold is a space which is not necessarily a Euclidean
space nor is it a domain in a Euclidean space, but which, from the viewpoint
of a short-sighted observer living in the space, looks just like such a domain
of Euclidean space. A case in point is the two-sphere S2. This cannot be
considered a part of the Euclidean plane E2. However our observer on S2
sees that he can describe his immediate vicinity by two coordinates and so
he fails to distinguish between this and a small domain on E2.
We have the technical problem of describing an w-manifold with sufficient
precision so that we can define functions, tensors, and differential forms on
such a space. The definition which follows is motivated in this way. Each
observer on the manifold has an immediate neighborhood (local coordinate
neighborhood) described by n coordinates. Each point of the space must
lie in at least one of these observed neighborhoods. Now if we consider
simultaneously two observers, their immediate neighborhoods may overlap,
and we must specify what happens in each such overlap. In the next three
sections we go over these matters with some care.
After this is accomplished we tackle the problem of defining the integral
of a differential form. In Sections 5 and 6 we lay the groundwork by
denning chains, the geometrical sets over which forms are integrated, and
in Section 7 we define the integral.
5.2. Manifolds
An w-dimensional manifold consists of a space M together with a collection
of local coordinate neighborhoods Ut, U2 , • • • such that each point of M lies
in at least one of these U. On each U is given a coordinate system
x\ -•- ,af
so that the values of the coordinates
(^(P),...,^(P)),
where P ranges over U, make up an open domain in Euclidean w-space En.
49

50 V. MANIFOLDS AND INTEGRATION
Suppose that U with coordinate system
x\ ••• ,xn
and V with coordinate system
overlap (intersect). We may express the V coordinates y of a point P in
terms of the U coordinates x of this point:
yl = yi(x\ ••• ,xn) (f = 1, ••• ,n).
As part of the definition, we assume that these functions are smooth
(differentiate as often as we please).
Having this formal definition out of the way, we explore some consequences.
First of all, on the overlap of U and V above we may interchange the roles
of U and V to write smooth functions
Substituting yields
y' = yV(y),-",*"(y))
and we may differentiate by the chain rule:
* ~ L dx* dyk'
»).
which has the matrix interpretation
1—1
dxj
•
\dxj\
= 1.
We take determinants by the product rule:
%', •".»") d(xl,---,x»)
d(x\---,x") %',■
It follows that the Jacobian
d(xl,---,x")
it is different from 0 at each point.
■ y")
l.
#0;

5.2. MANIFOLDS
51
A manifold is called orientable (two-sided) if it is possible to choose the
local coordinates in the first place so that each such Jacobian (on an overlap
of local coordinate neighborhoods) is positive.
Example. We make the two-sphere S2 into a manifold by using six
coordinate neighborhoods. We set
S2 = {(x, y, z) where x2 + y2 -f- z2 — 1}.
The neighborhoods are
coordinate system y, z.
coordinate system z, y.
coordinate system z, x.
coordinate system x, z.
coordinate system x, y.
coordinate system y, x.
In comparing the overlap of two of these, we shall not be pedantic and
introduce different letters, hoping the reader will forgive this sloppy notation.
On the intersection of U \ and U J we have the coordinate transformation
u+ =
ur =
u+ =
u2 =
u3+ =
u3 =
{x > 0},
{x < 0},
{y > o},
{y < o},
{z>0},
{z<0},
y
and so
i — z,
%> 2)
d(z, x)
x>0, y>0
V"
1
V"
o
On the intersection of U jf" and U 3 ,
d(y,x)
On the intersection of U^" and U3
x — x
1
y
V"
0
X
Z=-y/l
d(x, z)
■tr-**
-T>o.
x>0, z < 0,
x
T>o.
y < 0, z < 0,
d(y9 x)
0
y
V"
1
X
--f>0-
etc.

52
V. MANIFOLDS AND INTEGRATION
Thus the two-sphere is a two-manifold, and our choice of local coordinates
proves it to be orientable.
One could also cover the sphere S2 with a system of only two local
coordinate neighborhoods by taking two opposite hemispheres, each extended
slightly to make open overlapping neighborhoods.
The sphere S2 has two opposite orientations (outward or inward normal,
corresponding to counterclockwise or clockwise sense of rotation). Similarly
an orientable w-manifold has two opposite orientations. A definite one of
these is determined by the order in which local coordinates x1, • • • , xn are
given, up to an even permutation of this order. Making an odd permutation of
local coordinates gives the opposite orientation.
Let M be an w-manifold. To say that a real-valued function /on M is
smooth at a point P of M means the following. Let U be a local coordinate
neighborhood containing P with coordinates x1, • • • , xn. We require that
/(x1, • • • , x") be smooth near P. This restriction on/ is independent of the
particular U one chooses, since two coordinate systems whose neighborhoods
overlap on a region including P are themselves related by smooth functions
(from the definition of manifold). A real-valued function/ is smooth on M
if it is smooth at each point of M.
Similarly, if M and N are manifolds of dimensions m and n, respectively,
one defines a smooth mapping
</>: M—-> N
by the requirements that in local coordinates x1, • • • , xm on U in M and
y1, • * • , yn on V in N, we have </> represented by smooth functions
yl = y'(*V">*m) (* = li '•',*>)
on that part of U which </> maps into V.
A manifold M is called a submanifold of a manifold N provided there is a
one-to-one smooth mapping
j: M—+N
which has this regularity property: in local coordinates (as written above),
the matrix
3a>

5.3. TANGENT VECTORS
53
has (maximal) rank m at each point. We refer to j itself as an injection or
imbedding of M in N.
This applies in particular when N = En so that we may refer to submani-
folds of Euclidean spaces. It is an established result of manifold theory
that each ra-dimensional manifold which is not too large may be imbedded
in E" with n = 2m + 1.
5.3. Tangent Vectors
We study a manifold M and a point P on M. Our job is to define the
tangent space at P, an w-dimensional vector space whose elements are the
tangent vectors at P. Because we are not within the simple terrain of
Euclidean space we cannot merely draw arrows emanating at P. We need
a way of considering ordinary Euclidean vectors which depends in no way
on arrows, or directed line segments. The answer is simple. We may
identify Euclidean vectors with directional differentiations. Thus in case P
is a point of E3 and v = (a, b, c) is a vector at P, we may identify v with the
operator
(a — + b — + c-)\
\ dx dy dz/\P
This does the usual things to sums and products, which motivates the
following definition.
First some notation. If M is a manifold, we denote by
F°(M)
the space of all smooth real-valued functions on M.
Let P be a point on a manifold M. A tangent vector v at P is an operator
v: F°(M)—> R, the reals
satisfying
(i) v(a/+ bg) — av(f) + &v(gr), a,b constant,
(ii) v(/.9r)=9r(P).Y(/)+/(P).v(9r).
Thus v assigns to each smooth function/on M a real number v(/).
We shall first observe that if we take a constant function c, then v(c) = 0.
For setting f = g = 0 in (i) yields v(0) = 0, setting / = g = 1 in (ii) yields
v(l) = 0, and setting/ = 1, a =0, and g = 0 in (i) yields v(c) = 0. Observe
that
v(c/)=cv(/)
for any / and constant c.
Next, suppose x1, • • • , x" is a local coordinate system, valid in some
neighborhood of P. Then each of the operators
d

54 V. MANIFOLDS AND INTEGRATION
(the vertical bar means "evaluated at P") is a tangent vector, as one easily
verifies.
The totality of tangent vectors at P makes up a linear space, Tp, called
the tangent space to M at P. We shall show that these vectors v£ form a
basis of this tangent space. We set
(xl,--,a»)\P = {ci,••>,&*).
If v is any tangent vector at P, set
v(xl) = v^1 — cl) = a1.
Now if/ is any smooth function on M, we expand/ in a Taylor series up to
first-order terms with the integral form of remainder:
/(x)=/(c) + X(*'-c')0,(x),
Then
hence
8f
v(/) = v[/(c)] + £ gt(c) v(z' - c<) + £ (c; - c') v(g,)
= 0 + £a
P ^ dxl
which establishes the result. We refer to a1, • • • , an as the components of v
with respect to the coordinate system x. If y is another coordinate system
valid at P, and
we find, by the chain rule,
*'-Z
dy'
j dy'
aJ —:
dxJ
the usual transformation law for contravariant components of a vector.
Note here that we are working at a single point so that a and b are constant.
A vector field on M consists of a smooth assignment of a tangent vector to
each point of M. In local coordinates,
v-jyw^,
al(x) smooth.
On an overlap,
v-Z^x)^.
&'(y(x)) = Za'(x)
dxJ

5.4. DIFFERENTIAL FORMS
55
5.4. Differential Forms
The smooth functions on M will also be called 0-forms. They form a
space F°(M), the space of forms of degree 0 on M.
We now define a one-form at a point P of M. We must have an
expression
^atdx\ ai constant
for each local coordinate system (xl) valid in a neighborhood U which
includes P and such that any two such expressions
^atdxl9 Yshidyl
at P are related by
the usual transformation law for co variant vectors. Evidently this is
completely consistent with our local study in Chapter III.
Having this, we may form sums of exterior products of one-forms at P to
construct p-forms at P. Now we can define a p-form on M. This is a
smooth assignment of a p-form to each point P of M. If U is given with
local coordinates (xl), then on the neighborhood U we have the representation
oo = £ aH(x)dx*
with smooth functions aH(x) on U, H — [ht, • • • , hp}.
If we have the representation
o> = I<bK(y)dyK
with respect to a second coordinate system which overlaps the first, then the
relation between the fe's and a's is given by substitution of yl = yl(x) for yl
and
for dyl. As a consequence of our study of coordinate changes in Section 3.4,
we see that the space
P(M)
of p-forms on M is completely defined, that exterior multiplication
oo A rj
of two-forms on M is accomplished by operating with one point at a time, and
that the exterior derivative of a form on M is defined by working it out in
each local coordinate system.

56 V. MANIFOLDS AND INTEGRATION
All the rules of Chapter III are readily verified,
d(a> a rj) = dco a rj + (-l)<de8«» o> a drj
for example.
If M and N are two manifolds and
</>: M—> N
is a smooth mapping, then there is a natural induced mapping </>*,
0*: FP(N)—*FP(M)
which again is denned by applying the local construction in one local
coordinate system at a time and piecing together the results. As in the local
theory, we have the results
(i) </>* (a> + rj) = </>* o) + 0* rj.
(ii) 0*(Aaai) = (0*A)a(0*ji).
(iii) d(0*co) = 0*(dco).
The last of these can be expressed by means of a commutative diagram
R(M)
d
FP+1(M)-
P(N)
d
(j>*
t
F"+1(N)
Each of the two possible paths from FP(N) to FP+1(M) leads to the same
result.
In practice, one often constructs differential forms on a manifold this
way. One knows in advance several smooth functions f,g,m" on M.
From these one constructs one-forms df, dg, • • • and from these in turn forms
of higher degrees by taking exterior products.
Example 1. On the two sphere S2 considered in Section 5.2, the functions
x, y, z are smooth 0-forms. Thus dx, dy, dz are one-forms and dxdy, dydz,
etc., are two-forms. On the neighborhood U^ we have
x2 = 1 - y2 - z2,
dx =
-ydy — zdz
i i -ydy-zdz z
dx dy = dy — -dy dz,
x x
etc.

5.5. EUCLIDEAN SIMPLICES
57
Example 2. The circle S1 = [x2 -f- y2 = 1}. Here x and y are functions on
S1 and
xdx + ydy = 0.
This means we can define a one-form a. At a point where x ^ 0,
dy
a = — .
x
At a point where x = 0, we have 2/^0 and
dx
V '
On any arc of S1 which is not the complete circle, we can find a function 6
such that
x = cos 0, y = sin 0,
hence
a = dd.
It must be emphasized that no such function 9 exists on all of S1—it would
have to jump by 2n somewhere.
5.5. Euclidean Simplices
In this section we shall describe the standard building blocks which we
later piece together to form fields of integration, ^-dimensional spreads in a
manifold over which we can integrate p-forms. These building blocks will
be called Euclidean simplices of various dimensions—we shall omit repetition
of the adjective Euclidean in this section, but we understand that everything
takes place in Euclidean space.
A 0-simplex is a single point (P0).
A l-simplex is a directed closed segment on a straight line. It is
completely determined by its ordered pair of vertices (PqP^.
A 2-simplex is a closed triangle with vertices taken in some definite order.
It is completely determined by its ordered triple of vertices in the proper
order,
Similarly one has a 3-simplex based on an ordered quadruple
(p0) pv p2, p3)
of four points, no three collinear. Geometrically it represents a tetrahedron.
Finally, an n-simplex is the closed convex hull
(p0,---,pn)

58 V. MANIFOLDS AND INTEGRATION
of (n + 1) independent*)* points taken in a definite order. The geometrical
set so spanned consists of all points
P = t0P0+ -•+tHpH, tt*o9 £t, = i,
i.e., all possible centroids of systems of nonnegative masses t0, • • • , tn
located at P0, • • • , Pn, respectively.
The boundary ds of a simplex s is a formal sum of simplices of one lower
dimension with integer coefficients:
d(P0, P, , • • • , P„) = £ (-l)i(P0,Pl ,■■■, P,_t , P|+1 , • • • , P.).
> = 0
An examination of the lower dimensional cases convinces one that this is
consistent with the customary ideas on boundaries of oriented regions.
a(p0,p1) = (P1)-(p0),
d(P0, p,, p2) = (P,, p2) - (P0, p2) + (P0, pt),
d(P0,p1,p2,p3) = (P1,p2,p3)-(P0,p2,p3) + (P0,pl,p3)
-(p0,pltp2).
In the triangle, the ordering of the vertices gives a sense of rotation of the
triangle. In the tetrahedron, the ordering of the vertices gives a right-
handed screw sense in space and induces a positive sense of rotation in each
triangular face (outward drawn normal). One thinks of each minus sign
in ds as representing a reversal in this rotation sense. The result is that
d(P0 , - - - , P3) represents the oriented geometric boundary of the tetrahedron
according to the outward drawn normal.
t This means that the n vectors (Pi — Po), (P2 — Po), • • • , {Pn — Po) are linearly
independent.

5.5. EUCLIDEAN SIMPLICES 59
An n-chain is a formal sum
c = £a's,
where the a1 are constants and the sf are w-simplices. Its boundary is
denned by
dc = X o'(3s,).
A basic result is that the boundary of each chain itself has zero boundary:
d[dc] = 0.
It suffices to check this for simplices. Let us try low-dimensional cases:
d[d(P0, p, , p2)] = d(P,, p2) - d(P0, p2) + d(P0, pt)
= [(P2) - (Pi)] - l(P2) - (Po)1 + [(Pi) ~ (Pott = 0,
d[d(P0 ,■■■, P3)] = [(P2, P3) - (P,, P3) + (P,, P2)]
"[(^2. P3) -{P0,P3) + (P0,P2)]
+ [(P1,P3)-(P0,P3) + (P0)P1)]
- [(Pi, Pi) - (Po , Pi) + (Po , Pi)] = 0,
which illustrates the general idea; each face occurs twice with opposite signs.

60 V. MANIFOLDS AND INTEGRATION
More generally, in computing
a[5(P0> •••,?„)],
one obtains
(P0 , • • • , Pi-1, Pi+i, '' • , Pj-i> Pj+i> * *' j Pn)
twice, with opposite signs, once each from
d(P0,--,Pl-l,Pl+l,---,Pn)
and
v(Po> ' '' > Pj-i> Pj+i> ''' > Pn)>
so that everything cancels.
Given two w-simplices (P0 , • • • , Pn), (Q0 , • • • , Qn), there is a unique linear
correspondence between them which preserves the ordering of the vertices.
It is given by
thPi^-^thQi (h^o, tt, = i).
0 0 0
It is convenient for defining integrals to have standard models of the
simplices of each dimension. We define the standard n-simplex
s" = (i?0, ••-,£„)
as the simplex in Ew based on
#0 = 0
Rt = (10 • • • 0)
R2 = (010 • • • 0)
Rn = (00 • • • 01).
x
We must now agree on a certain convention for integration. Let a> be an

5.6. CHAINS AND BOUNDARIES 61
n-form defined on a domain U of E" which includes s". We wish to define
I-
We do this by writing co in the unique way
co = A(xl, • • • , xn)dxl dx2 • • • dxn
with the variables in their natural order, and then setting
co = I .
'sn J s"
A(x)dx1dx2 ---dx",
where the right-hand side is now the standard ordinary ^-fold integration,
which may be evaluated by any scheme of iteration, regardless of ivhat order
in which the variables are taken.
For example, if co = dzdydx, then
r r n ri-y ri
co = — dxdydz — — \ dy \ dx
J s3 J s3 J0 Jo Jo
-x-y
dz= -1/6.
5.6. Chains and Boundaries
Now we consider a manifold M and we shall define an n-simplex in M.
As a preliminary definition, this consists of three things: a Euclidean n-
simplex s", an 7i-dimensional neighborhood U of s" in Euclidean space,f and
a smooth mapping c6,
0: u—>M.
We denote this preliminary simplex by
(sn, U, <t>).
If we are given a second one,
(tw, V, ,/0,
it will be considered the same as the first provided
where
s" = (P0,Pi, •••,-?„), r = {Qo,Qi,-~,Qn)-
In other words, if we set up the natural order-preserving linear equivalence
t That is, a neighborhood in the smallest flat submanifold of Euclidean space
containing sn. This smallest flat submanifold is the totality of points 2o UPt> h real, 2 h = 1>
where sn = (Po, . .., Pn)>

62 V. MANIFOLDS AND INTEGRATION
between s" and tn:
s" < > t",
then (j)(P) = \jj(Q) whenever P and Q are corresponding points. This is also
expressed by the commutative diagram
The totality of these preliminary simplices (sw, U, 0) which in this way
are identified with a single one make up an object which we call an n-simplex
in M, denoted by a symbol on.
The open neighborhoods U we have introduced merely serve to eliminate
difficulties with differentiability on the boundary.
If an is a simplex represented by (sw, U, 0), then s" has faces t0, • • • , tn,
each a Euclidean (n — l)-simplex, where
w = E±t,.
By restricting cj) to the various tt, each extended a little in U to make open
neighborhoods Vf, we define the faces of a", each represented by
T,= (t„V',0)
and the corresponding boundary
u / \
/ s2 \
5 6- <> C
to
This is an (n — l)-chain in M. By an n-chain c of M we mean a formal sum
i
with constant coefficients at and w-simplices d". Chains may be added and
multiplied by constants. We denote by
C„(M)

5.7. INTEGRATION OF FORMS
63
the set of all w-chainst on M. We set
dc = Yjai drf f°r c = X ai(T"i •
Thus
d: CW(M)-^CW_1(M) (W=l,2,..-).
The basic property of the boundary operator d follows readily from the
corresponding Euclidean situation: for each n-chain c,
d(dc) = 0.
A cycle is a chain z whose boundary vanishes, dz = 0.
A bounding cycle (or simply boundary) b is a chain which is the boundary
of a chain of one higher dimension, b = dc.
Each boundary is a cycle, for if b = dc, then
3(b) = d(dc) = 0.
One further thing to be noted is this. In our preliminary definition of a
simplex (s", U, cj>) we do not require that the smooth mapping 0 on U
into M be a one-to-one mapping. Indeed, it may happen that it takes all of
s" into a lower dimensional space, even into a single point! A close analysis
shows that not only is there no harm in allowing such "bad" mappings but
that there are very great technical difficulties involved in attempting to avoid
them.
5.7. Integration of Forms
Our data is a manifold M of any dimension, a p-form co on M and a p-chain
c on M. We must define
\r
First we set
where the at are constants and the at are ^9-simplices and write
\ 0) = X ai\ 0)
so it remains to define the integral of co over a p-simplex a. Now we can
represent a in the form
(sp, U, (j>)
where sp is the standard p-simplex in Ep and 0 is a smooth mapping of the
neighborhood U of sp into M. Our definition is
f Precise topological terminology: ordered singular differentiate n-chains.

64
V. MANIFOLDS AND INTEGRATION
J<r Ji1
</>* CO.
Since $* co is a p-form on U, this is an ordinary p-fold integral, as discussed
in the next to last section.
In application, one often does not bother to spell out in detail how a given
geometrical region may be considered as a chain, but rather relies on the
usual combination of experience and intuition, the latter an excellent guide
in geometry. For example, suppose coisa 2-form on S2 = {x2 -f y2 -f z2 = 1}
and one seeks co taken over S2. There will usually be a more effective
procedure than using the coordinate planes to decompose the surface S2 into
eight spherical triangles, setting up mappings of the standard triangle onto
each of these, etc.
What then is the value of this rather long story on chains, boundaries, and
integrals? Tn this age, it hardly seems necessary to defend the placing on a
logical and rigorous basis things which are only understood in an intuitive
sense. In addition, we have here a powerful theoretical tool as we shall see
immediately in the following section on the general Stokes' theorem.
As an exercise, one could check that each of the standard tricks used to
evaluate surface integrals, etc., fits into the above scheme of things. It
hardly seems worth our time here.
5.8. Stokes' Theorem
The general result we establish now includes all known formulas which
transform an integral into one over a one-higher dimension spread.
Let co be a p-form on a manifold M and c a (p + \)-chain. Then
co = dco.
Jdc Jc
Since c is a sum of (p + l)-simplices with constant coefficients, it suffices
to prove
co = \ dco
J dtr J a
where a is a (p + l)-simplex. According to a representation
(s*+1,U,0)
of (j we have from the definition
|efco = 4>*(d(o)=\ d((j)*cj).
Jo Js* + i Js* + i
This reduces the problem to a Euclidean one. Let rj be a ^p-form on a

5.8. STOKES' THEOREM 65
neighborhood U of sp+1 in Ep+1. To prove
1=\ dV-
Now
rj = ^AiWdx1 ■••cfa'-1cfel+1 '-dxp+l
so that it suffices to check the formula in case rj is a monomial only. Since
we may permute coordinates provided we are careful about signs, it suffices
to take the case
r\ = A dx1 - - - dxp.
Then
dA
dr1 = (-iy——dx1 •••efop+1.
dxp+1
We remember that sp+1 consists of all points (a;1, • • • , xp + i) satisfying
p+1
xi ^ 0, £ a1 ^ 1.
i
We have
(i-Zf*0
-(-1)' | <**' •••<**'( J ^T^P+1)
{*£^0, S^jc^I}
A(x\ ••• ^Ojlcfo1 --dxp.
We must next investigate dsp+l. We write
2*! = (10 • • • 0) |
points in Ep+1.
*P+1 = (0---01)J
We have
31"+i = (R,, ■ ■ ■, Rp+1) + (-1)"+1(R0, R, ,---,Rp)
+ other faces,

66
V. MANIFOLDS AND INTEGRATION
where rj = 0 on each of the other faces since some one of re1, • • • , xp is constant
there. Thus
\ 1= | rj + (-iy+1 f
1= I rj + (-iy^ I rj.
dsP+l (Rw -~-,RP+i) (Ro, Rw -,RP)
The face (R0i Rlt - — , Rp) is the standard sp. On it xp+ l =0 and so
(-l)p + 1 f/ = (-l)P+1 ^(a;1,*2, •••,a*,0)efa1 • • • dxp
(Ro, '-,RP) s*
which is precisely the second term in the expression for I drj above. The
first term is obtained by projecting downward in the xp+1 direction:
dxp
(Ri,---,Rp+i) (Ri, • • ->Rp, Ro)
I n = [ a(x1,--,z', i - £Adx1
(Ru - •, RP, Ro)
= (-l)p a(x\ •-,zp, 1 - fvjcfo1 --dxp
(Ro,Rw • -,RP)
= (-1)p\a(x\ ••• ,xp, 1 - £a/W •••efa*,
■1<
and this is the first term in the expression for | drj. The proof is completed.
5.9. Periods and De Rham's Theorems
We consider an example. The manifold M consists of E3 with the origin
removed,
M = E3 - {0}.

5.9. PERIODS AND DE RHAM'S THEOREMS
67
Suppose co is a one-form on M such that dco = 0. Then is co exact? That
is, is it the differential of a function on M ? The proof in Section 3.6 will
not avail here because M cannot be shrunk to a point. Nonetheless, co = df,
where
/(x) = co,
J(i,0,0)
the integral taken along any path c which avoids 0. That this is independent
of the path follows from Stokes' theorem. For if c' is another path in M
from (1, 0, 0) to x, then the chain c — c' is the boundary of a piece of surface
£ (2-chain) in M and
co— co — \ co — \ co = \ dco = 0.
Jc Jc' Jc-c' JdZ Je
Next suppose a is a two-form on M such that da = 0. We seek a one-
form A on M such that a = dX. By the converse to Poincare's lemma in
Section 3.6, such a form X exists locally. But we are asking the global
question: Is there such a form X on all of M ? The answer to this one is no
in general, we shall have explicit examples later. For if there were such a
one-form X with dX — a we would have
fa=|eU=| A = 0
J S2 J S2 J dS2
since the unit sphere S2 has no boundary. But there is no reason a priori
for assuming that
I
a = 0.
s2
The correct result is this. If a is a two-form on M = E3 - {0} with da = 0
and
I
a = 0,
s2
then a = dX for some one-form X on M.
This result is contained in De Rham's theorems which we shall formulate
now without proofs.
We deal with a fixed manifold M about which we assume only some mild
limitation on its size, for example we may suppose it can be imbedded in a
sufficiently high dimensional Euclidean space.
A closed form is a differential form co on M satisfying dco = 0.
An exact form is a differential form co on M satisfying oo = drj for some form
f/onM.
Each exact form is closed:
doo = d(dr]) = 0.

68
V. MANIFOLDS AND INTEGRATION
Let co be a closed ^-form. To each p-cycle z on M corresponds a period^
of co,
00.
If z happens to be a boundary b = dc, the period vanishes,
co = co == dco = 0 = 0.
Jb J dc Jc Jc
Because of this there is a relation between periods:
(Whenever cycles zt, • • • are related by\
YJaizi=z boundary,
then
E°i[ *> = o.
De Rham's First Theorem. A closed form is exact if and only if all of
its periods vanish.
De Rham's Second Theorem. Suppose to each p-cycle z is assigned a
number, per(z), subject to the consistency relations
(whenever
£ a {Li = boundary,
then I
2>fper(zf) = oJ
Then there is a closed form co on M which has the assigned periods,
co = per(z) for each ^p-cycle z.
On many spaces one is able to apply these results because there is a finite
set of independent ^-cycles which spans all p-cycles, up to boundaries. For
example, on the ^-sphere S" it is known that each p-cyc\e is a boundary for
p > 0, p t*£ n, and that in dimension n there is a single n-cycle (Sw itself with
outward normal for orientation) such that each w-cycle is a multiple of this
one plus a boundary. These things are established by algebraic topology.
A complete analysis of De Rham's theorems reveals the following result,
which has considerable attraction in itself.
Suppose we consider only chains c = ]T atat which are sums of simplices
t The nomenclature derives from the periods of elliptic integrals and the corresponding
differentials for algebraic functions.

5.10. SURFACES; SOME EXAMPLES
69
with integer coefficients. Then we may talk of these as integer-chains and
have integer-cycles and integer-boundaries. The integer-periods
J.-
of a closed form a> are the periods taken over integer-cycles only.
Let a> and rj be closed forms of degrees p and q respectively. Suppose that
the integer-periods of a> and rj are all integers. Then the same is true of a> a rj.
5.10. Surfaces; Some Examples
It is shown in topology that each closed surface in E3 may be smoothly
deformed into a sphere with h handles, or alternatively, a button with h
holes. Let us consider the case h = 2 and orient this surface £ with the
outward drawn normal. The only significant two-cycle is £ itself. By De Rham's
First Theorem, a two-form a on this surface is an exact differential if and
only if
There are four significant one-cycles, cl5 c\, c2, c2 . Here ct and c^ intersect
once and cross, the same for c2 and c'2. But ct and c2 intersect once
without crossing. To see the geometric plausibility of the statement that
each one-cycle c on L is a sum of multiples of the ci and c't plus a boundary,
one cuts the surface L along these basic cycles. Having done this, L may be
smoothly deformed into a plane domain without holes.
De Rham's First Theorem now asserts that if co is a closed one-form on L,
then a> is an exact differential if and only if

70 V. MANIFOLDS AND INTEGRATION
CO = CO = CO = 0) = 0.
Jci Jc'! Jc2 Jc'2
Applied to dimension one, De Rham's Second Theorem asserts that if real
numbers ax, a\ ,a2,a2 are given, there exists a closed one-form co satisfying
\ co = at, co = a\, oo=a2, co=a'2.
Jet Jc'! Jc2 JC'2
It is also interesting to consider non-orientable closed surfaces. These of
course cannot be realized in E3. Perhaps the simplest is the projective
plane P2. This is denned by pasting the edges of a rectangle together in the
order indicated. The boundary relations are
S(P2) = 2c/-2c,
dc = (P) - (Q)
3c' = (P) - (Q).
This means first of all that there is no effective two-cycle, each two-form is
exact. The only effective one-cycle is c' — c, and this actually bounds,

±3P2
5.11. MAPPINGS OF CHAINS
Thus each closed one-form is exact.
71
O1
i
Q
Another interesting example is the Klein bottle K2, again defined by
pasting edges together. The boundary relations are
dK2= -2c
dc = dd = 0.
G>
P c P
The one independent one-cycle is c'.
5.11. Mappings of Chains
Suppose M and N are manifolds and / is a smooth mapping:
/: M—*N.
Then to each p-chain c on M there corresponds in a natural way a p-chain
/*con N.
It suffices to explain this when c is a simplex op. Such a simplex is
represented by (sp, U, </>) where U is a neighborhood of the Euclidean
simplex sp and <j>: U —► M. We merely compose / and </> so that /* c is
represented by
(S>, U,/o</>).

72 V. MANIFOLDS AND INTEGRATION
We illustrate the process for the case of a two-simplex
This induced map /* takes the space of chains onto the space of chains:
Cp(M)-fCp(N).
We observe that if c is a p-chain in M, then
f*{dc) = d(f*c)9
which leads to the commutative diagram
C,(M) ^ *-Cp(N)
3
Cp-xCM)
t
/*
■*-C._1(N)
which is certainly analogous to the corresponding diagram in Section 5.4
for /* and d. The validity of the result is established by looking at
individual simplices.
We now see what happens with two mappings. Let
m; ^n
Then the assertion is
(9 ° /)* — 0* ° /*

5.12. PROBLEMS
73
which again follows for a simplex almost directly from the definition of /* .
Finally we consider this situation. Let
/: M—>N.
Suppose that oo is a p-form on N and c is a ^-chain on M. Then/*oo is a
p-form on M and/* c is a p-chain on N. We have
f*CO = 00.
Jc J/«,c
This important result also follows directly from the definition for a simplex
and is obtained for a general chain by summation.
5.12. Problems
1. Show that the totality of unit tangent vectors to the sphere S2 is a
three-manifold. Construct local coordinates.
2. Show that the set of all directed lines in E2 is a 2-manifold. Discuss
orientation.
3. More generally, consider the set of all oriented r-dimensional planes in
E". Show that this is a manifold, compute its dimension, and discuss
orientation.
4. Projective w-space P" consists of all (n + 1)-tuples (a0 , • • •, an) of
real numbers not all zero, where proportional (n + l)-tuples are considered as
representing the same point. Show that Pn is an w-manifold.
5. Complex projective w-space CPW consists of all (n + l)-tuples
(a0, • • • , an) of complex numbers not all zero, where two such n-tuples are
considered the same if they differ by a (complex) proportionality factor.
Show that C Pn is a manifold and determine its dimension.
6. Show that the manifolds of Examples 4 and 5 are closed (compact).
7. Let M be the manifold of Example 3. Show that the set N of all
oriented r-planes in E" which pass through the origin is a closed submanifold
of M.
8. Denote by U the open region
U = {x\ + • • • + x\ > 1}
in E". Suppose co is an r-form in Ew which vanishes identically on U.
Under what conditions does there exist an (r — l)-form a on E" which also
vanishes identically on U and which satisfies dot = a>2.
9. Show by direct calculation (i.e., without De Rham's theorems) that
if co is a two-form on S2 whose integral over S2 vanishes, then co = da for a
suitable one-form a on S2.

VI
Applications in Euclidean Space
6.1. Volumes in E"
We denote by
co — dxt • • • dxn
the element of volume in Ew, an w-form, and set
7,= [ <0,
1*1
so that Vn is the volume of the unit ball. Next we denote by & the element
of (n — 1)-dimensional volume on the unit sphere S""1 = {x | r = l}, and set
4.-1= &'
JSn-l
Thus At=2n, A2 = 4tn, Vl =2, V2 = n, V3 = §7T. It is clear that the
volume of the sphere of radius r is rn~1An_l, hence
n 1
t»-lAn_1dr = -AH_i.
0 W-
One may evaluate F„ by integrating over slabs:
= v„.lJn
where
J.=
(l-x2)'"-^l2dx.
Integration by parts once leads to
</„=[' x(2x)^iy\ - x2y~^2 dx=(n- l)(-Jn+Ja_2),
J -n~1J
74

6.1. VOLUMES IN E*
75
These recursion formulae lead to the standard result
nn/2
•€♦•)
Next we obtain an explicit formula for o' in terms of the Euclidean
coordinates x1, • • • , xn. We begin with the form
rdr = £ x{dx{,
a one-form in E" which is invariant under rotations (orthogonal
transformations) of E". Consequently
/\
*rdr = Y<( — 1)' lxidxl • • • dx{ • • • dxn
(the "hat" denotes a missing factor) is an (n — l)-form in E" which is
invariant under rotations. It follows that on SM_1,
a' — c * r dr,
where c is a constant.
Next we note that
/\
d(* rdr) = £ (— 1)* l dxidxl • • • dxt • • • dxn
nco,
hence
An_t = a' = c
JS""i
= c no =
* rdr
s»-i
ft W~ 1
-J
d(* rdr)
c = 1, a' — * r dr on S"

76 VI. APPLICATIONS IN EUCLIDEAN SPACE
Summarizing, we set
a — * rdr = £ ( — 1)' lxidxi • • • efof • • • dxn,
defining an (n — l)-form a in Ew. Then c?<r = na>, and if or is restricted to
S""1, the result is the (n — 1)-dimensional volume form a' of S""1.
Next we consider the natural projection
7r: E"-{0}—► S""1
defined by n(x) = x/|x|.
We seek 7r* &\ an (n — l)-form on E" — {0} satisfying
d(n* a') = 0 since d(n* a') = tc* (dor') = tc* (0) = 0.
(da' is an w-form on S""1, hence 0.) We shall prove
t*t'
a
We could prove this by directly substituting
Vi = *i/r ^ or' = £ (-1)'" V^2/i --dyr- dyn,
but we prefer to proceed indirectly by exploiting the symmetries present.
We set
a
Then
7i , na> nr2a>
Now we observe that *(7r* or') is a one-form in E" — {0} which is invariant
under rotations, hence dependent on r alone. We may write
From this we have
f(r)
*(7r*<r') = — (rdr).
f(r)
df df
dr dr
a constant, n* a' = ex. To evaluate c, we simply note that on S""1, both
7T* g' and t collapse to <r, hence c = 1,
7r*<r' = t.

6.2. WINDING NUMBERS, DEGREE OF A MAPPING 77
6.2. Winding Numbers, Degree of a Mapping
A basic result of topology (Seifert und Threlfall [20], p. 283) asserts that
if M and N are closed oriented w-manifolds and /: M —► N, then the
chain /* M is an integral multiple of N plus a boundary. This integer
multiplier is called the degree of/ and written deg/.
Now suppose that L is a closed oriented (n — 1)-manifold in E" — {0}.
Then by the Jordan-Brouwer theorem of topology, £ decomposes E" into
exactly two regions. We assume £ is oriented by the outward normal. The
projection mapping n of Section 6.1 sends £ into S""1. It is true that
deg n = 0 or 1; our point is that this can be determined by an integral. Let
5 = deg 7C.
Then
hence
t = 7c* & —\ a' — S\ a' — SAn_ t,
J 2 J 2 Jn(2) JS""1
T.
/2
More generally, let M" 1 be a closed oriented manifold,
/: M""1—>Ew-{0}.
Essentially we are thinking of /(M""1) as a hypersurface in E" — {0} which
may intersect itself. We look on this hypersurface as winding around the
origin and we want to count how many times it encircles. This winding
number is given by the Kronecker integral
We may justify this as follows. Set g — n of: M" x—>Sn 1. What we
are after is deggr. Now
hence
g+ (M) = (deg^S""1 + (boundary),
\ g* a' =\ a'
Jm J<7*m
= (degg) (T/ = ^M_1deg9r,
de%9 = -A 9*<r'-
An-lJM

78
VI. APPLICATIONS IN EUCLIDEAN SPACE
M"
**- E" - {0}
g = n°f
n-l
l$utg*cr' = (/* o tz*)(t' =/*t, so we have
deg? = - /*t
^«-i Jm
The most general situation is this:
/: Mw—>N\
Let ft be the volume form on N taken so that
deg/= f*P-
For/* M = (deg/)N + (boundary), hence
Jm
U-
1. Then
f /*/*=f i» = (deg/)f jS = deg/.
J M J /• M J N
One interesting example: let "P be the n-torus, /: Sn —► Tn where n ^ 2.
Then deg / = 0.
Because the integrals involved are integer-valued, they remain constant
when the mapping in question is subject to a deformation. Precisely, let
ft: M —> N be a one-parameter family of maps. Then
deg/,
Jm
P
is a smooth function of t, always an integer, hence constant. It follows that
deg/0 = deg/^
One other remark. Suppose we have
/: M-
Then
N, g: N—>P so that h = gQf: M
degft = (deg/)-(deg0).

6.4. LINKING NUMBERS, THE GAUSS INTEGRAL 79
6.3. The Hopf Invariant
For each sphere Sw, let an denote the element of area, normalized so that
Is.*--1'
Consider first a map /: S3 —> S2. Then /*<r2 is a 2-form on S3. Also
^(Z*^) ==/*(^<T2) = 0- Since S3 has no nontrivial 2-dimensional cycles,
we deduce that
/*<72 = efa1
where the one-form at on S3 is unique up to the differential of a function.
The 3-form ax a /*<72 has an integral
a! a/*<t2,
Js3
which has the remarkable property of being an integer, called the Hopf
invariant of/. It is invariant under deformation of/. More generally, let
f: s2""1—>S\
Then /* an = dan_ x, and the Hopf invariant of/ is
<*„_! a/*<7„.
JS2n-l
We may represent S3 by pairs of complex numbers
(Z,W), |2|2 + H2 = 1.
The mapping (z, w) —> z\w provides a mapping of S3 into the closed complex
plane, i.e., the Riemann sphere S2. This map has Hopf invariant +1, hence
it is essential in the sense that it cannot be deformed to a trivial map,
everything going to a single point.
6.4. Linking Numbers, The Gauss Integral, Ampere's Law
Let Mr, Ns be oriented closed manifolds in Ew, where r + s = n — 1, and
suppose these have no common point. (Best example: two disjoint closed
curves in E3.) We want to count how many times they link. To do this,
we form the product space M ^ N which is an oriented manifold of
dimension r + s = n — 1. We consider the map/: M ^ N —> E" — {0} denned
by
/(*. y) = y - x
Now we set
link(M, N) = deg/.

80 VI. APPLICATIONS IN EUCLIDEAN SPACE
Thus if t is the w-form in E" — {0} we considered above,
link(M, N)=-i-f /*T = -J_f f /*r.
ViJmxn ViJmjn
We shall work this out in E3 for a pair of closed curves M, N:
1
1 (z x dz)-dz
|Z|3^ . J k |Z|3 2
We let x, y be the moving points on M, N, respectively. Then
*=/(x,y) = y-x
so that
/*T =
1
2|y-x|3
1
2|y-x|
l
[(y — x) x (dy — dx)] • (dy — dx)
3 {[-(X - x) x dy]-dx - [(y - x) x dx]-dy}
[y-ocp[(y"X)XcZy]^X'
link(M, N)=—I dx-| ^—-
47T JxeM JyeN ly-
— x) x c?y
(In this computation dy x dy = 0, etc., since dy involves only one variable.)
Imagine a steady unit electric current flowing around the closed loop N.
By Ampere's law, the magnetic field at a point x due to the current in a
segment dy is
1 (y — x) x dy
~4^ |y-x|3 '
hence the total magnetic field at x is
(y - x) x dy

6.4. LINKING NUMBERS, THE GAUSS INTEGRAL 81
It follows that link (M, N) = F(x) -dx is precisely the work done by this
Jm
field on a unit magnetic pole which makes one circuit of M.
In the next example, link (M, N) = 0, which seems surprising since the
curves cannot really be separated.

Applications to
Differential Equations
7.1. Potential Theory
We summarize the notation of Section 6.1. Space: E". Coordinates:
#1 > x2 > ''' * xn -
r2 = x\ + • • • + xl,
n
rdr = Yjxi d%i >
l
n n /\
a = *(rdr) = £#f*cfo(- = £ ( —l)1 lxtdxx • • • dxt • • • dxn.
l l
a
x = ?'
o) = dxt • • • dxn = volume element of E".
da = nco, dx = 0.
Let w be a smooth function on a domain in E\ Then
du = Y —-. eta\
*du = Y ( —l)1"1 --^cZa?1 ---d^'-dx",
d*du = 1^]—-tjlo) = (Aw)a>,
defining the Laplacian
(See Section 4.4 for details when w = 3.)
82

7.1. POTENTIAL THEORY 83
If u and v are functions on the finite domain R, the Dirichlet (bilinear)
integral is
D[u, v] = du a *dv = \ dv a *e£w = V [—:|(—Aco.
Jr Jr JrlWJWJ
Next we have by Stokes' theorem
u*dv = I d(u*dv).
JdR JR
But
d(u *dv) = du a *efa; + u d *dv
= du a *efo + w Av a>,
hence we have
Green's Formula
I u *dv = B[u, v] + I u Av a>.
JdR Jr
By reversing w and v and subtracting the results, we obtain
Green's Symmetrical Formula
(u *dv — v *du) = (uAv — v Au)a>.
JdR Jr
[One usually writes *du = (dujdv)X where X is the (n — 1)-dimensional volume
element on dR and dw/dv is the normal derivative.]
In case v is harmonic in the region R, Av — 0, and we have
Jr
(u *dv — v *du) -f | v Au o) = 0.
aR
By specializing further we have this result:
Let u and v be harmonic junctions in a region R. Then
I
u*dv =
dR
v*du.
dR
We derive further consequences by setting
1
v =
<&; = (rdr),
rn
*dv = — (n — 2)t,
d*cfa; = 0, Ar = 0.

84 VII. APPLICATIONS TO DIFFERENTIAL EQUATIONS
The function v is denned on En — {0}. We suppose the region R contains 0
and we apply the formula above to the punctured region
R - {r ^ e}
with e a small positive constant. We suppose u is harmonic on all of R.
Since
d[R - {r ^ c}] = dR - {r = e},
we have
u*dv — u*dv = v*du — v*du,
— (n — 2) wt + (ti — 2) ux = -^rj *^w "- -J—2 *du-
JdR Jr = e JdRr Jr=er
We evaluate the individual terms:
ux = — ua = — d(u<j) = — (du a (T + u-nco).
Now for |x| ^ e, w(x) = w(0) + 0(e), hence
L
u(o = u{0)Vn + 0(s)en.
Similarly
duAo=\ fcjjLAa, = 0(e)co = 0(s)sn,

7.1. POTENTIAL THEORY
85
hence
ux = nVnu{Q) + 0(e) = An_ ^(0) + 0(e).
J r = e
C i if if
I —^ *du = —^ I *du = —5 I d(*cfat)
Jr=£r"-2 £"-2Jr=£ £""2J^«
»* f (AM)o> = 0.
We substitute these results and let e —> 0 to get
u(0)
if if *du
which gives the value of a harmonic function at a point in terms of the
boundary values of it and its normal derivative.
Special case. Let R be the spherical region of radius a centered at 0,
R = {r <^ a}. In this case the second term on the right-hand side vanishes
for the same reason that the corresponding integral taken over {r = e}
vanishes. On dR = {r = a] we have
a 1
an an x^
where
1 /\
/* = J"x = " Z (-1)' ^l^i •••*»!••• **„
is the element of (n — 1)-dimensional volume on [r = a}. (For a = 1 this
reduces to a. Since there are n a;-terms in the numerator and a — |x| is in
the denominator, it is homogeneous of the right degree, n—\.) We have the
Gauss Mean Value Theorem
ujh
u(0)
^krl.r-f
v>
which tells us that the mean value of u over the sphere of radius a is the value
of u at the center.
Two important properties of harmonic functions follow from this result.
Maximum (Minimum) Principle. Let u be harmonic on the finite region R.
Then u never assumes its maximum {minimum) value at an interior point of
R unless u is constant.

86 VII. APPLICATIONS TO DIFFERENTIAL EQUATIONS
For suppose u assumes its maximum at an interior point #0 of R which, after
translation of coordinates, may be taken to be 0. Let £ be any (n — 1)-
sphere of radius a centered at 0 with a so small that [r ^ a] is in R. Since
u(x) ^ u(0) we have
«(0) = I W| li ^ I u(<W| Ji = u(0)
Je / Jl Jl / Jl
so we must have u(x) — u(0) for all x in S. Hence ?i is constant on the largest
spherical neighborhood of x0 we can draw in R. Evidently this means u is
constant in all of R since we can reach any other interior point by a sequence
of such overlapping spheres. The result for minima follows the same way.
Uniqueness Principle for the Boundary Value Problem. Let u and
v be harmonic on a finite domain R and coincide on dR. Then u — v on R.
For u — v vanishes on dR and is harmonic. By the Maximum Principle,
u — v gj 0 on R, u 5^ v. Similarly v ^ u, hence u — v.
The function (1/r""2) is ideally suited to the sphere. On other domains
it is inconvenient because of the term involving the normal derivative in the
expression above for u(0). Hence we introduce the Green's function.
Let R be a finite domain. A function v(x, y) defined for x and y distinct
points of R is called the Green's function of R if
(i) For each fixed y in R, v(x, y) is a harmonic function of x for x in
(ii) For each fixed y in R, v(x, y) = 0 for x in dR,
(iii) For each fixed y in R,
*(x, y) -
|x - y|"
is a smooth harmonic function on all of R.
Using the same method as above one proves the following:
If u is any harmonic function on a finite domain R and v = v(x, y) is the
Green's function for R, then
u(y)
-<^^L«x)*wx,y)-
In case R is the spherical domain r ^ a centered at 0, the Green's
function is
1 a-'2
1 yl |y|-2i
x lyl2

7.1. POTENTIAL THEORY
87
We note that
a*
y ^t^x
is the inverse of y with respect to the sphere R (reciprocal radii). For x on
dR, |x| = a and we have
a2 a*
|x-y12 = (x-y'),(x-y') = a2-2-IX7 + —j
\Y\2
\y\2
^l(|y|2--2x.y + |x|2) = ^|x-y|2,
lx-y'l = j7j|x"yl*
This explains why v vanishes for x on 3R.
Next
*e£
[«=-(w-2)[t(x-y)-^rl(x-y')]
where
t(x - y)
t(x - y') =
1
|x-y|
1
n Z (Xi - Vi) *dxi'
|x - y'|
TTnYt&i- Vi) *dxi'
We only need dv for x on dR. Recalling that y' = («2/|y|2)y for |x| = a we
have
*,-^M-S(-5F*)]-*
*dyV
-(n - 2) a2 - \y\2 -(n - 2) a2 - |y|2
<? = fix,
|x - y|-
|x - y|»
so that the representation of u in terms of its boundary values specializes to
u(y) =
a2 ~ l/l2
m(x)
ix|=Jx-y|J
This is the Poisson Integral Formula which provides an explicit solution
formula for the Dirichlet problem on the sphere—find a harmonic function
with prescribed boundary values.
Returning to a general finite domain, we mention the important symmetry
property of the Green's function:
v(x, y) = v(y9 x).

88 VII. APPLICATIONS TO DIFFERENTIAL EQUATIONS
(That this is the case for the sphere is not apparent from the unsymmetrical
formula above. But for the denominator of the second term we have
ly|2|x - y'|2 = |y|2(x - y')-(x - y') = |y|2(|x|2 - 2 ^ (x-y) + ^
= |x|2|y|2-2a2(x-y) + a4,
which turns out to be symmetrical after all.)
One of the many important consequences of the Poisson Integral Formula
is the
Liouville Theorem. Let ubea harmonic function on all of E" and u ^ 0.
Then u is constant.
We shall show that for each y in E", u(y) = u(0). We fix y with |y| = b
and select any a> b. Then
Now
hence
__a2 -b2 r u(x)
*(y)~l^Jlxl=ajx^^
|x-y|£|x| + |y|=a + 6,
|x - y| ^ |x| _ |y| = a - 6,
1 1 1
< <
(a + b)n ~ |x - y|" ~ (a - b)n
Since u(x) ^ 0 we may use these inequalities to estimate the integral:
(a2-b2) f (a2-b2) f
— — u(x) Uy, < u(y) < —- — u(x) ay.
aAn_x(a + bf)w=a ^x" KJ) -aAn.^a-bf)^ K ^x
But from the Mean Value Theorem,
J,
so we have
u(x)fi7C = (an-1An_l)u(0)i
|x|=«
(a2 -b2)an~2 (a2-b2)a"-2
— u(0) ^ u(y) ^ K— L— u{0).
[a + b)n v '~ v,/ - (a-b)n
By letting a —► oo we have
u(0) ^ u(y) ^ u(0),
and so for each y, u(y) — u(0), u is constant.

7.1. POTENTIAL THEORY 89
Remark 1. We return to the symmetrical Green's formula
(u *dv — v *du) = I (u Av — v Au) co.
JaR Jr
We apply this in this situation:
R = {e ^ r ^ a},
v harmonic in R for all e > 0,
v vanishes on r = a,
u smooth in \r fLa}.
We do not require that u be harmonic. The formula reduces to
I u *dv — I u *dv + I v *du + I v(Au) co = 0.
Jr=a Jr=e Jr=e Je^r^a
First case.
1 1
*dv = — (n — 2)t.
By the methods above,
f -(w-2) f
w*efo = —L- i w= -(7i- 2)An_lu(0) + 0(e),
Jr=e £ Jr=£
V*du = (^2 - ^l) d(*dw) = °(fi2)-
Substituting these in and letting e —> 0 we obtain
«" '4,-1 J r-a (*~ 2)^-1 J r^aV" * «" V
This gives information about a solution u of the Poisson equation Au = /
with boundary values of u assigned.
Second case.
r" a" '
*efa; = (—- 1 *dx: — n —^ a.
\rn anJ
<-i-"-^T2
One differentiates this to prove Av = 0. This also follows when one notes
that

90 VII. APPLICATIONS TO DIFFERENTIAL EQUATIONS
The end result in this case is
o= ^krlj*M" - irlA* ~ $(Au)l°-
du\
dx\{
Analogous formulas for higher derivatives are possible.
Remark 2. We have avoided n = 2. In this case the basic difference is
that the symmetrical harmonic function with singularity at 0 is In r rather
than r~(n~2\ Using this, results similar to those above follow.
7.2. The Heat Equation
We consider the parabolic equation
d2u d2u du
Suppose u is a solution, valid in a region of x, y, t space which includes a
region R and its boundary.
First we consider
a = (uxdy — uydx)dt — udxdy.
Then
dot — (uxx + uyy)dxdydt — utdtdxdy = 0,
hence
f~f
JdR JR
Next we consider
/? = 2u(uxdy — uydx)dt — u2dxdy.
Then
dp = 2(uxdx + uydy)(uxdy — uydx)dt + 2u(uxx + uyy)dxdydt — 2uutdtdxdy
= 2(u2x + u2)dxdydt.
It follows that
[2u(uxdy — uydx)dt — u2dxdy] — 2
JaR Jr
[2u(uxdy — uydx)dt — u2dxdy] — 2\ (u2 + u2)dxdydt.
Jr
Suppose R is taken in the special form of a cylinder T X [0> &] where T is a
region in the x, y-plane. We then have
<5R = (3T)X[0,&] + TX W-TX{0}.
We now assert the basic uniqueness theorem:
Ifu vanishes on the base T ^ {0} and on the lateral surface dT Y [0, b], then u
vanishes identically in R. For the integral formula above reduces to
2 (ul+ u2)dxdydt + u(x, y, b)2 dxdy = 0.

7.2. THE HEAT EQUATION
91
Since everything is positive, this implies
ux = uy = 0 in R,
tt = 0 on T X {&}>
which is more than enough to imply u — 0 in R. Because the heat equation
is linear, we deduce that two temperature distributions which coincide
initially at t = 0 and always coincide on the boundary of R must be identical
for all t at each point of R.
t
-*-y
We shall now do the same thing in n dimensions, where there is an
interesting sign change. Our variables are x1, • • • , xn, t and the heat equation is
Au — dujdt
where as usual Au = £ d2u/dxf. The operator * will apply to space
variables only.
This time we set
0 = 2u(*du)dt + (- lyVa),
where co = dxx • • • dxn . Now
du a (*du) = (gradw)2a> = Y — :
co
and we have
d£ = 2(gmdu)2codt + 2u(Au)codt + 2(-l)n~~ luutdto)
— 2(gmdu)2codt,
hence
JdR JI
(gradft)2a>e££.

92 VII. APPLICATIONS TO DIFFERENTIAL EQUATIONS
Now let T be a region in E", R = T X P>> bl Then
dR = dT X [0, 6] + (- 1)T X 3[0, 6]
= aTX [o,6] + (-i)wtx ft + t-ir'TX °-
Suppose w vanishes on dT X [0> &] an(* on T X ^* Since ^ = 0 on T Y 6
(i.e., £ = b = constant) we have
u*dudt = 0,
JaR
consequently
(_l)«-i u2o) = 2 (gradw)2a>efc,
JdR JR
that is
w2o> + 2 (grad u)2codt = 0
J TX* J R
and we conclude as before (gradw)2 = £ (du/dx^2 = 0 on R, du/dxi = 0 on
R, u is constant on R, u = 0.
7.3. The Frobenius Integration Theorem^
Everything is local in this section; we operate in a neighborhood of 0 in
E". Let co be a one-form which does not vanish at 0. We ask, under what
conditions are there functions / and g satisfying co = fdg ? In other words,
we seek an integrating factor for the differential equation co = 0. If co = fdg,
then / does not vanish in a neighborhood of 0, hence
dco — df a dg — df a /" 1a>,
d(o = 0 ag> (0=/_1d/=dln|/|)
and so
a)Ada) = a)A0Aco = O.
For a one-form co = Pdx + Qdy + i?dz in E3, this is the condition
P(Ry - Qz) + ©(Pz - Rx) + 5(0, - Py) = 0.
We note that if a> = /dgr, then the equations co — 0 and dgr = 0 are the same
and hence the solutions or integral surfaces of co — 0 are the hypersurfaces
g = constant.
Before passing on to precise statements we give two instructive examples.
Example 1. Let co — yzdx + xzdy + dz so that dco — ydzdx + xdzdy. It
follows that
tfco = l — I a co
t Material in this and the next section is taken from a University of California
Technical Report of June 1957, Seminar on exterior differential forms. This was prepared
for the U.S. Army Office of Ordnance Research Contract DA-04-200-ORD-456.

7.3. THE FROBENIUS INTEGRATION THEOREM 93
which is not so useful since dz\z is singular along the z-axis. A better choice
is 9 = — ydx — xdy and we have do = 6 a co. To determine the function g,
we use the fact that each integral surface g — constant will be cut by the
plane [x — at, y — bt} in a curve which intersects the z-axis in the solution
z of g(0, 0, z) — constant. The equation co = 0 on the plane x — at, y — bt
becomes
dz + 2abztdt = 0
with solution
z = cexj)(—abt2)
satisfying the initial condition z(0) = c. However abt2 = xy so these curves
span out a surface
z = ce~xy.
We now think of a, b, c as variables and make the transformation
(x —a
d(x, y, z) .
y = b with yy' } = e~ab # 0.
* d(a, b, c)
z = ce~ab
We have
dz = e~abdc — z(adb + bda),
which yields
co = e-abdc,
or in the original variables
<o = e~~xyd(ze*y)
and the integral surfaces are
zexy — constant.
It will be observed that we have arranged the function g so that g — c
intersects the z-axis precisely in z = c.
Example 2. This time we try the procedure on co = dz — ydx — dy. On
the plane x — at, y = bt, the equation co — 0 becomes dz — (abt + b) dt,
z = i abt2 + bt + c and we arrive at the surface
z = \xy + y + c.
But on the parabolic cylinders x = at, y = bt2 we have
dz = (abt2 + 2bt) dt, z = ±abt3 + bt2 + c,
a different family of surfaces. The reason for this failure to obtain integral
surfaces is seen from
doo = —dydx, oo A dco = —dzdydx ^ 0.

94 VII. APPLICATIONS TO DIFFERENTIAL EQUATIONS
Theorem. Let co = X/^1 be a one-form which does not vanish at 0.
Suppose there is a one-form 9 satisfying dco = 9 A to. Then there are functions
f and g in a sufficiently small neighborhood of 0 which satisfy co =fdg.
Note 1. Since co is given while 6 is certainly not uniquely determined, it
simplifies the proof if we avoid explicit use of 9 as long as possible.
Note 2. The condition on co is unchanged when we replace co by a multiple
of co. In fact, if h ^ 0, then
d(hco) = dh a co + hdco — dh a co + h9 A co — (dh + h9) A co,
d(hoo) = [{dh)h~l + 9](hco).
Proof. Since co #= 0 at 0, we may assume some one of the functions ft does
not vanish at 0. Since neither the hypothesis nor conclusion changes when
we multiply co by a nonvanishing factor, we may assume
n
oo = dz — £ Atdx1, A{ — ^4,(x, z).
a = (a1, ...,a")
We fix any point a in x-space and consider the equation co = 0 on the
hyperplane xl = alt, (i = 1, • • • , n):
We solve this equation with the initial condition z(0) = c. More precisely,
we seek a function F(t, a, c) satisfying
Ft(t,*,c) = X-4MJP(*,a,c)]o'
JF(0, a, c) = c.

7.3. THE FROBENIUS INTEGRATION THEOREM
95
The usual existence theorem of ordinary differential equations yields a unique
solution. We see that a change of scale is possible:
F{t, a, c)
4''H
since the function on the right is again a solution to the same problem. In
particular, setting k = 1/t,
P(*, a, c) = P(Ua, c).
We introduce the change of variables
(x = u
^z = F(l, u,v)
with
since
-P(l,u,<;)
dv
5(x, z)
d(u, v)
0
/
*
= ^-JP(l,0,t;)
o dv
dv
dv
V
= 0
: 1.
0
1
v=0
= 1
dv
F(0, a, t;)
v = 0
Thus the new variables u, v form a local coordinate system in a sufficiently
small neighborhood of 0. We suppose that in these coordinates we have
G> = Z Pidut + Bdv> Pi = Pi(U' V)> B = 5(U' v)'
Since a> vanishes identically on u = a£, v = constant, we have the relation
£P;(aMK = 0.
To continue, we consider the mapping </> on (t, a, v)-space to (u, v)-space
given by
(j)(t, a, v) = (to, v) = (u, v).
We have
</)*a> = £ p.(«a, v)(a'eft + *da£) + £(to, v)dv
= £ «P£(«a, v)**1 + J5(to, v)efo,
0* o> = £ P&, a, v) da1 + jB(«, a, v) dv,
where Pf(£, a, v) = tPt(ta, v) so that Pf(0, a, v) =0. The important point is
that $* a> is free of <ft.
The equation dco = 9 A co implies d((f)* co) = (0*0) A (</>*a>). We may
set
cf)*6 = H(t, a, a) eft + other terms.

96 VII. APPLICATIONS TO DIFFERENTIAL EQUATIONS
Then
d((f)*a>) = ]T -T-■ dtda1 + other terms.
We compare the dtda1 terms in this on the one hand and (</>*#) a (c/>*co) on
the other to obtain
dPl -
— =HP..
dt
But this combined with Pf(0, a, v) =0 implies, by the uniqueness theorem
for ordinary equations, that P{ — 0. Hence P{ = 0,
a> = JSefa;,
the desired result.
References. For this theorem and the generalization which follows, see
fi. Cartan [9, p. 46], [7, p. 367].
Example, co = xdy — ydx. Certainly co a dco = 0 since co a dco is a three-
form. However, the form co vanishes at 0 so one does not expect that the
integral curves of co = 0 will span out evenly a neighborhood of 0; in fact
these curves are just the lines ax -f by — 0 through 0. We note, however,
that dco = 9 a co is impossible in any neighborhood of 0. For dco = 2dxdy
so that if 9 = Adx + Bdy, then 2 = Ax + By which fails at x = y = 0.
Remark. From the theorem we easily deduce again that a one-form co
satisfying dco = 0 is exact. For consider 9 = dz — co where co is a form in
x-space. Then d9 = 0, hence there is a one-parameter family z = F(x, c)
of integral surfaces, jP(0, c) = c. For each choice of c, 9 vanishes on
z — F(x, c), i.e., co = dF. (We cannot proceed without passing to one
more dimension since co may vanish at 0.) This trick of introducing a new
independent variable for an unknown function is a useful one.
We now pass to the general problem. Let co1, • • • , cor be one-forms in
r + s space, linearly independent at 0. Set Q = co1 A • • • a cor. The
system is called completely integrable if it satisfies any of the conditions of the
following lemma.
Lemma. The following conditions are equivalent:
(i) There exist one-forms 9lj satisfying
n
dco1 = £ 9lj a coj (t = l, ••-,*■) (n=r+s)
(ii) dco1'a Q = 0 (; = l,---,r)
(iii) There exists a one-form X satisfying
dQ = X a Q.

7.3. THE FROBENIUS INTEGRATION THEOREM 97
Proof. That (i) implies (ii) is obvious (but unnecessary). Also (i) implies
(iii) with X = ]T dlt. Next, (iii) implies (ii) is the case since (iii) means
Et-l)'"1**)' AO)1 A ••• ACD4'"1 ACD*+1 A '"Of
— X A CD1 A • ' • A of
and we merely multiply by cd1 to deduce (ii).
It remains to prove that (ii) implies (i). Let of+ *, • • • , of be one-forms so
that cd1, • • • , of form a basis of all one-forms. We write
fa*1 = Z/^^ A <°k-
j<k
Since doo1 a Q = 0, we have
Z f'jk0*1 A • • • A Of A CDj A CD* = 0,
r<j<fc
hence / ljfc = 0 for r < j < Jc,
j=l *=j+l
Frobenius Integration Theorem. Let oo1, • • • , of be one-forms in E",
n = r + s, linearly independent at 0. Suppose there are one-forms 6lj satisfying
doo1 = £ 0'; a cdj* (i = 1, • • • , r).
7=1
jT&ett ^ere are functions fl ^ gj satisfying
j=i
Discussion. The hypothesis is certainly a necessary one. For if we
write
the conclusion is <o = dgF. The matrix F must be nonsingular in a
neighborhood of 0 and so
do = -dg a dF = -cd a F'1 dF = cd a 0
where
0= -F~ldF.
Next we note the hypothesis is invariant under a linear transformation of
the o)1. In fact, if r\ = cd^4 where i is an r x r matrix of functions, non-
singular near 0, then
a,i/ = a,cD^4 + CDAeL4 = cDA0^4 + cDAeL4
= 1/ a (^-10^ + ^"1^).

98
VII. APPLICATIONS TO DIFFERENTIAL EQUATIONS
We shall give two proofs of the theorem, each from a somewhat different
point of view. The starting point is always the same. We write
©* = £ h'jdz* (i = l, -"9r).
j=i
Since the col are linearly independent at 0, some r x r minor of \\hlj\\ is non-
singular in a neighborhood of 0. We multiply (co1, • • • , of) by the inverse
of this minor. On changing our notation slightly then we have
©' = dz1 - £ Alj{x\ • • • , x\ z1, • • •, zr)dxj (i = 1, 2, • • • , r).
j=i
First proof. For each point a = (a1, • • • , as) in x-space we consider the
system of equations col = 0 along the linear variety x = £a:
with initial conditions zl(0) = cl. By ordinary differential equations, there
is a unique solution in a sufficiently small neighborhood of 0, i.e., there exist
functions Fl(t, a, c) satisfying
dFl * .
— (t, a, c) = £ ^Pa, F(t, a, c)]a'
ot j = i
J"(0, a, c) = c1 (t = 1, • • •, r).
We shall write F = (F1, • • • , Fr).
Next, we fix & and set G(t, a, c) = F(ktt a, c). Then G(0, a, c) = c and
— (t, a, c) = & — (fc, a, c) = £ ii'/tfa, G) to>,
hence by uniqueness, G(£, a, c) = F(t, &a, c), i.e.,
F(Jfc, a, c) = F(f, la, c).
In particular, setting t = 1 and then replacing k by £,
F(«, a, c) =■ F(l, ta, c).
We pass to new variables u, v according to the transformation
x = u
z= F(l, u, v).
This is nonsingular in some neighborhood of 0 since
3(x, z)
3(u, v)
0
7 0
dx
* dv
0
dz
5v
1.

7.3. THE FROBENIUS INTEGRATION THEOREM
99
For
—
= -: P<(1, 0, V)
o SvJ
o = ^^a;v)
_ dv'
= 5).
In these new variables we may write
col = £ JB^u, v)dvk + £ P'y(u, v)duj.
fc=i i=i
The fact that each col vanishes identically along the curve u = £a, v =
constant implies
£ P'y(ta,v)a' = 0 (<=l,---,r).
We propose to show that the functions P'y(u, v) vanish identically. To do
this we consider the cone mapping </> on (t, a, v)-space to (u, v)-space
defined by
</>(*, a, v) = (*a,v)=(u,v).
We have
<£*a>* = £ P'y(£a, v)tdaJ + terms in efo*
= ^] Pl;(£, a, v)efaJ' + terms in dvk
where P'y(«, a, v) = P'/*a, v)« so that P';(0, a, v) = 0. It follows that
SJF£.
d 6*col = Y d£daJ + other terms.
r ^ dt
Finally we use the hypotheses dco1 = ]T 0'fc a a>*. We write
0* e\ = H\(t, a, v) dt + other terms
and compare the coefficients of dt daJ in the relation
d^a)1 = £(</>* 0'fc) A (</>*">*):
dPl _
-^i («, a, v) = X #'t(<, a, v) P*,(<, a, v).
We conclude from the uniqueness of solutions of ordinary systems together
with the initial conditions P';(0, a, v) = 0 that P1,. = 0, Plj = 0,
©' = £ £'fc(u, v)efr;fc
as required.
Our next proof is based on the sketch in fi. Cartan [10, pp. 188, 193],
We begin as before with the system
©' = dz{ - £ A'jdx* (i = 1, • • • , r)

100 VII. APPLICATIONS TO DIFFERENTIAL EQUATIONS
with the conditions dco1 = ]T 6lj a coj. We take any smooth curve from
the origin to a point a. We solve the system col = 0 on the cylinder this
curve spans in x, z-space, taking some definite initial point c on the z-axis.
We shall show that the point on this curve lying over x = a is independent
of the particular curve we start with in x-space. The point is that in a
sufficiently small neighborhood of 0 in x-space, any two smooth curves with the
same end points 0, a can be smoothly deformed, one to the other. Thus let
^
4-U-=>a
x = x(e, 0)
x = x(t, 1)
x = x(t, a) be a one-parameter family of curves from 0 to a, the time variable
t on each curve running from 0 to 1 and the parameter a taking all real values;
we are interested in the curves x(£, 0) and x(£, 1). We are assuming
x(0, a) = 0, x(l, a) = a.
Fixing a, the solution of col = 0 on the corresponding cylinder with initial
value c is given by functions Fl(t, a) satisfying
3F1 s
— = X ^-(x(^ a), F(*, a))
JF'(0,a) =c' (i = l,---,r).
dxj
~dt
We reduce the problem to a two-dimensional one by considering the mapping
(j) on (t, a)-space to (x, z)-space given by
Then
0(t,a)=(x(t,a), F(t,a)) = (x, z).
v 8t doc ^ J dt ^ J da
= H'da

7.3. THE FROBENIUS INTEGRATION THEOREM
101
where
dFl ^ . dxj
We set
^O^P^dt + Q^da
and compare coefficients of dt da in
d^co* = £(</>* 0'y) A((l>*(oJ)
to obtain
But
dt *■* J
*'<0,«)-^
da
-l^(0,a)
dx1
~8x
= _jp«(0,a) -j;4>,«)
r = o
^(0, a)
eft
It follows that Hl = 0,
3J1' v _, ,. drf
We apply this in particular at £ = 1; here
dxj
da
(t,a)
d . i .
= —xJ{\,a) = — aJ = 0,
r=l
da
da
hence
—- (1, a) = 0, Fl(l, a) = constant,
da
We next fix the notation more precisely; we write Fi(t) a; a, c) instead
of Fl(t, a) so as to specify the dependence of this function on the initial
conditions. Since Fl(l, a) is independent of a we may set
£<'(a,c) = i^(l,a;a, c),
and also
F = (*V ••,*"), G = (GV--,6r).
Then we have the following facts:
(i) G(0,c) = c
(ii) For fixed c, a < ► (a, G(a, c)) is a 1-1 correspondence on a
neighborhood of 0 in a-space onto a manifold Vc in a, c-space. (For we simply take
any curve from 0 to a and use it to define F and then G.)

102 VII. APPLICATIONS TO DIFFERENTIAL EQUATIONS
(iii) Each a>1 vanishes identically on Vc. (For a>1 vanishes on each
curve on Vc, since it is a one-form it vanishes identically.)
We consider the mapping
(a, c) — (a, G(a, c)) = (x, z)
on a, c-space to x, z-space. Because of (ii),
|3(x,z)| =1
|3(a, c)|o
hence we may use (a, c) for a new coordinate system in some neighborhood
of 0. Writing a>1 in these new coordinates and using (iii) shows us that wl
involves only the differentials dc1, • • • , dcr, which completes the proof.
The striking feature of these proofs is that we reduce the original system of
partial differential equations (with integrability conditions) to a system of
ordinary differential equations.
7.4. Applications of the Frobenius Theorem
Example 1: We begin with a question in matrix form which is motivated
by considerations in differential geometry centering around infinitesimal
transformations.
Let Q = || co/1| be an r x r matrix of one-forms defined in a neighborhood
of 0, say, in En. We ask when it is possible to find anrxr matrix A of
functions, nonsingular, satisfying
a = dAA-x.
To fix matters, let us require the initial value A0 — I. It is convenient to
set
0 = dCl - Q2.
Then the basic result is this.
There is a matrix of functions A defined in a neighborhood of 0 such that both
A0 = I and
Cl = (dA)A~1
if and only if
0 = 0.
When this is the case, then there is only one such matrix A. First of all suppose
there is a solution A. Then (dA)A~l — CI, dA — CIA, and we have
0 = d(dA) = dCIA - CldA = (0 + Q2M - Cl(QA) = 0.4.
Hence 0.4 = 0. Since A is nonsingular we have 0 = 0. If B is another
solution so that dA = CIA, dB = QB and A0 = B0 = /, then
d(B~lA) = -B~ldBB-lA + B'ldA
= -B~l{ClB) B'XA + B~l{ClA) = 0,
hence B~XA is constant, B~lA = (B~lA)0 = J, B = A.

7.4. APPLICATIONS OF THE FROBENIUS THEOREM 103
Now we come to the existence. We pass to (n -f r2)-dimensional space
with coordinates x1, • • • , x", zJ (1 ^ i,j _ r) and introduce the r2 forms
which are the coefficients of the matrix
A = dz-nz, z = ||Z/||.
We are assuming 0 = 0, hence we have
dA = -dQZ + QdZ = -Q2Z + Q(A + QZ),
d\ = QA.
It follows that our system A, which is already in standard form, is completely
integrable, hence there exists a matrix A of functions of x with prescribed
initial values at x = 0, so that Z = A is an integral manifold of A = 0, that is,
dA = QA.
We remark that if Q is skew-symmetric, then A is orthogonal (provided
the initial condition is A0 = I). For we set B — XA ~i, the inverse transpose
of A, and have
B0 = /, dB = - B(d *A)B = -B'A *QB = + QB.
It follows by uniqueness that B = A.
Example 2. We next consider another equation:
dA = QA -AQ.
Here Q is the same as before, an r x r matrix of one-forms in a neighborhood
of E" and A is the unknown matrix of functions. Again we set 0 = dQ — Q2.
Let us pose the problem this way. Can we find a solution A of the above
equation taking an arbitrary initial value ^40? We seek a necessary
condition by differentiating:
0 = d(dA) = dQA - QdA -dAQ-AdQ
= (0 + Q2) A - Q(QA - AQ) - (CIA - AQ) Q - A(G + Q2),
which simplifies to
GA = AQ.
Since we are assuming the initial values A0 may be arbitrarily prescribed,
the values of A at each point of a sufficiently small neighborhood of 0 will
fill an 7i-dimensional domain; the commutativity of 0 at such points with
so many different A evidently implies that the matrix 0 of two-forms must
be of type
0 = a/
where a is a two-form. From
dQ - Q2 = ocl

104 VII. APPLICATIONS TO DIFFERENTIAL EQUATIONS
we have, differentiating,
da I = -dQQ + QdQ
= -(a/ + Q2) Q + Q(a/ + Q2) = 0.
Thus, locally, a — da where a is a one-form. The necessary condition we
arrive at is this: There must exist a one-form a satisfying
dQ-Q.2 = (da)I.
This can also be expressed another way: The matrix
H = Q-aI
satisfies
dH-H2 = 0.
Under this condition, the sufficiency is easily demonstrated. As before
we form
r = dz - nz + zn
in x, z-space, and note that
dr= -dnz + Q.dz + dzn + zdn
= -(Q2 + 0) Z + Q(r + QZ - ZQ) +(r + QZ - ZQ) Q + Z(Q2 + 0)
= ze - ez + or + ro,
hence
dY = Qr + TQ,
which shows that the system T is a completely integrable one. The existence
proof now proceeds as in the last example. Uniqueness can be handled this
way. Since the system T = 0 is in normal form and is completely integrable
we know there is a unique integral surface passing through a given initial
point Z\Q = A0.
Example 3. We shall consider a type of system of partial differential
equations known as a system of A. Mayer (see C. Caratheodory [6, pp.
26-31]).
We work in a neighborhood of 0 in Er + S with coordinates x1, • • • , xs,
zl, • • • , zr as before and are given functions Blj(x, z),i = 1, • • • 9r,j = 1, • • • ,s.
The Mayer system is
dz1
£7 = *A*' ')•
We define

7.4. APPLICATIONS OF THE FROBENIUS THEOREM 105
We evidently have Aljk -f Alkj = 0. The Mayer system is called completely
integrable in a neighborhood of 0 provided to each choice of initial conditions
c there exists a solution z = F(x, c) of the system with F(0, c) = c. The
necessary and sufficient condition for complete integrability is precisely
Aljk = 0. The reason is the following. We set
©' = dz* - X BiM> z)dxJ (t = 1, • • • , r)
i=i
a system of one-forms in x, z-space in our standard form. The vanishing
Aljk = 0 implies, after a short calculation,
dB1,
<fc»'-l(E^)
a or
so that the system oo1,---,oor is completely integrable. The integral
surfaces z = F(x, c) solve the Mayer system. Conversely, suppose the
Mayer system is completely integrable. Then it is clear that the system
co1 = • • • = of — 0 has integral surfaces, one for each choice of initial
conditions c. Hence the necessary condition
do1 = £ 0*j a coj
must be satisfied. On the other hand, we directly verify that
dof = \ X Alikdx>dxk + X r\\ a o>a
where
<.-z%f»-
Since dxl, • • •
,dxs
Pj A OJJ =
Note. If cq
one-forms 0 =
or
= lie1.
, co1, • • • , of are
= Z^aA <D\
linearly independent
Y,AiJkdxJd3t? = 0,
• , of is completely integrable, there is
f\\ satisfying
dco1 =
do =
■ I elj a o>j,
= —CD A 0
we conclude that
A'1
an r x
= 0.
r matrix of
in matrix notation. However, from the solution
<0 = dgF, g = (y1, ••-,/), J'=||/,;ll
we conclude that c£cd = — cd a F~l dF so that we may always choose 0 in
the very special form 0 = F'1 dF. (This provides some motivation for the
problem in Example 1.)
We shall see further applications of the Frobenius theorem in our study of
local Riemannian geometry. Also cf. Problems 4-7, p. 194.

106 VII. APPLICATIONS TO DIFFERENTIAL EQUATIONS
7.5. Systems of Ordinary Equations
We consider a system
dx"
— = X»(t,xl,---,x»).
at
Closely associated with this system is the differential 7i-form
Q = (dx1 -XUt)--- (dx* - Xndt)
in (t, x)-space. By a short computation,
(dXl\
£ jefccfoi ...da*.
We make a change of variables
yl = y%x\-^xn) (i = l, ••-,*)
and suppose that the systems
^ = X% x) and J = Y% y)
are equivalent under this change. We set
Q = (dyl - Yldt)--- (dyn - Yndt)
and propose to determine how dQ and dQ are related. Now
dtf_ _ <V ^ dy'1 dxj
dt ~ dt ^ dxj dt '
Also
hence
dyl - Y'dt = Y^-. (dx> - XJdt).
ox1
We denote the Jacobian by J:
W
j = 8(y\---,yn)
d(x\--- ,.r")
dxJ

7.5. SYSTEMS OF ORDINARY EQUATIONS 107
and have by exterior multiplication,
Q = JQ.
We differentiate this:
dU = (dJ)Q+JdQ.
Now
/ dYl\ / dY{\
d& = (I ji) dtdvx ''' dvn = [J1 yr) dtdxl '' *dx"
and
-d( + ^-1 da;'IQ + Jl £ —j I dtdx1 -■•daf
^ fly' ~ J\a« + ^ 3a;' /'
A function / = /(£, x) is called a, first integral of the system if/is constant
along each trajectory, or solution curve. Since the usual existence and
uniqueness theorems guarantee a solution through each point of space where
the system is defined, we have the condition
dt
i.e.,
for a first integral.
Suppose that each of the functions yl, • • • , y" in the transformation above
is a first integral. Then
Yl = 0, Q = dyl • • • dy\
(Such a transformation is always possible in a small region of space because
of the existence of a general solution, one depending on arbitrarily prescribed
initial conditions.)
A function M = M(t, xl, • • • , x") is called a last multiplier of the original
system if
d(MQ) = 0,
i.e.,
dM _ d(MX'1)
dt ^ dxi

108 VII. APPLICATIONS TO DIFFERENTIAL EQUATIONS
Using the transformation above based on first integrals,
MQ = (y)^ = HU = Hdy1 • • - dy\
Hence
d(MQ) = (dH)dy1 • • • dy" = — dtdy1 • - dyn
ot
so that if is a last multiplier if and only if H is independent of t, i.e.,
MQ = H(y)dy1 --dy".
\iMx and if2 are two *ast multipliers, then
if XQ = H^dy1 • • • dy\ M2Q = H2(y)dyi • • • eft/",
hence
Jf n/Jfa = £My)/J5T2(y).
It follows that MljM2 depends only on the yl, hence is constant along
trajectories (as is each t/1) and consequently is a first integral. This proves
the important result:
The quotient of two last multipliers is a first integral.
7.6. The Third Lie Theorem
What is known as the Third Fundamental Theorem of Sophus Lie was
devised in order to reconstruct a continuous group given only its constants of
structure. These concepts will be explained in Chapter IX. For our present
purposes, we shall look upon this theorem as a result, and a rather deep one
at that, in partial differential equations.
We work in E". All indices run from 1 to n. First of all we are given n3
constants cljk subject to these constraints:
c'jk + c'*; = 0
J
The problem is to find n one-forms a1, • • • , a" which are linearly
independent on some neighborhood of 0 in E" and which satisfy the relations
The Lie Theorem asserts that this can be done.
The quadratic relations we have assumed for the constants (c) are easily
verified to be the same as d(dal) = 0, assuming our problem is solved, hence
they are necessary conditions. That they also are sufficient conditions will
now be seen. The proof we give is based on those in Cartan [10, p. 239] and
[7, pp. 280-283]. Because the proof is lengthy, we shall break it into several
steps.

7.6. THE THIRD LIE THEOREM 109
Step 1. We solve the system of ordinary differential equations
for functions hlj = hlj(t, x) satisfying the initial conditions
*£y(0, x) = 0.
Since this is a (nonhomogeneous) linear system there is an (analytic) solution
valid for all t and x. Note that here the variables x enter as parameters
and that there are n2 equations and n2 unknowns. At x = 0 we have
*&*-* mm-*.
hence
and we have
Step 2. We set
hlj(t, 0) = 5) t,
co1 = Y,hij(tix)dxj
so that the col are n one-forms in (t, x)-space. We also set
do1 = A* + dt a af
where the two-forms A1 are free of dt as are the one-forms a1. Thus
«' = I ^ ^ = I (*J +1c«** *fj) ^
Step 3. We shall establish a formula for dXl. We have
0 = d(d(ol) = eU1' - eft a da1',
dA1 = dt a da1'
= d£ a (£ clkld^ a en1 + Y, ciki xkda>1),
dkl = d* a (£ c «*** a^ + X c'H a* A1),
since
doj1 = A' + dt a (X1
and the term dt a dt drops out.
Step 4. We set

110 VII. APPLICATIONS TO DIFFERENTIAL EQUATIONS
We shall prove the decisive formula
dO1 = eft a (Z c{jk xj 9k) + (terms free of eft).
To do this we must use the skew-symmetry of the c's in the lower indices
several times and the quadratic relation on the c's once. We have
= dt a (Z clklda* a o>' + Z ciki *h *l) ~ Z cijkW + * a ocj) a cok.
Now
aj a cok = cftr7' a a>fc + £ c-^ x*of a cofc
so that
d0* = eft a QT c'kl a* A' - Z c^cJ'rs zro>s A o>fc) + (terms).
But
Z cijkCJrs<»a A CO* = J Z (C V« ~ ^j^^Q)' A CO*
= i Z (<V« + ^js^krW A CO*
hence
d0* = & A (Z clkl xk kl + £ X cV^sfc *ra)S A ">*) + (terms).
By changing a few indices around one comes to
ddl = eft A Z c'k| a*(A' - J Z cV/°>5 A 0)j) + (terms)
= eft a Z ciu xk & + (terms).
Step 5. We shall now prove that
0* = 0.
Since both the coJ and the Jl* are free of dt, so is 9l and we have
We may even assert that
<7<;fc(0, x) = 0.
For ^(0, x) = 0 which not only means that the coefficients of the co* vanish
at t = 0, but also that the coefficients of
*-»! $-£)**'
vanish at t = 0 according to the very definition of partial derivative,
_ gW'/O, x)
dh'j
cV
0.
From these facts, what we say about the initial values of the glJk follows.

7.6. THE THIRD LIE THEOREM 111
The result of Step 4 implies
This homogeneous linear system taken together with the initial conditions,
the vanishing of the gr's at t = 0, has the unique solution
and so 6l = 0.
Step 6. Now we can wind up this story. Since 6l = 0 we have
dct)1 = \ £ cijk0)J * a>k + dt a a1.
We consider this relation on the subspace t — 1.
Setting
al = (ot\t.1^h,J{l,x)dxl,
it becomes
*r'= i I c'yiy a <j*.
Since ^(1, 0) = <5j (Step 1), the one-forms a1, • • • , a" are linearly independent
at 0, which implies of course that they are linearly independent in some
neighborhood of 0. The proof is complete.

VIII
Applications to
Differential Geometry
8.1. Surfaces (Continued)
Everything in this section will be based on the local theory of Section 4.5.
Now we have integration at our disposal and we shall discuss a few global
results. Let £ be a closed surface in E3. For e3 we take the outward drawn
normal to £. The mapping
x—>e3
is a map on £ to the unit sphere S2. As x varies over £, e3 varies over S2 a
whole number of times, called the degree of the normal map (cf. Section 6.2).
The element of area of the normal map is
(Oi(o2 = Kala2
since
de3 = ct)lel + a>2e2-
Here K is the Gaussian curvature. Hence
I.
Kaxa2 = 47m
where n is the degree. The factor 4n is simply the area of the unit sphere.
In particular, if £ is a closed convex surface, then e3 covers S2 exactly
once as x covers £, hence
I
Kola1 = 47T
in this case.
After this, we shall limit our discussion to closed convex surfaces. Two
important invariants are the total area
A = J ai(T2
and the integrated mean curvature
M = Haia2.
112

8.1. SURFACES 113
Given a closed convex surface E and a fixed positive number a, we form
the surface E' parallel to E at distance a by marking off on the outward-drawn
normal at each point x of E the distance a and taking the locus of all points
so obtained. Thus the typical point on the parallel surface is
y = x + ae3
where e3 always denotes the normal at x. We have
dy = dx + ade3
= ((T^! + G2e2) + a(colel + co2e2)
= (at + aa)1)e1 + (g2 + aco2)e2 •
It follows that the normal to the parallel surface E' at y is again e3 and that
ex and e2 can be taken as a basis of the tangent space at y. Thus we have
dy = t^ + r2e2
with
rt = a1 + ao)1, t2 = <r2 + aa>2 •
It follows that the element of area of E' is
Tit2 = {g1 + aco1)(a2 + aa>2)
= (7^2 + a(Glco2 — a2(0i) +a2a>1CD2
= (1 + 2aiy + a2K)G1G2
so that the total area of E' is
r==lTlT2=J(1
A' = | T!T2 = | (1 + 2aH + a2^)^!^ ,
A' = A + 2aif + 47ra2.
(This formula can also be proved by first doing it for a polyhedron and then
taking limits in an approximation of E by a sequence of polyhedra. If one
examines what the formula means when E is.a convex polyhedron, one will
see that H measures dihedral angle and K vertex angle.)
By integrating with respect to a, one easily comes to a relation between the
three-dimensional volumes V and V enclosed by E' and E, respectively:
V = V + aA + a2M + f na3.
One can also verify the relations
M' = M + 47ra,
H + aK
K' =
1 + 2aH + a2K'
l + 2a# + a2iT

114 VIII. APPLICATIONS TO DIFFERENTIAL GEOMETRY
We next introduce the support function of our closed convex surface E.
This is denned by
2> = x-e3.
It is convenient to fix E in space so that the origin 0 is inside E. Then we
have p > 0 at each point of E.
The following method will be used to obtain several identities. Let X be
any one-form on E. Then
I eU = 0.
For 3E = 0 and Stokes' theorem gives us
First we consider the form
a = e3-(x x dx).
Here
Now
da — de3 • (x x dx) + e3 • (dx x dx).
de3-(x x dx) — — x-(de3 x dx)
= — x-(dx x de3) = — x-(2,Hr(T1(T2e3)
= — 2Ha1a2(x-e3) = — 2pHa1a2
e3-(dx x dx) = e3-(2(71(72e3)
= 2<71<72 ,
so that
da = 2[ala2 — pHa±a2\
Since the integral of da is zero,
and
'l/^'l
pHala2 -
Je J:
Next we set
j3 = x-(e3 x de3).
By a calculation similar to that used for a,
dp = 2(jpi£<71<72 — Hata2]
so that
ilf = Hata2 = pKata2 .

8.1. SURFACES 115
Since we have found the integrals of H and K weighted by p it is also
reasonable to seek the integral over E of the form pal<J2> We get this by starting
with the vectorial area
(<7i<72)e3 = \dx x dx = (dydz, dzdx, dxdy)
from which
pala2 — (x-e3)((T1(T2) = x-(dydz, dzdx, dxdy)
= xdydz + ydzdx + zdxdy.
Let R be the region of E3 bounded by the closed convex surface and let V
denote its volume. Then
(xdydz + ydzdx + zdxdy)
pala2 =
Je J(e=c
= d(xdydz + • • •) = 3 dxdydz = 3V,
Jr Jr
i V°i°2 = V.
We close this section with the following interesting theorem:
Let E be a closed convex surface of constant Gaussian curvature K. Then
E is a sphere.
To prove this, we recall the relations
{dx = alel + <72e2
de3 = co^! + o)2e2
(<ot = ^ + £<72
W2 = g^ + ra2
H = h(p + r)
K — pr — q2
developed in Section 4.5 for any moving frame. Since E is convex, the
matrix
fp q\
is positive definite, p > 0, r > 0, K > 0. (See p. 120-121 in the next section
for details.) Because of the arithmetic-geometric mean inequality we have
K = pr - q2 g pr g [£(p + r)]2 = H2.

116 VIII. APPLICATIONS TO DIFFERENTIAL GEOMETRY
We also note that there can be equality, K = H2, only when q = 0 and
p = r = H, which implies a^ = H g^ , co2 = ##2 > ^e3 = #^*-
But
(7^2 = pH<7lcr2 ^ p^/Kala2 = —^= pKaxa2
where each integral is taken over £ and we have exploited the hypothesis
K = constant. Because the quantities at the ends of this chain of
inequalities are equal, all integrands must be equal, H = yjK. By our remarks
in the last paragraph, this implies that de3 — Hdx with H constant,
Hx = e3 +Hc with c a constant vector, |x — c| = (l/H)\e3\ = l/H, S is a
sphere.
8.2. Hypersurfaces
We shall extend our study of surfaces to higher dimensions and at the same
time motivate some of the things in the next section on Riemannian geometry.
A hypersurface is an 7i-dimensional manifold M embedded in Ew+1. We
denote the moving point on M by x. Our study is local so we pick a definite
unit normal n at each point x of M. The map x —> n is a smooth map on
M into S". (This can be done globally on a hypersurface M precisely when
M is orientable.) The tangent space at x is an ^-dimensional Euclidean
space; we pick an orthonormal basis for it, et, • • • , e„. Thus at x, the
vectors et, • • • , e„, n make up an orthonormal basis of E"+1. Since dx is
in the tangent space we have
dx = <71e1 + • • • + crne„
where at, • • • , an are one-forms on M. From the relations
erefc = <5£fc, e,-n = 0, n-n = l,
we deduce that
det-ek + et-dek = 0,
de^n + Gi'dn = 0, ndn = 0
and so
\dn= £o>fef
where co^, ca,- are one-forms on M and
toy + COji = 0.

8.2. HYPERSURFACES
117
It is convenient to write all of these structure relations in matrix form. We
set
(<7l:
><o>
Q
CO:
a) = (a>i
co„
Then
dx = ae
■CH2 -?)0
Q + rQ = 0.
By taking exterior derivatives we obtain integrability conditions. (We
shall omit the symbol " a " in what follows. All products of differentials are
exterior.)
0 = d(dx) = (da) e - a(de)
= (da) e — (j(Qe — rcon)
= (da — (xQ) e + a 'con,
da = dQ, a rco = 0;
0 = d
KM
\dt» 0 j\n/ \
/dfl -d'e>\/e\ _ /:
[dm 0 )\n) \,
(da-
[dCl-
i - Q2 + '(0(0
cod
'(do) + Q
0
tra
dQ - Q2 + rfi)w = 0, efo> = coQ.
We define a skew-symmetric matrix of two-forms:
0= \\eu\\ = e*Q-Q2.
We sum up our results:
( da = dQ,
Q + rQ = 0,
dco = coQ,
\<d + rcow = 0,

118 VIII. APPLICATIONS TO DIFFERENTIAL GEOMETRY
or in terms of individual elements of the matrices,
do)j = Y,0)i0)U^
\9U + a>i(oj = 0.
The ai form a basis for one-forms on M, hence we have relations
Because Y,(7ia)i = ^>tne ^u must De symmetric,
The mean curvature H and Gaussian curvature K are denned by
H =-Y.hu, K = \bi}\
n
Since gx • • • an is the w-dimensional volume element on M and col • • • a>n is
the corresponding quantity for S", K represents the ratio of volumes, volume
of spherical image over volume of M, due to
G>1 ' ' ' <On = (E 6U*j) ' ' * (E 6«;<7j) = l6f>l * ' ' °n
Suppose one has a function v = v(y, z, • • •) of several variables where v is
always a tangent vector to M. For example, we might assign to each point
of a curve on M a tangent vector at that point, arriving at a vector-valued
function of one variable.
How does an observer constrained to M observe the motion of v? We
write
v = E c,e,
where the c{ are functions and have
dv = E<fc«ei + Ecfdei
= E dct ei + E c*(E «>tfij - ^n)
= E (dcj + E ci°>u)ej - (E ci®i) n
Our observer who is constrained to move in the hypersurface M cannot
"see" the motion of v which takes place in the direction normal to M; he
sees only the tangential motion of v. Consequently he believes v is
motionless provided

8.2. HYPERSURFACES
119
that is,
dc, + £cfo>l7 = 0 (j = l, •- ,n).
A vector function for which these equations are valid is said to move by
parallel displacement.
The following can be checked. If v = v(t/, z, • • •) and w = w(y, z, • • •)
are two such vector-valued functions which are compatible [for each point
(y,z, • • •) in the parameter space v and w are tangent at the same point of
M] and each moves by parallel displacement, then vw is constant. In
particular, |v|2 = vv is constant.
Let P = P(s) be a curve on M parametrized by its arc length s so that
dP
t = *(*)= —
as
is the unit tangent vector. The curve is called a geodesic provided t moves
by parallel displacement.
There is a geometric interpretation of the matrix ||6fj|! which is quite
fundamental. To each particular displacement dx of the position vector x
corresponds a displacement dn of the unit normal n. Both dx and dn are
in the tangent space at x so that we can look on
dx—>dn
as a linear transformation A of this tangent space. More precisely, let v be
any tangent vector at x. Pick any curve x = x(t) through x so that
dx
~dt
= v.
We follow the normal n = n(Q as x traverses the curve.
dn\
"eft"!
Then
Av
is our definition of A. We see that this is quite independent of the choice of
the curve x(t) so long as it has the prescribed tangent v at t = 0.
For suppose
v = c1e1+ ••• +cnen.
Then
so that
But
dx = Y.aiet
dt
ct.
dn
~dt
L dt

120 VIII. APPLICATIONS TO DIFFERENTIAL GEOMETRY
Now
so our result is
00
dt\
i = lbU°J>
-l*u
dt\
"Z*U°J
^(Z^^KZ^^e,
which establishes these points: (1) A is a well-defined function on the tangent
space at x to itself, (2) A is linear, (3) the matrix representation of A with
respect to the basis e is ||6;y||.
Since the matrix ||6^.|| is symmetric, the linear transformation A on the
(Euclidean) w-dimensional tangent space at x is self-adjoint: for each pair
of tangent vectors v, w,
(^4v)-w = v(Aw).
It is clear that our definition of A depends only on the hypersurface M and
the way it is embedded in Ew+1, not on our choice of the moving frame e.
Consequently the formulas
H = - trace (^1)
n
K=\A\
show that H and K are geometric quantities.
Since A is self-adjoint, its characteristic roots are all real and they are
called principal curvatures. The corresponding characteristic vectors define
(in general) n direction fields on M called principal directions whose
integral curves are lines of curvature.
Convexity of M can be interpreted in terms of a definiteness condition on
the transformation A. To make this precise, suppose M is convex and we
choose for n the inward unit normal.
We fix a point x0 on M and form a corresponding normal section, the curve
of intersection of M with any (two-dimensional) plane on the line n0. This

8.2. HYPERSURFACES 121
is a curve x = x(t) which is convex on its plane, which means that the function
(x - x0)-n
x0 = x(0)
is convex, hence satisfies
/(0 = (x-xo)-no
d2f\
dt2\t = n-
We have
Now
df (dX(t)\ n /Vff'*\ „
^•no = l(f)e,n0-(|)„.n0.
Since the tangent vectors e.(0) at x(0) = x0 are orthogonal to the normal
n(0) = n0, the condition reduces to
WfAl (fh\\ <0.
But a>t = Y, bijOj, so we have
z'«(2)l.(2)L**
Since the direction
(?i ... f^l
\dt' ' dt)\0
is arbitrary, we conclude from this that the matrix
ll&yll
is negative semidefinite so that the same is true for the transformation A
which this symmetric matrix represents.

122 VIII. APPLICATIONS TO DIFFERENTIAL GEOMETRY
(The correctness of sign can be verified in the simplest possible case, that
of a convex curve in the plane. The usual Frenet formulas are
' dx = ds et
de. = Kdsn
dn = — Kds et (s = arc length)
so that a1=ds,(ol = —Kds,bll = —k^ 0.)
We return to the general situation. The elements 9^ of the curvature
matrix 0 are curvature forms. We may write
Qij = 2 Z Rijki°k°i > Rijki + Rijik = °>
defining the Riemann curvature tensor Rijkl of the hypersurface. Because of
the relations
6ij + (OiCOj = 0,
and
(o.o. = £ bikbjlakGl = J X (6ik6yi - feiifejfc) <Tk^
we have
|6» 6»/|
*0*i +
bjk bji
:0.
Algebraic consequences of these formulas are the following:
Rijki + Rijik = °'
^ijfc/ + Riklj + ^i
o** ■
0,
Rijkl — Rklij>
all of which follow easily.
We shall see that the Riemann tensor is independent of how M is embedded
in Ew + 1 so that these relations are particularly interesting, connecting the

8.2. HYPERSURFACES
123
intrinsic Riemann tensor with the quantities b^, which clearly depend on
the embedding.
Indeed, it turns out that the JR's are determined by the <r's alone with no
reference to the normal n. This means that if two hypersurfaces tAx and M2
are in a one-one correspondence which preserves distance (i.e., preserves the
Euclidean geometries in each pair of corresponding tangent hyperplanes),
then Mj and M2 have the same Riemann curvature tensor.
What we shall show is that the equations
do = (jQ, Q + rQ = 0
determine Q uniquely, so that Q is completely determined by the <t's alone.
This is more in context in general Riemaniiian geometry so we shall postpone
the proof until p. 129 of the next section. Having this result, it follows that
0 = dQ - Q2
is also completely determined by the <r's, hence so is the Riemann tensor.
We shall now take up the special case in which our hypersurface M is
given in the form
u = u(xl, • • • , xn)
in x1, • • • , X*1, w-space.
u
k
-*-*"
It is convenient to set
du
d2u
Pi dxr r,v dxldxr
We have for the position vector
x = (xl, x2, - • • , x", v)

124 VIII. APPLICATIONS TO DIFFERENTIAL GEOMETRY
and so
dx = (dx1, dx2, • • • , dx", du)
= (dx\ -••,da»,'Epidzi)
where
The vectors tf, • • • , tn are tangent vectors and evidently form a linear basis,
but generally not an orthonormal basis, of the tangent hyperplane.
The vector
w = (-Pi, •••, -p„, 1)
satisfies
so it is a normal vector (with positive component in the w-direction). The
unit normal n is given by
w = wn
where
w2 = w w = 1 + Y,Pi2-
We note that
wdw = Y,PidPi = Y,Pirijdx3'
We shall now determine the matrix representation of the basic linear
transformation A with respect to the basis tt, • • • , tn of the tangent hyperplane.
This matrix is not symmetric in general because (tf) is not an orthonormal
basis. However, its trace and determinant are the trace and determinant
of A since any matrix representation yields a valid determination of these
quantities.
Suppose the matrix we seek is ||af..|[. Then this means
Ati = Y.aUtj'
Let v be any tangent vector. Because of these relations plus the fact that A
is self-adjoint we have
(4v)-tJ = (i4t£)-v = 5;ay(tJ-v).
Now going back to the very definition of A, symbolically,
A: dx —► dn,
we see that this relation means that
(dn)-t£ = j;«y(ix-ty).

8.2. HYPERSURFACES 125
It is from this set of equations that we shall determine ||a£j.||. We have
dvt = divn + wdn,
dwti = ^(cZn)-t;
since n • t, = 0. But
dwti = (-dpl9 ••• , -dpn,0)-(8ii9 -- ,$in,Pi)
= -dPi"= -Y,rudxJ
so we have obtained
dn • t, = Y r.:dxj.
w^ lJ
On the other hand,
dx-tj = (efa1, • • • , tfV, dw)-(5;i, • • • , SJn9Pj)
= aV + pj du = aV + pj £ pfc drc*
so we have
Z ay(fy* + PjPk)dx* = - - Z *■»***»
Zau((5ifc+^^)= "r»
Setting
*-IM, p-Im, 4-K».
we may write this as
A(I +pfp)= --J8, ^4= --U(/ + p'p)-1.
w w
Since
pfp = Z^2==w2-x
we see that
^ + p'p)(^-(i)p'p) = ^p'p-(i)p'p-^p'p=^
<J + p'p)-'=Z-(^)p'p,
hence

126 VIII. APPLICATIONS TO DIFFERENTIAL GEOMETRY
The mean curvature H is found by taking the trace:
# = -tr(.4)= - — trlR-^Rp'p]
n nw l w J
where Au is the Laplacian of u.
The Gaussian curvature K is found by taking the determinant:
JR: = MI = ^|i?l|/ + p'p|-1.
One finds by a short calculation! that
i/ + p'pi=«>2
so that
(-1)"
For the special case n — 2 of surfaces, we use the standard Monge notation
du du d2u d2u d2u
dx* dy' dx2' dxdy' dy2
and have
w2 = 1 +p2 + q2,
2ttT
-1
2t?
■1
= 7T5 K1 + ^2 + ?2Xr + 0 - (P2f + 2pg* + g2Q]
= 2^[(1 + ^)r " 2^5 + (1 + ^
and
A = S 5
both familiar formulas.
t Set ei = (1, 0,..., 0), e2 = (0, 1,..., 0),..., en = (0, 0,..., 0, 1). Then
\I + p 'p| = |ei + px <p,..., en + pn *p|
= jei en| + 2i*.|ei,...,e<-i,*p, e«+i,...,e,»|
= i + 2 p<2 =w2-

8.3. RIEMANNIAN GEOMETRY, LOCAL THEORY 127
8.3. Riemannian Geometry, Local Theory
The problem here is to deal with the inner geometry of a manifold which is
not part of a Euclidean space. If the manifold were part of Euclidean space,
it would inherit a local Euclidean geometry (distance function) from that of
the including space, as was the case for the hypersurfaces discussed in the last
section. However, it is not part of Euclidean space, so we must postulate
the existence of a local distance geometry. What we do in effect is to
presuppose that each tangent space possesses an inner product which is
smooth.
Thus we let M be an w-dimensional manifold. We suppose that an inner
product is given in the tangent space at each point P of M. Thus if
v and w are two tangent vectors at the same point P, v-w is a real number.
The inner product is supposed to be smooth in this sense: If v and w are
vector fields on M, then v-w is a smooth function on M. (Precisely, at
each point P of M, the values of the given fields v, w at P are vP, wp,
tangent vectors at P, and we are requiring that vP • wP be a smooth function
of P.)
The procedure in Section 2.5 (pp. 13-14) for finding an orthonormal basis
may be made constructive, smooth operations at each step. We know that
there exists on each local coordinate neighborhood on M a set of n vector
fields, forming a basis for the tangent space at each point of the
neighborhood. We convert these fields to orthonormal ones to arrive at smooth
vector fields
ei> "' ■ > e«
denned on the local coordinate neighborhood in question and satisfying
ei-eJ = 8iJ.
Pretending for a moment that we are observers constrained to the manifold
M, we set out on a somewhat symbolic voyage, hoping in doing so to motivate
the right steps. We let P denote the moving point on M and wish to think
of its arbitrary displacement dP as a tangent vector with differential form
coefficients; we hopefully write
dP = £<r,el
with (Tt, • • ■ , Gn differential one-forms on our neighborhood. (Since dP
is in no sense an exterior derivative of anything, we distinguish this d by
bold-face type from the usual d. The same applies to the def below.) We
expect to arrive at a basis Gt , • • • , an for the one-forms which in some sense
is orthonormal, dual to the basis et, • • • , e„ of vectors.
We must be guided by our experience in Euclidean space. There we
would take a coordinate system u1, • • • , un and without hesitation write

128 VIII. APPLICATIONS TO DIFFERENTIAL GEOMETRY
the vectors
_d_ d__
~du} ' ' " ' du11
being the natural frame associated with the coordinate system. But
precisely this can be done on M. The expression
is independent of the local coordinate system. Indeed, if ul, • • • , un is
another system, then
du1
[du'J L'dul\duk)'
and so
that is, dP is the same either way.
Having this, we express the natural frame in terms of the orthonormal one,
and solve for the Gj:
Y,duiaiJeJ = YjoJeJ,
aJ = YiaiJdui.
We have reached our first equation of structure for local Riemannian
geometry:
dP = £a,e,.
Our next venture is to attempt an analogue to the equations for the
displacements det of the vectors of the moving frame. Here we make an essential
departure from what we did with surfaces and hypersurfaces; we must
restrict our attention to "the tangential component" of det for the simple
reason that we are constrained to M and can see no happenings in any
"normal" direction. Thus we seek expressions

8.3. RIEMANNIAN GEOMETRY, LOCAL THEORY 129
with one-forms a>fj.. We try to find such cD,-y so as to be consistent with these
conditions:
(1) derek + erdek = 0,
(2) d(dP) = 0.
The reason for (1) is that d(e( -efc) = d(5ik) = 0 and we hope to "differentiate"
the dot product by the usual product rule. In choosing (2) we simply go
according to the Euclidean analogue. In the usual way, (1) reduces to
(1') G)tt + G)H = 0.
We explore condition (2):
d£>,.e,.) = 0,
so (2) is equivalent to
(2') dffj = Y,ai(0ij-
We have finally come to a well-formulated problem: given the basis
ffj, • • •, a„ of one-forms, find one-forms w^ satisfying
f(l') ©y + <0Jt = 0
1(2') dff, = 2>;a>y,.
We shall show that this problem has exactly one solution. (This completes a
point we left open on p. 123 of the last section.)
Since the at form a basis we may write
0>ij = Z Tijk Gk
where the (unknown) functions Tijk are the connection coefficients, or Christoffel
symbols. Equivalent to (V) is
(i") ryt + rm = o.
The dai are known since the g{ are known; we may write
d°i = 2 Z °ijk °j °k> Cijk + Cikj = 0.
We have
d°t = i Z cy*^ff* = Z *j ®yi
= Z tfy Z Fy**** = i Z (rj*fc ~ rkij)°j<rk t
and so (2') is equivalent to
(2") rjfk — rkij = c^.fc.

130 VIII. APPLICATIONS TO DIFFERENTIAL GEOMETRY
Our precise statement now is this:
Given functions cijk such that cijk + cikj = 0, the system of linear equations
(rijk + rjik = o
has a unique solution given by
*ijk = 2\ckij ~~ cjki ~~ Cijk)'
For if Tijk is any solution, then we use the equations alternately to derive
*ijk = ~" V/ifc = ~~*kij ~~Cijk
= *ikj — Cijk = *jki + Ckij — Cijk
— "^^kji + Ckij ~~ Cijk — ~^ijk — Cjki + Ckij ~~ Cijk »
hence
**ijk — Ckij ~~ cjki ~" C0'&
which establishes the uniqueness of solution. It is easily verified that the
asserted values of Fijk really are a solution.
We have completed our structure equations,
dP = (je
de = Qe
[Q + 'Q = 0,
where we have introduced the matrix notation
(T = ((7ii'",an)i e = I : J, Q= ||g>v||
and already have one integrability condition,
da = aQ.
There is no reason for believing that d(de) = 0 in any sense. We have
d2e = d(de) = d(Qe) = (dCl)e - Q(de)
= (dQ-Q2)e.
We set
0 = ||0O.|| =dn-n2,
the curvature matrix which appears from the symbolic equation
d2e = Oe

8.3. RIEMANNIAN GEOMETRY, LOCAL THEORY
131
as representing a "second derivative"—exactly how one thinks of curvature
in elementary differential geometry.
We derive further integrability conditions by differentiating. From
we have
hence
From
we have
hence
da = aQ
0 = d(d<r) = (da) Q - a(dQ)
= (aQ)Q-a(dQ),
0 = dQ - Q2
de = d(dQ) - d(Q2)
= 0 - (dQ.) Q + Q(dQ)
= -(0 + Q2)Q + Q(0 + Q2),
de = Q0 - 0Q,
which comprises the Bianchi identity.
We sum up:
Structure equations
' dP = ae ^
de = Qe
Q + <Q = 0
0 = dQ - Q2
Integrability conditions
da = aQ
>0 = O
d0 = Q0-0Q.J
The 7i form (Tj • • • an is the volume element of M. It is determined up to
sign. If M is oriented, one may fix it by choosing only moving frames
coherent to the orientation.
The 0tj are two-forms which may be written
6tJ = 2 X Rijkl°k°l
which defines the Riemann curvature tensor. We have
The relation aQ = 0, or
is equivalent to
Rijki + Ruik = °
Bijki + Rjiki = °-
■^i/fc/ + Riklj + ^iO'J
iO'fc '
0.

132 VIII. APPLICATIONS TO DIFFERENTIAL GEOMETRY
In the special case of a hypersurface we had the symmetry condition
Rijkl — Rklij
as an obvious consequence of the expression for Rijkl as a two-round minor
(see p. 122). Such a determinant representation is not possible in this
general situation, but it turns out that this symmetry of the Riemann
tensor is true anyway, an algebraic consequence of the other relations:
(™ijkl ~~ -"klij) — -^ijkl + (-"kijl + **kjli)
=
=
=
=
=
(Rijkl + Riklj) -
~^iljk ~" Rjkli
Rlijk + (Rjlik +
' Rjkli
Rjikl)
(^lijk + **ljki) + Rjikl
~^lkij + Rjikl
— ~~ (Rijkl — Rklij)>
and so
2(RUki - Rkiij) = °
which implies the symmetry in question.
Those who have been through the mill in Riemannian geometry a la
classical tensors will be anxious to see the connection. There one deals with
a natural frame
-— -JL
Vl " dul' ' " ' V" " gj?
due to a local coordinate system (u1, • • • , un). One sets
denning the positive definite symmetric matrix (metric tensor)
G=\\9ij\\-
Usually one looks instead at the corresponding definite quadratic form
where the brackets remind us that this is ordinary, not exterior,
multiplication of differentials. This is motivated by the formula
■-jM3$r*
for the arc length of a curve ul —u\t).
Now one has
dP = £dto'v£.
We try

8.3. RIEMANNIAN GEOMETRY, LOCAL THEORY
and then introduce Christoffel symbols by
133
^■ = 1
;>'
To have
and
requires
dgij = dvi-Vj + vrdvj
0 = d(dP) = -£e^£dv,
duk
ik
jk
Lowering indices:
the equations become
with the usual solution
[j,ik]+[ijk]=^k
d9ij
duk
and so it goes.
This digression out of the way, we briefly consider parallel displacement.
Let v be a tangent vector on M which is a function of one or more variables.
We write
v = £c,e£
where the c{ are functions and have
dv = Y<dciei + lLci0>ijej
= y(dCj + Y.ciajij)ej'
A vector v moves by parallel displacement if dv = 0, i.e.,
With this interpretation of dv one has for two compatible vector functions
v, w that
d(vw) = (dv)-w + v(dw).
Consequently if both v and w move by parallel displacement then v • w is
constant; its differential vanishes.

134 VIII. APPLICATIONS TO DIFFERENTIAL GEOMETRY
A curve P = P(s) on M where s is the arc length is called a geodesic if
the unit tangent dP/ds moves by parallel displacement.
Example. We consider the upper half plane with the (Poincare) metric
[dx]2 + [dy]2
Here
ds2 = l =-
y
dx dy
<j = — , <j.2 = — .
y y
Since the position vector is P = (x, y) we have
dP = {dx, dy) — aiel + <72e2
so that
et = (V, 0), e2 = (0, y).
We have
dxdy
hence
(dat, efo2) = (ai, 02)( _a ^ ]>
Q
The only significant component of the curvature tensor is
■"12 12 = !•
In our discussion of surfaces we wrote dw + Kala2 = 0 for the Gaussian
curvature K. Here the right interpretation is w = a>l2 = o^, K — — i21212
= — 1. For this reason we say that the upper half plane with the Poincare
metric has constant negative curvature.
We shall show that each semicircle
P = (a + r cos t, r sin t) 0 <t <n
orthogonal to the a;-axis is a geodesic. In coordinates it is given by
x = a + r cos t, y — r sin t,

ds _ R-sinO2 + (cost)2~\l/2 _ 1
dt L sin2£ J sin
8.3. RIEMANNIAN GEOMETRY, LOCAL THEORY 135
and we have for the tangent
dPjdt = r( — sint, cost)
r
= - [(— sin t) et + (cos t) e2]
y
l
= —— [(— sin t) et + (cos t) e2].
sin J
Hence for the arc length s,
1/2 =
sin£
t = dPjds = (— sint) ex + (cos £) e2 .
Along the curve we have
col2 = &1 = —dt,
det = — dte2 , de2 — dtet,
dt = d[( — sin^)e1 + (cos^)ej
= — (cost)dte1 — (sint)dte2 — (sin£)( —dte2) + (cost)(dtet)
= 0,
the unit tangent moves by parallel displacement, the curve is a geodesic.
We shall close this local study with an application of The Frobenius
Integration Theorem of Sections 7.3 and 7.4.
Let M be a Riemannian manifold with curvature tensor zero. Then M is
flat: there exists a local coordinate system u1, • • • , un for which the natural frame
A. ... JL
en1'"'9 8un
is an orthonormal frame.
This is proved as follows. We are assuming 0 = 0, i.e., dQ = Q2. By
the first application in Section 7.4, there is a matrix A of functions satisfying
{dA)A~1 = a9
and A is orthogonal. We define x = (x1 , • • • , x„) by x = a A. Then
dx = d(oA) = (do) A-(rdA = (oQ)A - a(QA) = 0.
Each of the one-forms xt is closed, dxt = 0, hence exact locally,
t. = du\

136 VIII. APPLICATIONS TO DIFFERENTIAL GEOMETRY
This defines our local coordinate system (ul, • • • , un). On the one hand we
have
dp-*>'(*?)
and on the other,
dP = (je= T^_1e,
hence
ldjdul\
\dldunJ
Since the frame e is orthonormal and the matrix A is orthogonal, the natural
frame (d/dul, • • • , d/dii") is also orthonormal.
8.4. Riemannian Geometry, Harmonic Integrals
In this section we shall sketch the remarkable results of W. V. D. Hodge
on the potential theory of closed Riemannian manifolds. This work pertains
to differential forms alone so we can forget all about vector fields. In this
spirit we had better make a fresh start and reformulate the pertinent facts
about Riemannian manifolds which we shall need. We shall presuppose
that the manifolds we discuss are orientable, so this will be built into the
structure.
Thus we have a manifold M. It is covered by a system of overlapping
neighborhoods Ux, U2 , • • • . On each U, there is a basis
°\ > *'' > fffi
for the one-forms. If g1 , • • • , an is this basis on U and at, • • • , dn is the
one on U, then wherever U and U intersect we must have
where A = ||a£j.|| is a proper (determinant one) orthogonal matrix.
The volume element tr1 , • • •, an is an intrinsic quantity, according to
Next, the star operator of Section 2.7 applies. To each p-form co corresponds
on (n — p)-form *a>. Locally
*(<V • 'Op) = <7p+1- • *(TM.
We recall that
**o) = (-l)p("-p)a).
We define a new operator d by
dco = (-l)np+n+l*d*(o.

8.4. RIEMANNIAN GEOMETRY, HARMONIC INTEGRALS 137
The significance of the sign will appear shortly. We note that doo is a
(p — l)-form when oo is a p-form. If oo —f is a zero-form, or function, then
5f = 0. The final operator we define is the harmonic operator (generalized
Laplacian) according to
A = d°d + 5°d.
We henceforth restrict attention to a closed (compact) manifold M. We
denote by t the volume element, an w-form on M which nowhere vanishes
and which satisfies t = ax • • • an locally.
We propose to turn the space of ^p-forms on M into an (infinite dimensional)
inner product space. If oo and rj are two p-forms then oo a *rj is an w-form
and we define
(a>, rj) = oo a * rj.
This is evidently linear in each variable and we have
(oo, rj) = (rj, oo),
a consequence of
Locally, if
then
hence
oo a *rj — y\ a *a>.
oo a *a> = (£ afyz
(oo, a>) ^ 0
and (oo, oo) = 0 if and only if oo = 0.
We shall now establish the fundamental formula:
// oo is a p-form and y\ a (p + l)-/orm ^ew
(do>, rj) = (a>, (fy).
For we integrate over the closed manifold M the relation
doo a *rj + ( —l)pa> a d*rj = d(a> a *f/):
efa> a *ff + ( —l)p oo a d*j/ = d(a> a *t/) = a> a *rj = 0,
JM JM JM JdM
(doo, t]) = ( — \)p~l oo A d*q.
Jm
Since d*rj is an (w — p)-form, we have
**(cZ*f/) = *(*d*r]) = ( — l)p(n~p)d*ri
so that
(-l)p-ld*ri = (-l)p~ V-l)p("_p)*(*d*i/)
= (-l)"*+1*(*d*f/).

138 VIII. APPLICATIONS TO DIFFERENTIAL GEOMETRY
But rj is a (p + l)-form, hence
drj = (-l)n(r+1)+n+l*d*rj = (-l)np+i*d*rj,
and so
( — l)p"1d*7/ = *drj,
(dco, rj) = (-l)p-* co Ad*rj
Jm
= co a *(dr]) = (co, c%).
Jm
A form co is called harmonic provided
Aco = 0.
It is clear that if the £>-form co satisfies the two equations dco = 0, dco = 0,
then co is harmonic. The converse is also true. Indeed, if co is any £>-form,
then
(Aco, co) = (ddco, co) + (ddco, co)
= (dco, dco) + (dco, dco).
Now if co is harmonic, then Aco — 0,
(dco, dco) + (dco, dco) — 0.
But each term is nonnegative, hence each vanishes, (dco, dco) = 0, (dco, dco)
= 0 and this implies in turn that dco = 0, dco = 0.
The operators d, d, act on the space of ^-forms. The relation (dco, rj)
= (co, drj) may be interpreted as saying that d and d are adjoint to each other.
We next see that A, which maps p-forms into p-forms, is self-adjoint,
(Aco, rj) = (co, Arj),
indeed, either side is (dco, dr\) + (dco, drj). Since (Aco, co) ^ 0 with equality
only when Aco = 0, we are entitled to call A a positive definite (or elliptic)
self-adjoint differential operator.
We may now state Hodge's main result, a deep theorem in harmonic
analysis:
// co is any p-form then there is a (p — l)-form <x, a (p + l)-form /? and a
harmonic p-form y such that
co = d<x + dp + y.
The forms da, dp, y are unique.
The proof that a, P, and y exist is difficult. We shall only settle the
uniqueness part. Suppose we have
da + dp + y = 0.

8.4. RIEMANNIAN GEOMETRY, HARMONIC INTEGRALS 139
We then have d{da) — 0 and also dy = 0 since y is harmonic. Hence
dSfi = 0,
(dSp, j8) = 0,
(5j8, dp) = 0,
da + y = 0.
Similarly da = 0, y = 0.
By an almost identical argument one shows that in case co is a closed
p-form, dco = 0, then the term dp in the Hodge decomposition of co is absent.
co = d(x + y.
It follows from this that if z is any p-cycle, then
J.—J/
that is, y has the same periods as does co. (See De Rham's theorems,
Section 5.9.) The result of this is that if co is any closed form, then there
exists a unique harmonic form y with the same periods as those of co.
We can also answer the following question. Given a p-form X, when is
there a p-form rj such that the equation
Arj = X
is satisfied ? The answer is: if and only if
for every harmonic form y.
For suppose X — Ay and y is harmonic. Then
(y, X) = (y, Ai|) = (Ay, rj) = (0, rj) = 0.
On the other hand, suppose X is a form satisfying (y, X) — 0 for each harmonic
form y. From the decomposition
X = d<x + dp + y
we have, using the particular y which is part of A,
0 = (y, X) = (y, efa) + (y, dp) + (y, y)
= (<5y, a) + (dy, P) + (y, y)
= (y> y)>
hence y = 0,
X = d<x + dp.

140 VIII. APPLICATIONS TO DIFFERENTIAL GEOMETRY
We shall set rj =fi + v and try to solve Afi — da, Av = dp, separately. We
take the first
AjU = da.
Decomposing a,
a = dax + Spt + y1,
da — ddpl.
Next,
pt = da2 + dp2 + y2 ,
dd/^ = A5da2 = (e&5 + 5d)(da2) = A(da2),
da = Apt with jU = da2 .
We find v similarly.
Example ,1. E". We shall compute the operator A in E". Contrary to our
previous notation in dealing with the standard Laplacian, we shall denote the
Laplacian by Lap,
The result is this: if
then
Aa> = — X (Lap aH) dxH.
It will suffice to establish this for the monomial
oo — A dx1 • • • dxp.
We shall abbreviate the calculation by these conventions:
(1) subscripts on A denote partial derivatives;
(2) a = 1,2, --,p,j=p + l, ••• ,n;
(3) each repeated index is summed over its range.
We also remark that in taking the star of a monomial, the choice of sign is
always governed by the rule rj a *rj — Bdx1 • • • dxn where B > 0, for
example,
/\ /\
*(dx"dxp+1 •••(&>••• dx") = (-l)'"+*+a+Jefo1 • • • dx* • • • dxpdxJ.
Another point: since oo is a p-form and doo is a (p + l)-form,
doo = (-l)np+n+l*d*oo,
ddoo=(-l)np+l*d*doo.

8.4. RIEMANNIAN GEOMETRY, HARMONIC INTEGRALS 141
We have
*a> = Adxp+i --dx",
d*a> = Aadx*dxp+l • • • dx",
/\
*tf *o) = {-\)np+n+<x+lAJxl • • • dx* • • • da",
/\
(5a) = (-lJM.efc1 • • • dx* • • • efa",
/\
d&» = -^aa^ ••>dxp + (-ly+'^A^dx1 • • • cfoa • • • dWefc'.
Next,
da> = Ajdx^dx1 • • • efop,
/\
*da) = ( —l)-/+1^Jdtep+1 • • • dxj • • • da/1,
/\
d*da) = (-l)Mj7d^+1 • • • dx" + (-l)J'+1^yaefoaefo*+1 • • • dx3 • • • dx",
/\
*d*dw = (-lyM^efa1 • • • dxp + (-l)^+p+a+1^;ada;1 • • • dx* • • • dxpdxj,
y\
8da>= -Ajjdx1 - • - dxp + (-l)p+*Ajadxi • • • dx* • • • dWefe*.
Combining these expressions,
Aa) = d(5a) + Sdoo = -[^4aa + Ajjjdx1 • • - dxp
— —(Lap^4)dx1 - • • dfep.
The minus sign seems strange in view of the relation
(Aco, 0)) ^ 0
on closed manifolds. An example may clear this point. Let us take the
zero-form sin # on the flat torus 0 :g x, y, z ^ 2n, where numbers are identified
if they differ by a multiple of 2n. Then
d(sin#) = (cosx)dx
*d(sin#) = (cos x)dydz
d*d(sm.x) = — (sin x)dxdydz,
*d*d(sinx) = —(sin a;),
A(sin#) = Sd(sinx) = -fsinrc,
(A(sin#), sin a;) = (2n)2
2n
,2,
sin2xdx = (2n) n > 0.
o

142 VIII. APPLICATIONS TO DIFFERENTIAL GEOMETRY
We have used the fact that eZ(sin:c) is a one-form, n — 3, p = 1,
S[d(sinx)] = (-l)np+n+1(*d*)d(smx) = -*d*tf(sina;).
Example 2. S2. If/ is a function on E3, we have the spherical coordinate
form of the Laplacian (Section 4.4, p. 40),
VJ r2 sin <j> [dr \ * 8rJ d<t>\ V d<j>) 80 \sin $ 86 J J
Suppose that Lap / = 0 and that / is of the form
f(r, <j>, 0) = r"g((t>, 9).
Then g is called a spherical harmonic and must satisfy
n(n + l)(sin</>)</ + ^ [(sin0)^]+ ^(—7) = 0.
The function g may be considered as a function, or zero-form, on S2, the unit
sphere. There we have
a1=d(f)i <72 = sin$d0,
= - (-^) d<t> + (sin^de,
\sm <p)
d*d9 = iL (-^j) W^ + ii; [(sin *W d<t>de
00 \sm<p/ o<j>
But
Ag = (dd + dd)g = ddg = — *d *^,
so the condition for gr to be a spherical harmonic is
Ag -n(n+ \)g = 0.
In other words, the spherical harmonics are eigenfunctions for the generalized
Laplacian on S2. Many of the usual facts about spherical harmonics
follow from our calculations. We take one example, the orthogonality

8.5. AFFINE CONNECTION
143
relation: // g and h are spherical harmonics of distinct degrees m and n,
respectively, then {g, h) = 0.
For Ag = m(m + \)g, Ah = n(n + \)h, hence
m(m + 1) m(m + 1)
.^±iLfo,4),
m(m + 1)
(<7, *) = 0.
8.5. Affine Connection
We shall approach the problem of affine connection this way. We seek
the weakest structure with which we can endow a manifold so that parallel
displacement of vectors along curves is possible. Considerably less than a
Riemannian structure is required.
Let M be a manifold. An affine frame (or simply frame) on a neighborhood
U of M consists of n vector fields ex, • • • , e„ on U which are linearly
independent at each point of U. Thus at each point P of U the vectors
(ei)p> "' ■ > (en)p furnish a basis of the tangent space T at P. There is a
dual basis a1, • • • , a" of one-forms on U so we may write
as we did for the Riemannian case in Section 8.3.
We now want to associate with each vector field v on M a vector field dv
with one-form coefficients. We must be able to do this for the vector fields
e^ of the basis, so we require
where the a>/ are one-forms on the neighborhood U. There are certain
consistency conditions which guarantee that the computation of dv will be
independent of any frame.
Locally we may describe an affine connection as follows. We are given
U, the affine frame et, • • • , e„, and the dual basis a1, • • • , a" of one-forms.
An affine connection consists of n2 one-forms a>tj subject to no constraints
whatever.
We shall develop some of the local geometry of an affine connection before
attacking the crucial problem of finding proper consistency conditions which
make the definition of an affine connection over a whole manifold possible.
By introducing matrix notation:

144 VIII. APPLICATIONS TO DIFFERENTIAL GEOMETRY
we may write our basic structure equations:
dP = ae, de = Qe.
We shall quickly point out the relation to the customary tensor formulation
of an affine connection. First we expand each a>/ in the basis ak:
defining the connection coefficients lYfc. In the usual tensor formulation,
the frame ex, • • • , e„ stems from local coordinates:
e*" dul'
and correspondingly, ai = dui. The T{jk — T^k(ul9 • • • , un) are then n3
arbitrary functions assigned on U.
We derive relations by differentiating the structure equations several
times. First
d2P = {da) e — a de = {da — <tQ) e = re.
Here
t = (t1, ••• , rn) = da-aQ.
The two-forms zl are the torsion forms. We may write
denning the torsion coefficients Tljk. Next,
d2e = {dO) e - Qde = {dQ - Q2) e = 0e.
Here
e = dQ-Q2 = \\eij\\
is the matrix of curvature forms 0/. The curvature tensor R(Jkl is obtained
from
0/ = 4 E Rhi** a or', *,»„ + iJ/n = o.
Integrability conditions are obtained by applying the exterior derivative to
the equations t = da — aQ, and 0 = dQ — Q2:
dx = {-da)Q + adQ
= -(t + (tQ)Q + (t(0 + Q2),
dt = a& - tQ,
and
d0 = (-dQ)Q + QdQ
= -(0 + Q2)Q + Q(0 + Q2),
dG = Q0 - 0Q.

8.5. AFFINE CONNECTION
145
If v is a vector field, then
v = Zfei = Fe, F = {f1, ••-,/"),
where the /l are scalars; we have
dv = (dF) e + F(de) = (dF + FQ)e.
Suppose the vector field v is defined over a submanifold. It is said to move
by parallel displacement if dv = 0, i.e.,
dF + FQ = 0.
If P = P(t) is a smooth curve on U defined over an interval t0 ^ t 5^ tt of
the t axis, if v0 is a tangent vector at P0 = P(£0)> then there exists a unique
assignment of a tangent vector v(t) at P(t) for each value of t such that
v(t0) = v0 and v(t) moves by parallel displacement.
For we write v = ^]/Ief. Along the curve the conditions for parallel
displacement become
£+*'•(£)-*
a first order linear system which taken with the initial data determines the
fl uniquely.
Now we tackle the global situation. We have a manifold M in front of
us and we must consider each conceivable moving affine frame et, • • • , e„
together with its neighborhood of definition U. With each one of these e
we have an affine connection, i.e., an n x n matrix Q of one-forms on U.
We want these to "fit together" whenever two such neighborhoods overlap.
Thus let (U, e, d, Q) be one such system and (0, e, a, Q) another, where
we assume the neighborhoods U and 0 overlap. The basic thing we require
is that for any vector field v defined on the intersection of U and 0, the
computation for dv in either connection must yield the same result.
On the overlap, where we shall always operate in what follows,
e = Ae
where A = ||a/|| is a nonsingular matrix of functions. Since
dP = (je = <re,
we have
ffAe = (je,
Now
de = d(Ae) = (dA) e + 4(de)
= (dA +^4Q)e
= (dA + ASl)A~le.

146 VIII. APPLICATIONS TO DIFFERENTIAL GEOMETRY
But also de = Qe, so we have the transformation law for Q:
n = AQA~1 + (dA)A~l.
This is quite different from the transformation law for a which is forced
by the very definition of manifold and frame. It tells us how the various
matrices Q we are associating with the various moving frames e must be
related if we are to define an affine connection on M as a whole.
From these formulas one can derive the transformation laws for t and 0:
x = xA~\
e = AeA~l.
If v = Fe = Fe is a vector field, then F = FA~\
(dF + FQ)e
= (dFA~l -FA-UAA-^Ae + iFA-'HAQA-1 +dAA~l)Ae
= (dF + FQ) e,
so that dv is the same, either way it is computed. (We have used the rule
d(A~l) — —A~1(dA)A~l, as we shall several times.)
We shall now have a second look, only in our present context, at the
considerations of Section 4.3. This will provide us with another way of
looking at affine connection. What we shall do fits into a general pattern:
quantities subject to a transformation law become absolute invariants when
considered on a suitably extended space.
We begin with a manifold M of n dimensions. We form a new manifold
F of dimension (n + n2). This frame manifold consists of all frames at all
points of M. Precisely, at each point P of M consider all possible bases
ex, • • • , ew of the tangent space TP at P, and do this for all P.
We obtain coordinates on F this way. First let U be a local coordinate
neighborhood on M with coordinates w1, •••,w". Pick a moving frame
ex, • • • , ew on F. Thus at each point Pof U, (ejp, ••• , (eJp *s one basis
of the tangent space Tp at P. The most general basis of Tp stems from this
given one by applying an arbitrary nonsingular transformation. It is
f t, • • • . f„ where
with ||6/1| an arbitrary n x n nonsingular matrix. It is clear from this
that the (n + n2) independent variables
u\ ••• ,un,btj
serve as a coordinate system for the neighborhood in F consisting of all
frames at all points P of U.

8.5. AFFINE CONNECTION
147
With the moving frame et, • • • , e„ on U goes the dual basis a1, • • • , <rn
of one-forms on U. The forms &ly • • • , dn denned by
9 = aB"\
where
a = (or1, • • •, a"), 5 = (a\---,d"), B = \\bJ\\,
are one-forms on the part of F lying over U. More precisely, the values of
the forms a1, • • • at the point f of F given by
are
*lf =(<r\P)B-1.
Now suppose that 0 is a second coordinate neighborhood, e a moving
frame on 0, etc.. Suppose that P lies both in U and in 0. We have
e = Ae
where A is a matrix of functions,
(X = a A ~ 1,
all worked out above. If the point f of F has coordinates B with respect to e
and E with respect to e, then
Be=Be=BAe,
B=EA.
Thus
d = (rB~1 = (aA)(SA)"1 =dB~K
This implies that the one-forms a1, • • • , on are defined on all of F and are
completely independent of the particular local coordinate neighborhoods and
moving frames used in their definitions.
This is of first importance. We began with an w-manifold M. We
constructed over it a new manifold F. On this new manifold we
automatically have, free of charge, the n (linearly independent) one-forms
3\ ■ ■ •, 5".
Now suppose an affine connection is given on M. Thus to the
neighborhood U with a definite affine frame e is given a matrix Q = ||co^|| of one-
forms. The matrix Q corresponding to the frame e on 0 is related to Q
(on the overlap of U and U) by a certain transformation law. This
transformation law means nothing more nor less than that the n2 one-forms co/
denned by
Q = IIS/II = BQB~l + (dB)B-\
apparently defined only on the part of F lying over U, are defined on all of
F, are completely independent of e and its U.

148 VIII. APPLICATIONS TO DIFFERENTIAL GEOMETRY
This statement may be verified by a calculation. According to the
notation above, one must prove the formula
BQB~l + (dB)B~l = BUS-1 + (dB)B~K
We leave the details to the reader, but point out that the result can be
motivated by a symbolic calculation:
f=Be,
df = (dB) e + B(de) = (dB + BQ) e
= {dB + BQ)B~li,
similarly
df=(dLB + lX5).B-1f,
hence
(dB + JBQJjr1 = (dB + BU)B~K
From the one-forms a1, • • • , co/ on F, one constructs two-forms t1', 5/
according to
t = (t1, ••• ,t") = d5 - 5Q,
&=||^||=rffi-fi2.
These are the general torsion and curvature forms respectively.
8.6. Problems
1. Let E be a closed convex surface with constant mean curvature H.
Prove that L is a sphere.
2. A surface E is given in the Monge form z =f(x, y), denned for all x, y.
We suppose this surface is convex from below. Show that this means that
is positive semidefinite.
3. (Continuation.) Show that the mapping
(z, y) —> (x + p, y + q)
increases distance and hence is one-one.
4. A point of a hypersurface is an umbilic if the transformation A at that
point has all of its characteristic roots (principal curvatures) equal. Let M
be a hypersurface all of whose points are umbilics. Prove that M is a
portion of a hyperplane or sphere.
5. Suppose on a Riemannian manifold M there is a scalar K such that
Prove that K is a constant.

8.6. PROBLEMS
149
6. Let M be a manifold with an affine connection given. Show that
the two-form a defined locally by
a = £ 0.' = trace 0
is actually defined on all of M, independent of local frames. Show also that
da = 0. It is even true that there exists a one-form k on M such that a = ett,
but this is difficult to establish.
7. (Continuation.) Prove
dGr = QGr-GrQ
and that
d[trace (0r)] = 0.
8. Let M be a manifold with affine connection. Given an affine frame
et, • • • , e„ with corresponding connection coefficients T and torsion
coefficients T, define a new connection by specifying the new connection
coefficients T*:
1 i k ~ l i k ^ 2 -1 ik'
Show that this indeed defines a connection on all of M and that this
connection is symmetric (no torsion). Investigate the meaning of "symmetric"
for a local coordinate frame.
9. Consider the flat torus T" which consists of all points (xt, • • • , xn)
where each xt is taken modulo one. That is, (xt, • • • , xn) — (y1, • • • , yn)
if xt — y{ — integer (i = 1, • • • , n). The flat metric is given by the ortho-
normal basis
al=dxl,--- , an = dxn
of one-forms. (In older notation, [ds]2 = [dxt]2 + • • • + [dxn]2.) Find all
harmonic differentials of all degrees.

Applications to
Group Theory
9.1. Lie Groups
A Lie group consists of a smooth manifold G which has a group structure
(x, y) —► xy.
We suppose that this group operation, which may be considered as a mapping
is smooth and also that the map x —► x~l on G —► G is smooth.
With each element a; in G, there is associated a transformation Lx of
G, called left translation:
Lx{y) = xy.
A differential p-form co is called left invariant provided
Lx* CO = CO
for all x in G.
Let e denote the unit element of G. The left translation Lx~x = Lx-t
sends x to e. If co is left invariant, co = Lx-i co is completely determined at x
by its value co0 at e. If co0 is any given p-form at e, then a left invariant
form co is denned by
These remarks serve to determine the existence of left invariant forms.
Let us begin with one-forms. We let n be the dimension of G. Since the
space of one-forms at e is an 7i-dimensional linear space, there are exactly n
linearly independent left invariant one-forms on G. Let
G , • • ' , G
be such a system. Any other left invariant one-form is a linear combination
of these with constant coefficients.
More generally, if co is any left invariant p-form on G,
150
IX

9.2. EXAMPLES OF LIE GROUPS 151
where the cH are constants and a11 = ahl • • • ahp. Any p-form co can be
expanded in this way and the coefficients cH will, in general, be scalars on G.
Supposing co left invariant forces each of these scalars to be left invariant.
This means that each cH takes the same value at each point of G, hence is
constant.
Next, if co is left invariant, so is dco, since
Lx*(d(o) = d(Lx* co) = dco.
It follows that there are constants of structure cljk such that
*rf = i I c'jk **<*> cijk + c1*; = 0.
Substituting this into the relations d(dal) = 0 eventually yields
I (C>^ + c1^^ + c£y-0 = 0.
Particularly important is the w-form
<7 • * • (J
which defines a left invariant volume element on G. It is clear from this
that G is orientable.
The right translation Rz associated to a group element z is the mapping
y —► RzV = y*
on G to G. From the associative law
x(yz) = (xy)z
we deduce that
LxoRz = RzoLx,
hence
J8r* o L* = Lx* o #2*.
Suppose co is a left invariant p-form. Then for each a; and z,
Lx*(R2*co) = R*(Lx*co) = i2r*o>,
hence i22* cy is also a left invariant p-form.
9.2. Examples of Lie Groups
Example 1. ra = 1. We shall determine the local structure of all one-
dimensional groups. Let t be a parameter on G, chosen so that t = 0 is the
identity e. Let <r be a nontrivial left invariant one-form; locally,
a = f(t) dt, never zero.
We integrate a to get a new parameter for G,
(ff(t)dt.

152 IX. APPLICATIONS TO GROUP THEORY
Thus we may assume we have started with a parameterization of a
neighborhood of e by a single variable t such that
a = dt
is a left invariant form.
We next express the group product analytically. The product of the
point with coordinate s with that of coordinate t will have coordinate u
given by
u — p(s, t)
with
p(s, 0) = s, p(0, t) = t
according to xe — x, ey — y. In coordinates,
Ls: t—>u—p(sf t).
The left invariance of a, Ls* a = <r, means
dp
dt — — dt,
dt
hence
^=1, P(8,t)=t + <l>(8).
Setting £ = 0:
S = p(s, 0) = (j}(8)9
and so
p(s, t) = s +t.
It follows that the group operation corresponds to nothing more than ordinary
addition of coordinates. As a corollary, G is abelian (commutative).
Example 2. G is a group for which the constants of structure cljk all vanish.
Thus
da1 = • • • = dan = 0.
In a small neighborhood of e,
vl = du\
taken so that (0, 0, • • • , 0) < > e. The product of points with coordinates
u, v, respective^, is a point with coordinates w given by
wl = pl(u, v)
with
p'(u, 0) = u\ pl(0, v) = v\
The invariance of a1 under the left translation
v —► w, wl = pl(u, v)

9.3. MATRIX GROUPS
is expressed analytically by
153
*>' = Iyi(u,v)£fo>
—' ovJ
which implies
dp1
dvJ
= /,
Setting v = 0 yields
hence
dp1
pl(u, v) = vl + </>''(u).
ul = </>''(u),
p'(u, v) = w1' + v1'.
If P(u) denotes the point with coordinates u, this says
P(u)-P(v) = P(u + v)
so that locally the group looks like a neighborhood of 0 in E".
Corollary. G is abelian.
9.3. Matrix Groups
Now we shall consider a group G which is a smooth subgroup of the group
GL(m) ofmxm nonsingular matrices. (The notation stems from the
common name, general linear group.)
Suppose ul, • • • , un is a coordinate system on G in some neighborhood of/,
the identity matrix, and that X = X(ux, • • •, un) is a typical point in this
neighborhood. The matrix dX of one-forms certainly contains n linearly
independent one-forms because the ^-dimensional group G is smoothly
imbedded in GL(m). Consequently the matrix
Q = X~ldX
of one-forms contains n linearly independent ones. But each element of Q
is left invariant. For if A is any fixed element of G, the left translation
by A is given by
X—>AX,
while
(AX)~ld(AX) = (X-lA~l)(AdX) = X-'dX.
This allows us to make explicit calculations in many important groups.
Next we note an important geometric interpretation of Q. We interpret

154 IX. APPLICATIONS TO GROUP THEORY
each element X of G as a linear transformation on the space Ew of row
vectors v = (vt, • • • , vn). Thus
v—► w =vX.
We ask, how does dw grow out of w under the group action? Here v is
fixed and X varies over G. We have
dw = vdX = {wX'^dX,
dvf = wQ.
This means Q can be interpreted as an "infinitesimal group element."
(Cf. Section 4.2.)
One final remark. The constants of structure can often be explicitly
obtained from these considerations:
n=x~ldx
dX = XQ,
0 = d(dX) = dXQ. + XdQ
= (XQ)Q + XdQ.
Hence,
da + q2 = o.
9.4. Examples of Matrix Groups
Example 3.
the proper affine group on the line. One easily sees the isomorphism between
G and the transformation group
t —► xt + y.
Here
x = (x yV x-> = l-(l -A
\o 1/ x\o xy
Q - -(l ~y\(dX dy\ - \(dX dy\
~ x\0 x)\0 0/ "" x\0 0/
Hence a1 = dxjx, a1 = dyjx are left invariant. The left invariant volume
element is

9.4. EXAMPLES OF MATRIX GROUPS 155
Since da1 = 0, da2 = — dxdy/x2 = —a1 a <t2, the only significant constant
of structure is
ci2 = ~c2i = ""!•
If we seek right invariant forms, we find them in
1 \(dx dy\ (\ — y\ l/dx — ydx + xdy\
<dX)X-1 = Ao o)(o J'io o )'
so a
The
basis is
1 1 2
a1 = or1 = — , a2
right invariant volume element is
1 2
a1 a or =
—ydx +
dxdy
xdy
X
very different from the left invariant one. Also
Cfa* - £*! - «' A «*.
We shall compute the effect on a2 of the right translation EA, where
A " (o ')'
We have
w-iA-d^i |)-(7te+»).
d(&a + y) & efo 1 dy 6 x 1 2
a# a a; a a; a a
Example 4. The step transformation group of all matrices
! 1/* -y\/dx dy\ _ \ Ixdx xdy - ydx\
We may take
1 ** j/i x 2 xdy-ydx (y\
a; ar \ay
The choice of new coordinates
y
u = ln#, v = -
x
follows the procedure in Example 2.

156 IX. APPLICATIONS TO GROUP THEORY
Thus
* - r, :■") - '(J :)•
is another, then
The invariant volume element is dxdy/x2 = dudv.
Example 5. G =GL(?i), the general linear group of all nonsingular n x n
matrices. The general element is X = \\xtJ\\ where the xtJ are independent
variables (subject to the inequality A = det (xtJ) ^ 0). Set
7 = cofX, X"1 = - 7.
A
We have
a = x-ldx = \\oi%
where
The n2 left invariant forms ak are necessarily linearly independent. We
compute the volume element
* = IW
i,k
in two steps:
t = v1 a •• • a v",
vk = a* A • • • a Gnk = — (Zyijdxjk) a • • • a (£ y,/cfe/)
From X 7 = AI we have det (X) det (7) = A", hence
det(7) = Aw"1,
vfc = — dxf - • • dxnk,
* = h n (<v • • • <*o-

9.4. EXAMPLES OF MATRIX GROUPS
157
It is clear from this that the right invariant volume element will also be t,
which is an unusual feature of this group since it is highly non-abelian.
Example 6. G = SL(?i), the special linear or mdmodular group of all n x n
matrices of determinant one. The special feature we shall note is
trace (Q) = 0.
This follows from a general formula for a matrix function X of any number
of variables. Set A = det (X). The formula is
dA
truce (X~1dX).
This is proved as follows. Denote by
C =
^
the jih column of X and consider
A = det(c1, --,cn)
as a function of the columns. Then
dA = £ A(c1, • • • , c'"1, dd, cJ+1, • • • , C")
j=i
= Z X y/dxij = trace [(coiX)dX]
j=i *=i
= trace [AX~ * dX] = Atrace (X~x dX).
For G =SL(w) we have A = 1, dA = 0, hence trace (Q) = 0. For n = 2 we
have
X = \ y\, A = zv — yu = l,
\u v)
Q = X-'dX = ( V -y\(dx dy\
\ — u x) \du dv)
(vdx — ydu vdy — ydv\ _ la1 g2\
— udx + xdu —udy + xdvj \cr3 a4/'
Differentiating A = 1 yields
xdv + vdx — ydu — udy = 0, a1 + <74 = 0.

158 IX. APPLICATIONS TO GROUP THEORY
For the left invariant volume element we may take
t = (T1 a <73 a a2 = dxdu(vdy — ydv)
= vdxdudy — ydxdudv.
Example 7. G = 0+(n), the proper orthogonal group of all n x n matrices
X for which
lX = X~\ det(X) = +1.
Here the superscript t denotes transpose. The essential feature about Q is
that it is a skew-symmetric matrix,
Q + 'Q = 0.
Because the group G has* dimension n(n — l)/2, it follows that the elements
above the main diagonal in Q form a basis for left invariant one-forms and
their product is the left invariant volume element. We establish this
property of Q as follows:
X <X = I, (dX) *X + X \dX) = 0,
X'1dX + \dX)tX'x = 0,
X-l dX + <(X- ldX) = 0, Q + rQ = 0.
For n = 2,
Y — I COS^ sm#\
~~ \ — sin 9 cos0/'
\sin0 cos6J\ — cosv — sm0/ \ —1 0/
For n > 2 the calculation becomes complicated and hinges on explicit
parametrizations of G. The cases of even and odd n are rather different.
9.5. Bi-invariant Forms
We take a Lie group G with identity element e. Because G is a manifold,
there is a coordinate neighborhood U of e with a local coordinate system such
that the coordinates of e are (0, 0, • • • , 0). Suppose x and y are very near
to e and in U. Then z — xy is in U. If the coordinates of these three
points are (x1, • • • , x"), (y1, • • • , yn), (z1, • • • , zw), respectively, we may write
Since xe = x and ey = y, we have
zi(xli--,xn,0,-'i0) = xi,
2'(0,---,0,y1,---,y") = y',

9.5. BI-INVARIANT FORMS 159
and because of these facts,
zl = xl + yl + (higher order terms in the xj and t/fc).
In particular, if y = x~1, then z = e, and
0 = xl + yl + (higher order terms).
We apply these simple remarks as follows. Let i// denote the mapping
\j/(x) = x'1,
$: G—>G.
If we write
y = \j/(x),
yl = y\xl, • • • , a/1),
then by the relation just discussed,
—• = -#.
3^ |o
This means that
yl — — x{! + (higher order terms in the #J).
We may also express this another way:
\j/: (a:1, • • • , xn) —► — (xl, • • • , xn) + (higher order terms).
Since ij/(e) = e"1 = e, the induced mapping i/j* takes each differential form
at e to another form at e. Evidently we have
\lt*(dxl) = -dxl (at e),
^♦(efc1 • • • dW) = (- l)p(dxl • • • dxp) (at e).
Thus if o)e is any p-form at e, then
*l'*(<Oe) = (-l)P<De.
For each yinG, the right translation Ry was defined by
Ry(x) = xy.
A form a> is called right invariant if
Ry* a> = a>
for each y in G. Our first application of the map \j/ is the following:
A form a> is left invariant if and only if \j/* a> is right invariant.
For
il/(Ry(x))=\l/(xy) = (xy)~l =y~lx~l
= Ly-l(x-')=Ly.l(il,(x))i

160
IX. APPLICATIONS TO GROUP THEORY
hence
\JJ O By = Ly-l O \j/,
Ry* 0 tff* = ^* o Iri*.
If co is left invariant, then for each y in G
Ry*(il/*co) = il/*(Ly-*oj) = ^*co,
hence ^* co is right invariant. Similarly, if co is right invariant then ty* co is
left invariant. Since (aT1)"1 = x, \j/ o \j/ = i and so i^*(^*co) = co. It
follows that if \j/* co is right invariant then co is left invariant.
Next we shall see that using right invariant forms instead of left invariant
ones does not give additional constants of structure. For let or1, • • • , a" be a
basis of the left invariant one-forms. Then the corresponding constants of
structure are read from the equations
da1 = \Y<ciJk°* A <**•
The forms t1 = \j/* g1, • • • , t" = \j/* a" are now a basis for the right invariant
one-forms. But \]/* applied to our equation yields, since the c's are constants,
Now we pass on to the study of bi-invariant forms, i.e., forms which are
both left and right invariant. We derive one important result.
Let co be a bi-invariant p-form. Then
dco = 0.
For \j/* co is left invariant since co is right invariant. We know from our
calculation on the previous page that at the point e,
**(©.) = (-l)PG>..
But co and ij/*(co) are both left invariant, hence what is true at e is true
everywhere,
i/,*(co) = (-l)pco.
On the other hand, dco is a (p + l)-form, also bi-invariant, so the same
conclusion applies:
ll/*(dco) = (-l)p+1dco.
But,
^(dco) = d(il/*co) = d[(-l)pco] = (-l)pdco.
From these equations follows dco = 0.

9.6. PROBLEMS
161
We apply this to the case in which G is a commutative group. Then the
left and right translations are the same thing so that each left invariant form
is bi-invariant. In particular if or1, • • • , cf1 is a basis of left invariant one-
forms, each da1 — 0; the constants of structure all vanish. In Example 2
of Section 9.2 we showed that any group with vanishing structure constants
has the local structure of Euclidean space (and incidentally is commutative).
Here is one more result on bi-invariant forms which goes in a different
direction.
Let G be an n-dimensional closed {compact) Lie group and let co be a left
invariant n-form on G. Then co is bi-invariant.
For each x in G, Rx* co is also left invariant. Assuming co ^ 0, we have
Bx* co — f{x)co, where f(x) is a real number, since the space of left invariant
w-forms has dimension one. Because Rx* 0 Ry* = Ryx*, we have f(xy)
= f(x)f(y). (Real numbers commute!) Now f(x) never vanishes since
1 = /(e) = /(#)/(#" *)• Thus / maps G into the reals R with 0 removed.
Since G is compact the image of G under / is a bounded interval in R,
bounded away from zero. It is also a subgroup of the multiplicative group
of positive reals since / preserves multiplication. [Positive because the
image/(G) of the manifold is an interval and it contains 1 =f(e).] If/(G)
contains any real number a^l, then an —> oo or an —► 0, both impossible,
since an must remain in the interval/(G) which is closed under multiplication.
Hence /(G) consists of 1 alone, f(x) = 1 for each a; in G, Rx*co — oo, co is
bi-invariant as asserted. (We have used the fact that G is a manifold, hence
connected, to conclude that /(G) is an interval.)
9.6. Problems
1. Let C* denote the multiplicative group of nonzero complex numbers.
Find the invariant volume (area) element.
2. Consider the 4-dimensional group of all matrices
(z w\
o l)
where z and w are complex numbers, z ^ 0. Determine constants of
structure. Show that the left invariant volume element is
— (dz a dz a dw a dw)/\z\*.
Here dz = dx — idy if dz — dx + i dy.
3. Discuss other groups of complex matrices analogous to the examples
of Section 9.3. For example, discuss the relation between unitary matrices
and skew-hermitian ones.
4. Extend the coordinate considerations of Section 9.5 by showing that
zi = xl + yi + Y, u'jk^l/1 + (terms of order three and higher).

162 IX. APPLICATIONS TO GROUP THEORY
Show also that
are the constants of structure for a suitable basis of left invariant one-forms.
Compare xy and yx.
5. We know that each left invariant p-form can be expressed in terms of
left invariant one-forms. Does a corresponding result hold for bi-invariant
forms ?
6. Let cljk be constants of structure of a group and set
i,k
Show that gjt is a symmetric tensor. Now set
cUk — L, c U^ik -
i
Show that cijk is a skew-symmetric tensor.
7. Let z be a p-cycle on G. Show that for each closed £>-form co and
each g in G,
(We must assume that G is connected, i.e., consists of one piece only.)

X
Applications to
Physics
10.1. Phase and State Space
We propose to study a holonomic mechanical system with a finite number
of degrees of freedom, avoiding collision phenomena. In this section we
formulate the geometry of such a system.
The position space is simply an ^-dimensional manifold M.
We next define the phase space attached to M. This is the space of all
covariant vectors at all points of M. To make this precise, we consider a
coordinate patch U on M with local coordinates
q1,'-', f-
At a point P of U, a covariant vector is simply a one-form at P, hence is
given by its components
Pi > *'' > Pn > Pi real
(where the one-form itself is £ Pidq1).
If
is another local coordinate system valid at P, then the components of the
same covariant vector with respect to the ql are
The totality of all such covariant vectors at all points of M constitutes the
(2n)-dimensional phase space P. To each coordinate neighborhood U on M
with local coordinates q1, • • • , qn corresponds the coordinate neighborhood
U ^ Ew with local coordinates
q\ •- ,qn,Pi, -' ,Pn-
It follows that the one-form
a = Y^Pidqi
is a one-form on P, entirely independent of local coordinates. We have
da = Y^dPidq1
163

164 X. APPLICATIONS TO PHYSICS
so that the phase density (see Section 2.3)
dpt • • • dpndq1 • • • dqn
is a 2n-form on P, never zero, defined by
±(da.)n = (n\)(dpl---dpndql---d<f),
and serves us as a volume element on P.
We shall derive some useful relations from the transformation of
coordinates
SqJ (i = 1, • • •, n)
valid on the overlap of local coordinate neighborhoods U and 0.
First we differentiate with respect to ql the relation
d2q> dqi dq> dV df
L 8qJ dq' dqk L dqJ 8qk dqr dq'
We multiply by dqs/dql and sum:
ydf d2g< dgJ ay ay
L dq1 dqJ dq1 dq" dq* flf dq'
We permute indices according to
(s r k I i j\
j r i his)
to write this relation as
ay dqr _ v dqJ d2ql (hf
2- As> 3«>- 3„fc — 2- fl*.
dq'dq'df ^8q'dqsdqk8q
Next we differentiate the formula for pt with respect to g* and substitute in
this last relation:
^Pi _ y dy dqr
dqk~ LPjdq-idq-r8jc
a2?' 5j*
-I ft
3^ ftj* 5g£

10.2. HAMILTONIAN SYSTEMS
165
But
hence
v . 8ql
dq{ LP' 8qkdqsdqi'
We have derived the important symmetry relation
* + **.- 0.
dqk dql
From
it is evident that
so in total we have
dpt dqJ
dpj ~ dq''
'd£i
dq*
dq1
\dpi = dq*_
[dpj dq' '
Finally, the state space is the product
s = pXe1
a (2n + 1) dimensional space. We think of E1 as the time axis. Local
coordinates for S are
qx
,?", Pl,-",Pn> L
10.2. Hamiltonian Systems
We wish to consider a dynamical system in Hamiltonian form. We begin
by tracing the evolution of this from Lagrange's equations of motion, which
in Euclidean coordinates reduce to Newton's law of motion. We deal only
with conservative holonomic systems.
The treatment first of all is local. We deal with a coordinate patch in
g1, • • • , qn space. For each instant of time, there is a point (position)
(ql(t), • • • > fit))
which represents the trajectory of the system.
u — dujdt. The kinetic energy is a function
T(q\---,q",q\ ■■•,?)
As is customary, we set

166 X. APPLICATIONS TO PHYSICS
which is supposed to be a positive definite quadratic form in the variables ql.
The potential energy is a function
V=V{q19---,f9t)
and the Lagrangian function, or kinetic potential, is
L = T - V.
The differential equations of motion are then
d (dL\ dL
For the first term we have
It W) ~ 8qldt + dqldqj Q + dql dqk 9
so that the Lagrange equations are a system of n second order ordinary
equations for the unknowns q1, • • • , qn. We now convert these to a system
of 2n first order equations in 2n unknowns.
We introduce the generalized momentum components
by
_ dL _ OT
Pi~dqri~"dqri'
Because the quadratic form T is definite, the transformation of variables
to1, • • •,qnA\ - • • >in)<—> to1, • • •,qn,Pi, • • • ,Pn)
is a smooth one both ways.
To reach the Hamilton form, we shall follow tradition and use a rather
confusing notation. The matter was better expressed in Section 3.5.
The function T is always considered as a function of the 2n variables
tf1, • ••,$", g1, •••,$".
The function V which involves the ql (and t) alone may be considered as a
function on the space of variables q1, • • • , qn, q1, • • • , qn, t or on the space of
variables q1, • • • , qn, px, • • • , pn, t.
We introduce the Hamiltonian
always considered as a function on the space of variables

10.2. HAMILTONIAN SYSTEMS
Since T is homogeneous quadratic in the ql we have
dT
hence H = 2T - L = 2T - (T - V),
H=T + V,
and so H represents total energy.
From
167
we have
But
Subtracting,
From this,
2T = ZPiq>
2dT = YlP>dqi + Z4qidpi.
dT
= -I^i^' + I «'4Pi-
Thus
/'air _ az, _ <* /di,\ _
5? ~ ~ dp ~ ~ * W/ *"
This gives us the equations of motion in Hamilton, or canonical, form:
( ., _dH_
9 ~ dPi
(t = 1, •••, w).

168 X. APPLICATIONS TO PHYSICS
We next check what functions H qualify as Hamiltonians. We write
T=hIctij(q)qiqJ
where ||a^(<z)|| is a symmetric positive definite matrix function of the position
variables q. It is convenient to set
ll&y(ff)ll = KteJir1,
also symmetric, positive definite. Then
8T
which we invert to
This gives us
T=il4aiJbikbJ'pkPl
From this,
B(q,P,t) = iY.biJ(q)pipJ+V(q,t).
This shows us the form of any Hamiltonian function.
We now wish to formulate Hamiltonian mechanics globally. To discover
the correct approach, we compare two Hamiltonian systems
T-V 1
i^iJ(q)PiPj\
?% t)
dH
dpi
dql J
which are supposed to be defined on intersecting regions U, 0 and be
equivalent on the common part U pj 0 of U and 0. The coordinate
transformation
(q, P> t) < ► (q, p, t)
is given as in Section 10.1:
- v &
H=T- V
T = h"Lbii(<i)PiP}
v = Via, t)
.. dH
dpt
8H
~H
T
V
il
Pi

We have
Hence
10.2. HAMILTONIAN SYSTEMS
L Pj Bfi q L8gkq
~Ldqkdpk~Ldqk P'-
and so
on the common part of U and 0, hence also
H - H = ? - V.
Using the symmetry relation,
dqk dql
derived in the last section, and the remaining equations of
^ 8ql dpj ^ 8ql dq*
_ _8H_
~dq~1'
8H _dH_
~8q7i~W
From this,
dql
It follows that V — V is a function of t alone,
V=V+f(t),
H = H +f(t),
this taking place on the intersection of U and 0.

170
X. APPLICATIONS TO PHYSICS
Having this, we can formulate what we mean by a global Hamiltonian
system.
We begin with a position space M and form its derived spaces, phase
space P, and state space S. We are give a function T on P such that over
any local coordinate neighborhood U on M with coordinates ql, • • •, qn we
have
a positive definite quadratic form. Here (q1, • • • , qn, pt, • • • , pn) are the
derived coordinates on the neighborhood U Y E" of P lying over U.
Now let Ua, U^, • • • denote the various local coordinate neighborhoods
on M. For each one of these Ua we have a function
V«=Va(q,t)
defined on
Whenever Ua overlaps U^, then on the common part Ua n U^ we have
r.-v, =/„((),
a function of t alone. (Clearly /a/? + ffia = 0, and
fa, +ffiy +/y« = ° on UanUpn Uy.)
On the part of state space S lying over Ua, i.e., on Ua ^ Ew ^ E1, set
HX=T+VX.
The equations of motion are given by
I*
dH
dpi
dH
Ikp
on Ua^ En X E1, where (ql) are local coordinates on Ua. These are
independent of the local coordinate systems (consistent) and define a motion
(or flow) on all of S, always moving forward in the t, or time, direction.
Remark. It is customary in mechanics to exhibit the potential function
so we have set down the functions Va in our formulation. Actually the
equations of motion merely require the gradient of the potential which is an
intrinsic quantity on S. Precisely, there is a differential form w on S so
that locally

10.3. INTEGRAL-INVARIANTS
171
We can free ourselves altogether of reference to the functions Fa by requiring
that there be given a one-form w on S satisfying
(1) w is free of dt,
(2) dm
= 0.
dt = 0
By the converse of the Poincare Lemma (Sections 3.6 and 3.7, especially the
last remark on p. 31) this implies the existence of the functions Fa.
A trajectory of the motion is any particular solution of this system of
differential equations. From the theory of ordinary differential equations
we know that there is a unique trajectory through each point, so that state
space S is smoothly filled with these curves. Along each one, t steadily
increases, but not necessarily to arbitrarily large values. (For example, a
particle may run off to infinity in finite time.)
Finally we note the energy law:
Along any trajectory,
dH _ dH
~dt ~ ~dt '
For
dH . ^dH . ^dH dH
^ = * = L^ + %^ + ^
dH
dt '
Remark. Whether or not the functions fap(t) can be removed altogether
by redefining each Fa so that there will be a single potential function V on all
of M Y E1 depends on two things, the manner in which the applied forces
vary with time, and the topology of M.
The topological difficulties are easily seen from the standard example of
the steady magnetic field in the manifold M, which consists of E3 minus the
z-axis, due to a steady electric current in the z-axis. The trouble is that
there are closed loops in M which are not boundaries of surfaces. (Cf. the
situation in De Rham's theorems, Section 5.9.)
10.3. I ntegral-i nvariants
Over a local coordinate neighborhood U = Ua in position space M we
consider the one-form
0) = a>a = Y^Pidq1 — Hdt.

172
X. APPLICATIONS TO PHYSICS
This is denned on the portion U X E")( E1 of state space which lies over
U. These forms coa do not necessarily fit together to make a one-form on S
because on an intersection Ua n U^ we have
If the Va can be chosen so that all/a/,(£) = 0, then we do have such a one-form
on all of S. This is exactly the case for a globally conservative system, the
case in which the external forces are derived from a single potential function
F. While this cannot be expected in general, we do see that
da> = dcoa = £ dpidq1 — dHdt
is a 2-form on all of S, independent of local coordinates. This simply means
that
da>a = d(Dfi
on Ua n Up, which is true because
dHp dt = d(Ha - fap) dt = (dHa - fap dt) dt = dHa dt.
We shall call this 2-form da>, even though there is no one-form co on all of S
which it is the "d" of.
Suppose we have on some portion of S an r-parameter family of solutions
of the equations of motion. Let us denote by #x, • • • , xr the parameters.
What this means is that we have a mapping </> on a region W of (t, x) space
of the sort indicated, a cylinder in the t direction with top and bottom
curved r-chains,
<j>: w—>S
x
trrzzi

10.3. INTEGRAL-INVARIANTS 173
where in local coordinates
Pi = 9i(t, xl9">9xr)
t = t.
For each (xt), this represents a trajectory, hence
The mapping <f> is supposed smooth and one-to-one, so that an
(r-f-1)-dimensional region in S is evenly filled up by these trajectories.
Now we compute </>* (dco). We have
doo^Ys dVidq1 - £ —. dqldt - £ t— dpi*
Now
and similarly
so that
which establishes our first point, <£* (dco) w independent of dt.
Since d(dco) = 0, we have d[<j)* (dco)] = 0. But
dAjk
d[(j)* (dco)] = £ dtdxjdxk -f (terms in dxtdxjdxk) = 0.
We conclude that
dAJk
-^7- = 0,

174 X. APPLICATIONS TO PHYSICS
each Ajk — Ajk(x) is independent of t, and we may write
</>* (do) = £ Ajk(x) dzjdxk.
A differential form a of degree s on the state space S is called an (absolute)
integral-invariant (historical terminology) if for each r-parameter family of
trajectories given by such a mapping </>, </>* (a) is an s-form on the x-space
alone (no t nor dt terms) and if in addition da — 0.
Each of the forms
dco, (dco)2, • • • , (dco)n
is an integral-invariant. For
and
0
0* (dco)s
is independent of t and dt.
d(do)f
[4>* (dco)]' = [ltAJk(x)dxJdxkY
-c,
Consider in state space S a small piece c2 of surface which is filled by a
one-parameter family of trajectories. We may describe c2 analytically by
ql = q% y)
Pi = Pi(t, y)
My) £ t S Hy)
y0^y^yi-
Our reasoning above shows that the two-form we get by substituting these
expressions for p and q in do is a two-form in y and dy only, hence vanishes.
This means in particular
J.
do = 0.

10.3. INTEGRAL-INVARIANTS
175
Next, suppose one has a piece of volume, or three-chain c3 in S which is
the span of a two-parameter family of trajectories:
ql = q% y, z)
Pi=Pi(t,y,z)
a{y, z)^t<^ b(y, z)
(y, z) in a domain D.
P
z
A
->^y
Then
3c,
£i - S0 + c2
where El5 E0 are the terminal and initial surfaces respectively and where the
lateral surface c2 is spanned by a one-parameter family of trajectories
corresponding to the parameter point (y, z) on 3D. Now
J 0c3 J c3
d(da>) = 0,

176
X. APPLICATIONS TO PHYSICS
and we showed above that
hence
do — 0,
J.
I dco = I
dco.
We have established the very striking property of the form dco:
// E0 is any 2-chain in S transversal to the trajectories, and if one displaces
each point of L0 any amount along its trajectory to form a new surface 2^ , then
dco = dco.
J El J E0
It should be clear that we have not used any special properties of dco in
proving this result so that any integral-invariant satisfies a corresponding
property.
Let a be an integral-invariant of degree r on S. Let cr be any r-chain on S
transversal to the trajectories. Let c'r be a second such r-chain so that the points
of cr and c'r may be put into one-one correspondence with corresponding points
on the same trajectory. Then
J cr J cr
It is possible to reverse our steps to prove that the property expressed in
this result actually characterizes integral-invariants. This is done in ]£.
Cartan [8].
We pass on to relative integral-invariants.
An r-form a on S is a relative integral-invariant provided da is an integral-
invariant. The basic result about relative integral-invariants is this.
r +1
*^t

10.3. INTEGRAL-INVARIANTS
177
Let a be a relative integral-invariant of degree r. Let br and b'r be two
r-dimensional boundaries which are in one-one correspondence in such a way
that corresponding points are on the same trajectory. Then
Jb„ Jbv
To prove this, we select (r + l)-chains cr+1, c'r+1 so that
br = dcr+1, b'r = dc'r+1
and do this in such a way that cr+1 and c'r+1 correspond one-one with
corresponding points on the same trajectory. Then
a = a = da = dec = a = a_.
J b'r Jflc'r+i Jcv+i Jcr+i Jflcr+i J br
The differential form
0) = YjPidtf — Hdt,
which is defined locally, is a relative integral-invariant of degree one where
it is defined. Consequently so are the forms
ojdeo, oj(da>)2, • • •, oj(dco)n
smce
d[(o(dco)r] = (do)
r+l
We shall specialize by considering chains which exist at a single instance
of time. Let a be any differential r-form and c an r-chain in S lying in a
hyperplane t = constant. Then clearly
a =J «l*-o-
*^t
0

178 X. APPLICATIONS TO PHYSICS
Let us apply this to do in particular. We have
d^\dt=o = 1LdPi(ki,
which leads to this result:
Let E0 be a 2-chain in S at t — t0 and Hj the 2-chain obtained by moving each
point o/L0 along its trajectory to time t = tl. Then
J Eo J Ei
In this result we may think of E0 and Ht as 2-chains in phase space.
If we apply the procedure to the (2?i)-form
(do)n = ± n\(dpt --dpndql • • • dqn) + Xdt
which gives us the phase density
pi = dpt - • • dpndql • - • dqn
we have the
Liouville Theorem. // a 2n-dimensional region D0 in phase space at
time t0 moves to a region Dt at time tt, then
J Di J D0
We shall close this section with a result which will be needed in Section
10.5. It shows that the integral-invariant do completely determines the
equations of motion, in itself an important mechanical principle.
Let
[Pi = Bt{t, q, p)
be a system of equations on a region of state space S which has
do — Yj dptdql — dHdt
as an integral-invariant. Then
dPt ' dq1
For let
(ql=-f% xl9 --,x2n)
be a general solution, so that do must be expressible in the terms of dxt
alone. From this, and the differential equations
A'>
dt
§H

10.4. BRACKETS 179
we deduce
dql = A*dt + A', dpt = Bidt + fjLi,
where A1, ^ are one-forms in the dxj alone, and so
dco = £ dp^q1 — dHdt
= (I ^ - E J*0* + E **' - (Z % *' + E ^ /«.) *•
Since dco is free of dt,
I (^ - ^)(#, - B,*) - I (b, + ^(itf - 4'*) - 0,
s(*'-£)*.-E(a. + £)<!' + <->*-o.
We conclude that
as asserted.
,i SH ^ dH ^
A1 - — = 0, Bt + —| = 0
3p< 5g»
10.4. Brackets
In the transformation theory of classical mechanics one uses the bracket
expressions of Poisson and Lagrange. In this section we shall show how
these expressions relate to differential forms.
Before doing this, it is a good idea to digress on the subject of Lie brackets.
We take any differentiable manifold N and recall the definition in Section
5.3 of tangent vector and the definition of vector field at the end of that
section. We may consider a vector field v on N as an operator which takes
each function on N to another function on N :
v:F°(N)—>F°(N).
If xl, • • •, xn is a local coordinate system in which v has the representation
then
v-Ia'M^,
v(/) = I«'(x)|

180 X. APPLICATIONS TO PHYSICS
shows, locally, what v does to a function/. One sees from this formula, or
from the very definition of v as the assignment of a tangent vector (directional
differentiation) at P to each point P of N, that
v(/-<7) = v(/).<7 + /.v(<7)
for any two functions / and g on N. If v and w are two vector fields on N,
we define the Lie bracket of v and w by
[V, W] = V o W — W o V.
This is another vector field on N. If in local coordinates
v = Y a1 —- , w = Y bj —.,
^ dx1' ^ dxjt
then a computation shows that
(The main point to notice is that the second partials cancel each other.)
The following algebraic identities are easily established:
(i) [v,v] = 0.
(ii) [w, v] + [v, w] = 0.
(iii) [V! + V2 , W] = [V! , W] + [V2 , W].
(iv) [[u, v], w] + [[v, w], u] +[[w, u], v] = 0 (Jacobi identity).
Now we return to mechanical systems. As before, let P be the phase
space associated to a position space M. We denote by a the differential form
on P so that
da — Y^dVidq1
as in Section 10.1.
Poisson Brackets. To each pair /, g of real functions on phase space P
we associate a new function (/, g) defined by
n{dfdg)A(da)m-1 = (f9g)(da)n.
In local coordinates
(da)--i = [(n - 1)!] £ {dp, dql) - • • (dp.dq1) • • • (dpndqn)
and
(da)n = (n\)dpt dql • • • dpndqn,

10.4. BRACKETS 181
from which we deduce the local expression for (/, g):
From the definition one derives these relations:
(i) (/,/) = 0.
(") (f,9) + i9,f) = 0.
(iii) (/^i+^) = (/^i) + (/^2).
Using [df)d(gig2) = gi(dfdg2) + g2(dfdg1), one has
(iv) (/, gtg2) = 9i • (/, g2) + 92* (/> 9i)-
The identities (iii) and (iv) taken together may be expressed by saying that
for fixed /,
9 —► (/, g)
is a vector field vf on P:
vf(9) = (/, g).
The basic connection between the Lie and Poisson brackets is given by
(y) v(/.f) = tv/'vJ-
One has
*(/..)(*> = ((/.?>.*)
and
[v,,v,](A) = v/[v,(A)]-v,[v/(A)]
= V/((flr, A)) - v9((/, A))
= (/, (flr, *)) ~ (9, (f, *))
= -(to.*),/)-((*,A?)
so that (v) is equivalent to JacobVs relation
(vi) ((/, g), h) + (to, h),f) + ((*,/), </) = 0.
One proves this relation [(v) or (vi)] either by a lengthy direct calculation,
or by the following more sophisticated argument based on the fact that a
vector field is completely determined locally by its effect on each member of a
local coordinate system.
First of all,
(/,?')
■ d{f,ql)
Ib£
dpt
1.
\8PJ
o
df
dqJ
%

182 X. APPLICATIONS TO PHYSICS
which may be interpreted as
Vqi ePl
Similarly
</,*>--35, and vpi = ^
Because of these relations (vi) easily follows when both g and h are taken from
the set of coordinate functions [q1, • • • , pn}. This means that
v(/,x) = lyf> yx\
when x is one of these coordinate functions since the vector fields on both
sides agree when applied to any coordinate function qJ or Pj. Hence for
any h,
((/, *), h) + ((*, *),/) + ((h,f), x) = 0.
This is now established for any functions / and h and any coordinate function
x. But this may bp interpreted as saying
W(h,f)X = [Vft> yf]x
so that the vector fields
must agree since they agree on all of the coordinate functions. Hence
v(fcf/)to) = [vfc,v/]flr
for all/, g, h. This completes the proof of (v) and (vi).
One applies brackets to a pair of functions on the state space S by simply
treating Usa parameter.
A function / on S is called a first integral of the equations of motion if it
is constant on each trajectory. If this is the case, then
df df . df 8f
L 8q' dp, ^ dpt dq' 8t
= <*•'>+ i
so that the partial differential equation for first integrals is

10.5. CONTACT TRANSFORMATIONS 183
Lagrange Brackets. Let
<j>: E2 —*P,
where E2 is the Euclidean plane with rectangular coordinates u, v. Then
(j)*(da) is a 2-form on E2 (where as before da — ]T dptdql) and we write
$* (doc) = [u, v]dudv
defining the Lagrange brackets which have the local expression
o(u, v)
10.5. Contact Transformations
Here we can but touch briefly on an extensive topic. For simplicity we
shall only treat the local problem and shall look on contact transformations
as coordinate changes.
First we take the case of a time independent change. As usual, let ql9 pt
denote local coordinates in phase space P. We consider new coordinates
?*> Pi>
qi = qi(q\ • • • , qni pl , • • • , p„)
Pi = Pi(ql> --,f>Pi,"-,Pn)
which are unrelated to the old except for one requirement. We set
cc = Yjpidqi, a = Y,Pid?
and require
da = da.
This defines what we shall call a homogeneous contact transformation.
Since d(a — a) = 0 and we are only working locally, the condition may be
expressed as
a = a + dcj)
where </> is a real function on P. In the relation da = da,
one replaces dpi and dq1 by their expressions in terms ofdpj and dqk to obtain
( d(pi,qi)_Q yd{p„ql) _Q yflft.g')./]
\Ldtf,f) ' Ld(Pj,Pk) ' Ld(Pj,qk) T
(These relations may be expressed in terms of Lagrange brackets.)
Similarly,

184 X. APPLICATIONS TO PHYSICS
If we have equations of motion
/
r
\.
k
dH
dpt
dH
dq'
we shall show that they transform into equations of the same type. For
suppose after changing the coordinates according to our contact
transformation we arrive at
Pi
B;
so that
dqJ dpj dpj dqJ
I =_ydp1dH dptM
{ ' L dqJ dpj L dpj dqj '
If we multiply the first by
and the second by dq1 similarly and sum, we find after some simplification
lAidfi + ZBidq^Z^dqJ + Z^dp,
= dH-dJLdt.
dt
Hence
dpi'
B,
so that the equations of motion in the new coordinates are precisely
i = dH
dpi
l dH

10.5. CONTACT TRANSFORMATIONS
185
Now we pass to the general situation. We begin with a mechanical
system
dH
on the state space S and consider a coordinate change on S:
fqi = qi{t,q1,-'-9<f,p1,~-,PH)
t = t.
We set
co = YtPidq* — Hdt
as usual. The coordinate change is called a contact transformation if there is a
function H and a function </> so that if
co = Y,Pi d? — H dt,
then
or what is the same thing,
co = co + d(j>,
dco = dco.
The first basic result is that the equations of motion in the new coordinates
are precisely
I 8Pi
L=-—
For whatever the new equations of motion are, they admit
dco = £ dpidq1 — dHdt
as an integral-invariant. But the final result in Section 10.3 tells us that
the only equations with this integral-invariant are the stated ones.
(Compare this slick proof with a direct computation!)
A particular type of contact transformation is obtained as follows. Let
be a function of 2m + 1 variables satisfying the independence condition
d24>
dxl dyJ
*0.

186 X. APPLICATIONS TO PHYSICS
Set 0 = ^, J1, •••,«", J1, •••,}").
_d$ _ _ _#
Pi " 8qi ' Vi~~ dql '
For fixed g and p, the g are determined by the first set of equations (since
the determinant above does not vanish) and then the p are determined by
the second set of equations.
We have
co - co = - (£ ft d$* - ifcft) + Q>,dg1 - J5fd»)
= + # - -^ eft + Hdt - #«&,
and so
co = co — d(j>
provided we set
The most important case is that in which cj) is a solution of the Hamilton-
Jacobi equation
H „ -, . rrL - 8<t>
t,q,q) + H\t,q,---,-^,,--\=0.
In this situation H = 0 and the new equations of motion are simply
U = o
with solutions
gl = constant, ft — constant.
The original system is said to be transformed to equilibrium.
One other point we shall notice is this. If a contact transformation is
stationary, i.e., independent of time, then it is equivalent to a homogeneous
contact transformation. For suppose
q* = q\q, p)
Pi = ftfo P)
and
(Z PM ~ Hdt) = fcPidq1 -Hdt) + cty.

10.5. CONTACT TRANSFORMATIONS
Equating coefficients:
187
p
kp
dql
8q>
'dpk~
H-.
-Pj +
dip
~ dPk
= H-
Hq1
8<j>
'Tt
Since q, p, q, p are independent of t, we deduce from the first two equations
that
d2(f)
0,
d2(f)
0.
dtdqJ dtdpk
Hence dcfr/dt is a function of t alone. This evidently implies
(l>(t,q,p)=f(t) + iHq,p)
and so
YsPiW^YsPiW + W
which means we have a homogeneous contact transformation as asserted.
Let us briefly examine what are called infinitesimal contact transformations.
We ask when
r *' = ?' +tf'
Ift = Pi + egt
is a contact transformation up to first order terms in e. We restrict attention
to the homogeneous (stationary) case. We have
I (Pi + £9i)(<k' + edf) - 5>,dg' = £#.
e(X0,<¥ + E ^ <*/') = «#.
so the condition is
If we set
this becomes
or
5>i<#+ £*«#'= **•

188 X. APPLICATIONS TO PHYSICS
We finally have
t:1 ~l
dip
dpi
Here ij/ is called a generating function for the infinitesimal contact
transformation.
10.6. Fluid Mechanics
We consider a fluid moving in a region of E3. The position vector as
usual is
x = (x,y,z) = (xl,x2,x*).
At each time t, the velocity at x is v,
v = v(t, x) = (u, v, w) = (v1, v2, v3).
The density of the fluid is a scalar
P = P(t, x).
In this section we shall denote the vectorial area element of a surface by d,
(j = (dydz, dzdx, dxdy).
(See Section 4.5, p. 43.)
If c3 is a three-dimensional region which is fixed in space, the change in
mass at each point x of c3 per unit time is
— dx dy dz
ot
and so the total time derivative of mass in c3 is
— dxdydz.
We assume conservation of matter, so this must result from flux of fluid over
the boundary, hence
\Sdxdydz=~\>
pv-ff.
dc3
By Gauss' theorem,
pv-(x= div(pv) dxdydz.
J 5c3 J c3

10.6. FLUID MECHANICS 189
By taking c3 arbitrary we deduce from the equality of these integrals the
continuity equation
dp
-^ + div(pv) = 0,
a necessary condition the flow must satisfy. We shall deduce some
consequences of this. We set
Q = p(dxl - vl dt)(dx2 - v2 dt)(dx3 - v3 dt).
To compute dQ, we set
fi = (dx1 - v1 dt)(dx2 - v2dt)(dx3 - v3 dt)
so that
dv1 (dv2 \ /dv3 \
dB = - —T dx1 dtdx2 dx3 + dx1 | —t dx2 dt dx3 - dx1 dx2\ —r- dx3dt \
H dx1 \dx2 J \dx3 J
= (div v) dt dx1 dx2 dx3,
dQ = d(pp) = dp a p + pc^yS
= (-^ * + Z Tl **') A ^ + ^(div v)(*cfal dx2dx*)
[dt ^ dx1
+ p(divv)
(dtdx1 dx2dx3).
Thus the continuity equation is equivalent to the relation
Suppose we express the flow in terms of initial conditions (or other
parameters) by
x = x(t, a1, • • • , a3)
so that the <xl are the parameters and
dx
St V'
Thus
(dx1 dx1 \
— dt + Y—.da.j\ -vldt
dt *-* d(xJ ]
d(xJ
so that
d(x\x2,x3) ^A
d(a.\ a2, a3)
^ = P \] / / \[ da1 da2 da3

190 X. APPLICATIONS TO PHYSICS
Since dQ = 0, we deduce that dAjdt = 0,
Q = A(ac)d<xld<x2d<x3.
This means that Q is an integral-invariant for the flow as explained in
Section 10.3. We consequently have the following result.
Let c3, c'3 be three-chains in the four-dimensional (t, x) space which are in
one-one correspondence in such a way that corresponding points lie on the same
trajectory of the flow. Then
J°-J.a
JC3 JC'3
In particular, if all the points of c3 exist simultaneously, i.e., at a fixed
time t0, then
Q= O|r = f0 = pdxdydz
Jc3 Jc3 Jc3
Thus if we take a region c3(0) at time t0 and follow it to c3(1) at time tx, we
have
J.
pdxdydz —
pdxdydz
C3<1>
which says that mass is preserved in the flow, another form of the
conservation of mass.
We now proceed to the dynamic situation. We suppose our fluid is
nonviscous so that the pressure is a force per unit area at each point acting
normal to any surface element through the point, always with the same
magnitude. Let
p — p(t, x) = pressure
F = F(£, x) = body force per unit mass.
Let c3 be a fixed region in space. The total acceleration of all matter in
c3 is
p — dxdydz.
I
c3 dt
At the instant of time in question, this must equal the total force on the
matter in c3 which is
I pFdxdydz — pa.
J c3 J dc3
By a variantf of Gauss' theorem,
(gr ad p) dxdydz,
P<r = (
J dC3 J C3
f The usual simple proof is to dot both sides into a constant vector a and apply Gauss'
theorem plus the fact that
div (pa.) = (grad p) • a.

10.6. FLUID MECHANICS 191
hence
lp- pf + gr&dpldxdydz = 0.
We conclude that
dv _ 1
— = F - - grad p,
at p
the Euler equation of motion.
Here the interpretation is
dv dv ~ dv ,
dt dt ^ dxl
Let us suppose that the body force F is conservative,
F = — grad V
where
V = V(t, x)
is the force potential.
We shall add the hypothesis that p and p are functionally related, i.e.,
dp a dp — 0,
as is the case, for example, with an isothermal motion. In this case we
can define a function q = q(t, x) by
so that
Jo P
aq = — .
P
The equations of motion may be written
dv
— = -grad(F + g)
or
i.e.,
We set
dt
du d „
dt ox
du du du du d
Tt+uT* + vTy + wJz=-s-*{V + q)> etc-
* = i(vv)+ V + q
= \(u2 + v2 + w2) + V + q,

192 X. APPLICATIONS TO PHYSICS
the energy per unit mass. Finally we define the vorticity
/dw dv du dw dv du\
\dy dz dz dx dx dy/
We compute
dE du du / dv dw\ du d
dx dt dx \ dx dx] dt dx
(dv dw\ / du du\
dx dx) \ dy dz]
— V^—WY]
and similarly
(dE du
1 = v£
dx dt
fdE dv
WY\
«f
dE dw
(These are equivalent to the vector formula
dv
grad# + — = v x {.)
Now we consider the differential form
CD = udx + vdy + wdz — Edt.
We have
dco = (£dydz + r\dzdx + £dxdy)
+ dt (ut dx -f vtdy + wt dz)
— (Ex dx + Eydy + Ez dz) dt,
da> = (£dydz + rjdzdx + (,dxdy)
+ dt^v^ — wrj)dx + (w£ — uC)dy + (ur\ — v£)dz].
Actually, dco is an integral-invariant so that co is a relative integral-invariant.
One sees this indirectly by making a comparison to Hamiltonian systems.

10.7. PROBLEMS
193
For if one thinks for the moment ofx,y,z,u,v,w,ta,s independent variables,
the equations of motion are
/' d \
= u ii= - —(V + q)
ox
<y = v
z — w
or
(x = dE/du
J y = dEjdv
= dEjdw
v=--(V + q)}
oy
oz
v = -dE/dx\
v= -dEjdy^
w= -dEjdz)
which makes our assertion clear. It follows that if c2, c'2 are two-chains in
(t, x) space which are in one-one correspondence so that corresponding points
are on the same trajectory, then
f d©= f
J C2 J C
dco.
In particular if c2(0) is a 2-chain in E3 at time t0 and it moves to c2(1) at
time tt according to the motion, then
1
(£dydz + t]dzdx + £dxdy)
(£dydz + rjdzdx + ^dxdy).
This is the Helmholtz theorem on conservation of vorticity. In the first
integral we must understand £ = £(t0 , x), etc., and in the second, £ = <J(^ , x),
etc. An important consequence is this further result of Helmholtz. Suppose
at a fixed time t0 , the vorticity vanishes identically. Then it always vanishes.
For by the formula above,
I
(£dydz +
0
for each 2-chain c2(1) at each time tu which evidently means that the
integrand must vanish.
10.7. Problems
1. On p. 54 and again on pp. 179-180 we defined a vector field v on a
manifold M as a mapping which takes each function / on M to another
function v(/) and satisfies the rules
v(/+<7) = v(/) + v(<7)
v(/?)=/-v(<7) + v(/).<7.

194 X. APPLICATIONS TO PHYSICS
We know that locally
v = V a\x) —..
^ v ' dxl
We have alluded several times to the duality between vector fields and one-
forms. (Cf. pp. 127, 143 for example.) Now we shall make this more
explicit by associating with each vector field v and each one-form donM
a scalar (a, v). Locally, if v has the representation above and
then we set
Show that
a ~ X ht(x)dx\
(<7,V) = X «<>)&•(*).
((Tj + <72 , V) = (Gl , V) + (<72 , V),
(<7, Vx + V2) = (<7, V^ + (<7, V2),
(g<r,v) = g-(<T,v) = (<T,gv)
for each scalar g. Show also that this operation has the intrinsic
characterization
(dg, v) = v(g).
2. Show similarly that two-forms may be paired with two-vectors by
|(*i>Vi) (*i>v2)|
(at a <r2, vx a v2) =
|(^2 , Vl) (^2?V2)
3. Now prove the formula
(da, v a w) + (<r, [v, w]) = v{(<7, w)} - w{((7, v)},
which relates all these things to the Lie bracket.
4. Combine this with the Frobenius integration theorem (Section 7.3) to
establish this result: Let vt, • • • , vr be vector fields on a neighborhood of
0 in E" which are linearly independent at each point. Suppose that for each
i and j, [vf, Vj] is a linear combination of vt, • • • , vr. Then there is a
coordinate system xl, • • • , x" of some neighborhood of 0 such that
d
v
= !>/(*) ^J (<=1. ••-.').
5. Let vx, • • • , vr be vector fields on a neighborhood of 0 in Ew which are
linearly independent at each point. Suppose that each bracket [v;, Vy]
vanishes. Prove that there exists a local coordinate system re1, ••-,#"
such that

10.7. PROBLEMS
195
6. Let ux, • • • , un be functions of x1, • • • , xn and set
the Jacobian matrix. Write J = A + B where A is symmetric and B is
skew so that A = ||ay||, B = \\bu\\
_ \/dut duA _ 1/diii duA
(i) Suppose B = 0. Prove there is a function / satisfying df = £ utdx\
(ii) Suppose ^4=0. Prove that J is constant so that the ut are linear
functions of x1, • • • , rr". (This comes up in the theory of strain.)
7. Let a;j. be n2 functions of xl, • • • , xn. Show that the integrability
conditions
d2*ij d2aik = d\j S2alk
dxkdxl dxj dxl dxkdxl dxj dxl
are necessary and sufficient for the existence of functions ul, • • • , un
satisfying
\ldui duA
Investigate n — 3 and n =2. (Due to Saint-Venant. See Love [18, p. 49].)
8. Let v be a vector field in E3. Show that there exists a function </> and
a vector field w with div (w) = 0 so that
v = grad (</>) + curl (w).
(Hint: use the mechanism of harmonic differentials. Assume that if g is
any function in E3, the equation Lap (f)=g has a solution. See Abraham-
Becker [1, pp. 37-38] and Love [18, p. 47].)
9. Consider the transformation
The problem is to give a new proof of the relation
^ + ^ = 0
d<f dq'

196 X. APPLICATIONS TO PHYSICS
derived in Section 10.1. Set
and prove
(i) £^ACV = 0
(ii) Xj = ^gJkd^
from which you conclude that gjk — gkj. Show that this is equivalent to the
desired relations.

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There is no attempt at completeness nor historical accuracy. In general
we have tried to avoid references to less readily accessible material.
The most elementary introduction to our subject is found in [5]. More
advanced introductions will be found in [7], [12], [13], and [19]; the algebraic
aspects are treated fully in [4]. Items [2], [20], and to some degree [30]
supply all one needs in topology. The principal sources of our material on
physics are [1], [8], [14], [17], [18], [21], and [22]. Further applications of
exterior differential form theory to physical science will be found in [25],
[27], [28], [29], and [31]. An important source for material on differential
equations is [6].
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metriques, Act. Sci. et Ind. 994, Paris (1945).
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traitees par la methode du repere mobile, Paris (1951).
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12. De Rham, G., Varietes differentiables,formes,courants, formes harmoniques,
Paris (1955).
197

198
BIBLIOGRAPHY
13. Goldberg, S. L, Curvature and homology, New York (1962).
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(1960), esp. pp. 105-115.
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New York (1945).
18. Love, A. E. H., Treatise on the mathematical theory of elasticity, 4th ed.,
Cambridge (1927), reprinted New York (1944).
19. Nickerson, H. K., Spencer, D. C, and Steenrod, N., Advanced calculus,
Preliminary edition, Princeton (1959).
20. Seifert, H., und Threlfall, W., Lehrbuch der Topologie, Leipzig (1934),
reprinted New York (1947).
21. Sommerfeld, A. J. W., Electrodynamics, Lectures on theoretical physics,
Vol. 3, New York (1952).
22. Whittaker, E. T., Treatise on the analytical dynamics of particles and
rigid bodies, 4th ed., Cambridge (1937), reprinted New York (1944).
23. Willmore, T. J., Introduction to differential geometry, Oxford (1959), esp.
pp. 189-192 and 202-205.
II. Papers
24. Chern, S. S., Some new viewpoints in differential geometry in the large,
Bull. Amer. Math. Soc. 52 (1946), pp. 1-30.
25. Debever, R., Espaces de Velectromagnetisme, Colloque de geometrie
differentielle, Louvain (1951), pp. 217-233.
26. Flanders, H., Development of an extended exterior differential calculus,
Trans. Amer. Math. Soc. 75 (1953), pp. 311-326.
27. Gallissot, F., Application des formes exterieures du 2e ordre a la dynamique
Newtonienne et relativiste, Annals de l'institut Fourier 3 (1951), pp. 277-
285.
28. Gallissot, F., Formes exterieures en mecanique, ibid. 4 (1952), pp. 145-297.
29. Gallissot, F., Formes exterieures et la mecanique des milieux continus,
ibid. 8 (1958), pp. 291-335.
30. Hadamard, J., Sur quelques applications de Vindice de Kronecker,
appended to J. Tannery, Introduction a la theorie des fonctions d'une
variable II, 2nd ed., Paris (1910).
31. Misner, C. W., and Wheeler, J. A., Geometrodynamics, Ann. of Physics 2
(1957), pp. 525-603. To be reprinted shortly with several related papers
by Academic Press.

Glossary of Notation
This is a list of special terminology, definitions, and symbols, especially
from the early chapters.
A. Spaces
E" Euclidean w-space.
R The set of real numbers, also considered as E *, the Euclidean line.
U, V, • • • Open sets (in Ew, or on a manifold).
L, M, • • • Vector spaces.
A p L The space of p-vectors on L.
M, N, • • • Manifolds.
Fp( U) The collection of all p-forms on U.
Cartesian product. If S and T are arbitrary sets (collections of objects),
their cartesian product is the set
consisting of all ordered pairs (s, t) where s belongs to S and
*to T.
X2S This is the cartesian product S ^ S of S with itself. Similarly
X3S = S X S X«. etc.
SnT This is the intersection of the sets S and T. For example, if
S = {1, 2, 5, 7} and T = {2, 3, 7, 9}, then SnT= {2, 7}.
I The unit interval O^t^l.
B. Functions
Mapping. A mapping is a smooth function </> from one space M to another
N. We write
0: M—->N.
Composite mapping. If </>: M—► N and \jj\ N—> P, then we may
form the composite mapping ij/ 0 <t>: M —> P. It is defined
by
(ij/ o <j))(x) = il/[(j)(x)] for x in M.
See pp. 3, 22, ff.
199

200
GLOSSARY OF NOTATION
Linear functional. A linear transformation on a linear space L to the one-
dimensional space R of real numbers.
Jacobian. If ul = ul(xl, • • • , xn)(i = 1, • • • , n), the Jacobian of this mapping
is the determinant
\dulldxj\.
<f>* The mapping on differential forms induced by the mapping <j>
between spaces, p. 23, ff.
</>* The mapping on chains induced by the mapping $ between
spaces, p. 71, ff.
C. Special symbols
%A The transpose of the matrix A, obtained from A by
interchanging rows and columns.
| A | The determinant of the linear transformation (matrix) A, p. 21, ff.
dim L The dimension of the linear space L.
gh Here H = {h1, • • • , hp], a set of indices in increasing order,
1 <; hl < h2 < • • • < hp <; n, and aH = crhi a Ghl a • • • a ahp.
H' This is the complementary set of indices. For example, if
n = 8 and H = {2, 3, 5, 6}, then H' = {l, 4, 7, 8}.
sgnrc If n is a permutation on {1, 2, • • • , n], then sgnrc = 1 if n is
effected by an even number of interchanges (of two numbers)
and sgn n = — 1 if 7T is effected by an odd number of
interchanges.
* The star operator, p. 15.
(a, P) The inner product, p. 12, ff.
d The boundary operator, p. 58.
dx1 a • • • A dxl a • • • A dxn means dx1 a • • • A dxl 1 a dxl + 1 a • • • A dx".
The circumflex indicates omission.

Index
A
Abraham, M., 195, 197
Affine connection, 143 if
Affine frame, 143
Affine group, 154
Alexandroff, P., 197
Ampere's law, 45, 80
B
Becker, R., 195, 197
Beltrami operator, 44
Bianchi identity, 131
Bi-invariant form, 160
Blaschke, W., 44, 197
Boundary of a simplex, 58
Bounding cycle (boundary), 63
Bourbaki, N., 197
Brackets, 179 ff
Buck, R. C, 197
G
Caratheodory, C, 104, 197
Cartan, £., 3, 4, 96, 99, 108, 176, 197
Cauchy problem, 4
Chains, 61 ff
Chern, S. S., 198
Chevalley, C, 197
Christoffel symbols, 129
Closed form, 67
Commutative diagram, 56
Completely integrable system, 96
Complex projective space, 73
Compound (of a transformation, matrix),
11 ff
Connection coefficients, 129, 144
Constants of structure, 151
Contact transformation, 183 ff, 185
Continuity equation (fluids), 189
Convex surface, 112
Curvature forms, 122, 144, 148
Curvature matrix, 122, 130
Cycle, 63
Cylinder construction, 27 ff
D
Debever, R., 198
Degree of a mapping, 77 ff
DeiRham, G., 197
De Rham's theorems, 66 ff
Dirichlet integral, 83
E
Electromagnetic field, 16, 44 ff
Euler equation of motion, 191
Exact form, 67
Exterior derivative, 20 ff
Exterior multiplication, 8 ff
F
Faraday's law, 45
First integral, 107, 182
Flanders, H., 198
Flat Riemannian manifold, 135
Fluid mechanics, 188 ff
Flux, 43
Frame manifold, 146
Frenet formulas, 122
Frobenius, G., 4
Frobenius integration theorem, 92 ff
G
Gallissot, F., 198
Gauss, C. F., 43
Gauss integral, 79
Gauss mean value theorem, 85
Gaussian curvature, 42, 118, 126
General linear group, 156
Geodesic, 119, 134
Goldberg, S. I., 198
Golomb, M., 198

202 SUBJECT INDEX
Goursat, £., 4, 26, 198
Grammian (Gram determinant), 12
Green's formulas, 83
Green's function, 86
H
Hadamard, J., 198
Hamilton-Jacobi equation, 186
Hamiltonian, 166
Hamiltonian systems, 165 ff
Hamiltonian equations of motion, 167
Heat equation, 90 ff
Helmholtz law of conservation of
vorticity, 193
Hodge, W. V. D., 15, 136, 138, 198
Homogeneous contact transformation,
183
Hopf, H., 197
Hopf invariant, 79
Hypersurfaces, 116 ff
I
Infinitesimal contact transformation, 187
Infinitesimal transformation, 37
Integral - invariant, 174
Integral surfaces, 92
Injection (imbedding), 53
J
Jacobi identity (relation), 180, 181
Jordan-Brouwer theorem, 77
K
Kahler, E., 4
Kinetic potential, 166
Klein bottle, 71
Kronecker integral, 77
L
Lagrange brackets, 183
Lagrangian, 166
Lamb, H., 198
Laplace expansion (determinant), 10
Laplacian, 38 ff, 44, 82
Last multiplier, 107
Left invariant form, 150
Left translation, 150
Lie, S., third theorem of, 108
Lie bracket, 180
Lie group, 150 ff
Line of curvature, 120
Linking number, 79 ff
Liouville theorem (harmonic functions),
88
Liouville theorem (phase space), 178
Local coordinate neighborhood, 49 ff
Lorentz inner product (or metric), 12,
46
Love, A. E. H., 195, 198
M
Manifold, 49 ff
Maximum principle, 85
Maxwell's equations, 16, 44 ff
Mayer, A., system of, 104
Mean curvature, 42, 118, 126
Minimal surface, 44
Misner, C. W., 198
Momentum components, 166
Monge notation, 126
Moving frames, 4, 32 ff
N
Nickerson, H. K., 198
O
Ordinary differential equations, 106 ff
Orientable manifold, 51
Orientation (linear space), 15
Orthogonal coordinates, 39
Orthogonal group, 158
Orthonormal basis, 13 ff
P
Parabolic equation, 90
Parallel displacement, 119, 133, 145
Parallel surface, 113
Periods, 66 ff
Phase density, 10, 164
Phase space, 163
Poincare, H., 4
Poincare lemma, 2, 4, 27 ff
Poincare metric (half-plane), 134
Poisson brackets, 180
Poisson integral formula, 87
Position space, 163
Potential theory, 82 ff
Poynting vector, 47
Principal curvatures, 42, 120
Principal directions (hypersurfaces), 120
Projective plane, 70

SUBJECT INDEX 203
Projective space, 73
Proper affine group, 154
Proper orthogonal group, 158
£>-vector, 5 ff
R
Relative integral-invariant, 176
Riemann curvature tensor, 122, 131
Riemannian geometry, 4, 127 ff
Right invariant form, 151, 159
Right translation, 151
S
Saint-Venant, B. de, 195
Seifert, H., 77, 198
Signature (inner product), 13
Simplex, simplices, 57 ff
Sommerfeld, A. J. W., 198
Special linear group, 157
Spencer, D. C, 198
Spherical coordinates, 34
Spherical harmonics, 142
Standard w-simplex, 60
Star operator, 15 ff
State space, 165
Steenrod, N., 198
Step transformation group, 155
Stokes' theorem, 2, 64 ff
Submanifold, 52
Support function, 114
Surfaces, differential geometry of, 40 ff,
112 ff
T
Tangent vectors, 53 ff
Tensors, 2 ff
Threlfall, W., 77, 198
Torsion, 144, 148
Trajectory, 171
U
Umbilic point, 148
Unimodular group, 157
V
Vector field, 54, 179 ff
Vorticity, 192
W
Wheeler, J. A., 198
Whittaker, E. T., 198
Winding number, 77 ff
Willmore, T. J., 198