Functions Of Several Variables [1 ed.] 9781114126862, 1114126861

Table of contents :
Front Cover
Front Flap
Title Page
Copyright
Forword
Preface
Contents
1 The Topology of R^n
1 Fundamental structure of R^n
2 Open sets, closed sets, and neighborhoods
3 Sequences
4 Compact sets
5 Continuity
2 Differentiation on R^n
6 Differentiation
7 Higher-order derivatives. Taylor series expansions
8 The inverse function theorem
9 Change of variables in multiple integrals
10 The implicit function theorem
11 Local coordinates
12 Maps of R^n into R^m
3 Vectors and Covectors
13 Vectors
14 Vector fields
15 Covectors
4 Elements of Multilinear Algebra
16 Introduction
17 Multilinear maps and the antisymmetrization operator
18 The exterior product
19 k-vectors
20 The inner product
5 Differential Forms
21 Differential forms
22 The scalar product
23 The standard m-simplex
24 m-chains. The boundary operator ∂
25 Stokes' theorem
26 Volume, surface area, and the flux of a vector field
27 Green's identities
28 Harmonic functions. Poisson's integral formula
6 Vector Fields and Differential Forms
29 Flows and vector fields
30 Frobenius' theorem
31 The operator θ_x
32 Homotopy and Poincaré's lemma
7 Applications to Complex Variables
33 Complex structure
34 Analytic coordinates
35 Analytic functions of one variable
36 Taylor series
Answers to Selected Exercises
Index of Symbols
Index
Back Flap
Back Cover

Citation preview

HARBRACE COLLEGE MATHEMATICS SERIES

Functions of Several Variables

Funct ions of Sever al Varia bles by JOHN W. WOLL, JR. University of Washington

in the HARBRACE COLLEGE MATHEMAT ICS SERIES

Salomon Bochner and W. G. Lister, Editors

This series presents textbooks for a flexible program for the undergradua te mathematics curricula. Its purpose is twofold : first, to provide basic textbook materials in compact units; second, to make available a variety of supplementary textbooks, each covering a single topic.

FUNCTIONS OF SEVERAL VARIABLES treats selected topics from the calculus of several variable s that provide the undergradua te background for graduate courses in differential geometry and complex variables. It is designed for students in the third- or fourth-year analysis program who have completed the standard freshman-so phomore calculus sequence plus an introduction to linear algebra.

The topics included in this textbook were selected according to two objectives : they cover the notions usually called "vector analysis" and they are concepts that can be easily generalized to differential manifolds in a relatively coordinate-f ree manner. Except for the funda(continued on back flap)

Functions of Several Variables JOHN W. WOLL, JR. UNIVERSITY OF WASHINGTON

Ill Harcourt, Brace & World, Inc. New York

I

Chicago

I

Burlingame

To Patricia, Holly, and Heather

@ 1966 BY HARCOURT, BRACE & WORLD, INC.

All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. LIBRARY OF CONGRESS CATALOG CARD NUMBER: PRINTED IN THE UNITED STATES OF AMERICA

66-14924

Foreword

The Harbrace College Mathematics Series has been undertaken in response to the growing demands for flexibility in college mathematics curricula. This series of concise, single-topic textbooks is designed to serve two primary purposes: First, to provide basic undergraduate text materials in compact, coordinated units. Second, to make available a variety of supplementary textbooks covering single topics. To carry out these aims, the series editors and the publisher have selected as the foundation of the series a sequence of six textbooks covering functions, calculus, linear algebra, multivariate calculus, theory of funtions, and theory of functions of several variables. Complementing this sequence are a number of other planned volumes on such topics as probability, statistics, differential equations, topology, differential geometry, and complex functions. By permitting more flexibility in the construction of courses and course sequences, this series should encourage diversity and individuality in curricular patterns. Furthermore, if an instructor wishes to devise his own topical sequence for a course, the Harbrace College Mathematics Series provides him with a set of books built around a flexible pattern from which he may choose the elements of his new arrangement. Or, if an instructor wishes to supplement a full-sized textbook, this series provides him with a group of compact treatments of individual topics. An additional and novel feature of the Harbrace College Mathematics Series is its continuing adaptability. As new topics gain emphasis in the curricula or as promising new treatments appear, books will be added to the series or existing volumes will be revised. In this way, we will meet the changing demands of the instruction of mathematics with both speed and flexibility. SALOMON BOCHNER W. G. LISTER

Preface

This book is an exposition of selected topics from the calculus of functions of several variables. It is intended for undergraduate mathematics students in the third or fourth year analysis program, who have had several semesters of the calculus and at least an introduction to linear algebra. Specifically, the prerequisites include knowledge of the real numbers and functions of one variable plus some introductory experience with multivariate calculus of the type that is usually included in the first two years of college mathematics. The linear algebra needed, which is approximately the content of Linear Algebra by Ross A. Beaumont, includes the concept of a finite dimensional vector space, some experience with the idea of a basis for a vector space, and some elementary concepts and properties associated with linear transformations, such as those of rank and determinants. Aside from the fact that the fundamental existence and uniqueness theorem for ordinary differential equations is used without proof, the results used are proved in the body of the text. The topics treated in this book were selected with two primary objectives: (1) these topics cover the notions usually referred to as "vector analysis," and (2) they cover concepts that can be easily generalized to differentiable manifolds in a relatively coordinate-free manner. The book divides naturally into three sections. The first two chapters are rather standard, treating respectively the point set topology of R" and differentiation on R". In the second chapter the inverse function theorem and the theorem on change of variables in multiple integrals are proved and several important implications are discussed in detail. The latter include the concept of local coordinates and the rank of a differentiable map from Rm to R". Basically preparatory, these two chapters constitute the theoretical foundations of the material developed in the remainder of the book. Chapters Three, Four, and Five constitute the next unit. They are basically manipulative. In Chapter Three the notion of a (tangent) vii

Functions of Several Variables JOHN W. WOLL, JR. UNIVERSITY OF WASHINGTON

Ill Harcourt, Brace & World, Inc. New York

I

Chicago

I

Burlingame

X

I

CONTENTS

CHAPTER THREE

Vectors and Covectors

47

13 Vectors

47

14 Vector fields

52

15 Covectors

53

CHAPTER FOUR

Elements of Multilinear Algebra

58

16 Introduction

58

17 Multilinear maps and the antisymmetrization operator

66

18 The exterior product

68

19 k-vectors

70

20

The inner product

72

CHAPTER FIVE

Differential Forms

74

21

Differential forms

74

22

The scalar product

78

23

The standard m-simplex

84

24 m-chains. The boundary operator a

87

25 Stokes' theorem

102

26 Volume, surface area, and the flux of a vector field

107

27

Green's identities

28 Harmonic functions. Poisson's integral formula

111

116

CONTENTS

/

xi

CHAPTER SIX

Vector Fields and Differential Forms

120

29 Flows and vector fields

120

30 Frobenius' theorem

129

31 The operator Bx

136

32 Homotopy and Poincare's lemma

139

CHAPTER SEVEN

Applications to Complex Variables

143

33

143

Complex structure

34 Analytic coordinates

146

35 Analytic functions of one variable

149

36 Taylor series

156

Answers to Selected Exercises

159

Index of Symbols

165

Index

167

Functions of Several Variables

CHAPTER ONE

Topology of R n

1

Fundamental structure of R"

n-dimensional euclidean spaceR" is the set of all n-tuples p = (a\ . . . , a") of real numbers, ak represents the kth member of the n-tuple (not a to the kth power), and the letters p and q are used to represent elements of R". R" is a vector space, two points p = (al, . . . , a") and q = (bl, . . . , b") having the sum p + q = (a 1 + b 1, • • • , a" + b"). If ~ is a real number, ~p = (~a 1 , • • • , ~a"). The length of an element p = (al, . . . , a") of R" is given by

and it satisfies the important relations

liP + qll ::;; liP II + llqll j II~PII = I~IIIPII-

The euclidean distance d(p, q) between p and q is the length

2

liP - qll-

Open sets, closed sets, and neighborhoods

The e-ball centered at q or, equivalently, the ball of radius e centered at q is the subset B.(q) of R" consisting of those points p for which d(p, q) ::;; e. B.(q) = {p E R":d(p, q) ::;; e).

q is an interior point of the set A if A contains some ball of positive radius centered at q as a subset. The set of interior points ,of A is denoted by 1

2

I

CHAPTER ONE

Topology of R"

interior (A). The set U is open if U = interior (U), while U is a neighborhood of q if q E interior (U). So a set is open if and only if it is a neighborhood of each of the points it contains. The empty set f2f does not contain any points and accordingly is equal to its own interior and open. Every point of R" is an interior point of R", so that R" is also open. A set F is closed if its complement F•, the set of points in R" which are not members of F, is open. For example, the complement of the empty set f2f is f2f• = R" which is open, so that f2f is closed. (f2f is both open and closed as is R".) A few of the more important properties of open and closed sets are established below as examples; many other properties are left as exercises. In general, of course, most sets are neither open nor closed. (2.1)

If U,, . . . , U,. are open sets, their intersection U, f"\ is open. In fact, if p belongs to u I ("'\ • • • ("'\ u"'' then for each i = 1, . . . , m there is a number £; > 0 such that p E B.,(p) C U;. The intersection of these concentric balls B.,(p) is the ball Ba(p) where 6 = minimum I£" £2, . . . , £,.}, so that p E Ba(p) C B.,(p) C U; for each i. That is, Ba(p) C U, f"\ · · · f"\ U,. and p belongs to interior (U 1 f"\ · · · f"\ U,.). Sincepwas an arbitrary point of U, f"\ · · · f"\ U,., this intersection is open.

u2

("'\

Example. • • • ("'\

um

(2.2) Example. The union of an arbitrary class of open sets is open. Let I Ua : 01 E r} be a class of open sets and let S be the union of all the sets Ua, 01 E r. If p E S, then p E UfJ for some {j E r, and since UfJ is open, B.(p) C UfJ C S for some e > 0. Thus p E interior (8) and S is open.

(2.3) Example. The intersection of any IFa:OI E r} be a class of closed sets whose to each Fa if and only if it does not belong V F~ is known to be open by Example (2.2)

class of closed sets is closed. Let intersection is D. A point belongs to any of the sets F~, 01 E r. S = and D is consequently closed.

a

A subset V of the set Din R" is called relatively open in D if for each p E V there is a ball B.(p) centered at p such that B.(p) f"\ V = B.(p) f"\ D. Corresponding to this, a subset F of the set D in R" is called relatively closed in D if D f"\ F• is relatively open in D. The consequences of these uefinitions are left to the exercises. Exercises 2.1 Show that V is relatively open in D if and only if V = D f"\ W for some open subset W in Rn. 2.2 Give an example of a set which is neither closed nor open.

A sequence of sets A 1, A 2, • • • , A,., ... is monotone decreasing or just decreasing if A,:::> A2 :::> · · · :::>A,.:::> An+!:::> · · • •

To Patricia, Holly, and Heather

@ 1966 BY HARCOURT, BRACE & WORLD, INC.

All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. LIBRARY OF CONGRESS CATALOG CARD NUMBER: PRINTED IN THE UNITED STATES OF AMERICA

66-14924

4 I

CHAPTER ONE

2.14

Topology of R"

.

(Based on Exercise 2.13.) Show that F = ('\ F 11 k. That is, show that k-1

each closed subset of R" is a G1• 2.15 For each subset A C R" let U(A) = { q E A :B.(q) r\ A is at most countable for some E > 0}. Show U(A) is a countable subset of A.

A class 'U of subsets of R" is a covering of B or covers B if V { U: U E :::>B. 'U is a subcovering of the preceding covering if 'U C 'U and 'U covers B. 'U is a finite subcovering or countable subcovering of B if the class 'U is finite or countable, respectively. 'U}

2.16 Let 'U be a covering of B and suppose each U E 'U is open. Show that 'U has a countable subcovering 'U of B. [HINT: See Exercises 2.9 and 2.11.]

3

Sequences

A sequence p1, p2, . . . , pk, . . . of points of R" converges to the point p-in symbols, limk pk = p or lim Pk = p-if and only if for each neighborhood V of p the set {k: Pk e V} is finite. In this case p is called the limit of the sequence {Pm} :_1• (The reader is cautioned that the points p1 need not differ for different values of j. It is even perfectly possible that P1 = Pk for all k.) Since each B.(p) is a neighborhood of p, lim Pm = p if and only if for each e; > 0, IIPk - Pll < e, except possibly for a finite number of k's. Stated alternatively, lim Pm = p if and only if the limit of the numerical sequence {IIPk - Pll} is zero. The concept of convergence can be phrased in another manner. The sequence {Pk} ::_1 is ultimately in the set J if and only if {k: Pk E J} is finite. In this terminology a sequence {pm}:_1 converges to the point p if and only if it is ultimately in each neighborhood of p. R" is the space of n-tuples of real numbers, so that a sequence whose kth term is Pk = (a~, . . . , a~) gives rise ton sequences of real numbers {a~, k = 1, 2, . . . }, . . . , {a~, k = 1, 2, 3, . . . }. The inequalities (3.1)

max {Ia~

- a1 1, ... , Ia~

-

a"l}

IIPk - Pll {Ia~ - a 11, ... , Ia~

~

~ n max

- a" I}

show that lim Pk = p where p = (al, . . . , a") if and only if limk a~ = ai for each j = 1, 2, . . . , n. This last observation can be exploited to reduce many properties of sequences in R" to corresponding properties for sequences of real numbers. (3.2) Example. {Pk I is a Cauchy sequence in R" if for each E > 0 the set {j:for some m ~ j, IIPm - p 111 > e} is finite. Every Cauchy sequence in R" con-

SECTION

3 Sequence&

I 5

verges. In fact an inequality like the first inequality in (3.1) shows that each of the subsidiary sequences {a~ l, . . . , {a~ l is a Cauchy sequence of real numbers. Each Cauchy sequence of real numbers converges (this is one of the basic properties of R); so lim al = a 1, • • • , lim ai: = a". According to the observation preceding this example, lim Pk = p where p = (at, ... , a").

The sequence {q,.} is a subsequence of the sequence {Pk} if q,. = Pk(m = 1, 2, . . . ) where m ~ k(m) is a map which assigns to each positive integer m another positive integer k(m) subject only to the requirement that k(m + 1) > k(m). The point q is an accumulation point of the sequence {pk} if for each neighborhood W of q the set {k :pk E W} is infinite. The sequence {pk} is eventually in the set J if and only if {k: Pk C J} is infinite. In this terminology q is an accumulation point for {pk} when and only when {pk} is eventually in each neighborhood W of q. As a further criterion: q is an accumulation point of {Pk} if and only if {pk} has a subsequence {q,.} which converges to q. To see this suppose first that {q,.} is a subsequence of {pk}, q,. = Pk };:,.1 converging to some limit a 1• Now {a~c;Jlj".. 1 may not converge, but it certainly has a convergent subsequence {a~c;c~:»}:_ 1 with limit a 2 and lim~r. a~c;ck» = a 1• Passing in this manner to successive subsequences, one obtains after n steps subsequences {a~ c...>}, . . . , {afcm>} converging respectively to a', ... , a". The observation following (3.1) then shows that lim,. Pk = p where p = (a', . . . , a"). To prove Theorem (4.4) when n = 1 suppose F c [-M, M] and {p~:} is a sequence of points of F. Define q" by 2Mq~r. = p~; M and note that 2Miq~: - q,.l = IP~r. - p,.i, so that a subsequence {P~r.w} is a Cauchy sequence and hence converges if and only if {q~r.(J) } converges. q~; E [0, 1]; so without loss of generality one may as well assume from the outset that {p~r.} is a sequence of points from the closed interval [0, 1]. Set [,. = [(j - 1)/2"', j/2"'] where j is the smallest integer, 1 ~ j ~ 2"', for which

+

Foreword

The Harbrace College Mathematics Series has been undertaken in response to the growing demands for flexibility in college mathematics curricula. This series of concise, single-topic textbooks is designed to serve two primary purposes: First, to provide basic undergraduate text materials in compact, coordinated units. Second, to make available a variety of supplementary textbooks covering single topics. To carry out these aims, the series editors and the publisher have selected as the foundation of the series a sequence of six textbooks covering functions, calculus, linear algebra, multivariate calculus, theory of funtions, and theory of functions of several variables. Complementing this sequence are a number of other planned volumes on such topics as probability, statistics, differential equations, topology, differential geometry, and complex functions. By permitting more flexibility in the construction of courses and course sequences, this series should encourage diversity and individuality in curricular patterns. Furthermore, if an instructor wishes to devise his own topical sequence for a course, the Harbrace College Mathematics Series provides him with a set of books built around a flexible pattern from which he may choose the elements of his new arrangement. Or, if an instructor wishes to supplement a full-sized textbook, this series provides him with a group of compact treatments of individual topics. An additional and novel feature of the Harbrace College Mathematics Series is its continuing adaptability. As new topics gain emphasis in the curricula or as promising new treatments appear, books will be added to the series or existing volumes will be revised. In this way, we will meet the changing demands of the instruction of mathematics with both speed and flexibility. SALOMON BOCHNER W. G. LISTER

SECTION

5

5 Continuity I

9

Continuity

A map p ~ f(p) defined on the subset D of R" with values in Rm is continuous at p E D if whenever I Pk} is a sequence of points of D converging top it is true that limkf(pk) = f(p), a statement which is usually abbreviated limf(q) = f(p). f is continuous or continuous on D if it is q-+p

continuous at every point of D. As a point of terminology, if m = 1 so that the values of f are real numbers, f is called a real-valued function or just a function. Among the most primitive examples of continuous functions on R" are the cartesian coordinate functions x 1, • • • , x" associated with the underlying product representation of R" and defined by xi[(at, . . . , a")] = ai, j = 1, 2, . . . , n. Given any map f of D into B, usually described as a map f: D ~ B, several notations are in common use. If E is a subset of D, f(E) = lf(q) :q E E} is used to denote the image of E under f. f is onto or a map of D onto B if f(D) = B. IfF is a subset of B,J- 1(F) = IP ED: f(p) E F} denotes the inverse image of F. f is one-one if f(p) = f(q) implies p = q and f is one-one onto iff is one-one and onto. In this case f has an inverse map f- 1 mapping the points of B onto those of D and described by: f- 1 (b) = p if and only if f(p) = b. Note that if such an inverse map exists, each of the two meanings attributed to f- 1 (F) defines the same subset of D. Iff: D ~ B and g: B ~ E are two maps, their composition is the map go f: D ~ E defined by go f(p) = g(f(p)). (5.1) Example. The mapf: D--+ R"' is continuous J-1 (U) is relatively open in D whenever U is open in R,.. PROOF. (~): Supposef- 1(U) is not relatively open in D; then for some open subset U of R"' there is a p E/- 1(U) such that p (l interior /- 1(U) and

B.(p) r"\f- 1 (U)

C B.(p)

f"\ D

P'

for any £ > 0. Choose for each k a point Pk E Bllk(P) f"\ D, Pk (l/- 1 (U). limk Pk = p; U is a neighborhood of f(p) and yet {k:f(pk) (l Ul includes each k > 0 and is infinite. So {f(pk) lh- 1 does not converge to f(p) and f is not continuous at p. ( 0 so that Ba(p)

r'lf- 1(U)

= Ba(p) f"\

D

it follows (see Figure 1) that {k:pk (l Ba(p)}

=

{k:pk (l Ba(p) f"\ D}

:::> {k:pk eJ-l(U) = fk:pk er~cu)J = fk:f(Pk)

e u1.

f1

Dl

10

I

CHAPTER ONE

Topology of R"

FIGURE I

The first set above is finite because limk Pk = p. The last set is then finite and since U is an arbitrary open neighborhood of f(p), limk f(pk) = f(p). (5.2) Example. Iff: D-+ R"' is continuous and F is a compact subset of D, then f(F) is compact in R"'. That is, the continuous image of a compact set is compact. PROOF. Suppose that I qk} is a sequence of points in f(F) and for each k choose Pk E F so that f(pk) = qk. IPk} has a convergent subsequence IPkl, lim,. Pk = p E F, and by continuity lim,. f(Pk) = f(p). Thus the subsequence I qk I of {qk} converges and f(F) is compact.

(5.3) Theorem. If D is a compact subset of R" and f: D __... B is continuous, one-one, and onto, thenj- 1 : B __...Dis continuous. Let {qk} be a sequence of points of B converging to the point PROOF. q. Suppose {j- 1(qk)} ;:_1 converges top. By the continuity of the mapj, J(p) = lim,.J o J- 1(qk(m)) = lim,. qk(m) = q;

so p = J- 1(q) is the only accumulation point of {j- 1(qk)} and by Example (4.1) {f- 1(qk)} converges to J- 1 (q). That is, J- 1 is continuous. For the sake of illustrating the possibilities a second proof is given for Theorem (5.3). Let g = j- 1• It suffices to show according to Example (5.1) that g- 1 (U) is relatively open in B whenever U is open in R". Now g- 1(U)

= J(U n D) = B - J(U•

fl

D).

Since U• fl Dis compact, so isj(U• fl D) by Example (5.2), and g- 1(U) = B fl W where W = [f(U• fl D)]• is open. According to Exercise 2.1 g- 1 (U) is then relatively open in B. + There is another important set property besides compactness which is preserved under continuous mappings. A set H is connected if it is not the union of two nonempty, disjoint, relatively open subsets. That is, H is connected if whenever H = J 1 u J 2, J 1, J 2 relatively open and J 1 fl J2 = f25, it follows that either J1 = 0 or J2 = fZf. The continuous image of a connected set is connected. Indeed if j: H __...

SECI'ION

5 Continuity

/

11

R"' is continuous and f(H) is not connected,

where J 1, J 2 are relatively open inf(H); J 1 n J 2 = f2f; and neither J 1 nor J 2 is empty. The same is true for the sets J- 1 (J 1) and J- 1 (J 2), so that the set H = J- 1(J 1) v J- 1(J 2) cannot be connected either. Very few additional facts about connectivity are needed for this book. With minor omissions they are: (1) The closed unit interval [0, 1] is connected. To prove this suppose J 1 and J 2 are disjoint, relatively open sets in [0, 1], and to be specific suppose 0 E J 1· If J 2 is not empty there is a point c E J 2· In this eventuality put (5.4)

b = sup {a E J 1: a

< c},

and consider the two possibilities (i) b E J 1 and (ii) b E J 2· Each of these leads to a contradiction with (5.4). Elaborating on this, note first that since J 2 is relatively open (c - 6, c

+ 6) n [0, 1] c J2

for some 6 > 0; sob ~ c - 6 < c. If b E J 1, (b - e, b + e) n [0, 1] c J 1 for some e > 0 and b is not the supremum of the set of a's in (5.4). If bE J2, (b- e, b +e) n [0, 1] C J2 for some e > 0 and b is not even a limit point of J 1, let alone the supremum of the set of a's in (5.4). (2) If each pair p, q of points of H belongs to a connected subset A,.,, of H, then H itself is connected. If H were not connected under these conditions, H = J 1 v J 2 as above with J 1 ¢ f2f and J 2 ¢ f2f. Choose p E J1 and q E J2; then A,.,,= (A,.,, n J1) v (A,.,, n J2) and these last two sets are nonempty, disjoint, and relatively open in A,.,,, violating the hypothesis that A,.,, is connected. A continuous map g: [0, 1]-+ His called an arc in H joining g(O) to g(1), and a set H is called arcwise connected if each pair p, q of points of H can be joined by an arc in H. (3) An arcwise connected setH is connected because an arc g: [0, 1]-+ H joining p and q provides a connected subset A,.,, = g([O, 1]) satisfying the conditions of (2) above. Exercises 5.1 In the following let j: A VB-+ R and /lA: A-+ R, /IB: B-+ R denote the restrictions off to the sets A and B respectively.

(a) Show that f need not be continuous if /lA and /IB are continuous. (b) If A and B are open and /lA, JIB are continuous, show f is continuous. (c) If A and B are closed and /lA, JIB are continuous, show f is continuous.

12

I

CHAPTER ONE

Topology of R"

5.2 Let f: R 2 --+ R be given by !( a b) _ ab(a - b) ' - (a2 b2)3/2

if (a, b)

+

~

(0, O);

j(O, 0) = 0.

Let fa: R--+ R be the map X--+ f(a, X) andjb: R--+ R the map X--+ j(X, b). Show that for each a, b E R the maps fa and fb are continuous but that f is not continuous. [HINT: j(ljn, 1/n) = O;j(1/n, -1/n) = 1/v/2.) 5.3 Show that if J is a connected subset of R containing a and b with a < b, then J :::> [a, b). 5.4 (Based on Exercise 5.3.) Show that each connected subset of R has one of the forms (a, b), [a, b), (a, b), (a,+ co), [a,+ co), (-co, a), (-co, a], [a, b),(- co,+ co) = R where a .:5 b. 5.5 Show that the map j: B --+ R" is continuous at p E B if and only if for each s > 0 there is a 8(p) > 0 such that llq - Pll < 8(p), q E B, implies that IIJ(q) - f(p) II < s. The 8 needed here depends in general on p.

Definition. A map f: B ~ R" is uniformly continuous on B if and only if for each e > 0 there is a 8 > 0 (that does not depend on p) such that llq - PI! < 8, q, p E B implies llf(q) - f(p)ll < e. 5.6 Show by example that not every continuous function is uniformly continuous. 5. 7 Prove that each continuous function f on a compact set B is uniformly continuous on B.

Recall that the diameter of the subset A C R" is the number diameter (A) = sup {liP - qll: p, q E A j. If q,: B

~

R", the oscillation of q, at the point p E B is

oscillation, (q,) = inf {diameter [f(B n B.(p))]: e

> 0 j.

5.8 Show that the map q,: B--+ R" is continuous at p if and only if oscillation, ( q,) = 0. 5.9 Let ,P: R"--+ R" be described by xm

0

r~>

n

=

'L arxk(p) k=l

where (ak') is a real n X n matrix. (a) Show that q, is linear, that is, q,(Xp + f.l,q} = X,P(p) + f.Lr/>(q) whenever p, q E R" and X, f.L E R. (b) Show that each linear map ,P: R"--+ R" has the above form for some matrix (a~).

(c) Under what conditions on (a~) is q, continuous? (d) Under what conditions on (a~) is q, one-one? (e) When is q, an open mapping? That is, when is it true that ,P(W) is open in R" whenever W is open in R"?

14 /

CHAPTER TWO

Differentiation on R"

The proof is based on (6.2). In fact (6.5) for some qt E Pt-IPt• and (6.4) is obtained by summing (6.5) from k k = n.

=

1 to

Formula (6.4) shows that if the first partial derivatives off exist and are bounded in a neighborhood of the point p, then f is continuous at p. In general the mere existence of first partial derivatives in a neighborhood of p does not imply continuity at p. (6.6)

Example.

The function f defined by

/(z, y)

=

r z' _ yt cos--2z1+y'

when z 1

+ y 1 > 0,

/(z, y)

=

0

when z 1

+y

1

=

0

has first partial derivatives at every point of R 1, but it is not continuous at (0, 0). In fact the sequence Pt = (1/k, 1/k) converges to (0, 0) while 0

= J(O, 0)

'# limt/ >

(!k .!)k

=

1.

Contrast this with the one-dimensional case where the mere existence of a derivative at p is sufficient to insure continuity at p.

A major consequence of equation (6.4) is the chain rule for partial derivatives. The hypothesis of this theorem concerns a map tf>: W--+ U of an open set W of Rm into the open ball U of R" and a function/ defined on U (see Figure 2). For convenience let x 1, • • • , x" denote cartesian coordinate functions on R" and yl, . . . , ym cartesian coordinate functions on Rm. (6.7) Theorem (The chain rule). Suppose that (1) the partial derivatives Dd, . . . , D,.f exist on U and are continuous at tf>(q); (2) the functions x 1 o tf>, •.. , x" o tf> have partial derivatives D;(xk o tf>)(q) at

6 Differentiation I

SECTION

q (j, k = 1, . . . , m). Then D;(f o til)(q) = (aj o af o til ~ a:x/' o q, af ay; (q) = '-' ~ (q) ax• [q,(q)],

(6.8)

ti~/iJyi)(q)

15

exists, and

(j = 1, 2, . . . , m).

k-1

Remark.

Equation (6.8) is equivalent to n

D;(f 0 til)(q) =

L

D;(xk

0

til). (q)D,J[til(q)].

A:-1 PROOF. Let e; be the m-tuple with a one in the jth position and zeros elsewhere. Since . x• o q,(q + he;) - x• o til(q) ax" o til hm h = --"-.- (q) ~0

u~

it follows in particular that lim xk

o

til(q

11-->0

+ he;)

=

xk

o

til(q), so that when

h is small all the points ,P(q + he;) together with the line segments connecting them belong to U, and formula (6.4) can be applied to give (6.9)

f

o

q,(q

+ he;)

- f

o

h

,p(q) = ~ x• o til(q '-' k=1

+ he;)

- x•

o

h

q,(q) aj [ (h)] ax• Qk '

where for the present argument the only fact needed about qk(h) is that lim Qk(h) = q (k = 1, 2, . . . , n). Taking the limit in (6.9) as h--+ 0 and

11-->0

using the continuity of ajjaxk (k = 1, . . . , n) at q gives (6.8).

(6.10)

Corollary.

+

Suppose the first partial derivatives of the function q E R" and are continuous at q. Then for

f exist in a neighborhood of each pER", (6.11)

d

(jjf(q

+ tp) 1,-o =

aj L x•(p) ax• (q). n

k=1

Apply (6.8) with q, equal to the map t --+ q + tp of R 1 into R", and the "q" of (6.8) corresponding to the point t = 0 of R 1. Note that xk o t~~(t) = xk(q + tp) = xk(q) + txk(p); so axk o til/at= xk(p). • PROOF.

One consequence of (6.11) is a multivariate mean value theorem: If

f has continuous partial derivatives in an open set V which contains the line segment joining p and q, then

(6.12)

f(p) - f(q)

=

L n

.lo=1

aj

(xk(p) - xk(q)] ax" (qo)

Preface

This book is an exposition of selected topics from the calculus of functions of several variables. It is intended for undergraduate mathematics students in the third or fourth year analysis program, who have had several semesters of the calculus and at least an introduction to linear algebra. Specifically, the prerequisites include knowledge of the real numbers and functions of one variable plus some introductory experience with multivariate calculus of the type that is usually included in the first two years of college mathematics. The linear algebra needed, which is approximately the content of Linear Algebra by Ross A. Beaumont, includes the concept of a finite dimensional vector space, some experience with the idea of a basis for a vector space, and some elementary concepts and properties associated with linear transformations, such as those of rank and determinants. Aside from the fact that the fundamental existence and uniqueness theorem for ordinary differential equations is used without proof, the results used are proved in the body of the text. The topics treated in this book were selected with two primary objectives: (1) these topics cover the notions usually referred to as "vector analysis," and (2) they cover concepts that can be easily generalized to differentiable manifolds in a relatively coordinate-free manner. The book divides naturally into three sections. The first two chapters are rather standard, treating respectively the point set topology of R" and differentiation on R". In the second chapter the inverse function theorem and the theorem on change of variables in multiple integrals are proved and several important implications are discussed in detail. The latter include the concept of local coordinates and the rank of a differentiable map from Rm to R". Basically preparatory, these two chapters constitute the theoretical foundations of the material developed in the remainder of the book. Chapters Three, Four, and Five constitute the next unit. They are basically manipulative. In Chapter Three the notion of a (tangent) vii

sEcTION

xk o q,-I, 1 ~ k

~

n,

7 Higher-order Derivatives I

17

have continuous first partial derivatives on their respective

domains. Show that

~t

n

=

L a~x: q,

(p)

ax~:kq,-1 [r/>(p)J,

m=l

~ j, k ~

n, p E W. 6.5 Generalize Theorem (6.7) to cover second-order partial derivatives of the type (iJ~f,P/iJyiiJyi)(q). 6.6 Verify that the function f of Example (6.6) has first-order partial derivatives at the origin. 6.7 Suppose thatjis a C 1function on the cubeD = f (a 1, . . . , a"): iaki < e, 1 ~ k ~ n} and that iJfjiJxk+ 1 = 0, . . . , iJjjiJx" = 0 on D. Show there is a function h depending only on k variables such thatj(p) = h[x 1(p), . . . , xk(p)], p ED. 1

7

Higher-order derivatives. Taylor series expansions

The following theorem on the interchange of the order of differentiation is seldom used in its full generality, but it does simplify both the notation and computations involving higher-order partial derivatives, and it has many important consequences. (7.1) Theorem. Suppose the function f is defined on the open set U of R" and that Dk.f, Dif, and Dk(Di.f) exist on U. Suppose further that Dk(Dif) is continuous on U. Then Di(Dk.f) exists on U and

(7.2)

q E U.

PROOF. For any function g on U, set !l.~g(p) = g(p +hem) - g(p) when h is small enough so that p + hem E U. These difference operators satisfy two easily verified identities: (1) 6.76.~ = 6.~6.~ (they commute), and (2) 6.7Dkg = Dk!l.;g when Dkg is defined on U. Furthermore according to the mean value theorem, whenever Dif exists on U,

(7.3)

where 0 ~ 8(h) ~ 1 and 8(h) depends only on h if q is considered fixed. Applying e- 1 6.~ to (7.3) and using the mean value theorem on the term e-16.~. (Di.f)[q + h8(h)ei] gives

where 0

~ r(h,

e)

~

1. Taking the limit as e- 0 in (7.4) and using the

18

I

Differentiation on R"

CHAPTER TWO

fact that

D~c(D;f)

D~c(D;!) · [q

is continuous gives

+ htJ(h)e;]

= D~c(h- 1ii~f) · (q) = h- 1 11~ · (D~cf) · (q)

and (7.2) itself follows by then passing to the limit ash--+ 0.

+

Partial derivatives of higher order than the first order are denoted in the conventional manner. For example

Many of these higher-order partial derivatives can be identified using the result of Theorem (7.1) once it is known that they are continuous on an open set. A function f is C"" on U or continuously differentiable of class k on U if the function f and all its partial derivatives of orders less than or equal to k are continuous on U. The function f is C"" on U if all its partial derivatives are continuous on U. The phrase "continuously differentiable of class k on U" or "C" on U" is also applied to a map q,: U--+ R"'. There it means that each of the functions x 1 o q,, . . . , x"' o q, is C" on U when xl, . . . , x"' are cartesian coordinates on R"'. The material in this book is not in general concerned with questions of just how differentiable a certain function or map is. Sometimes the degree of differentiability is mentioned, but where it is not the reader can assume the objects under discussion are differentiable enough to allow the computations indicated and still preserve continuity. Little is lost by assuming all functions and maps which occur to be C"" unless otherwise mentioned. In several variables Taylor's theorem with remainder is a straightforward consequence of the corresponding theorem in one variable together with Corollary (6.10). If h denotes the jth derivative of h with respect to the variable s, Taylor's theorem in one variable states that for a function h which is (m + 1) times continuously differentiable on an interval including [0, s], (7.5)

h(s)

= h(O)

+ shC1>(0) + 2!s 2 h 2 (0) + · · · + m! s"' h..9•.iD;Dif(q) ;::: 0 i,;-1 for each choice of >.. 1,

8

.•• ,

>..n.

[HINT:

Use (7.14).]

The inverse function theorem

The inverse function theorem adequately summarizes the results obtained in proving it, so that it is almost never necessary to actually use details of the proof itself. With this in mind the reader who is in a hurry can

SECTION

8 The Inverse Function Theorem I

21

learn the content of the theorem by reading it without proof and then pass on to the following sections, where some of the many consequences of this important theorem are developed. The proof of the inverse function theorem used here proceeds by induction, switching variables one at a time, so that the central argument focuses on the one-dimensional case. The heart of the argument is contained in the lemma below. (8.1) Lemma. Suppose tf>: U-+ R" is a C1 map of the open set U of R" into R" described by

(8.2)

X" o

tf> = j,

where/ is C 1 on U. If (iJf/iJx")(qo) '¢ 0 at some point q0 E U, then (i) q0 has a neighborhood A on which tf> is one-one, and (ii) tt>(U) is a neighborhood of tf>(qo). PROOF. To be specific suppose that (ajjiJx")(qo) > 0 and L is some positive number chosen so that (iJf/iJx")(qo) > L > 0. Using the continuity of iJffiJx" on U and the fact that U is open, choose e > 0 so small that the closed n-cube A(e) of side 2e centered at qo,

A(e) = {(tt, ... , t"):jti- x'(qo)l

~

e, 1

~

i ~ n},

is contained in U and (iJf/iJx")(q) 2::: L, q E A(e) (see Figure 3). tf> is one-

x"

x"

[JJ

~

(p(O)) = \ ... , f(p(O))) I

..• ,/(tAJ))

I

p = (... 't") B

.I( e)

- - 1 1 - - - - - - - • xi, ... , xn-1

- - - - i - - - - - - - - - x1, ... 'xn-1 FIGURE 3

22 I

CHAPTER TWO

Differentiation on R"

one on A(e). In fact if ,P(q) = ,P(p) for two fixed points p, q E A(e), then 1

~

k

~

n- 1,

by (8.2) and using this in (6.12) shows that 0 = x" o ,P(p) - x" o ,P(q) = f(p) - f(q) = [x"(p) - x"(q)] iJf (q') dX"

where q' is on the line segment pq. Since (iJf/iJx") (q') ~ L > 0, it follows that x"(p) = x"(q) too, and consequently p = q and cp is one-one on A(e). To prove assertion (ii) of the lemma choose 8 > 0 so small that (a) if(q) - f(qo)i ~ eL/2 if llq- qoll ~ 8 yn, and (b) 8 < e. It suffices to show that every point in the rectangular neighborhood B = {(tl, . . . , t"):jti- xi(qo)i

~

8,1

~j ~ n- 1; it" -f(qo)i ~

e;}

of ,P(qo) belongs to ,p[A(e)]. Let p = (tl, . . . , t") E B. The points p(s) = (tl, ..• , t"-1, x"(qo)

+ s),

s E [-e, e],

belong to A(e) and ,P(p(s)) = (tl, . . . , tn- 1, f[p(s)]), so that p E ,p(A(e)) if f[p(s)] = t" for somes E [-e, e]. By the intermediate value theorem f[p(s)] takes on all values between f[p( -e)] and f[p(e)]; so it suffices to show that

(8.3)

f[p( -e)]

~

t"

~

f[p(e)].

Note that iJf d iJx" [p(s)] = ds f[p(s)] ~ L

> 0,

so that f[p( -e)] ~ f[p(e)]. Inequality (8.3) can be established through the following argument: Since p(O) = (tt, . . . , t"-1, x"(q 0)), .. -1

llp(O) - qoll 2 =

L it'- x'(qo)i2 ~

(n- 1)8 2

i=-=1

and according to the choice of 8, (8.4)

if[p(O)] - f(qo)i ~

eL

2'

Since p E B, it"- f(qo)i ~ eL/2, and this together with (8.4) shows that

(8.5)

f[p(O)] - eL ~ t" ~ f[p(O)]

+ eL.

nu 1

PREFAcE

vector at p E Rn and the dual notion of covectors at pare developed. With this introduction, Chapter Four is devoted to exposition of the multilinear algebra necessary to construct and verify the properties of exterior multiplication. This chapter actually includes a little more than is needed, however, since the exterior product is constructed by antisymmetrization of multilinear forms rather than by the somewhat more elementary method of giving a multiplication table with respect to a specific basis and showing that the resulting properties of the product imply uniqueness. Chapter Five treats differential forms on Rn, k-chains, Stokes theorem, and some related integral expressions involving the metric, such as Green's identities and Poisson's integral formula for harmonic functions. Chapter Six treats the concept of a flow with velocity field X and the related derivations on vector fields and differential forms. It includes Frobenius' theorem on completely integrable systems of first-order partial differential equations and Poincare's lemma that a closed differential form is locally exact. Chapter Seven shows how the notation and ideas developed earlier can be used in the theory of functions of a complex variable. After a discussion of terminology and of the concept of an analytic coordinate system in the first two sections, the remainder of the chapter is devoted to developing some of the standard material centering around Cauchy's integral formula and power series expansions. The nature of these last two chapters is again somewhat more theoretical than manipulative. JOHN W. WOLL, JR.

Seattle, Washington

24

I

Differentiation on R"

CHAPTER TWO

FIGURE 4

or

(8.8)

X" o

q,- 1(p

+ he;)

-

X" o

q,- 1(p)

h = {

n-1

L ~: ax;x: q, [q,-1(p)]}.

ax;x: q, [,B(h)J} -1 { ~'l -

k-1

Equation (8.8) shows that

(ax" o q,- 1jax;)

(p) exists and is equal to

(8.9)

ax"a::-1 (p)

=

{ax;x: q, (q,-1(p))r1 {~;-

:t: ~:

ax;x: q, [q,-1(p)] }.

obtained by taking the limit as h - 0 in (8.8). It follows immediately from (8.9) plus the previously mentioned fact xk o q,- 1 = xk, k ~ n - 1, that q,- 1 is C1 on q,(W). To get higher-order differentiability for q,- 1 notice that

a {ax" o q, ax; ~

0

q,

_

1}

_ ~ ~

(p) -

a2xn o q, _ axi q,-1 . axi axk [q, (p)] ax; (p), 1

0

•=1

so using the C1 differentiability of q,- 1 the right-hand side of (8.9) can be differentiated provided q, is C 2 to show that q,- 1 is also C 2• This procedure can be continued to give an expression for the kth-order partials of x" o q,- 1 involving the kth and lower-order partials of x" o q, and partial derivatives of xi o q,- 1, 1 ~ i ~ n, of order less thank. In this manner it can be shown that q,- 1 is Ck so long as q, is. • With the aid of Theorem (8.6) it is now possible to establish The inverse function theorem. Suppose 1/t: U- R" is a map of the open set U of R" into R" and that the functions x 1 o 1/t, ... , x" o 1/t are

SECTION

8

The Inverse Function Theorem

I 25

> 1. If det [(ax• o if;jaxi) (q 0)] ;oil 0 at some point qo E U there is an open set W containing qo such that (i) the map !/llw: W-+ if;(W) obtained by restricting if; toW is one-one onto; (ii) if;(W) is open; and (iii) the functions x 1o (!/llw)- 1, • • • , x" o (!/llw)- 1 are Ck on if;(W). That is, the map if;: W-+ if;(W) and its inverse if;- 1 : if;(W)-+ Ware both c~c. Ck on U with k

PROOF. Let (A{) be the matrix inverse of [(ax; o if;jaxi) (qo)] and let A: R"-+ R" be the linear transformation defined by n

xk

o

L A7xi(p),

A(p) =

i=l

Put q, = A o if;. if; satisfies (i), (ii), and (iii) for W if and only if q, does; so it sufficies to prove the theorem for q,. The chief advantage of switching to q, is that

.

.

(8.10)

~

ax• 0 q, ----;}Xi (qo) =

. axm 0 "' . axi (qo) = 5j.

LJ A:,.

m=l

, q,,. be the R"-valued maps defined on U by

q,,(p) = (x'(p),

1

x"-'(p), x"

t/>2(p) = (x'(p),

1

xn-2(p), xn-1

0

q,(p)); 0

q,(p),

x"

o

q,(p));

(8.ll)

q,,.(p) = (x 1 o q,(p), . . . , x"

o

q,(p)) = ,P(p).

Since i ::::; n - k, i > n- k,

it follows immediately from (8.10) that (8.12)

1 ::::; i, j ::::; n.

Let Sk stand for the statement: "There is an open set Wk containing qo such that (i) the map tf>klw.: Wk-+ tf>k(Wk) obtained by restricting t/>~c to Wk is one-one onto; (ii) tf>k(Wk) is open; and (iii) the map (t/>~clw.)- 1 : tf>~c(Wk)-+ Wk is Ck.'' S,. is the desired conclusion for ,P; so it suffices to establish these statements by induction. S 1 follows from Theorem (8.6) because q,, has the form (8.2) of the q, in Theorem (8.6) and (ax" o q,Ifax") (q 0 ) = 1 ;oil 0. Suppose now that Sk has been established. It is evident from (8.11) that the map tPk+l o q,;': tf>k(Wk)-+ R" has the form (8.2) of the q, in Theorem (8.6) if n - k and n are interchanged in (8.2), and with this interchange tPk+l o t/>'k1

26

I

CHAPTER TWO

Differentiation on R"

will satisfy the hypothesis of Theorem (8.6) at tPk(qo) provided (8.13)

Several computations are necessary to establish (8.13). First

by (8.12), and accordingly (8.14)

Next axn-k 0 tPk+1 0 t;-;;1 [A. ( )] = ~ axn-k 0 tPk+1 ( ) iJxi 0 t;-;;1 [A. (q )) 'l'k qo LJ 8xi qo axn-k 'l'k 0 8x" k i=l n

=

" LJ

~~-k ' M.n-k

= 11

i=l

which proves (8.13). Theorem (8.6) applied to tPk+ 1 o q,;1 shows that there is an open set Uk+l C tPt(Wk) containing tPk(qo) such that (i) fPk+l o c;; 1 is one-one on Uk+1i (ii) tPk+1 o t;; 1(Uk) is open; and (iii) (tPH1 o q,;1jui+,)- 1 is Ck. Sk+l is then true with Wk+l = q,;1 (Uk), and the proof of the inverse function theorem is completed by finite induction to show that S,. is true. + Exercises 8.1 Let cp: R-+ R be the map described in terms of the cartesian coordinate x by x o cp = x 3• Show cp is one-one and cp(U) is open whenever U is open, but cp does not satisfy the hypothesis of the inverse function theorem and q,- 1 is not differentiable. 8.2 Let cp: U-+ Rm and suppose cp(U) is open in Rm. If cp is one-one and q,- 1 is differentiable (at least C1) show that det [(iJxk o cpfiJxi)(p)] ~ 0, p E U.

9

Change of variables in multiple integrals

This section uses Rome integration theory not developed in this book and can be skipped by the reader. It is designed to show how the proof of the inverse function theorem given above lends itself to establishing the

SECTION

9

I 27

Change of Variables in Multiple Integrals

well-known formula for changing variables in multiple integrals, which can be formulated as follows: (9.1)

Suppose that U is an open subset of R" and that

Theorem.

1/1: U- R" is a C1 map whose Jacobian 01/1 = det [ iJ:I;i ---axr

J(!/1) (q)

(q)

]

¢

q E U.

0,

Then (9.2)

J · · · Jh dx

1 • • •

J · · · Jh

dx" =

o

y;- 1 !J(!/I- 1)1 dx 1 •

• •

dx"

~(U)

U

for every integrable function h on U. PROOF. The proof is divided into two sections-first a section recalling some properties of the integral and then a series of steps belonging to the proof proper. The essential facts needed from integration theory are: (a) It suffices to prove (9.2) for h's which vanish outside of a compact subset D of U. This comes from the fact that if D 1 C D2 C · · · C Dk C Dk+l C · · · is an increasing sequence of compact subsets of U

.

whose union is U itself, U Dk = U, and I D• is the characteristic function k=l

of Dk, ID.(p) = 1 if p E Dk and ID.(p) = 0 if pEl Dk; then (9.3)

lim .t-oo

J · · · J hiD. dx

1 •

•

•

U

dx" =

J · · · J h dx

1 •

•

•

dx",

U

together with the fact that 1/!(Dk) forms a similar sequence for 1/!(U) and (hi D.) o y;- 1 = (h o y;- 1)I ~· If (9.2) is valid with h replaced by hiD,., passing to the limit on both sides as k - oo and using (9.3) gives (9.2). (b) The formula for changing variables in one dimension:

£ g(s) ds

(9.4)

Jw go j(t)if'(t)i dt

=

in which f is a one-one C1 function with nonvanishing derivative on W. (c) Fubini's theorem in the form which relates then-fold integral over U to an iterated integral, a one-dimensional integral followed by an integral over the projection of U onto the (n - I)-dimensional subspace {p:x"(p) = 0} in R". (9.5)

J u · Jh(s

=!

1, • . •

,

s") ds 1

•

•

•

ds"

28

I

Differentiation on R"

CHAPTER TWO

Note that the function of (s 1 , • • • , s"- 1) in brackets on the right-hand side of (9.5) vanishes whenever (s 1 , • • • , s"- 1, 0) is not in the projection of U on the set lp:x"(p) = 0}. The remainder of the proof breaks into a sequence of steps: 1°. If ~ 1 : U --+ ~ 1 ( U) and ~ 2 : ~ 1 ( U) --+ ~ 2 o ~ 1 ( U) satisfy the hypothesis of the theorem on their respective domains and the theorem is true for the maps ~ 1 and ~2, then it is valid for the map ~ 2 o ~ 1 • The details of this argument involve just two facts other than the validity of (9.2) for ~2 and ~ 1-namely, the chain rule for partial derivatives and the product rule for determinants [see Example (16.18)], which are used to show that

J(~11 o ~21)(q)

det {iJxi o ~~~ o ~21 (q)}

=

= det { ~ iJxi o ~~1 [~-1(q)] ax,. o ~21 (q)} 2 ax' iJxk i..J k=l

J(~1 1 H~2 1 (q)JJ(~2 1 Hq),

= so that

f

u

Jh dx

1

•

•

•

J · · · Jh

dx" =

o

~~ 1 IJ(~1 1 )1

dx1

>1-•(U)

f ··· f h = J · · · Jh

=

o

~~ 1 o ~2 1 IJC~1 1 ) o ~2 1 IIJ(~2 1 )1

o

(~2 o ~ 1 )- 1 IJ[(~ 2 o ~ 1)- 1 ]1 dx 1 · · · dx".

dx 1

•

•

•

dx"

>1-••>1-•(U)

(>/-••>1-•HUl

2°. It suffices to show that each point p E U has an open neighborhood UP on which the theorem is true. The argument for this is topological. Assume the open sets Up exist and, following (a), that there is a compact set D C U with the property that h o ~(p) = 0 if p fl D. The class I UP: p E D} forms an open covering of D, and (see Exercise 4.8) there is a number e > 0 such that whenever diameter (B) ~ e, BCD, the set B belongs entirely to one of the sets UP in the covering. Then it is easy to Ree that h = h1 + · · · + hm, where hk vanishes except on a set of diameter less than e-just chop D up using a small enoughrectangular mesh and suitably apportion the resulting rectangles together with their faces. By assumption the theorem will be true for the hk's and hence for h itself. 3°. Because of step 2° there is no loss of generality in assuming that U is so small that the maps ,P 1 , • • • , q,,. = A o ~ of (8.11) are one-one on U with C1 inverses, ~

= A-1 o (q,,. o q,;;-_:1 )

·

·

·

(q, 2 o r/>11)

o

q, 1 ,

and step 1° now shows that it suffices to prove the theorem for the maps

SECTION

I 29

9 Change of Variables in Multiple Integrals

c/J1, c/Jk o q,;!.h 1 < k ~ n, and A - 1• Except for a renumbering of coordinates the maps c/J1, c/Jk o q,;!.1, 1 < k ~ n, all have the form q,: V--+ R" with

(9.6) xn-l

0

X" o

q, = xn-t, cp

= j,

wheref is C1 and (iJf/iJx") (q 0 ) :F- 0 on V. As a nonsingular linear transformation A - 1 can also be written as a product of transformations of the type (9.6) mixed with transformations which permute two variables of the type x"' o q, = x"', ifm~j,k; (9.7) xi o q, = xi', xi' o q, = xi. This last remark is nothing other than the statement that A - l is a product of elementary row divisors (see the appendix to this section). 4°. It remains to show that (9.2) holds when 1/; has either of the types (9.6) or (9.7). The trivial verification when 1/; has type (9.7) is left to the reader. When 1/; has type (9.6), so does q, = 1/;-1 : 1/;(V)--+ V and

J ·v· · Jh

o

c~JIJ(q,)l dx 1

• • •

dx"

f [(

=!

}(a•:(a', ••• ,a•)oV)

X

h(sl,

I:!n (s

' s"- 1, f(s 1, 1,

•••

,

•••

'

s"))

J

s") Ids" dsl · · · dsn-l.

Using (9.4), we obtain

f[( f ... R•-•

}(l•:(a 1,

which equals

•••

,a•-•,l•)oq,(V) l

h(s 11

•••

f . . . f h dt

1 •

'

•

sn-t, t") dt"] ds 1

•

•••

dsn-l,

dt"

q,(V)

and yields (9.2) on making the substitutions q, = 1/t-t, V = 1/;(U), and q,(V) = U. + Exercises 9.1 Let cp: R"-+ R" be the linear transformation

(9.8)

cp(q) = x1{q)pi

+ · · · + x"(q)p,.

Contents

CHAPTER ONE

The Topology of Rn

1

1

Fundamental structure of Rn

1

2

Open sets, closed sets, and neighborhoods

1

3 Sequences

4

4 Compact sets

6

5 Continuity

9

CHAPTER TWO

Differentiation on Rn

13

6 Differentiation

13

7 Higher-order derivatives. Taylor series expansions

17

8 The inverse function theorem

20

9 Change of variables in multiple integrals

26

10 The implicit function theorem

33

11 Local coordinates

35

12 Maps of Rn into R"'

39 ix

SECTION

Exercise Exercise of B except Exercise

9 Appendix

I 31

Show that M(X, J) has type (9.6). Show that the matrix of M(>-., j) o B is the same as the matrix that the jth row has been multiplied by X. 7. Describe the matrix of B o M(X,J).

5. 6.

Shears. The matrix of the shear S(i, >-., k), X E Rand i ¢ k, is the same as the n X n identity matrix except that X times the kth row has been added to the ith row. Show that S(i, X, k) has type (9.6). Show that the matrix of S(i, X, k) oBis the same as the matrix of B except that X times the kth row of B has been added to the ith to get the ith row of S(i, X, k)B. Exercise 10. Describe the matrix of B o S(i, X, k). Exercise 11. Show that S(i, X, k) = M(X -t, k)S(i, 1, k)M(X, k) if >-. ¢ 0. What is S(i, 0, k)? Exercise 12. Show that M(>-., k) = P(k, n)M(X, n)P(k, n). Exercise 13. Show that det {P(i, j) I = det {S(i, X, k) I = 1 and det {M(X, J) I = X. Exercise 14-. Show that every shear, multiplication, and elementary permutation is a product of transformations of the form P(i, i + 1), 1::; i < n; M(X, n), X ¢ 0; and 8(1, n, 1). Use the results of Exercises 11 and 12. Exercise 8. Exercise 9.

According to the above definitions and exercises, in order to show that A- 1 = E 1E 2 • • • EJ where the Ek's correspond to transformations of type (9.6) and (9.7), it suffices to show that E 1E 2 • • • EJA = I, the n X n identity matrix, where the Ek's are shears, multiplications, or elementary permutations. Exercises 3, 6, and 9 show how the matrix EB differs from that of B when E has one of these forms, so that in operational language it suffices to show that the matrix A can be reduced to the identity matrix by a sequence of permutations of two rows, multiplications of a row by a nonzero scalar, and operations which add a multiple of one row to another. An effective procedure is outlined below. Step 1. Since det (A) ¢ 0, not all elements in the first column of A are zero. Use a permutation if necessary to be sure the element in the first column first row is not zero. Step 2. Using a sequence of shears of type S(k, X, 1), reduce all other elements in the first column to zero. Step 3. Again, since det (A) ¢ 0, not all elements in the second column rows 2 through n are zero. Using a permutation if necessary, make sure that the element in the second-row second-column position is not zero. Step 4. Using shears S(k, X, 2), reduce all other elements in the second column to zews. Continue this procedure until all the off-diagonal elements are zero. No diagonal element can be zero, since so far the determinant of the matrix has not been changed. Last step. Using multiplications, reduce all diagonal elements to l's.

32

I

Differentiation on R"

CHAPTER TWO

(9.9) Example. The procedure outlined above is illustrated below in a manner which demonstrates how it can be used effectively to compute the inverse of a matrix A. 3 4 -1

Multiply both on the left by 8(2, 1, 1) and then by 8(3, -1, 1) to get

[: _: J

~ :J

ud [ _:

respectively.

Multiply these on the left by M(t, 2) to get

[~

0

~ ~] ;

:] and [

: -4

-1

-3

•

0 1

Now apply 8{1, -3, 2) and 8(3, 4, 2) on the left to get

[~ ~

:] and [

!

-t

-t

0 0 1

As a final step apply 8(1, -2, 3) and 8(2, -1, 3) to get

I

~ ~ :J ~d ~ ~ [:

A-•

[

--:

=]

Recording the above steps in order gives A- 1 = 8(2, -1, 3)8(1, -2, 3)8(3, 4, 2)8(1, -3, 2)M(t, 2)8(3, -1, 1)8(2, 1, 1). Exercise 15.

Find the inverse of the matrix

using row operations as above. Exercise 16. Using the results of Exercises 4, 7, and 10, describe a method of calculating A - 1 using column operations and use this method to calculate the inverse of the matrix A in Exercise 15.

SECTION

1Q

10

The Implicit Function Theorem

I 33

The implicit function theorem

This theorem gives one answer to the question: When can the equation F(x, y) = 0 be solved for yin terms of x or, more generally, when can the system of m equations

(10.1)

be solved for the yi's in terms of x 1 , • • • , x"? Just as the inverse function theorem establishes the existence of a local inverse without ever exhibiting it, the implicit function theorem will establish that under suitable conditions the yi's can be expressed in terms of the x''s in (10.1) without showing how to effect this solution. The two theorems are almost different versions of the same theorem, each being easily established from the other. In order to match the formulation (10.1) the letters yt, ••• , ym are used in this section as other names for the cartesian coordinate variables x"+t, . . . ' xn+m on ftn+m. (10.2) Theorem (The implicit function theorem). Suppose them functions Ft, ... , Fm are Ck (k ~ 1) on the open set U C Rn+m and vanish at the point q E U where iJFI iJF1 iJy1 (q) iJym (q) (10.3)

¢0.

det aFm iJyl (q)

aFm aym (q)

Then the equations (10.1) can be solved uniquely for the yi's in terms of the xk's at q. More precisely, the point (x 1(q), . . . , x"(q)) E R" has a neighborhood V on which there are m functions jl, . . . , Jm such that yi(q) = Ji[xl(q), . . . 'x"(q)], (10.4)

1

~

j

~

m;

F'(x 1 (p), . . . , x"(p);

fl[x 1 (p), ... , x"(p)], ... , Jm[x 1 (p), ... , x"(p)])

= 0, 1 ~ i

~

m, p E V.

The functions jl, . . . , Jm are Ck on V and uniquely determined by the conditions (10.4). PROOF. This theorem will appear as a by-product of other results later on [cf. Example (11.5f)]. To prove it let 4> be the map of U--+ ftn+m

34

I

CHAPTER Two

Differentiation on R"

defined by , x"(p), F 1(p),

= (x1(p), .

q,(p)

---axr-

det {ax•oq,} (q) = det

p E U.

, F"'(p)),

1

0

0

0

0

1

0

*

*

iJFI iJyl (q)

iJFI iJy"' (q)

*

*

iJF"' iJy1 (q)

aFm iJy"' (q)

0

¢0,

so that according to the inverse function theorem there is an open set W, q E W c U, such that q,: W- q,(W) and q,- 1: q,(W)- Ware both Ck maps. For V take any connected open subset of B

=

{(at, . . . , a"): (a 1 ,

•••

,

a", 0, . . . , 0) E q,(W)}

containing (x 1(q), . . . , x"(q)) and put ji(a1, . . . , a") = yi 0 q,-1(at, .

(a 1,

, a", 0, . . . , 0), •••

,

a") E V, 1

~

j ~ m.

(10.5) ji[x 1(q), .

. ,x"(q)] = yi o q,- 1(x1(q), . . . , x"(q), F 1 (q), . . . , F"'(q)]

= yi 0 ,p-1 0 q,(q) = yi(q), F'[x 1(p), . . . , x"(p); fl(x 1(p), . . . , x"(p)), . . . , f"'(x 1(p), . . . )]

=

x"+i

0

q,

0

,p-1(p) = xn+i(p) = 0,

pEV,

so that the functions jl, . . . , f"' on V satisfy (10.4). Equations (10.5) show they are Ck on V. To establish uniqueness let g1 , • • • , g"' be another set of continuous maps which satisfy (10.4) and put

(10.6)

G(p)

=

(x 1 (p),

. ' x"(p), gl(p),

. ' g"'(p)),

pE V;

F(p)

= (x 1 (p),

, x"(p), fl(p),

. 'f"'(p)),

pE V.

ObservethatifG(p) E W, then q,(G(p)) = q,(F(p)) = (x 1(p), ... ,x"(p), 0, . . . , 0) and since q, is one-one it follows that G(p) = F(p). As a consequence the sets {p E V:G(p) = F(p)} = {p E V:G(p) E W} and {p E V:G(p) ¢ F(p)}

SECTION

Local Coordinates I

11

35

are both open. They are disjoint and cover V, so that one of them must be empty because V is connected. By (10.4) G(x 1 (q), . . . , x"(q)) = F(x 1 (q), . . . , x"(q)). Thus the second set is empty, G(p) = F(p), p E V, and uniqueness has been established as asserted. + Exercises 10.1 Under what conditions can the system

+ a~x 2 + b~y 1 + b~y 2 = a~x 1 + a~x 2 + b~y 1 + b~y 2 = a~x 1

0, 0

be solved for the y's in terms of the x's? How is this related to Cramer's rule? 10.2 Use the implicit function theorem to prove the inverse function theorem.

11

Local coordinates

The inverse function theorem has many applications, among the most useful of which is the freedom it pr mits in choosing a suitable system of local coordinates. (11.1) Definition. A family zl, . . . , z" of Ck differentiable functions on the open set U in R" forms a (Ck) local coordinate system for U if (i) The map z: p---+ (z 1(p), . . . , z"(p)) of U---+ R" is one-one. (ii) D 1z.1(q)

det [

D1z."(q)l

·

.

D,z 1 (q)

D,.z"(q)

~

0,

qE U.

Several consequences of this definition are immediate: (a) From (ii) and the inverse function theorem it follows that z.( U) is a neighborhood of z(q) for each q E U and hence that z(U) is open in R". (The map z plays the role of the map q, in the inverse function theorem. Note x' o q, = z'.) (b) The map z- 1: z(U)---+ U is C"'. Because of (a) and (b) whenever his a continuously differentiable function on U, h o z- 1 is a continuously differentiable function on z( U) and it is customary to use the notation (see Figure 5) (11.2)

iJh iJzk (p)

to mean

Dk(h o z- 1)[z(p)].

X

I

CONTENTS

CHAPTER THREE

Vectors and Covectors

47

13 Vectors

47

14 Vector fields

52

15 Covectors

53

CHAPTER FOUR

Elements of Multilinear Algebra

58

16 Introduction

58

17 Multilinear maps and the antisymmetrization operator

66

18 The exterior product

68

19 k-vectors

70

20

The inner product

72

CHAPTER FIVE

Differential Forms

74

21

Differential forms

74

22

The scalar product

78

23

The standard m-simplex

84

24 m-chains. The boundary operator a

87

25 Stokes' theorem

102

26 Volume, surface area, and the flux of a vector field

107

27

Green's identities

28 Harmonic functions. Poisson's integral formula

111

116

SECTION

11

Local Coordinates I

37

which is not zero since neither of the factors on the right-hand side of (11.4) is zero. + (II.5) Example. Consider the set S = fp E R":Fl(p) = 0, . . . , F"'(p) = 0} of simultaneous zeros of the m functions F 1, • • • , F"', and suppose that at the point q E S the m X n matrix

iJFl (q) iJxl

.

(11.6)

iJFm iJxl (q)

iJFm () dX" q

has rank m. (If this last condition is not fulfilled Scan be quite complicated at q.) Without loss of generality one can as well assume that the m X m submatrix consisting of the last m columns of (11.6) has a nonzero determinant. Then it is easily checked that the functions zl = xl, . . . ' zn-m = xn-m, zn-m+l = F 1, • • • , z" = F"' form a local coordinate system in some open neighborhood U of q and (11.7)

U (\ S =

fq

E U:zn-m+l(q) = 0, . . . , z"(q) = 0}.

If, as usual, z: U--+ R" is the map p--+ (z 1(p), . . . , z"(p)), z(U f\ S) is just the slice cut from z(U) by the linear subspace of points whose last m coordinates are zero. These arguments lie very close to the implicit function theorem. Since xk(p) = zk(p) if 1 :::;; k :::;; n - m,

xi(p) = xi o z- 1(zl(p), . . . , z"(p)) = xi o z- 1(x 1(p), . . . , x"-"'(p), zn-m+l(p), . . . , Z"(p)},

p

E U,

which becomes xi(p) = xi o z-I(x 1(p), . . . , x"-"'(p), 0, . . . , 0),

and solves F'(x 1, xl, . . . 'xn-m,

••• ,

x") = 0, 1 :::;; i :::;; m, for xn-m+I,

p E U (\ S, . . , x" in terms of

We turn now to specific examples of local coordinate systems. Four examples are given (besides cartesian coordinates themselves)-namely, polar coordinates, cylindrical coordinates, spherical coordinates, and a special set of coordinates adapted for problems with toroidal symmetry. These serve to illustrate the possibilities and furnish a base on which later examples can be built. Each of these examples except for polar coordinates arises in the same manner as an inverse of a map h: R 3 - R 3 restricted to an open subset U of R 3 on which his one-one. To fit the usual notation it is convenient to take two copies of R3, namely R! and Rg, and to regard h as a map h: R!- R:. x 1, x 2 , x3 denote cartesian coordinates on Rg, while

38

I

CHAPTER TWO

Differentiation on R"

the letters used to designate cartesian coordinates on R! vary with the example. In the case of polar coordinates h: R!--+ R: but other details are similar. (11.8) Example (Polar coordinates). Let r, 8 be cartesian coordinates for R! and let h: R! --+ R~ be the map defined by

x1 o h

= r cos 8;

x2 o h = r sin 8.

h is one-one on the subset U = {q E R!:O< r(q), -r < 8(q) < rl and the functions roh- 1, 8oh- 1 defined on h(U) = {p ER::o < x 1 (p) or x 2 (p) ~01 are known as polar coordinates on R 2 .

(ll.9) Example (Cylindrical coordinates). Let r, 8, z be cartesian coordinates for R! and let h: R!--+ Rf be the map defined by

x1 o h

= r cos 8;

x3 o h

x2 o h = r sin 8;

=

z.

his one-one on the subset U = {q E R!:O < r(q), -r < 8(q) < rl and the functions r o h- 1, 8 o h- 1, z o h- 1 defined on h(U) are known as cylindrical coordinates on R 3•

(11.10) Example (Spherical coordinates). Let p, 8, for R! and let h: R! --+ R: be the map defined by

q, be cartesian coordinates

x 1 o h = p cos 8 sin q,; x 2 o h = p sin 8 sin ,P; xs o h = p cos q,. his one-one on the subset U = {q E R!:O < p(q), - r < 8(q) < r, 0 < ,P{q) < r} and the functions p o h- 1, 8 o h- 1, q, o h- 1 defined on h{U) are known as spherical coordinates on RB. (ll.ll) Example. Let w, 8, fJ be cartesian coordinates for R! and let h: R! --+ R: be the map defined by (b > 0)

+ w cos fJ) cos 8; (b + w cos fJ) sin 8;

x1 o h

= (b

x2 o h

=

x3 o h = w sin fJ. h is one-one on the subset U = { q E R!: 0 < w(q) < b, - r < 8(q) < r, - r < fJ(q) < 1r I and the functions w o h- 1, 8 o h- 1, fJ o h- 1 form a set of coordinates for h(U) particularly suited for studying the tori generated by rotating circles in the x 1,x3-plane centered at x 1 = b, x3 = 0 about the x3-axis. In fact 8 is the variable 8 of cylindrical or spherical coordinates; w is the distance from the circle about which these tori center; and fJ is the angle of elevation or depres-

SECTION

12

Maps of R" into Rm

/

39

sion above the x 1,x 2-plane measured looking outward from the nearest point on this central circle.

The h of Example (11.11) is also one-one on the subset 'II" = I q E R! :0 -71" < fJ(q) < 71", -71"/2 < {3(q) < 71"/2} and the functions w o h- 1 , fJ o h-', {3 o h- 1 form a set of coordinates for h(V) which is the exterior of the circular cylinder of radius b whose axis of symmetry is the x3-axis. In particular if b = 0, then wh- 1, fJh- 1 , h- 1 + ('11"/2) are just the ordinary spherical coordinates for R 3•

< w(q),

Exercises ll.l Suppose that u and v are C1 functions on the open subset U of R 2 which satisfy

au

av

ax' = ax2;

au

av

az2

ax 1

Show that whenever one of the above first partial derivatives is not zero each point p E U has an open neighborhood on which u, v form a local coordinate system. ll.2 Let f be C1 on U and suppose p belongs to the surface described by f = 0 in U. Show that the functions z1 = j, zk = xk, 1 < k ~ n form a local coordinate system in some neighborhood of p if and only if (aj;ax 1 )(p) ¢ 0. What is the equation of the surface f = 0 in the coordinate system z1, • • • , z"? ll.3 Show that a map ,P: U-+ R" is Ck if and only if for each q E ,P(U) there is a Ck local coordinate system z1, • • • , z" on an open neighborhood V of q in R" such that the functions z1 o q,, ... , z" o q, are Ck on q,- 1 (V) C U.

12

Maps of R" into Rm

At the heart of any study of continuously differentiable maps q, of the open set U of R" into Rm is the whole concept of rank. There is a good reason for this: In the small any map q, behaves at q very much like the associated affine transformation, xk

o

q,(p)

= xk o q,(q)

+~ LJ

o q, ----;}Xi (q)[xk(p) - xk(q)]

axk

+ small terms,

i=1

so that it is quite reasonable to expect that the local properties of q, will reflect those of this affine transformation. The reader will notice that the rank of q, at q is by definition just the rank of this affine transformation. (12.1) Definition. Suppose that y 1, • • • , y" is a local coordinate system for the open set V and that z1 , • • • , zm is a local coordinate system for some open neighborhood of ,P(p). Then the rank of q, at pis the

40

I

CHAPTER TWO

Differentiation on R"

rank of the (n X m) matrix iJzl o q, ( ) iJyl p iJz 1 o q, ( ) iJy2 p

iJz2 o q, ( ) iJyl p

iJzm o iJyl

q, ( )

iJz 2 o q, ( ) iJy2 p

iJzm o q, ( ) iJy2 p

iJzl o q, ( ) iJy" p

iJz2 o q, ( ) iJyn p

iJzm o q, ( ) iJy" p

p

(12.2)

It is the maximum number of linearly independent columns (or equivalently of linearly independent rows) in the matrix (12.2). More specifically, the rank of q, at p is k if there is a (k X k) submatrix iJzi• o q, ~ (p)

iJzio o q, a:y;;(p)

iJzi• o q, iJyj• (p)

iJzio o q, iJyj• (p)

az••

0

q,

az••

0

q,

~(p)

ay;•

(p)

with nonzero determinant and the determinant of every (k + 1) X (k + 1) submatrix formed in this manner is zero. In particular the rank of q, cannot be greater than morn. (12.3) Theorem. Suppose fl, . . . , fk (k ~ n) are CN functions on the open subset U of R" and the map q-+ (fl(q), . . . ,jk(q)) of U-+ R" has l'ank kat the point p E U. Then the set of functions jl, . . . , fk can be enlarged to a set jl, . . . , f\ fk+ 1 , • • • , f", which is a CN local coordinate system for some open neighborhood of p. PROOF. Let x 1, • • • , x" be a local coordinate system for an open set V which contains p and is contained in U. By renumbering the xjls it can be assumed without loss of generality that

ap

iJxl (p)

iJfl' iJxl (p)

¢0.

det

ap ax" (p)

iJf"

ax" (p)

CONTENTS

/

xi

CHAPTER SIX

Vector Fields and Differential Forms

120

29 Flows and vector fields

120

30 Frobenius' theorem

129

31 The operator Bx

136

32 Homotopy and Poincare's lemma

139

CHAPTER SEVEN

Applications to Complex Variables

143

33

143

Complex structure

34 Analytic coordinates

146

35 Analytic functions of one variable

149

36 Taylor series

156

Answers to Selected Exercises

159

Index of Symbols

165

Index

167

42

I

CHAPTER TWO

PROOF.

Differentiation on R"

According to Exercise 11.3 it suffices to show that for each

q E V one can find a local coordinate system yl, . . . , y" for some open neighborhood W of q,- 1 o h(q) E U such that the functions y1 o q,- 1 o h, • • • , ylc o q,- 1 o h are CN in a neighborhood of q. This can be done as

follows: Let z1 , • • • , z" be a local coordinate system valid in some open neighborhood of h(q). By renumbering the zi's if necessary, one can assume without loss of generality that the map p-+ (z 1 o q,(p), ... , z" o q,(p)) has rank k at q,- 1 o h(q) and thus [by Theorem (12.3)] that y1 = z1 o q,, • . . , y" = z" o q, is a local coordinate system for some neighborhood of q,- 1 o h(q). But with this choice yi o q,- 1 o h = zi o q, o q,- 1 o h = zi o h is CN at q because the map h: V-+ R" is CN by hypothesis. • (12.6) Definition. A subset M C R" is called a regular k-dimensional CN submanifold of R" if each point q EM has a neighborhood U together with a map q,, such that

(i) q,: V-+ R" is a CN map of an open subset V of Rk into R". (ii) q, is one-one. (iii) q,(V) = M f1 U. (iv) q, has rank k at each point of V. It is important to observe that if r/>t, V 1 and r/> 2, V 2 are two pairs satisfying conditions (i)-(iv) above and if r/>1(V1) f1 ,P 2(Y2) ¢ JZJ, then the map r/>2 restricted to the subset W = rJ>21[q,1(V1) f1 r1>2(V2))] of V2 is CN while r/>1 both is CN and has rank k at each point of V1 (see Figure 7). This means according to Theorem (12.5) that the map

(12.7)

is CN. The concept of a differentiable manifold is based upon axiomatization of this property, but it is not pursued further here. In a descriptive sense the map q, provides a parametrization of the set M f1 U. Two different local parametrizations q, 1 and q, 2with overlapping ranges are tied together by the condition that the map r/>11 o ,P 2 is CN where it is defined. For practical purposes a regular submanifold of R" is usually described in one of two ways: (1) by giving an explicit parametrization forM or (2) as the set of zeros of a finite set of CN functions Ft, . . . , Fm. To be more explicit there is an alternative definition for a regular k-dimensional CN submanifold of R" given below as Theorem (12.8).

FIGURE 7

SECTION

12

Maps of R" into R"'

I 43

xk+t, ... , x"'

w x', ... 'x"

FIGURE 8

(12.8) Theorem. M is a regular k-dimensional CN submanifold of R" if and only if each point q EM has an open neighborhood U with a CN local coordinate system z1, • • • , z" such that

Mr. U = {p E U:

(12.9)

zH 1(p) =

0, . . . , z"(p) = 0}.

PROOF. Suppose z', . . . , z" and U exist at q as stated in the theorem and as usual let z: U--+ R" be the map p--+ (z 1 (p), . . . , z"(p)). Let

V

=

{(a 1 ,

••• ,

ak) E Rk:(a', . . . , ak, 0, . . . , 0) E z(U)}

and define the map q,: V--+ R" by q,(a 1 ,

••• ,

ak) = z- 1 (a 1 ,

.•.

,

ak, 0, . . . , 0).

• • • , ak) = (a 1 , • • • , ak, 0, . . . , 0); so q,(V) c M r. U and conversely if p E M r. U, p = z- 1 (z 1 (p), . . . , zk(p), 0, . . . , 0) = ,P[z 1 (p), . . . , zk(p)] E q,(V). Thus q,(V) = Mr. U. The other conditions of Definition (12.6) are easily checked, showing that M is a regular k-dimensional CN submanifold of R". Suppose now that M is a regular k-dimensional CN submanifold of R" and let q, q,, V, and U be as in Definition (12.6). In outlining the argument to be used here, Vis considered as a subset of R" and the map q, extended from V to a map 1/; of an open subset W of R" into U (see Figure 8). The extended map 1/; has rank nat q,- 1 (q) and the zi's can then be defined by zi = xi o 1/;- 1 in some neighborhood of q. In detail, since q, has rank k at q,- 1 (q), by renumbering coordinates in R" if necessary it can be assumed that the map p--+ (x 1 o q,(p), . . . , xk o q,(p)) of V--+ Rk has rank k at q,- 1 (q). In this case let

z o q,(a 1,

1/;(p) = ,P(x 1 (p), . . . , xk(p))

+

(0, . . . , 0,

xH 1(p),

. . . , x"(p)),

44

I

CHAPTER TWO

Differentiation on R"

when (x 1 (p), . . . , xk(p)) E V, p E R", and put W = {q E R": (x 1 (p), . . . , xk(p)) E V, 1/l(p) E U}.

A check shows that the n X n matrix

reduces to 0

0

0

0

----------------------------------r-i------------o 0 0 0 0 0 0 0 0

i0

1

and has rank n at the point q,- 1(q) because the upper left k X k matrix has rank k there. If U 0 is a neighborhood of q on which 1/1- 1 is one-one and has rank n, then the local coordinates z1 = x 1 o 1/1- 1, • • • , z" = x" o 1/1- 1 on U o satisfy (12.9). In fact if p E U 0 and ?f+i(p) = 0, 1 ~ j ~ n - k, then xk+i o l/l- 1 (p) = 0, 1 ~ j ~ n - k, and

where (al, . . . , ak) E V. That is, p = l/l(a 1 , • • • , ak, 0, . . . , 0) = q,(al, ... , ak) E q,(V). Hence p E q,(V) r1 U 0 = M r1 U o. On the other hand if p EM r1 Uo, p = q,(al, . . . , ak) = l/l(a 1, . . . , ak, 0, . . . , 0) for some (a 1, • • • , ak, 0, . . . , 0) E Wand 1 ~ j ~ n- k.

+

SECTION

12 Maps of R" into R"' /

45

(12.10) Example. If F 1, • • • , F"' are CN functions, the subset M of points q E R", where F 1(q) = 0, . . . , F"'(q) = 0 and the matrix

iJFm iJx' (q)

iJFm iJxn (q)

has rank m, form a regular (n - m)-dimensional CN submanifold of R". In fact at each point q EM the map q--+ (F 1(q), . . . , F"'(q)) of R"--+ Rm has rank m, and by Theorem (12.3) the set Ft, ... , Fm can be enlarged to a set z1, • • • , zn-m, zn-m+l = F 1, • • • , zn = Fm, which is a CN local coordinate system for some open neighborhood U of q. Theorem (12.8) then shows that M is a submanifold of R" as stated. In euclidean three-space, for example, an equation of the form F = 0 describes a regular two-dimensional submanifold or surface, S = lp:F(p) = 0}, if the derivatives iJF jiJx 1, iJF jiJx 2 , iJF jiJx 3 are not all zero at some point of S. This condition can be restated in the usual vector terminology as grad F(q) ¢ 0, q E S. Two equations F = 0, G = 0 describe a regular one-dimensional submanifold or a curve if the matrix [

iJF iJx 1

iJF iJx 2

iJF] iJx 3

iJG iJG iJG -2 -3 1 iJx

iJx

iJx

has rank 2 at each point of C = I p: F(p) = 0, G(p) = 0 }. Using vector notation this can be rephrased as grad F(p) X grad G(p) ¢ 0, p E C.

Exercises The first three exercises below illustrate some of the behavior that can be expected when the matrix (11.6) does not have maximum rank. Describe S and check the rank of (11.6). 12.1 F = (x 1) 2 + (x 2) ' - (z3) 2 on Ra; q = (0, 0, 0). 12.2 F 1 = (x 1 - 1)' + (x')2 + (z3)2 - 1; F2 = (x 1 - 2)2 + (x 2) 2 + (x 3) 2 - 4; q = (0, 0, 0). 12.3 F(p) = exp [1/(1 - IIPI! 2)] if liP II < 1, F(p) = 0 if IIPII 2:: 1. This function is actually C"' on R 3 • Its set of zeros is the exterior of the open unit ball centered at the origin. 12.4 Describe the set / (e1 cos (t), e1 sin (t)): t E RJ. Is this set a regular one-dimensional submanifold of R 2 ? 12.5 Describe the set I (t, sin (1/t)): t > 0} VI (0, t) :t E R}. Is this set a regular one-dimensional submanifold of R 2 ? 12.6 If M is any subset of R" and k any nonnegative integer, show that the set M k of points q E M which have a neighborhood U and a map cp satisfying

CHAPTER THREE

Vectors and Covectors

13

Vectors

A vector at pis an operator XP of the form

aI+ X p = b1 -ax 1 p

(13.0)

a· · · · + b" -I ax" p

(IP indicates evaluation at p) whose domain is the set of all functions which are C1 at p.

X

f

f = bl aj1 (p) + ... + bn aj (p).

p

ax

ax"

If xl, . . . , x" are the coordinate functions used in (13.0), the preceding equation shows that Xpx; = b;, so that

aJ (p). + (Xpx ") ax" XP satisfies the two relations

a, b constants,

(13.1)

and (13.2)

It is an automatic consequence of (13.1) and (13.2) above that if Cis a constant, CXPC = XpC 2 = CXpC + (XpC)C = 2C·XpC; so XPC = 0. More can be said: If J P is any operator on those functions which are CH 1 (k ~ 0) at p satisfying Jp(af

+ bg)

= aJpf + bJpg

and 47

48

I

Vectors and Covectors

CHAPTER THREE

there is a unique vector XP at p such that Jpf = Xpf whenever f is Ck+ 2 at p. Thus equations (13.1) and (13.2) are almost characteristic of vectors at p. The argument for this last assertion is based on the observation that (13.3)

f(q) = f(p)

+

,.t

1

[xk(q) - xk(p)] [

:~ (p) + Rk(q) J

where R,.(q) =

fo :~ [p + t(q 1

{

:~ (p)} dt

p)] -

whenever f is C1 on B.(p) and q E B.(p). If J P is now applied to both sides of (13.3) and it is observed (i) that Jp[f(p)]

=

Jp[xk(p)]

=

JP

[:~ (p)]

=

0

because J P applied to a constant function is zero, and (ii) that R,. is CH 1 on B.(q) when f is Ck+ 2 there, the result obtained is Jpf =

,.t

1

(Jpxk) [

:~ (p) + R,.(p)] +

,.t

1

[xk(p) - xk(p)][JPR,.]

Thus

is the only vector for which XPf = Jpj whenever f is CH 2 at p. The sum XP + YP of two vectors at p is a vector at p, (XP + Yp)f = Xpf + Ypj, and a scalar multiple aXP of a vector at pis a vector at p, (aXp)f = a(Xpf). With this addition and scalar multiplication the vectors at p form an n-dimensional vector space naturally isomorphic with the space of n-tuples (bl, . . . , b") through the association (13.0), usually called the tangent space toR" at p and denoted by TP(R"). (13.4) Example. defined by

If the curve C: ( -1, 1)

C'(O)f = df o C dt

It-o =

--+

R" is C1 at 0, the vector C'(O)

n

~ i..J k=l

dxk o C (0) iJf [C(O)) dt iJxk

is called the tangent vector to the curve C at 0. n

C'(O) = ~ dxk o C (0) _i_ i..J dt iJxk k-1

I . C(O)

Functions of Several Variables

50

I

CHAPTER THREE

Vectors and Covectors

The euclidean length of the vector XP at p E U is the number IXPI defined by (13.9) where x 1 , , x" are cartesian coordinates for U. Closely associated with the notion of length is the euclidean scalar product (or dot product) of two vectors XP and YP. This scalar product (Xp, YP) is defined by n

(13.10)

(Xp, YP) =

L (XPx;)(Ypx;) j=l

where, as in (13.9), it is essential that x 1, • • • , x" are cartesian coordinates. The relation between the scalar product and length is 1Xpl 2 = (Xp, Xp).

The angle fJ between two vectors XP and YP at p is defined by the two conditions: (13.ll)

(1) 0 ~ () ~

71'j

(2) (Xp, Yp) = IXPIIYPI cos fJ. Exercises 13.1 The vector X P is said to be tangent to the level surface S = { q :f(q) = f(p), p fixed} of the function fat p if XJ = 0. (a) Find all vectors tangent to the sphere (x 1) 2 + (x 2) 2 + (x 3)2 = 1 at the point (1, 0, 0). (b) Find all vectors tangent to the paraboloid x3 = (xi) 2 + (x 2) 2 at the point (1, 2, 5). 13.2 Suppose xi, . . . , x" are cartesian coordinates for R" and the curve C: (0, oo)-+ R" is described by xi o C(t) = t, x 2 o C(t) = (t) 2, • • • , x" o C(t) = (t)". Find tangent vector Xc(to> = (iJiiJt 1,.) to C at C(to) in terms of iJiiJxi lc(toh •.• ' iJiox" lc(to)· Calculate IXC(to>l and find the cosine of the angle

c.

between Xc and iJiiJx 3 lc· 13.3 Let xi, x 2, x 3 be a coordinate system for R 3 and yl, y 2 a coordinate system for R 2• Let the map q,: R 3 --> R 2 be given by

(a) Find the vector Z = q,. (iJiiJx 1 lo. 2,-o) and evaluate Zfwheref = yi sin y 2 • (b) Find all vectors X = a (iJiiJxi) + b (iJiiJx 2) + c (iJiiJx 3) at (0, 1, O) such that q,. (X) = 0. 13.4 Show that (iJiiJxi IP, XP) = XPx; when xi, . . . , x" are cartesian coordinates. 13.5 Suppose xi, x 2, x 3 are cartesian coordinates in R 3 and zi = xi + x 2, z2 = xi - x 2, z3 = xi + x 2 + x 3, so that the z's form a coordinate system too. Find vectors Y I, Y 2, and Y a such that (a I ()zi, Y ;) = li;;.

SECTION

13.6 (i) (ii) (iii) (iv)

13

Vectors

/

51

Prove that the scalar product satisfies

+

+

(l"p, Zp). Yp, Zp) = (Xp, Zp) (Xp (aXp, Zp) = a(Xp, Zp). (Xp, Zp) = (Zp, Xp). (Xp, Xp) > 0 unless Xp = 0.

13.7 If ui, . . . , u" is a coordinate system for R" and vectors (at p) satisfying (iJ/iJu', Y;) = 8;;, show that

Y~,

. . . , Y,. are

for each vector X at p. 13.8 Under the hypothesis of Exercise 13.7 note that iJ

n

Y,.=

l:cr..

,Yk)a-;;

u

k=1

and use this to show that

I

k= 1

(_i_, ~)

(Y,., Yk)

iJuk ou'

= 8,.;.

That is, the matrices [(L., l"k)] and [(iJjiJuk, iJjiJui)] are inverses of each other. 13.9 Show that if ui, . . . , u" is a coordinate system for R" then det [(iJjiJui, iJjiJui)] ~ 0 at p, and use this to show that there exist vectors YI, . . . , Y" at p such that (iJjiJu', Y 1) = 8;;. [HINT: Define Y k as suggested by the results of Exercise 13.8.] 13.10 If ,P: U---> V and Y,.: V---> W show that (Y,. o ,p).(Xp) = Y..•[tf>•(Xp)] for each XP E Tp(R"), p E U. In particular if C: ( -1, 1)---> U is a curve in U and ,P: U---> V, show that the tangent vector (I/> o C)'(t) to the curve q, o C at t is just tf>•l C'(t) }. 13.11 (The Gram-Schmidt orthonormalization process.) If X I, . • • , X k are linearly independent vectors at p define the vectors Y ~, . . . , Y k recursively as follows: where j

Z;+I

=

xi+ I

-

~ (X;+" Y,) v,. i=l

(a) Show that the vectors r ~, ... , Y k are orthonormal. That is, show that (Y;, Y;) = 8;;, 1 ~ i,j ~ k. (b) Show that xi= 2;!-1 (X;, Y;) l";, 1 ~ j ~ k, so that YI, . . . ' Y; span the same subspace of TP(R") as XI, ... , X;. (c) If XI=

_i_- _i_ + g_i_, iJxi

iJx2

iJxa

iJ iJ iJ X2=-+-+-, iJxa iJx2 iJxi iJ iJ xa=-+-, iJxa iJx2

with evaluation at p being understood, find YI, Y2, and Ya.

52

I

CHAPTER THREE

14

Vectors and Covectors

Vector fields

A vector field X on the open set U C R" is a map p ~X, which assigns to each point p E U a vector X, at p. If xl, . . . , x", denote cartesian coordinate functions for R", pE U.

The vector field p ~ X, is a Ck vector field on U if all the functions bl, . . . , b" are Ck on U. The sum X + Y of two vector fields and the product JX of a vector field by a function on the left are again vector fields defined in the natural manner. The notation Xj is reserved to denote the function (Xf)(q) = X qf defined whenever f is C 1 on U. Exercises In the first two exercises below x 1, x2 , x3 are cartesian coodinates for R 3 and U is the set U =

fp E R 3 : x 1(p) > 0 or x 2(p)

'¢

0}.

(Cylindrical coordinates.) The three functions, r, 8, z defined on U by 8; x 2 = r sin 8; x 3 = z; -7r < 8 < 1r; and r > 0 form a coordinate system for U. Show that (fJ/fJr, fJjfJr) = 1; (fJ/fJr, iJjiJ8) = 0; (iJjiJr, iJjiJz) = 0; (iJjiJ8, iJjiJ8) = r 2; (iJ/iJ8, fJ/fJz) = 0; (iJ/oz, a;az) = 1 on U. 14.2 (Spherical coordinates.) The three functions p, 8, q, defined on U by x1 = p cos 8 sin ,P; x2 = p sin 8 sin ,P; x 3 = p cos ,P; -7r < 8 < 1r; p > 0; and 0 < q, < 1r form a coordinate system for U. Show that (ojop, ojop) = p2 sin 2 q,, (ojiJq,, a;aq,) = p 2, and the other scalar products are zero. 14.3 (a) If X and Y are C1 vector fields on the open set U in R" and z, is the operator f--+ ZJ = X,(Yf) - Y ,(Xj), show that z, satisfies (13.1) and (13.2) whenever f and g are C 2 on U, and hence that Z 11 determines a unique vector at p. The vector field Z determined at each p E U in this manner is denoted by Z [X, Y]. It is a C0 vector field. (b) If 14.1

x1

= r cos

.

X =

I

n

(Xxi)

_!__

and

Y=

OX'

j=l

find the components of Z

=

I

X = sin x 1 _i_ ox!

-

cos x2 _i_ fJx2

and

find [X, Y].

ax•

(X, Y] in terms of the Xxi and the Yx 1•

(c) If

Y =

') a

(Yx• -.•

i=l

,

a +x 1 -a• fJx2

e~-

fJxl

54

I

Vectors and Covectors

CHAPTER THREE

a covector at q,(p), p E U, then ,P*Iw

I is the covector at p defined by tf>•IW(p)}(Xp) = W(p)(q,.[Xp]). In particular, (15.7)

q,*([dfl

)(Xp) = (df)

tP• (Xp) = tf>• (Xp)f = Xp(f o q,) =

ld(f o q,)]p(Xp),

which, if q,*jis defined by ,p*f(p) = f o ,P(p), can be abbreviated q,*d = dq,*. Note that q,* and tf>• carry objects in opposite directions. (15.8) Example. Suppose y 1, • • • , ym is a system of coordinates for the open set U C Rm; x 1, ••• , X" is a system of coordinates for the open set V C R"; ,P: U--+ Vis C1 ; and wp = C1(dx 1)P + · · · + Cn(dx")P is a covector at q = ,P(p). Then {15.9)

,P*(wq) =

n

n

k=l

k-1

L Ck,P*(dxk)P = L Ck[d(xk

o

,p)]P

(15.10) Example. There is an important manner in which covectors can be obtained from vectors. If Y P is a vector at p, the map (15.11)

Xp E Tp(R")

is a covector at p. Moreover every covector at p can be obtained in this way.

(15.12) Theorem. vector Y P such that

For each covector wp E T*(R") there is a unique

(15.13)

and conversely for fixed YP E TP the left side of (15.13) defines a unique covector wp. The correspondence Y P ~ wp is a linear isomorphism between Tp(R") and T!(R") and this relationship is denoted in the future by YP = w!orwp = Y!. PROOF.

The theorem is proved by the two formulas

(15.14)

in which xi, . . . , x" are cartesian coordinates. To check the first, if

15 Covectors I

SECTION

55

YP is given by (15.14), (Xp, Yp) = ( Xp,

;t

;tt

wp

wp

(a!; IP) a!; IP)

(a!; lp) (

xp,

a!; lp)

If Y~ were another vector satisfying (15.13) then IYP - Y~l 2 = (Yp- Y~, Yp) - (Yp- Y~, Y~) = wp(Yp- Y~) - wp(Yp -

Y~) = 0;

so Y P = Y~, establishing the uniqueness. The converse is proved similarly. + The results of Theorem (15.12) can be exploited to define a scalar product for covectors, using one of the three equalities (15.15)

(wp, ap) = (wt, a!) = wp(a!)

= ap(w!). All three of these possibilities yield the same value for (wp, ap). If x 1 , . . . , x" are cartesian coordinates, since (dxi)* = ajax; IP it follows that (15.16)

The vector-space isomorphism #: Tp(R") ~ r;(R") and its inverse (which is also denoted by#) is sometimes used to eliminate the concept of covector. If this choice is made, the gradient of the function! is the vector field (df) *rather than the covector field dj, and all other covector concepts are changed accordingly. Although conceptually sound, this procedure has difficulties of the following sort: If q,: U ~ Vis a differentiable map of U into V and f is a function on V, then it is reasonable to consider f on V and f o q, on U as corresponding objects and by extension the vector field d(f o q,)*, representing the gradient off o C, should transform into (df) 1 under the map q,. used to transform vectors at pinto vectors at q,(p). In general this does not happen, as the following example shows. (15.17) Example. Suppose ,P: (-1, 1)--+ (0, 6) is given by x o q, = 3x + 3. Then if f(p) = x(p), p E (0, 6), fo q, = 3x + 3 on (-1, 1). Calculations show that d(f o ,P) = 3 dx; [d(f o ,P))* = 3 a;ax. (xis a cartesian coordinate variable.)

CHAPTER ONE

Topology of R n

1

Fundamental structure of R"

n-dimensional euclidean spaceR" is the set of all n-tuples p = (a\ . . . , a") of real numbers, ak represents the kth member of the n-tuple (not a to the kth power), and the letters p and q are used to represent elements of R". R" is a vector space, two points p = (al, . . . , a") and q = (bl, . . . , b") having the sum p + q = (a 1 + b 1, • • • , a" + b"). If ~ is a real number, ~p = (~a 1 , • • • , ~a"). The length of an element p = (al, . . . , a") of R" is given by

and it satisfies the important relations

liP + qll ::;; liP II + llqll j II~PII = I~IIIPII-

The euclidean distance d(p, q) between p and q is the length

2

liP - qll-

Open sets, closed sets, and neighborhoods

The e-ball centered at q or, equivalently, the ball of radius e centered at q is the subset B.(q) of R" consisting of those points p for which d(p, q) ::;; e. B.(q) = {p E R":d(p, q) ::;; e).

q is an interior point of the set A if A contains some ball of positive radius centered at q as a subset. The set of interior points ,of A is denoted by 1

SECTION

covectors w1,

••• ,

15 Covectors

/

57

wn in T!(Rn) such that wi(x;) = ~J and that n

L

Y =

wm(Y)Xm

m=l

for each Y E Tv(Rn). (b) Show that w\ . , wn are linearly independent and hence form a basis for then-dimensional space T!(Rn). 15.8 The vector X v is normal to the level surface of h at p if (X p, Y p) = 0 for each Yvwhichistangenttothehypersurface {q:h(q) = h(p)l atp. Show that (dhp)* is normal to the level surface of hat p. 15.9 Show that (dx 1)p, . . . , (dxn)v are linearly independent if xl, . . . , xn is a coordinate system in some neighborhood of p. 15.10 Show that if yl, . . . , yn is a coordinate system for U C R 3 then any vector field X on U can be written as

L (X*, n

X =

k= 1

15.11

a dyk) -k . iJy

If y\ . . . , yn is a coordinate system for U C R 3 show that the

n X n matrices a;; = (iJjiJyi, iJjiJyi) and b;; = (dyi, dyi) are inverses of each other

at every point of U. 15.12 Using Exercise 15.11, show that if p, u, q, are spherical coordinates for R 3, then (dp, dp) = 1; (dB, d8) = { p sin q, l- 2 ; (dq,, dq,) = p- 2, and that all other scalar products between two of the covectors dp, d8, dq, are zero.

CHAPTER FOUR

Ele:ments of Multilinear Algebra

16

Introduction

This chapter is primarily algebraic in nature, treating the exterior algebra of a finite-dimensional vector space V together with its dual space V*. If V is an n-dimensional vector space over the field of real numbers the space V* of linear maps of V into R is called the dual of V. The sum w* + v* of two elements of V* and multiplication of v* by the scalar a E R are defined in the natural manner as maps by

(16.1)

(w*

+ v*)(v)

= w*(v)

+ v*(v),

(av*)(v) = a[v*(v)],

v E V;

v E V.

With this addition and scalar multiplication V* is a linear space (vector space), and in keeping with the applications intended for this section the elements of V will be called vectors and those of V* covectors. If e1, . . . , e,. is a basis for V, each v E Vis uniquely expressible as a linear combination of the e/s,

(16.2)

ak E R all k,

and ifj1, ,j,. are elements of another linear space W there is precisely one linear map ,P: V--+ W for which ,P(ek) = jk, k = 1, 2, . . . , n. If v is given by (16.2), ,P(v) = a 1fi + · · · + a"f,.. As an application of this result, suppose k is fixed and W = R. Let ek be the linear map of V--+ R determined by ek(e;) = 11;. ek E V* and the set {e1, . . . , e" I constitutes a basis for V* called the dual basis to e1, . . . , e,.. The proof is deferred until Theorem (16.11). (16.3) Example. The covectors (dx1) , • • • , (dx")p are a basis for r:(R"), which is dual to the basis o/ox 1 lp, . . . , ojox" ip of Tp(R"). 58

SECTION

16

Introduction /

59

A k-covector is a map (v1, . . . , vk)

--+

w(v1, • • . , vk)

of (ordered) k-tuples of vectors into R which satisfies two relations: (1) It is multilinear in the sense that for each m the map (16.4)

is linear when v1, . . . , Vm- 1, Vm+l, . . . , Vk are fixed vectors in V. (2) It is alternating or skew-symmetric, which means (16.5)

w(vl, . . •

1

V;1

•••

1

V;, . . •

1

V,~:)

so that w changes sign when two arguments are interchanged. It is worthwhile recalling at this point some properties of permutations of integers. Denote by Sk the set of permutations u of the integers 1, 2, . . . , k. Each u E Skis a one-one map of the set {1, 2, . . . , k} onto itself and is completely described by the ordered k-tuple (u(1), u(2), . . . , u(k)). Sk has k! elements corresponding to the k! such k-tuples. In a finite sequence of the type u(1),u(2), . . . , u(k), each time that a larger integer precedes a smaller, u(i) > u(j) when i < j, an inversion occurs in the natural order 1, 2, . . . , k. For example, the sequence 4 1 3 2 has four inversions and the sequence 1 3 4 2 two. A permutation u is called odd or even according to whether the number of inversions in the sequence u(1), . . . , u(k) is odd or even, and the sign of a permutation, sign (u), is +1 if u is even and -1 if u is odd. An important consequence of (16.5) is that (16.6)

w(v1, . . . , Vk)

= sign (u)w(Vv(l), . . . , Vv(k)),

Before indicating why (16.6) is valid, consider as an illustration for the permutation 4 1 3 2, w(v1, v2, va, v4)

= -w(v1, v2, v4, va) = w(v1, v4, v2, va) = -w(v4, v1, v2, va) = w(v4, v1, va, v2)

where each term is obtained from the term which precedes it by applying (16.5) to interchange vectors in adjacent positions. In general (16.6) can be derived in the same manner by first moving v.,< 1> to the first position by a sequence of adjacent interchanges each of which creates one inversion, then moving v.,' Next if

then by the multilinearity and (16.9),

. .

Wl(Vt, • • • ' Vk)

=

L L m,=l mo=l

.

:L ai' · · ·

.

:L ai' · ·

m•=l

concluding the proof.

+

If al, . . . , ak E V* the map a 1 (v 1)

(16.10)

(v1, . . . , vk) -

;, det

a;o•wl(em.,

.. , em.)

m•=l

[

: ak(vl)

ak'w2(em., . . . , em.)

16

SECTION

Introduction

/

61

is a k-covector which, anticipating later definitions, is denoted by (a 1 1\ · · · 1\ ak)(v 1, . . . , vk) = :! det {(ai(v;))\.

(16.11) the

e

Theorem.

1,

••• ,

k-covectors (ei• 1\ · · · 1\ ei•), 1 1\ kV*. If w is any k-covector,

PROOF.

e,, ... , e,.;

(n) = n! k

(16.12)

e" is a basis for V* dual to

w

=

L

•·< ..

If 1 ~it

k!(n- k)!

~ i1

< · · · < ik

~

n, form a basis for

k!w(e;, . . . , e;.)(ei, 1\ · · · 1\ ei•). ), ..ESm+>

and this product satisfies the relations (14.2) (i)-(iv). !-vectors and vectors are usually identified (cf. Exercise 19.3) and the linear space of k-vectors is appropriately denoted by AkV. (19.1) Theorem. There is a unique map (F, w) -- (F, w) which associates a real number (F, w) with each pair (F, w) consisting of a k-vector F and a k-covector and satisfies: (1) The map w-- (F, w) is linear on NV* for each F. (2) The map F-- (F, w) is linear on NV for each w. (3) (v 1 A · · · A vk, a 1 A · · · A ak) = det {ai(v;)} when the ai's are covectors and the v/s vectors. PROOF. If e1, • • • , e,. is a basis for V and et, . . . , e" the corresponding dual basis for V*, then, according to Theorem (16.11) and its

SECTION

19

k- Vectors

I 71

k-vector version, F and w can be uniquely represented in the forms

F = (19.2) w

Suppose (F, w) calculation (19.3)

--+

=

;, < i•
is the m-chain in V described by at ( cf> o Ut) + · · · + ak(cf> o uk) and denoted by cf> o em. The boundary operator a. The boundary iJflm of the standard m-simplex is the singular (m - 1)-chain m+t

iJflm =

(24.4)

L (-l)ku:'

k=I

where the u:''s are the singular (m - I)-simplices given in Example (23.4). If r is a singular CK m-simplex in U its boundary iJT is the (m - 1)-chain defined by

(24.5) and if e = atrt (m - 1)-chain.

+ a ri

+

1

is an m-chain in U, its boundary is the

iJe = al OTt

(24.6)

+ . . . + a; OTjo

If e is any m-chain in U iJ(iJe) = 0.

(24.7)

Preliminary reductions show that when e = atrt

+ · · · + akrk,

+ ·. · · + ak iJrk) at iJ(iJrt) + · · · + ak iJ(iJrk) atTt o (iJ[iJflm]) + ' ' ' + akTk

iJ(iJe) = iJ(at ort =

=

0

(iJ[iJflm]);

SECTION

24

m-Chains. The Boundary Operator

a

I

89

so it suffices to prove a(a.!1m) = 0. But m+l

a(ai1m)

:L

=

k(m). The point q is an accumulation point of the sequence {pk} if for each neighborhood W of q the set {k :pk E W} is infinite. The sequence {pk} is eventually in the set J if and only if {k: Pk C J} is infinite. In this terminology q is an accumulation point for {pk} when and only when {pk} is eventually in each neighborhood W of q. As a further criterion: q is an accumulation point of {Pk} if and only if {pk} has a subsequence {q,.} which converges to q. To see this suppose first that {q,.} is a subsequence of {pk}, q,. = Pk..C'(to) :0 ~ >.. ~ 1} in Tc(lol(R") is just the length of the tangent vector C'(t 0 ). It is the square root of the determinant of the 1 X 1 matrix (C'(to), C'(to)). {1-volume O(C'(to)) } 2 = (C'(to), C'(to)) =

- ]2 L" [dxioC

.

J=

1

dt

(to)

if xl, . . . , x" are cartesian coordinates. (26.3)

Example.

If 6 is the angle between e, and e2 then 2-volume O(e,, e2) =

je,lle2l sin 6 as is evident from Figure 24.

FIGURE 24

108

I

Differential Forms

CHAPTER FIVE

Now

le,l 2e2l 2sin 2 fJ

cos 2fJ)Ied 2le,l 2

= (1 =

le,l 2le2l 2

-

(led le2l cos 8) 2

or

Equation (26.1) has an infinitesimal analog which is taken in the following as the definition for the k-volume of a large class of objects. More specifically, (26.4)

Definition.

Let ,P: flk-+ R" be an imbedded k-simplex. Then

k-volume (q,)

(26.5)

=

J ·~. · j

{det ,P* (q,*

[~;] • ,P* [~;J)r' 2 dt

1 • • •

dtk,

where q,*(q,*[a/clt'], q,*[ajati]) is the function on flk whose value at p E flk is the scalar product of q,*[ajatilp] and q,*[ajatilp]. The integrand in (26.5) is closely related to (26.1), so that (26.5) can be regarded as an infinitesimal analog of (26.1). It is important from geometrical considerations that the k-volume of the imbedded k-simplex q, really only depends on ,P(flk). More precisely, (26.6) ,P(flk)

Theorem.

If q, and 1/1 are both imbedded k-simplices in R" and

= 1/l(flk), then k-volume (q,) = k-volume (1/1).

In the proof given here use is made of the concept of a k-form on ,P(flk). Strictly speaking, until now differential forms have been defined only on open subsets of R", so that it is necessary to elarify this term. Let W = interior (flk). A "k-form won q,(flk)" is a map p-+ wp E 1\ kT;(R"), p E q,(flk), with the property that q,*w, defined as usual, is a k-form on W. The initial argument in the proof of Theorem (23.7) shows that when 1/1 is another imbedded simplex with 1/l(flk) = ,P(flk), h = (t/JI w)- 1 o (1/llw) and h- 1 are both differentiable, and it follows that the property of being a k-form on ,P(flk) really depends only upon the point set ,P(flk). Theorem (23. 7) is valid as it stands for such w's. PROOF.

It suffices to construct a k-form won ,P(flk) with the property

that (26.7)

k-volume (1/1)

=

I £w I

26

SECTION

Volume, Surface Area, Flux

/

109

for each imbedded k-simplex 1/1 with 1/l(ll"') = q,(ll"'). Because, given such an w, (26.8)

by Theorem (23.7), and Theorem (26.6) then follows from the relations k-volume (1/1) =

I

L I = I L I = k-volume (q,). w

w

Construction of w: Let q, be a fixed imbedded k-simplex and let Z 1, • • • , Zk be orthonormal vector fields on q,(flk) which at each p E q,(fl"') span the linear space generated by q,*(o/fJt 1), . . . , q,*(ajat"'). [Z1, . . . , z,. can be obtained, for example, by using the Gram-Schmidt process to orthonormalize q,*(ajat 1), . , q,*(iJ/iJt"')-cf. Exercise 13.11.] Put ai

=

z~J'

w

=

a 1 1\ · · · 1\ a"'

1 ~ j ~ k,

and It remains to show that w satisfies (26.7) for each imbedded k-simplex 1/1 with 1/l(flk) = q,(flk). Using the notation above, where h = q,- 1 o 1/1 restricted to W = interior (ll"'), h*(a;ati)lq is a linear combination of (ajat•)lq, 1 ~ i ~ k, when q E W. Consequently, 1/l*(o/oti)IP = q,*h*(o/oti)IP can be expressed as a linear combination of q,*(ajoti)lp, 1 ~ j ~ k, and hence of Z1lp, .. , Zklp·

(26.9)

1 ~ j ~ k,

on 1/I(W), where the coefficients in (26.9) are easily checked. k

(26.10)

(1/1*

(~i). 1/1* (~j)) = m~l (1/1* (~j). am) (1/1* (~j), am);

so (26.11)

det(""*

(~i).""* (~j))

= {det\1/1*

(~,} am)r·

From this it follows that

(26.12)

where h(t 1, . . . , tk) = ± 1. Since h is continuous it must be constant on W, and (26.7) follows on comparing (26.12) with (26.5). +

SECTION

4 Compact Sets

I 7

gives a contradiction: W is a neighborhood of q = p and I m: q,. E W} is not infinite. In fact it is empty, because q,. El W for each m. (4.2) Example. A compact set Dis always closed. According to Exercise 3.2 if Dis not closed there is a sequence IPk) of points of D which converges to a point p El D. The only accumulation point of I Pk) is p and it does not belong to D; so D is not compact. (4.3) Example. If D is compact there is a number M such that llqll ~ M whenever qED. Otherwise D would contain points p,. such that IIPmll ~ m for each m = 1, 2, 3, . . . . The sequence IPml cannot have an accumulation point q, because if W is the neighborhood B 1{q) of q, every point p of W satisfies liP II ~ llqll + liP- qll ~ llqll + 1 and consequently the set lk:p" E W) is certainly finite.

A set F for which there is an M with IIPII ~ M whenever p E F is called bounded. The preceding examples show that a compact set D is always closed and bounded. More is true. (4.4) Theorem. The set Dis compact in R" if and only if Dis both closed and bounded. PROOF. Because of Examples (4.2) and (4.3) it is sufficient to show that a closed bounded set F is necessarily compact. As F is closed, it suffices to show that every sequence of points ofF has a convergent subsequence. This can be done directly, but it is schematically a little simpler (1) to assume the theorem is true when the dimension n = 1 and establish the general result from this, and (2) to establish the result when n = 1 separately. If{p~:},p,. =(a~, . . . ,a::,),isasequenceofpointsofF,thesequences of real numbers {a~}, ... , {a::,} all belong to the closed bounded interval [ -M, M] on the real line. (M is chosen so that IIPII ~ M when p E F.) Assuming for the moment that Theorem (4.4) is true when n = 1, the first of these sequences has a subsequence {a~c;> };:,.1 converging to some limit a 1• Now {a~c;Jlj".. 1 may not converge, but it certainly has a convergent subsequence {a~c;c~:»}:_ 1 with limit a 2 and lim~r. a~c;ck» = a 1• Passing in this manner to successive subsequences, one obtains after n steps subsequences {a~ c...>}, . . . , {afcm>} converging respectively to a', ... , a". The observation following (3.1) then shows that lim,. Pk = p where p = (a', . . . , a"). To prove Theorem (4.4) when n = 1 suppose F c [-M, M] and {p~:} is a sequence of points of F. Define q" by 2Mq~r. = p~; M and note that 2Miq~: - q,.l = IP~r. - p,.i, so that a subsequence {P~r.w} is a Cauchy sequence and hence converges if and only if {q~r.(J) } converges. q~; E [0, 1]; so without loss of generality one may as well assume from the outset that {p~r.} is a sequence of points from the closed interval [0, 1]. Set [,. = [(j - 1)/2"', j/2"'] where j is the smallest integer, 1 ~ j ~ 2"', for which

+

SECTION

27 Green's Identities I

111

That is, the integral of the flux of X over any imbedded (n - I)-simplex q, is to within a sign (depending on the normal chosen) just the integral of the normal component of X over the surface q,(.:::\n-l) with respect to (n - I)-dimensional volume. Formula (26.I5) will be reinterpreted and rederived later, but for the moment consider the following special case: (26.16) Example. The vector field (dh) 1 is normal to the level surfaces of h (Exercise I5.8), and where dh does not vanish N = (dh, dh)-ll 2 (dh)* represents a unit vector field normal to the level surfaces of h. Its flux N _j dx 1 A · · · A dxn = (dh, dh)- 1' 2 •dh

has a ready interpretation. When integrated over an imbedded (n - I)-simplex q, lying on one of the level surfaces of hit gives the surface area [(n - I)-volume] of q, to within a sign.

Exercises 26.1 Express the 2-form (dh, dh)- 112 •dh which gives surface area in R 3 in terms of dx 1, dx 2, dx 3, and the partial derivatives of h, where the x's are cartesian coordinates. 26.2 Find a unit normal vector field N on the surface described by the equation x 3 = (x 1) 2 - 3(x')' + x1x'. 26.3 Suppose w is a I-form on the open subset U of Rn and that the total flux,

r w'

}a~

dz1 I\ . . . I\ dxn

_j

'

of w through the boundary of any n-simplex q, in U is zero. Show that d*W = 0 on U. 26.4 Suppose X is a vector field and h is a differentiable function and both are defined on the open subset U of Rn. Suppose dh ¢ 0 at any point of U and Xh = 0 on U. Show that there is an (n - 2)-form a on U such that X _j dx 1 I\ · · · I\ dxn = dh I\ a. 26.5 (Based on Exercise 26.4.) If, in addition to the conditions in Exercise 26.4, q, is an (n - I)-simplex with h o q, = a constant on ,:::\n- 1, show that

L

flux of X = 0.

Give a geometrical interpretation of this result.

27

Green's identities

For the moment let p = (a 1 , • • • , an) be a fixed point m Rn. The homothetic transformation H 1 centered at pis the map Ht(q) = p

+ t(q -

p),

t > 0, q ERn,

ll2 I

CHAPTER FIVE

Differential Forms

of R" onto itself. Suppose that b [or b(p)] is ann-chain in R" representing integration over the 1-ball centered at p, so that in particular the integral of the n-form dx 1 I\ dx 2 I\ · · · I\ dx" over b gives the volume of this 1-ball-x 1 , • • • , x" are cartesian coordinates for R". (27.1)

so if

p2 = (xl _ al)2 2p dp = 2(x 1

a 1)

-

+ ... + (x" _ a")2; dx + · · · + 2(x" - a") dx"; 1

x 1 - a1 dp = - - - dx 1 + p

•dp =

xl- al p

· · · + x"-p

dx 2 I\ · · · I\ dx"

a"

dx"·

'

+ ···

+ x"- a" ( -1)"-1 dxl I\

. . . I\ dxn-1;

p

and using Hi(x 1 - a 1) = t(x 1 - a 1), Hip = tp, this gives (27.2)

These calculations also yield dp I\ •dp = dx 1 I\ · · · I\ dx"; thus from Theorem 21.11 it follows that (dp, dp) = 1 and then using Example 26.16 that

the surface area of the unit sphere in R". Let h,., n = 1, 2, . . . , be the function defined on (0, + oo) by hl(t) =

1

2 t; 1

h2(t) = 2'11" log t; h,.(t)

= (2 -

1

n)w,. t2-n,

n

> 2.

Put (27.3)

Sp(q) = h,(IIP - qll) = h,(p)

for q E R""" {p} [R" with the point p deleted]. Sp is differentiable on R""" {p) and possesses the fundamental property (27.4)

SECTION

(27.5)

Lemma.

27 Green's Identities

I ll3

*dSP has in addition to (27.4) the properties:

(i) Hi( *dSp) = *dSp. (ii) }ab *ds P = 1· (iii) d*dSp = 0.

r

PROOF.

Property (i) follows from (27.2) and the observation that

Hi p1-n = t1-npl-n. Property (ii) is a consequence of

( *dp }ab

= Wn

together with the fact that p = 1 on the support of iJb. To establish property (iii), calculate directly (27.6)

1

d( *dSp) = - [(1 - n)p-n dp 1\ *dP Wn

+ p -n d*dp]. 1

Now from the equations preceding (27.2) d(p *dp) = n dx 1 1\ · · · 1\ dxn = n dp 1\ *dp;

so dp 1\ *dp

+ p d*dp

= n dp

1\ *dp;

pi-n d*dp = p-np d*dp = (n - 1)p-n dp 1\ *dp,

which yields (iii) when combined with (27.6).

+

(27.7) Theorem (Green's third identity.) If c is an n-chain in Rn representing integration over the region G which contains pin its interior, then for each differentiable function f on Rn (27.8) The first integral on the right-hand side of (27.8) is a convergent improper integral. lim Sp(q) = + oo if n > 2 and - oo if n = 2; its convertJ-+P

gence is verified in the proof. The main difficulty in the proof arises because Stokes' theorem cannot be applied to such improper integrands. PROOF. For each t > 0, c - Ht o b is an n-chain in R" {p }-if t is small enough it represents integration over G (the t-ball centered at p)-and the forms involved are perfectly regular on R" Ip}, so that Stokes' theorem can be applied to yield ro.J

ro.J

ro.J

1-H,•b d(f *dSP - Sp *df)

(27.9)

1-H,ob (f d*dSp - Sp d*df) c-H,ob = Sp d*df,

f

114

I

CHAPTER FIVE

Differential Forms

because df I\ *dSP = dSP I\ *df [(22.14), property (4)] and d *dSP = 0. 1aH ••J *dSp = lab Hi(f *dSp) = lab (Hif) *dSp,

because Hi(*dSp) = *dSp. Since Hif· (q) = f[p formly on the support of ob, it follows that

+ t(q-

p)]-+ f(p) uni-

(27.10) Using the fact that Hi dxi = t dxi for each i, we obtain S *df = ( ]aHo•b P

~

~

j=1

f t"- 1(H*S ) (H*1 of.) ( -1)i-1 dx 1 A · · · 1

lab

P

ox'

I\ dxi- 1

I\ dxi+l I\ · · · I\ dxn.

Now as t-+ 0, Hi(ofjiJxi)-+ (ofjaxi)(p) uniformly on the support of ob, while for each n, t"- 1 (HiSp)-+ 0 uniformly on the support of ob. Thus (27.11)

lim t-+O

r

H b Sp *df = 0. ]a ••

Passing to the limit in (27.9) using these relations then gives

r

r

H b sp d*df, }ac (f *dSp - Sp *df) - f(p) = -lim t-+O )c- ,a which is just a rephrasing of (27.8).

+

There is a more primitive identity then (27.8). It states that whenever f is differentiable and the n-chain c represents integration over the region G containing p, then (27.12) To prove it let b represent integration over B1(p) as before and apply Stokes' theorem to get lace-Hob/ *dS P = 1-Hob df I\ *dSp, because p e support (c - H 1b) and d*dSp = 0. This equality can be rearranged to read laH,bf *dSP = laJ *dSp - 1-Hob df I\ *dSp which yields (27.12) when we let t-+ 0 and use (27.10).

SECTION

27 Green's Identities

I 115

Exercises 27.1

Derive Green's first identity:

27.2 Derive Green's second identity:

27.3

where

r

Show that

denotes the gamma function.

A C2 function f is subharmonic on the open subset G of Rn if d•df 2::: 0 on G. In the following exercises suppose c is an n-chain in the open set U in Rn representing integration over the open subset G of U and oriented so that Jc dx 1 1\ · · · 1\ dxn > 0. 27.4 Iff is subharmonic on G and C2 on U, show that

27.5 Iff is subharmonic and C2 on U and b,(q) is an n-chain representing integration over B,(q), the ball of radius r centered at q, show B,(q) CU. [HINT: 8 4 = >.,a constant, on support [b,(q)] and 8 4 - >. observations and (27.8).] 27.6 Show in the notation of Exercise 27.5 that

~

0 on B,(q). Use these

using only the properties of •dS11 given in Lemma (27.5). 27.7 (Based on Exercises 27.5 and 27.6.) Show that iff is C2 and subharmonic on the open subset U of Rn, thenf attains its maximum on an open subset of U. Show that if U is connected, either f is constant or it does not attain its maximum at any point of U. [HINT: Iff attains its maximum at q E U, then f - f(q) is subharmonic and 0 = f(q) - f(q)

~

hb,(ql [f - J(q)] •d811

implying/- f(q) = 0 on {p: liP- qi! = r}.]

~

0,

B,(q) C U,

ll6

I

28

CHAPTER FIVE

Differential Forms

Harmonic functions. Poisson's integral formula

A function f is harmonic on the open set U if d•df = 0 on U. In this case if c is an n-chain representing integration over the region G in U and p is an interior point of G, Green's third identity (27.8) becomes (28.1) If hP is another harmonic function on U, Green's second identity yields

Adding this to (28.1) gives (28.2)

where gp(q) = Sp(q) + hp(q). Suppose it is possible for each p E G to choose hp so that gp = 0 on the support of oc, that is, on the boundary of the region G. Then the function (p, q) ~ gp(q) of p and q is called Green's function for the region G. From (28.2) it follows then that (28.3)

whenever fis harmonic on G. Theorem (28.5), known as Poisson's integral formula, illustrates (28.3) when G is the ball B,(O) of radius r centered at the origin. Let p = 1/q - pi/ and p' = llq - >. 2pll with>. = r/IIPII· In the notation of (28.3) the function hn(P' />.) is harmonic except at the point >. 2p where it is not defined and (28.4)

gp(q)

= hn(P) - hn ( ~) = Sp(q) - hn ( ~)

is a candidate for Green's function for B,(O). Now, referring to Figure 25, the triangles 6.qop and 6.(>. 2p)oq are similar when llqll = r because>. = r/IIPII = I/>- 2PII/r.

FIGURE 25

SECTION

28

Harmonic Functions. Poisson's Integral Formula

r/IIPII =

In particular p' / p =

llqll =

117

r, so that

= h,.(p) - h,. (~) = 0

gp(q)

on the sphere sphere.

X when

/

{q: llqll = r},

and gp(g) is indeed Green's function for this

(28.5) Theorem (Poisson's integral formula.) If cis ann-chain representing integration over the ball B,(O) in R" and f is a harmonic function on B,(O), then (28.6) where s 2 = (x 1) 2 + · · · + (x") 2 ; so *dS describes surface area on the sphere {q: llqll = r} which is the support of ac. PROOF.

gp(q) dgp

It suffices to use (28.3) with gp given by (28.4). If n

= (2 - 1n)w,. {p2-n -

(p'X)2-n} ;

= (2 _ 1n)w,. { (2 - n)p 1- " dp - (2 - n) (

f')1-n x-1 dp' }

')1-n x-1 dp' }.

1 {p!-n dp - ( ~ = Wn So

Since p' = Xp when

llqll =

r it follows that on the sphere,

(28.7) Just as in the computations preceding (28.2) 1\ dx"

+

· 1\ dx"

and on the sphere

llqll =

r

(>-21) *ds _X_2_

= rp

> 2,

+

llqll =

r,

8 I

CHAPTER ONE

Topology of R"

the set lk:pk E [(j- 1)2-m, j2-"']} is infinite. Note that

and that length Im = 2-m. Choose k(m) (m = 1, 2, . . . ) inductively so that (1) k(m) > k(m- 1) and (2) Pk Elm. This can always be done because Ik : Pk E I m} is infinite. Contention: I Pk(m) } is a Cauchy sequence and hence converges. In fact IPk - Pkwl ~ (!)m when i, j ;;::: m. This completes the proof of Theorem (4.4). + Exercises 4.1

If D 1

:::)

.

D2 :::) · · · :::) D... :::) · · · is a decreasing sequence of non-

emptycompactsets,showthat

n Dk ¢ k=l

fZ).

[HINT:

Letqk E Dk,k

=

1,2,

and consider q = limi qk for some subsequence {qkW I of {qd .]

A class ct of subsets of R" has the finite intersection property if each intersection A 1 f\ · · · f\ Ak of a finite number of sets in ct is nonempty, At f\ · · · f\Ak ~ fZ), A; E ct. 4.2 (Based on Exercise 4.1.) If ct has the finite intersection property and is a countable class of compact sets, show that f\ {A: A E ct I ¢ fZ). 4.3 (Based on Exercises 4.1 and 4.2.) If B is compact show that each countable open covering 'U of B has a finite subcovering. (For the definition of covering and subcovering see the insert before Exercise 2.16.) [HINT: Let ct = {B f\ Uc: U E 'UI and show that ct cannot have the finite intersection property.] 4.4 (Based on Exercises 4.1, 4.2, 4.3, and 2.16.) Show that each open covering 'U of the compact set B in R" has a finite subcovering. This is the Heine Borel property. 4.5 If B is not compact in R" show that there is an open covering 'U of B which does not have a finite subcovering. That is, the Heine Borel property is equivalent to compactness in R". 4.6 (Existence of the Lesbesgue number.) Let 'U be an open covering of the compact set D. Show that there is a number E > 0 such that if d(p, q) < E, p, q E B, then it is guaranteed that p, q E V for some V E 'U. The largest E for which this is true is called the Lesbesgue number of the covering 'U. 4.7 Show by an example that if 'U is an open covering of a closed set B, there may not be any E > 0 satisfying the conditions of Exercise 4.6.

The diameter of a set B C R" is the least upper bound of the numbers d(p, q), q, p E B. 4.8 Let 'U be an open covering of the compact set D. Show that there is a number E > 0 such that whenever diameter (B) ::::; E and B C D, then B C V for some V E 'U.

SECTION

28

Harmonic Function&. Poisson's Integral Formula

I

119

his constant or it does not attain its maximum at any point of U. (For a more general statement see Exercise 27.7.) 28.8 (Based on Exercise 28.6.) If b,(q) represents integration over B,(q) and h is harmonic on U, show that h(q) = {volume [B,(q)]}- 1

f

]b,(q)

•h.

Jo

Show rf bt(q>f •dSq] dt = f br(q) •f.] 28.9 (Based on Exercise 28.8.) Suppose h is harmonic on R" and bounded, ih(q)i :::;; M, q E R". Show that his a constant. [HINT: If p, q E R",

[HINT:

volume [B,(O)][h(p) - h(q)] = f

]b,(p) -bt(q)

•h;

so nM ih(p) - h(q)i :::;; -volume {[B,(p) V Bt(q)J fl [B 1(q) fl B 1(p)Jcl,

w,.t"

which approaches zero as t--+ oo .] 28.10 Show that a C2 function his harmonic on U if and only if (28.9) holds. [HINT: Use the hint to Exercise 27.5.]

CHAPTER SIX

Vector Fields and Differential For:ms

29

Flows and vector fields

If U is an open subset of the open set V, a flow of U into Vis a differentiable map q,: (-e, e) XU--+ V where e > 0

with the restrictions that (1) q,(O, q) = q, q E U, and (2) the map ,Pt: q--+ ,P(t, q) of U into V is one-one for each fixed t E ( -e, e). The terminology is taken from fluid mechanics where points correspond to positions in a moving fluid and the particle in position q,(t, q) at timet traverses the path t--+ ,P(t, q) = Cq(t) as time unfolds (see Figure 26). Condition (2) can be interpreted as stating that distinct positions in the fluid remain distinct throughout the motion. K ote that in the notation used here q,(t, q) = ,Pt(q) = Cq(t); Cq is the curve traced by a point during the flow, and ,P1 is the displacement of U created by the flow. The velocity vector field X(t) of the flow q, is the vector field on tPt(U) whose associated vector at tPt(P) is the tangent to the curve CP at Cp(t). That is, p E U, ltl > = C~(t),

u

FIGURE 26

120

SECTION

Flows and Vector Fields

29

121

/

In other terms, when q = q,(t, p),

of o q, = fit. (t, p).

X(t)f · (q)

The flow is stationary if X(t)lv = X(s)lv whenever p E 1/Jt(U) 11 q,.(U). In most discussions of stationary flows there is a vector field X on V with XIP = X(t)lv. p E 1/Jt(U), and in this situation in keeping with the usual terminology X is also called the velocity field of the flow q,. If x 1, • • • , x" is a coordinate system for V and p = q,(t, q), a comparison of the two equations n

X(t)lv =

L [X(t)xk](p) 0~k lv

k=l

and X(t)f. (p)

I :fk

=

(p)

ax~; q, (t, q)

k=l

shows that X(t) is the velocity vector field of the flow q, if and only if

(29.1)

oxk

0

q,

- 0 t - (t, q) = [X(t)xk]

o

q,(t, q),

k = 1, 2, . . . , n.

The system of differential equations (29.1) can be used to determine the flow from the velocity vector field X(t). The technique is illustrated below when X(t) does not depend on tin Theorem (29.2), which is a rephrasing of the fundamental theorem on the existence and uniqueness of solutions for ordinary differential equations.

(29.2) Theorem. If X is a C1 vector field on the open set V, then for each p E V there is an open neighborhood U of p, a number e > 0, and a flow q,: ( -e, e) X U---+ V of U into V with velocity field X. PROOF. Let x 1, • • • , x" be a coordinate system for V and let the function defined on the subset x(V) of R" by

(29.3)

X;[x 1(p), . . . , x"(p)]

= (Xx;)(p),

x; be

p E V.

The fundamental existence and uniqueness theorem for the solutions of ordinary differential equations ensures for each p E V the existence of an e > 0 and a C1 map y: ( -e, e) X N---+ x(V), y[t; u 1,

•••

,

u.r.]

= (y 1[t;

u 1,

•••

,

u"], . . . , y"[l; u 1,

•••

defined on the subset ( -e, e) X N of R"+l described by ltl


0 and is infinite. So {f(pk) lh- 1 does not converge to f(p) and f is not continuous at p. ( 0 so that Ba(p)

r'lf- 1(U)

= Ba(p) f"\

D

it follows (see Figure 1) that {k:pk (l Ba(p)}

=

{k:pk (l Ba(p) f"\ D}

:::> {k:pk eJ-l(U) = fk:pk er~cu)J = fk:f(Pk)

e u1.

f1

Dl

SECTION

z2 ,

••• ,

29 FlowB and Vector FieldB z"

= u 2, ••• ,

u" when z1

I 127

= 0 or u 1 = u 1(p)

--1-------;-.zl

q(O)....__ _ _....,_ _ _ q(t) p --4-----------~lV

FIGURE 28

are both integral curves of X agreeing when t = 0, so by the unstated uniqueness of such curves (cf. Exercise 29.5) they agree for all values of t for which they are both defined. (29.21)

C11(b - 6 + t) = t;(t, C11(b - 6)),

0 ,:5; t

< 6,

= q,(t- 6, q)

(cf. Exercise 29.6 for this section) and the parameter set for CP can be extended to (a, b + E) by Cp(t) = Cp(t) =

as previously defined when t

t;(t - b, q),

b ,:5; t

< b,

< b + E.

(29.22) Example. To close this section a result is proved about the possibility of choosing a coordinate system simultaneously adapted to the k vector fields X,, ... , Xk. Besides its inherent interest it serves a pedantic objective, because the proof given here is similar to the proof of the Frobenius theorem in the following section and, being simpler, serves as a natural forerunner to that proof. The result can be stated as follows: Suppose X 1, • • • , X~c are C 1 vector fields on the open set U in R". Then the point p E U has a coordinate neighborhood W with local coordinates z1, • • • , z" in which X 1 = iJjiJz 1, • • • , X~c = iJ/iJz" if and only if

(i) X,lp, ... , X~cl 11 are linearly independent in Tp(R"). (ii) [X,, X;] = 0, 1 ,:5; i, j ,:5; k, on a neighborhood of p. (Recall that [X, f) is the vector field defined by [X, Y]g

=

X(Yg) -

Y(Xg).)

PROOF. First one can assume by shrinking U if necessary that X,, . . . , X~c are linearly independent in U and [X;, X;] = 0 on U. The proof proceeds by induction on k. If k = 1 this result is a weak restatement of Theorem (29.11). Now suppose the result holds when k - 1 vector fields are involved. In keeping with this choose local coordinates u 1, • • • , u" in a neighborhood of p so that X,= iJjiJu 2, • • • , X"= iJjiJu". Since X 1lp is not linearly dependent on X2lp, ... , X~clp it follows that one of the terms (XIu 1)(p), (X,u"+l)(p), . . . , (X,u")(p) is not zero and without loss of generality one can assume (X,u 1)(p) ~ 0 (Figure 28). Now from Theorem (29.11) there is a coordinate system z1, • • • ,

128

I

Vector Fields and Differential Forms

CHAPTER SIX

z" on some neighborhood of p such that z'(q) = 0 implies that u'(q) = u'(p);

(29.23) 2

zi(q) = ui(q),

~

j

~

n,

when z1(q) = 0.

One consequence of (29.23) is that when z1 = 0 the other zi's coincide with the corresponding ui's, so that (29.24)

aj

aj

uz1

uU1

2

..,---: (q) = ,..-. (q) = Xif(g),

~

j, z1(q) = 0.

Now let W be a cubical neighborhood of pin the z's and suppose q(t) has z-coordinates (t, a2, • • • , a"); then a 1.zk)- [q(s)] ds (X;zk)[q(t)) - (X1zk)[q(O)) = --

ho' exaz' Jot X

1

(X;zk) [q(s)) ds

/o' {[X,, X;)zk + =

0,

j

X;(X 1zk)l[q(s)) ds

> 1.

From (29.24) j

which shows X; =

a;azi, j >

> 1,

1, on all of W.

Exercises 29.1 (a)

(b)

Find coordinates z', z2 in a neighborhood of (1, 0) adapted to X when x1 a x2 a X=--+--

ax r ax a a X=-+-· ax ax r

1

1

2

where r 2 = (x 1) ' + (x 2) 2, r

> 0.

2

29.2 If z1, • • • , z"are adapted to X on U and Xh = 1/fwheref ¢ Oon U, show that y 1 = h, y 2 = z2, • • • , y" = z" form a local coordinate system at each point of U adapted to fX. 29.3 Let r, fJ be polar coordinates on R 2 • Show that the differential equation XF = 1 does not have a solution on the annulus 1 < r < 2 if X = a;ae. 29.4 If X = -x 2 (a;ax 1) + x1 (a;ax 2), show that the equation Xf = 1 does not even have a local solution in a neighborhood of (0, 0). 29.5 Suppose X is a vector field on U and C4 : (a, b)--+ U, D4 : (a, b)--+ U are integral curves of X satisfying C4 (t 0 ) = D 4 (t 0) for some to E (a, b). Show that C4 (t) = D4 (t) for all t E (a, b). [HINT: Use a uniqueness theorem from ordinary differential equations.)

SECTION

29.6 velocity q E W. 29.7 Suppose

30 Frobenius' Theorem

I 129

(Based on Exercise 29.5.) If ,P: ( -e, e) X W---+ V is a flow with field X on V and lsi, iti < €/2, show that ,P(s, ,P(t, q)) = ,P(s + t, q),

c.:

Suppose (a, b)---+ U is an integral curve of the vector field X on U. s.. j band lim C0 (s .. ) = p E U. Show that lim C.(s) = p. m-+oo

aTb

29.8 Let f be a differentiable function on the open set U in Rn and suppose that X is a vector field on U satisfying Xf(p) = 0 whenever p E S = I q E U: f(q) = 01. If C: (a, b)---+ U is an integral curve of X with C(t 0) E S for some to E (a, b), show that C(t) E S for all t E (a, b). 29.9 (Based on Exercises 29.7 and 29.8.) If S is compact in U, show that each integral curve c.: (a, b) ---+ U starting at a point C.(O) = q E S can be extended to an integral curve c.:(- oo, oo)---+ S. 29.10 If X 1, • • • , X 2 are vector fields on U which are independent at p and [X;, X 1] = 0 all i and j, show that p has a neighborhood with local coordinates zl, . . . , zn on which the partial differential equations Xd = g1, . . . , Xkf = gk are equivalent to iJf/dz 1 = g1, . . . , iJf/iJzk = gk.

30

Frobenius'

theore~

There is a very important generalization of Theorem (29.11) which encompasses one of the basic theorems for systems of first-order partial differential equations. The system X 1, • • • , Xk of k vector fields on U is said to be involutive if (1) the vectors X 11., ... , Xkl., are linearly independent (in T.(U)] at each point q E U, and (2) the Lie bracket of any two of the X's can be written as a linear combination of X1, . . . , Xk,

,.

(30.1)

[X;, X;] =

L CijXm, m=l

i,j = 1, . . . , k

where the Cij's are differentiable functions on U. If the vector fields Y1, • • • , Yk are nonsingular linear combinations of x1, ... , xk with differentiable coefficients,

,.

Y;

=

L

ajX.. ,

det [aj(q)]

¢

0,

a=l

and the system X 1, ... , Yk. Namely, [Y;, Y;]

=

.. , Xk is involutive, then so is the system Y1,

,.

L a,ll=l k

(30.2)

L a,ll=l

[a7X .. , afX~] {a7af[X,., X~]

+ a7(X,.af)X~- af(X~a7)X,.}.

130

I

Vector Fields and Differential Forms

CHAPTER SIX

If we use (30.1), this last expression is certainly a linear combination of the X/s, and then if we use

X;=

L"'

[b;'(q)] = [aj(q) ]-I,

b;'Ym,

m=l

[Y;, YiJ can be written as a linear combination of the Y/s with differentiable coefficients. If the functions zl, . . . , z" form a coordinate system for U, the system ajaz 1, ... , ajaxk is involutive on U and, as the following theorem shows, all involutive systems are closely related to this simple example.

(30.3) Theorem (Frobenius' theorem.) Suppose the systemX1, . . . , Xk of vector fields on U is involutive. Then each p E U has a coordinate neighborhood V with coordinates z1, . . . , z" such that each Xi is a linear combination of ajazl, .. , ajazk on V with differentiable coefficients. Remark.

The proof shows that one can even require that X 1 = ajaz 1.

PROOF. The proof is by induction on k. When k = 1, Theorem (30.3) is a consequence of the existence of a coordinate system adapted to X 1 (X 1 = a;az 1) on some neighborhood of p. In general let y 1, • • • , y" be local coordinates at p such that X 1 = ajay 1. Put

Y1 = X1; n

yj =

L (X,ym) a~m = xj -

(Xjyl)X !,

j

>

1.

m=2

The X/s are linear combinations of the Y/s: XI= Yl;

j

>

1.

So it suffices to show the existence of a coordinate system z 1, • • • , z" at p in terms of which each Yi is a linear combination of ajax!, ... , ajaz". Note that for each q E U the sets X 1lq, . . . , Xklq and Y1lq, ... , Yklq, span the same subspace in Tq(U) of vectors at q. Since X1lq, .. , Xklq are linearly independent by hypothesis, the vectors Y1lq, ... , Yklq are linearly independent as are the vectors Y2 lq, . . . , Yklq· The system Y1, • • • , Yk is involutive according to the argument that precedes the statement of the theorem and when expressed in terms of the ajayi's none of the vector fields Y2, • • • , Yk involves ajay 1; so none of the brackets [Yh Y m] (j, m = 1, . . , k) involves Y1 ·= a;ay 1. That is, (30.4)

[Y;, Y,]

=

L"'

m=2

D;jYm,

In parti 1, is a linear combination of fJ/fJu 2, • • • , iJjiJuk and Y 1u 1(p) ¢ 0. [Since Y1 IP is independent of ajiJu 2 lp, . . . , iJ/iJuklp, certainly Y1u"'(p) ¢ 0 for some m = 1, k + 1, . . . , n and this u"' can be relabeled as u 1.1 Theorem (29.11) guarantees the existence of a coordinate neighborhood V C W of p with coordinate functions z1, • • • , z" such that (1) Y 1 = fJ/ az 1 on V and (2) if z 1(q) = 0, q E V, then u 1(q) = u 1(p), z2 (q) = u 2(q), . . . , zn(q) = u"(q). Without loss of generality one can assume that the range z(V) of z is a cube centered at z1 (p), . . . , zn(p), (30.5)

I (z 1(q),

. . . 'zn(q)) :q E V} = l(at, . . . , an):la;- z;(p)l

< e,j

= 1, 2, . . . , n}.

From the fact that (z 2 (q), . . . , zn(q)) = (u 2 (q), . . . , un(q)) and u (q) = u 1(p) when z 1(q) = 0, q E V, it follows immediately for each differentiable function f on V that on the sliceS= lq E V:z 1(q) = 0} 1

j

>

1.

In particular azmjau; = lij on S when j > 1. Since Y2, . . . , linear combinations of ajau 2, • • • 'ajauk on w :::> v, (30.6)

k ~

azm Y;z"' = 1..; (Y;u') au• = 0,

Y.~:

are

m>k?:.j>1

i=2

on S. To extend (30.6) to V note that (30.4) together with the relation Y1z"' = az"'fiJz 1 = 0 (m > k) implies

a

(30.7)

azl (Y;z"')

=

[Y1, Y;1z"' k

L

=

+

Y;Y 1z"'

D~;(Y..z"').

a=2

Fix a 2, • • • , an with Ia;- z;(p)l < e, j = 2, . . . , n, and let q(t) be the point in V whose z-coordinates are (t, a 2, • • • , an), -e < t < e. Put f;(t) = ( Y;z"')[q(t) 1and B~; (t) = D~;[q(t) 1; then (30. 7) and (30.6) state that (30.8)

1:

k

(t)

=

L B~;(t)j..

(t),

j = 2, 3, . . . 'k,

a=2

and (30.9)

f;(O) = 0,

j = 2, 3, . . . 'k.

Now the solutions of (30.8) satisfying (30.9) are unique, so that (30.10)

f;(t) = 0,

t E (-e, e), k ?:.j

>

1.

132

I

CHAPTER SIX

Vector Fields and Differential Forms

Since every point in V has z-coordinates of the form (a', a 2, • • • , a") with a' E (-e, e) and iai- zi(p)l < e = (j = 2, , n), it follows that (30.6) is valid throughout V. This means

a Y; = (Y;z 1) ax'

+ (Y;z

a ax 2

2)

+ · · · + (Y,zk) azka

and the proof of Theorem (30.3) is finished.

on V,

•

(30.ll) Corollary. If Xt, . . . , Xk is an involutive system of vector fields on U, each point p E U has a coordinate neighborhood V with local coordinates z 1, • • • , z" such that (30.12)

Xd = 0,

Xd = 0,

.. ' Xd= 0

on V

if and only if

(30.13) PROOF.

af az 1 = o,

0

0

0

on V.

'

According to Theorem (30.3), p has a coordinate neighborhood

V such that k

a

X·= ~ (X·z"') J i..J J azm· m=l

Since the X/s are linearly independent, the (k X k)-matrix (X;zm) must be nonsingular on V, so that

a~;=

k

L bjXm,

j

< k.

m=l

The equivalence of (30.12) and (30.13) follows immediately from these relations. • Exercises 30.1 Suppose that w 3 is a 1-form on the open set U C R 3 whose associated covector w! ~ 0, p E U. Let wl, w 2 , w 3 be linearly independent !-forms and let X,, X2, X a be the dual basis of vector fields, that is, (X;, wi) = c5{ on U. (a) Show that ([X,, X2], w3) = -(X 1 1\ X 2, dw 3). (Cf. Exercise 21.3.) (b) Show that the vector fields X 1, X 2 are involutive if and only if dw 3 = w 3 1\ a for some 1-form a. (c) Under the condition in (b), dw 3 = w 3 1\ a, show that each point p E U has a neighborhood W with a local coordinate system u 1, u 2, u 3 in terms of which w 3 = f du 3 for some differentiable function f. (d) Show that w 3 = f du for some functions! and u if and only if w 3 1\ dw 3 = 0. 30.2 (Based on Exercise 21.7.) Suppose w1, • • • , w" and X 1, • • • , X,. are bases for the !-forms and vector fields on U C Rn, respectively, which are

SECTION

30 Frobenius' Theorem

I

133

dual in the sense that at each p the associated covectors and vectors form a basis for r;(Rn) or T,(R"), and (X;, w8 ) = ~~. Show that (a) d(w 1 1\ w2) = a 1\ w1 1\ w 2 for some 1-form a if and only if X 3 , X 4 , • • • , X,. is involutive. (b) d(w 1 1\ · · · 1\ wk) = a 1\ w1 1\ · · · 1\ wk for some 1-form a if and only if xk+l, ... ' x,. is involutive. 30.3 (Based on Exercise 30.2.) Suppose that at each p E U C R" the covectors associated with the 1-forms w 1, • • • , wk are linearly independent. Show that each point p E U has a neighborhood V with functions g, f', Jk such that w 1 1\ · · · 1\ wk = g df' 1\ · · · 1\ dfk if and only if

d(w 1 1\ · · · 1\ wk) = a 1\ w1 1\ · · · 1\ wk for some 1-form a on U.

Applications of Exercise 30.3. equations

To find a solution of the system of

(30.14)

1 :::; j :::; n,

with the initial condition g(x~, . . . , xi)) = go when the functions f,, . . . , f,. are defined on the open neighborhood W of (x~, . . . , xi), go) in R"+l, one can proceed as follows. Let (30.15)

w = dx"+l - (h dx 1

+ · · · + f,. dx").

If dw = a 1\ w or equivalently dw 1\ w = 0, Exercise 30.3 shows that (x~, . . . , x~, g0) has a coordinate neighborhood with coordinates z 1, , z"+l such that w = f dz"+l for some f. Now it is easily checked that f dx 1 1\ · · · 1\ dx" 1\ dz"+l = dx 1 1\

= dx 1 1\ at (x~, . . . , xi), g0 ). So (Exercise 21.4) the functions y 1 = x 1, • • • , y" = x", y"+l = zn+l form a coordinate system in some neighborhood of (x~, . . . , xi), g0 ). In terms of these coordinates, remembering that dxk = dyk when k < n + 1, w = f dz"+l = f dy"+l

= dx"+l - (/1 dx 1 ~~I = --dyn+l ayn+i

+ · · · + f,. dx")

+ (~~~ -iJyi- -ft)

dy 1

+ ''' + (~~~ -"y" u

-fn ) dy".

This implies axn+~jay; = fi, j < n + 1. If (x~, . . . , xi), b) are the y-eoordinates of the point (x~, . . . , x~, g0 ), the required g can be taken as the function which assigns to (a 1, • • • , a") the value of x"+t at the point withy-coordinates (a 1 , • • • , a", b). To elaborate on this let y denote the

10

I

CHAPTER ONE

Topology of R"

FIGURE I

The first set above is finite because limk Pk = p. The last set is then finite and since U is an arbitrary open neighborhood of f(p), limk f(pk) = f(p). (5.2) Example. Iff: D-+ R"' is continuous and F is a compact subset of D, then f(F) is compact in R"'. That is, the continuous image of a compact set is compact. PROOF. Suppose that I qk} is a sequence of points in f(F) and for each k choose Pk E F so that f(pk) = qk. IPk} has a convergent subsequence IPkl, lim,. Pk = p E F, and by continuity lim,. f(Pk) = f(p). Thus the subsequence I qk I of {qk} converges and f(F) is compact.

(5.3) Theorem. If D is a compact subset of R" and f: D __... B is continuous, one-one, and onto, thenj- 1 : B __...Dis continuous. Let {qk} be a sequence of points of B converging to the point PROOF. q. Suppose {j- 1(qk)} ;:_1 converges top. By the continuity of the mapj, J(p) = lim,.J o J- 1(qk(m)) = lim,. qk(m) = q;

so p = J- 1(q) is the only accumulation point of {j- 1(qk)} and by Example (4.1) {f- 1(qk)} converges to J- 1 (q). That is, J- 1 is continuous. For the sake of illustrating the possibilities a second proof is given for Theorem (5.3). Let g = j- 1• It suffices to show according to Example (5.1) that g- 1 (U) is relatively open in B whenever U is open in R". Now g- 1(U)

= J(U n D) = B - J(U•

fl

D).

Since U• fl Dis compact, so isj(U• fl D) by Example (5.2), and g- 1(U) = B fl W where W = [f(U• fl D)]• is open. According to Exercise 2.1 g- 1 (U) is then relatively open in B. + There is another important set property besides compactness which is preserved under continuous mappings. A set H is connected if it is not the union of two nonempty, disjoint, relatively open subsets. That is, H is connected if whenever H = J 1 u J 2, J 1, J 2 relatively open and J 1 fl J2 = f25, it follows that either J1 = 0 or J2 = fZf. The continuous image of a connected set is connected. Indeed if j: H __...

30 Frobenius' Theorem

SECTION

I

135

30.6 If n = w 1 1\ · · · 1\ w"' where the covectors associated with w 1, • • • , w"' are independent at each point of U C RN, that is, n ~ 0 on U, show that dn = a 1\ n for some 1-form a if and only if each dwk is a sum of terms of the type w' 1\ {3' where the {3''s are 1-forms. This is often expressed dwk = 0 mod (w 1, . . . , w"'). That i>1, dwk = 0 for each k when the relations w 1 = 0, . . . , w"' = 0 are taken as valid equations for the purposes of substitution. (Cf. Example 30.18 below.) 30.7 Show in R 2 , with variables x, y instead of the usual x 1, x 2, that each of the following systems can be prolonged to a system of type (30.16) by introducing extra variables [see Example (30.18)] and investigate whether or not the prolonged system satisfies the integrability conditions an = a 1\ g or dwk = 0 mod (w 1, • • • , w"') of Exercises 30.5 and 30.6. (i) of = y2, of = 3x. oy ax (ii) ()2f

ax2

+ iJ2f =

e"'.

()y2

(iii) ()2f - ()2f + 3x of + e"' = 0, of + of = 0. ()y ax ax ox 2 oy 2 (iv)

~

()x2 oy

3f2+x.

=

+

(30.18) Example. o 2f/iJx 2 3f = x; ofjoy pendent variable p and consider the system

+ ofjox =

ap

of ax= p,

OX= X- 3f,

e"'. Introduce the de-

of oy = e"'- p,

which is equivalent to the original one. This new system does not have the form (30.16) because iJpjoy is not specified. To allow all possibilities add the equation opjoy = h where his an as yet unspecified function of x, y, f, and p. w1

=

df - (p dx

+ [e"' -

p] dy);

w2

=

dp- ([x- 3f] dx

+ h dy).

Now dw 1

=

+ dp 1\ dy 1\ dy + (x - 3f) dx

-dp 1\ dx- e"'dx 1\ dy

= h dx

1\ dy - e"' dx

= (h -

e"'

+x

1\ dy

- 3f) dx 1\ dy.

dw 1 = 0 mod (w\ w2) only if h is chosen to be h = e"' 3 df - dx and dh = e"' dx

+

+ 3f- x.

In this case

dw 2 = 3 df 1\ dx - dh 1\ dy

= 3(p - e"') dy

= (3e"' -

1\ dx - (e"' dx

3p - e"'

+ 3 df -

+ 3p + 1) dx

dx) 1\ dy

1\ dy,

which is not zero in x, y, f, p space. Consequently the system in the example will not satisfy the integrability conditions above for any choice of h.

136

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CHAPTER SIX

Vector Fields and Differential Forms

30.8 Show directly by differentiating the equations in Example (30.18) and substituting to arrive at a contradiction that this system has no solutions.

31

The operator fJx

If X is a vector field on U the operator fJx, extending the action of X on functions to differential forms and vector fields, is defined by fJxw = d(X

(31.1)

_j

w) +X

_j

dw;

fJxY = [X, Y].

If the function/is regarded as a 0-form, 8xf = d(X

_j

f)

+X

_j

df = Xf.

(31.2) Theorem. fJx is the only linear operator defined on differential forms and vector fields and satisfying: (i) fJx(a A fJ) = (fJxa) A fJ

+aA

(fJxfJ).

(ii) fJx da = dfJxa. (iii) 8xf = Xf. (iv) fJx(Y, w) = (fJxY, w)

+ (Y, fJxw) when w is a 1-form.

PROOF. Straightforward computations, which are left to the exercises, show fJx satisfies (i)-(iv). If 0~ is another linear operator satisfying (i)-(iv), it is readily seen using the representation (21.3) and (i)-(iii) that Ox agrees with fJ~ on differential forms, and if this is applied to (iv)

(fJxY, w) = (OxY, w)- (Y, Oxw)

= (O~Y, w) - (Y, O~w) = (O~Y, w) for each 1-form w; so OxY = O~Y, completing the proof.

+

There is a geometric way of defining Ox. It depends on the notion of the derivative of a family of forms or vector fields with respect to a parameter. If Iw(t): t E (- 8, 8) l is a family of k-forms on U, the k-form aw(t)jat is determined by the relation (31.3)

for any k vector fields X 1, • • • , X k· Similarly if I Y(t) : t E (- 8, 8) l is a family of vector fields on U, aY (t) j(Jt is defined by (31.4)

aY(t) f = ~ I Y(t)fl at at '

31

SECTION

The Operator 8x

I

137

or equivalently ;aY(t)

\--at,

(31.5)

\ a;=

a

at (Y(t), a)

for each 1-form a.

(31.6) Theorem. Suppose that q,: (- 8, 8) X V---+ U is a flow of the open set V into U with velocity field X(t) and X = X(O); then fJxw

a rPt*w It=O' = at

(31. 7) on V. PROOF. The proof consists of showing that the operator fJx defined by (31.7) satisfies (i)-(iv) of Theorem (31.2). 8x is clearly linear; (iii) is just a rephrasing of the statement that q, has velocity field X when t = 0; while (i), (ii), and (iv) are obtained, respectively, by differentiating the relations rl>i (a A {3) = rl>i a A rl>i {3,

rl>i da = dq,i a,

and

• One consequence of Theorem (31.6) is that

(31.8)

on

V.

To derive (31.8) note that for fixed t and smalls the map 1/;: (s, q,(t, p))---+ q,(t + s, p) is a flow of r/>1 V into U whose velocity field is X(t) when s = 0. Thus from Theorem (31.6) on r/>t V.

*

_ rPt* . as!/;• a *W Ia=O -_ 08 a rPt*1/1.*W Ia=O -_ as a rPt+•W * Ia=O -_ aq,jw at

rPt (JX(t)W -

on V. In applications the form w often depends on t directly, and a somewhat generalized version of (31.8) is needed to cover this situation.

(31. 9) Theorem. If q,: (- 8, 8) X V ---+ U is a flow of V into U with velocity field X(t), then whenever {w(t): t E (- 8, 8)} is a family of k-forms

138

on

I

U

(31.10)

CHAPTER SIX

~ 1/Jiw(t)

Vector Fields and Differential Forms

+ a~~t)}

=

1/Ji { (JX(t)W(l)

=

q,j {d[X(t) _j w(t)]

+ X(t)

_j dw(t)

+ a~~t)}

on V.

PROOF. If Y 1, • • • , Y k are vector fields on U the function g( u, v) = ( Y, I\ · · · I\ Y k, q,: w(v)) is continuously differentiable in both u and v, so that when u and v are regarded as functions of t,

Putting u = t,

1'

=

t yields (31.10).

•

Exercises 31.1

If I w(t) :t E (- Cl, Cl) l is a family of k-forms on U and c is a k-chain

in U, show that

~ { w(t)

{ aw(t) .

=

dt }c

}c

at

31.2 (Based on Exercise 31.1.) If Iw(t): t E (- Cl, Cl) l is a family of k-forms on U, q,: ( -e, e) X V--+ U is a flow with velocity field X(t), and c is a k-chain in V, show that (31.ll)

31.3

.!!:_

f

dt )q,,oc

w(t) =

f

X(t) _j w(t)

}il(t/>t•C)

Show that Leibnitz's formula [a -d

dt

J.b F(t, s) ds = a

+ }tf>,oc f [x(t)

_j dw(t)

+ aw(t)] · at

= a(t), h = b(t)]

F(t, b) -db - F(t, a) da dt

dt

+

J.b -aF (t, s) ds a

at

is a special case of (31.11). [HINT: Let 1/Jt o c represent integration over [a(t), b(t)).] 31.4 (Equation of continuity.) Let q,: ( -e, e) X V--+ U describe the flow of ann-fluid with velocity vector field X(t) and density w(t). Thus if c represents integration over the region G, c w(t) represents the amount of fluid in Gat time t. If the fluid has no sources or sinks, ••c w(t) is a constant since it represents the amount of a common batch of fluid. Show in this case that

J

J

d[X(t) _j w(t)]

+ aw(t)

=

0.

at

This is the equation of continuity. 31.5 (Invariant k-forms.) A k-form w is invariant under the flow q, if q,jw = w for each t. Show that w is invariant if and only if 8xw = 0 where X is the velocity field of the flow q,.

SECTION

32

32

Homotow and Poincare's Lemma

I 139

Homotopy and Poincare's lemma

Two differentiable mapsf0 and !I of Vinto U are (differentiably) homotopic ifthereisadifferentiablemapq,:[O, 1] X v~ Usuchthatj0 (p) = q,(O, p), p E V, and fi(q) = 1/>(1, q), q E V. The map q, is called a homotopy of fo and h. It describes a differentiable deformation of the mapf0 into the map !I, the intermediate maps f 1 being given by ft(p) = q,(t, p), t E [0, 1], pE V. A simple example of a homotopy stems from a flow 1/;: ( -~. ~) X V ~ U of V into U. The two maps p ~ 1/;(0, p) and p ~ 1/;(s, p) are homotopic, and all that is required to show this is a change of scale in the flow parameter s. The indicated homotopy is the map q,: [0, 1] X V ~ U given by

q,(t, q) = 1/;(ts, q). In the space [0, 1] X V, whose points are ordered pairs (a, p) where a E [0, 1] and p E V, the letter t is used to represent the function t(a, p) = a describing projection onto the first factor [0, 1] and 1r to describe projection onto V, 1r(a, q) = q. The symbol ajat by itself is ambiguous, of course, unless the remaining coordinate variables are given, but in such a product space aI at will henceforth be used for the unique vector field satisfying 11"*

(ft) =

;a

\

\fit, dt; = 1,

0,

at every point of [0, 1] X V. If yt, ... , yn is a coordinate system for V and t, x 1 = y 1 o 1r, . . . , xn = yn o 1r is used as a coordinate system for [0, 1] X V, ajat has its customary interpretation. The map p ~ (s, p) injecting V into the sth coordinate slice of [0, 1] X V is denoted by i,, so that i,(p) = (s, p). Thus if q,: [0, 1] X V ~ Uisahomotopyoff0 andf1,f1 = q, o it. Sincet o i,(p) =the constant s,i:(dt) = 0. Ifjisadifferentiablefunctionon V,i:(d[jo1r]) = dfbecause 1r o i, is the identity map on V. Thus i:(dx;) = dy; in the coordinate systems of the preceding paragraph. (32.1)

Lemma.

If w is a differential form on [0, 1] X V, on

(32.2)

V.

Using the (t; x) coordinate system introduced above, we find PROOF. two types of terms in w: those containing dt and those not containing dt. If w

= f(t;

x) dxit 1\ · · · 1\ dx••, 1\

dx'•

+ aj at dt 1\ dx'• 1\

· · · 1\

dx'•,

140

I

CHAPTER

srx

and since ajat ..J w

Vector Fields and Differential Forms

=

Batatw =

0,

iat

..J dw = aj (t· x) dxi, 1\ · · · 1\ dxi•; at '

so

= :s {f(s; y) dyi•

1\ . . . 1\ dyi•l

a ·• w. = as~. Whereas if w = g(t; x) dt 1\ dxh 1\ dw =

~ ag. dxi 1\ dt LJ ax' j

1\ dxi• 1\ · · · 1\ dxi•-•,

and

while

1\ dxi•-•

+ agat dt 1\ dx;, 1\

· · · 1\ dxi•-•;

so

and both

"*"uatatW = o

~.

because

an d

a as

·•W =

~.

o

•

With the aid of this lemma it is easy to prove the following important result. (32.3) Theorem. If /o and /1 are homotopic maps of V into U, for each k = 0, 1, . . . , n, n + 1, . . . there is a linear map Dk of k-forms on U into (k - 1)-forms on V such that (32.4) whenever w is a k-form on U. If n is the dimension of U, Do = D,.+l = 0.

SECI'ION

5 Continuity

/

11

R"' is continuous and f(H) is not connected,

where J 1, J 2 are relatively open inf(H); J 1 n J 2 = f2f; and neither J 1 nor J 2 is empty. The same is true for the sets J- 1 (J 1) and J- 1 (J 2), so that the set H = J- 1(J 1) v J- 1(J 2) cannot be connected either. Very few additional facts about connectivity are needed for this book. With minor omissions they are: (1) The closed unit interval [0, 1] is connected. To prove this suppose J 1 and J 2 are disjoint, relatively open sets in [0, 1], and to be specific suppose 0 E J 1· If J 2 is not empty there is a point c E J 2· In this eventuality put (5.4)

b = sup {a E J 1: a

< c},

and consider the two possibilities (i) b E J 1 and (ii) b E J 2· Each of these leads to a contradiction with (5.4). Elaborating on this, note first that since J 2 is relatively open (c - 6, c

+ 6) n [0, 1] c J2

for some 6 > 0; sob ~ c - 6 < c. If b E J 1, (b - e, b + e) n [0, 1] c J 1 for some e > 0 and b is not the supremum of the set of a's in (5.4). If bE J2, (b- e, b +e) n [0, 1] C J2 for some e > 0 and b is not even a limit point of J 1, let alone the supremum of the set of a's in (5.4). (2) If each pair p, q of points of H belongs to a connected subset A,.,, of H, then H itself is connected. If H were not connected under these conditions, H = J 1 v J 2 as above with J 1 ¢ f2f and J 2 ¢ f2f. Choose p E J1 and q E J2; then A,.,,= (A,.,, n J1) v (A,.,, n J2) and these last two sets are nonempty, disjoint, and relatively open in A,.,,, violating the hypothesis that A,.,, is connected. A continuous map g: [0, 1]-+ His called an arc in H joining g(O) to g(1), and a set H is called arcwise connected if each pair p, q of points of H can be joined by an arc in H. (3) An arcwise connected setH is connected because an arc g: [0, 1]-+ H joining p and q provides a connected subset A,.,, = g([O, 1]) satisfying the conditions of (2) above. Exercises 5.1 In the following let j: A VB-+ R and /lA: A-+ R, /IB: B-+ R denote the restrictions off to the sets A and B respectively.

(a) Show that f need not be continuous if /lA and /IB are continuous. (b) If A and B are open and /lA, JIB are continuous, show f is continuous. (c) If A and B are closed and /lA, JIB are continuous, show f is continuous.

142

I

CHAPTER SIX

Vector Fields and Differential Forms

Then the map q,: [0, 1] X x- 1(J.) xi o q,(t, q)

~

= (1 -

U defined by t)xi(p)

+ txi(q)

is a homotopy showing that N P = x- 1(J.) ~ ( p} in U and Theorem (32.4) applies to yield the desired (k - 1)-form a on N P· + Exercises 32.1

If fo and f 1 are homotopic maps of V into U, cis an m-chain in V with

ac = 0, and w is an m-form on U with dw = 0, show that

fcfriw = fcfiw.

32.2 Suppose it is known that the two open sets U 1 and U2 are each homotopic to a constant and that any two points p, q E U 1 rl U2 can be joined by a 1-simplex Cpq: [0, 1]---+ U1 rl U2, Cpq(O) = p, Cpq(1) = q. If w is a 1-form on U 1 V U 2 and dw = 0, show that w = df for some function f on U 1 V U 2· 32.3 Let U be the annulus 1 < (x 1) 2 + (x 2) 2 < 2 in R 2• If w is a 1-form on U with dw = 0, show that w = Xa + df for some function f on U and some X E R, where a = - (x 2)- 1 dx 1 + (x 1)- 1 dx 2 • a is the form usually denoted by d8 when r and 8 are polar coordinates. 32.4 Show that a 1-form w = ~ g; dx' has the form w = df locally if and only if 8g;/8xi - 8g;/8x' = 0, i < j.

CHAPTER SEVEN

Applications to Com.plex Variables

33

Complex structure

The topic "complex structure" as used here divides naturally into two subheadings. The first, and in a way the more superficial of these, is associated with the problem of using the complex numbers C as the scalar field rather than the real numbers R. The results obtained by doing this do not differ in any outstanding manner from the previous case, but since they form the framework for an exposition of the second subheading, the concept of holomorphy or analyticity, it is worthwhile to consider briefly the modifications needed in the definitions used so far. Let U be an open subset of R" with cartesian coordinates x 1, • • • , x". A function! on U is understood now to be a C-valued functionf: U--+ C as opposed to the previous interpretation of a function as being real-valued (R-valued). Such a function can be written uniquely as the sumf = u + iv of two real-valued functions u and v known, respectively, as the real and imaginary parts of f. The conjugate of f is the function J = u - iv, so that f is real-valued if and only if J = f. f is differentiable of class k or C"' on U if both u and v are C"' on U in the previous sense. Alternatively, f is C"' on U if the map f: U--+ Cis a C"' map when Cis identified with R2k in the natural manner. In this setting a vector ZP at p is a C-valued operator of the form (33.1)

Zp =

aL~ _j_ Ip• ox 1 Ip + · · · + a" iJx"

a; E C allj,

whose domain is the set of all (complex-valued) functions which are 0 at Associated with each vector ZP is its conjugate

p.

(33.2)

Z

P

=

-a, + · · · + -al a1 iJx 1

P

a"iJx" P ·

The vector Zp is called a real vector if Zp = Zp. Each vector Zp can be 143

144

I

Applications to Complex Variables

CHAPTER SEVEN

represented uniquely in the form Zp = Xp + iYp where Xp and Yp are real vectors at p. As before, the vectors at p form an n-dimensional vector space (over the scalar field C) which is denoted by CT P(R"). The real vectors at p form a subset TP(R") of CTp(R") which is closed under addition and multiplication by real scalars, but it is not a vector subspace since it is not closed under scalar multiplication by complex scalars. The dual space of CTP(R") is the space of covectors at p and is denoted by CT!(R"). It consists of all C-linear maps of CTp(R") into C and has a basis (dx 1)p, . . . , (dx")p where for each differentiable function h at p, the covector (dh)P is defined by (33.3) Thus a typical element

wp

E CT;*(R") has the form

(33.4)

bk E Call k.

Technically speaking these (dxi)p's are not quite the same (dxi)P's used earlier because they are defined on the larger space CTp(R"), but their restrictions to T p(R") are the same as the (dxi)p's used earlier. The conjugate wp of wp, the concept of a real covector at p, exterior products, vector fields, differential forms, the mapsf* andf*, and the exterior derivative are defined as before and satisfy the same relations with linearity over C replacing linearity over R. It is worth noting that iff: U ~ Rm is a differentiable map of the open subset U of R" into Rm, then the maps f*: CTp(R") ~ CT1

(Rm) and f*: cr;

(Rm) ~ CT!(R") preserve the realness of a vector or covector, respectively. On the real subsets of CTP(R") and cr;

(Rm) they coincide with the f* and f* used earlier. The scalar product of two vectors Xp, YP belonging to CTp(R") is given by n

(Xp, Yp) = ~ (XpXi)(Y pXi)-

(33.5)

;=t

where the xt, , x" are cartesian coordinates for R~ and (****)- is used to denote the conjugate of(****). It is a positive definite Hermitian scalar product satisfying the identities (subscripts p omitted): (i) (aX+ bY, Z) = a(X, Z)

(33.6)

+ b(Y, Z);

(ii) (X, Y) = (Y, X) = (Y, X)-; (iii) 0


.{(p)(dwk),,

p E U, 1

~j ~

m,

k=l

where the m X m rna trix (>.{ (p)) is the in verse of [(awkI azi) (p)] ; so (34.13)

(dzi), =

L"' x{(p)(dwr.), k=l

[because d] = (df)-]. Equations (34.12) and (34.13) show that (dw 1) ,, . . . , (dwm),, (dw 1),, • • • , (dwm), form a basis for CT!(Cm), and in

148

/

CHAPTER SEVEN

Applications

to Complex Variables

keeping with this notation the dual basis for CTP(Cm) is denoted by

a~' \P•

· · · ' a!m \P•

a~' \P•

· · · ' a!m \P·

Let 1 =::; k =::; m,

(34.14)

where the ui's are real-valued. Then dwk = du 2k-! + i du 2k; dwk = du 2k-! - i du 2k; so that (du 1)p, . . . , (du 2m)P also form a basis for CT!(Cm). This fact together with 2m

(dui)P =

(34.15)

.

L ::~ (p)(dxk)P k=l

implies (34.16)

oui 1 =::; j, k =::; 2m } ,e. 0, det { oxk'

p E U.

Since condition (i) of Definition (34.10) is equivalent to saying that the map p----+ (u 1 (p), . . . , u 2m(p)) of U----+ R 2m is one-one, it follows that the functions u', ... , u 2m form a local coordinate system for U in the sense of Definition (15.1). Conversely suppose that (dw 1)p, . . . , (dwm)P span the same subspace of CT!(Cm) as (dz 1 )p, . . . , (dzm)P. Then relations of the type (34.12) and (34.13) exist and the above chain of arguments can be followed from there (with p fixed) to yield (34.16), so that u 1, • • • , u 2m form a local coordinate system in some neighborhood of p by Theorem (15.3). This ensures that (i) and (ii) of Definition (34.10) will be satisfied in some neighborhood of p, and if the wi's are known to be analytic they will form an analytic coordinate system there.

+

iu 2i, 1 =::; j =::; m, be analytic (34.17) Summary. Let wi = u 2i- 1 parts u 2i-l and imaginary parts real functions on the open set U C Cm with 2 u i, respectively. Then the following conditions are equivalent: • • • , wm form an analytic coordinate system valid in some neighborhood of p E U. (2) det {(owijiJzk)(p), 1 =::; j, k =::; m l ,e. 0. (3) u 1, • • • , u 2m form a differentiable coordinate system valid in some neighborhood of p E U. (4) det f (oui/iJxk), 1 =::; j, k =::; 2ml ,e. 0.

(1) w 1,

Exercises (k

34.1 Show that there is a unique operator + I)-forms on U and satisfying: (i) o(aa + b(3) k-forms.

=

a oa

+ b o(3

a mapping

k-forms on U into

when a and b are constants and a and (3

SECTION

35

Analytic Functions of One Variable

/

149

(ii) aca " [j) = aa " {j + (-l)deg(a)a " a{j. a2 = 0. (iv) (af)P = ~f=l (ajjawi)(p)(dwi)P whenever w1, . . . , w" is an analytic coordinate system valid in a neighborhood of p. (iii)

a

34.2 (Based on Exercise 34.1.) Show that the operator = d - a satisfies (i), (ii), and (iii) of Exercise 34.1. Show that a function f is analytic if and only if o. 34.3 Show there is a unique operator J mapping k-forms into k-forms and satisfying:

a,=

(i) J(aa + b{j) = aJa + bJ{j. (ii) J(a 1\ {j) = Ja 1\ J{j. (iii) J 2 = -1. (iv) Iff is an analytic function, J(f)

= f,

J(df)

=

i df, and J(dJ)

=

- i dj.

+ ···+

34.4 Let w be the 2-form w = dz 1 1\ dz 1 dzm 1\ dzm on Cm. Put wi = w 1\ · · · 1\ w(j factors). Show that wm = m! dz 1 1\ dz 1 1\ · · · 1\ dzm 1\ dzm and express this 2m-form in cartesian coordinates. Show that dzk 1\ rlzk " wm-l = (1/m)wm. 34.5 (Based on Exercise 34.4.) If the functions h1, . . . , hm are analytic on U and c is a 2m-chain in U, show that

35

Analytic functions of one variable

Let U be an open subset of the complex numbers C. In this case put = x 1 + ix 2, so that the superscript on the z is omitted. A complex function f = u + iv on U is a map f: U-+ C (see Figure 29). The condition for f to be analytic on U, ajjaz = 0, becomes

z

0 =

(-~ j!!.._) + i (j!!.._ au) , 2 2 ax 1 - ~l __i_) ax 2 (u + iv) = (au ax 1 _ ax ax 1 + ax

or (35.1)

au

av

av ax 1

ax 1= ax ·2

FIGURE 29

U

= -

au ax 2 "

150

I

CHAPTER SEVEN

Applications to Complex Variables

Equations (35.1) are known as the Cauchy-Riemann equations and they have an important geometrical interpretation. Notice that

( a)

au a f* axt = axt axt

+ axt ax2;

a) au a f* ( ax2 = ax2 axt

+ ax2 ax2.

av a

av a

It follows from the Cauchy-Riemann equations that

a J [ a ]) au au (f* [ axt 'f* ax2 = ax I ax2

av av + axt ax2

= 0;

Thus when the Jacobian of fis not zero the vectors!* (ajax 1) andh (a!ax 2) are orthogonal, and have the same length and the same relative orientation as ajax 1 and ajax 2 (see Figure 30). In this case if the basis vectors a;ax 1lp, ajax 2lp are identified with a;ax 111.. is a scalar >0). A mapfwith the property that at each point!* is a rotation followed by a

f.

(a:2) f.

(a:~) FIGURE 30

12

I

CHAPTER ONE

Topology of R"

5.2 Let f: R 2 --+ R be given by !( a b) _ ab(a - b) ' - (a2 b2)3/2

if (a, b)

+

~

(0, O);

j(O, 0) = 0.

Let fa: R--+ R be the map X--+ f(a, X) andjb: R--+ R the map X--+ j(X, b). Show that for each a, b E R the maps fa and fb are continuous but that f is not continuous. [HINT: j(ljn, 1/n) = O;j(1/n, -1/n) = 1/v/2.) 5.3 Show that if J is a connected subset of R containing a and b with a < b, then J :::> [a, b). 5.4 (Based on Exercise 5.3.) Show that each connected subset of R has one of the forms (a, b), [a, b), (a, b), (a,+ co), [a,+ co), (-co, a), (-co, a], [a, b),(- co,+ co) = R where a .:5 b. 5.5 Show that the map j: B --+ R" is continuous at p E B if and only if for each s > 0 there is a 8(p) > 0 such that llq - Pll < 8(p), q E B, implies that IIJ(q) - f(p) II < s. The 8 needed here depends in general on p.

Definition. A map f: B ~ R" is uniformly continuous on B if and only if for each e > 0 there is a 8 > 0 (that does not depend on p) such that llq - PI! < 8, q, p E B implies llf(q) - f(p)ll < e. 5.6 Show by example that not every continuous function is uniformly continuous. 5. 7 Prove that each continuous function f on a compact set B is uniformly continuous on B.

Recall that the diameter of the subset A C R" is the number diameter (A) = sup {liP - qll: p, q E A j. If q,: B

~

R", the oscillation of q, at the point p E B is

oscillation, (q,) = inf {diameter [f(B n B.(p))]: e

> 0 j.

5.8 Show that the map q,: B--+ R" is continuous at p if and only if oscillation, ( q,) = 0. 5.9 Let ,P: R"--+ R" be described by xm

0

r~>

n

=

'L arxk(p) k=l

where (ak') is a real n X n matrix. (a) Show that q, is linear, that is, q,(Xp + f.l,q} = X,P(p) + f.Lr/>(q) whenever p, q E R" and X, f.L E R. (b) Show that each linear map ,P: R"--+ R" has the above form for some matrix (a~).

(c) Under what conditions on (a~) is q, continuous? (d) Under what conditions on (a~) is q, one-one? (e) When is q, an open mapping? That is, when is it true that ,P(W) is open in R" whenever W is open in R"?

152

I

CHAPTER SEVEN

iJif; = {p} -

Applications to Complex Variables

{po}, where {q} denotes the map F(p) = tfdz,

fq}(~ 0 ) =

q, define

f generalized analytic on U.

Then F is well defined, C\ and analytic. Moreover dF = f dz; so f = aF I az. 3. If F is analytic on U, the function G defined on U by

(35.5)

F(p) - F(q) ) z(p) - z(q) ' G(p) = aF iJz (q),

p E U, p

o;6

q,

p = q,

is generalized analytic on U. 4. Cauchy's integral formula. IfF is analytic on U, then FisC"" on U, iJkF/(iJz)k is analytic on U for each k ;:=: 0, and

(35.6)

iJkF ( ) [ [ dz J = k 1 [ F dz (iJz)k q }c z - z(q) · }c [z - z(q)]k+ 1

whenever c = aq, is the boundary of a 2-chain in U. In particular, if cis the 1-chain z o c(t) = z(p) + re2Trit, t E [0, 1], bounding the disk centered at p of radius r, lz(p) - z(q) I < r, and this disk belongs to U, then

(35.7)

iJ"F n! ( Fdz (iJz)" (q) = 2'11"i }c [z- z(q)Jn+l'

n

= 0,

1, 2, . . . .

Results 4 and 2 show that a generalized analytic function f is analytic, because f = iJF / iJz, which is analytic by 4. These four results are now proved in order. Proof of 1. The first result is a combinatorial observation. Suppose that h: [0, 1] X U ~ U is a differentiable map which deforms U to the point Po in the sense of Section 31; that is, h(1, p) = p, p E U and h(O, p) = p 0 , p E U. Then the associated cone operator, q, ~ cone (q,), carrying k-chains into (k + I)-chains is defined by two conditions:

(35.8)

cone (a,P

+ bif;)

=

a cone (q,)

+ b cone (if;)

when a and bare integers and q, and if; k-chains. If T: ~k cone (T) is the (k + I)-simplex

(35.9) Figure 31 should aid in following the argument.

~

U is a k-simplex,

SECTION

35

Analytic Functions of One Variable

cone

r =

(ooner)•uf

I 153

..

To u~

oone!TOul)

=

l

="

u FIGURE 31

Two computations are needed. (conerou7+')(tl, . . . ,tk)

= (cone r)(tl, . . . , 0, . . . , tk), (35.10)

= h(tl

+ tk, r(tl, . . . + tk, To u~(tl,

+

= h(t 1 +

=

cone (r o u7){t', . . . , tk),

(cone r o u!t~)(t 1 ,

(35.ll)

• •

,

= (cone r){t 1,

0 in the jth position,

, 0,

.. , tk-1))

.. , tk-1)) j


(p) = 0 all n, Taylor's series expansion shows f vanishes identically on some neighborhood of p. If j(p) ~ 0 for some k, use Taylor's finite development [Theorem (36.1)] to show (ii) holds.) 36.2 (Based on Exercise 36.1.) Iff and g are analytic on the open set U, show that every accumulation point of the set S = I q :f(q) = g(q) I either (i) is an interior point of S or (ii) is in the complement of U.

Answers to Selected Exercises

Section 2 2.2 IP E R:O ~ p < 11. 2.3 Un = (0, 1/n) in R. 2.4 Fn = [n, oo) in R .

..

2.5

(i) f1 (-1/n, 1/n) = 101; (ii) f1 (-1/n, 1) = [0, 1); n=1

n=1

(iii) f1 ( -n, n) = ( -1, 1). n=1

2.6

..

V [0, 1 - (1/n)] = [0, 1). n=1

2.ll Let 'Uo = I B.(q): e is a rational number, q is a rational point of Rn, and B.(q) C some W E '0 1. 'U0 is countable. For each U E 'Uo let W(U) be a specific element of '0 chosen so that U C W(U). Put 'Oo = I W(U): U E 'Uol· 2.12 B,((O, . . . , 0))

.

=

f1 B~+Oik>((O, •.. , 0)) . .1:=1

2.13

If p E F., d(p, q)

d(p', q) ~ d(p', p)

< e-

~for

some q E F, ~ C F •.

. + d(p, q) < e; so B1(p)

2.14 If q E f1 F 11 k, let qk E F with d(q, qk)

> 0.

< 1/k.

If p' E B,(p), then

qk--+ q, which implies

.1:=1

q E F since F is closed. The other inclusion follows from F C FI/k· 2.15 If p E U(A) let D(p) = B:(q) be an open ball with rational center q and rational radius E containing at most a countable number of points of A. From Exercise 2.9(b) the class 'W = ID(p): p E U(A) I is a subclass of 'U and hence at most countable. U(A) = V ID(p) fiA:D(p) E 'WI is countable because it is a countable union of countable sets.

Section 3 3.2 If Dis not closed its complement D• is not open; so there exists a p El D such that Buk(P) f1 D ~ fZf for each k. Choose Pk E B 11 k(P) f1 D. 3.3 Pk = k in R. 3.4 Let Ip,, P2, .•• 1 be any enumeration of the rational numbers in (0, 1). 159

160 /

Answers to Selected Exercises

3.5 d(p, qk) :::; d(qk, Pk) + d(pk, p) ---+ 0 as k---+ oo. 3.8 Let IPk} run successively (with repetitions) through the finite sets A2, . . . , where An= lm/2n:-n2":::; m:::; n2", d(m2-",F) < 2-"}.

A~,

Section 4 4.5 Let I qk I be a sequence of distinct points in B with no accumulation point in B. Let C = I q11 q2, . . . } and for each k let Uk be an open set with U.. (\ C = lq .. }. Put 'U = ![closure (C)]c, U1, U2, . . . , U,., . . . }. 4.6 Suppose the assertion is not true. For each k choose Pk, qk ED, d(pk, qk) < 1/k and Pk, qk fl V for any V E 'U. Since D is compact, by passing to a subsequence one can assume Pk---+ p and then qk---+ p too. But then p E some l' in 'U and for large enough k, pk, qk E V too, contradicting the original assumption. 4.7 Let B = I (a 1, a 2) E R 2 :a 1 > 0 and a 1a 2 = 1 or a 2 = 0}; 'U = I U1, U2l where U 1 = l(a 1,a 2 ):a 2 > 0}, U2 = l(a 1,a 2 ):a 1 > O,a 1a 2 < 1}. Ifpk = (k, 1/k), qk = (k, 0), then d(pk, qk) = 1;k, qk El U1, Pk El U2. 4.8 Let E = one-half the Lesbesgue number of 'U as a covering of D (cf. Exercise 4.6).

Section 5 5.3 If c E [a, b) (\ Jc, then the relatively open sets J 1 = ( - oo, c) (\ J, J 2 = (c, + oo) (\ J, would show that J is not connected. 5.6 The function f(X) = 1/X, X E (0, 1), is not uniformly continuous on (0, 1) because 1/(1/n) - f(l/2n)l = n while 11/n- 1/2nl-+ 0. 5.7 For each p E B let U(p) be an open set such that U(p) f'l B = lq:lf(q) - f(p) I < e/2} and let 5 be the Lesbesgue number of the open covering 'U = I U(p) :p E B} of B (cf. Exercise 4.6). Alternatively, iff is not uniformly continuous there exist points Pk, qk E B such that lf(pk) - f(qk) I -r+ 0 while d(pk, qk) ---+ 0. Since B is compact, by passing to subsequences one can assume Pk---+ p. Then qk ---+ p too and

because f is continuous at p. This contradicts the assumption that f is not uniformly continuous on B. 5.9 (c) Always continuous. (d) det (a~) ~ 0. (e) det (a~) ~ 0.

Section 6 6.3

True for n

=

1 only.

Section 8 8.2 This is a consequence of the formula in Exercise 6.4 and the fact that the determinant of a matrix product is the product of the determinants of the matrix factors.

CHAPTER TWO

Differentiation on R n

6

Differentiation

The value of a function! on R" at the point p = (a 1, • • • , a") is usually denoted by f(al, . . . , a"). Regarded as a function of ak alone with all the aPs except the kth being treated as constants, the derivative off with respect to ak is denoted by Dd or af I axe and called the partial derivative off with respect to xk. xl, . . . , x" here are the cartesian coordinate variables. Many of the arguments involving partial derivatives are most conveniently expressed in vector notation. For this purpose let e1 = (1, 0, . . . , 0), . . . , e, = (0, 0, . . . , 1), ek being the n-tuple which has a one in the kth position and zeros elsewhere. Thus Dkf(p) = lim f(p

(6.1)

+ hek)

- f(p).

h

ll-+0

The ordinary mean value theorem of differential calculus implies that if Dkf(q) exists at every point of the line segment joining p to p + bek, that is, at every q = p + 8bek, 0 ~ 8 ~ 1, then

(6.2) where q = p

+ 8bek for some 8 E [0, 1].

Given two points p and q of R", note that q - p = h1e1 + xi(q) - xi(p). Let Po = p, P1 = Po+ h 1e1, . . . , Pk = p,_ 1 + h"e, = q. Suppose f is a function defined on an open set containing the polygonal path PoP1 · · · p, made up of the line segments Pk-IPk joining Pk-I to pk, and that Dk/ exists at all points of the segment Pk-IPk for each k. Then for some choice of points qi E Pi-IP 1,

(6.3)

Example.

· · · + h"e, where hi = Pk-I + hkek, . . . , p, =

n

(6.4)

f(q) - f(p)

=

L

(xk(q) - xk(p)) (Dkf)(qk).

k=l

13

162

/

Answers to Selected Exercises

Section 14 14.3

(b) [X, YJ

=

I [I [cxxk) a(Y~i)-

i=l

(c) [ex' sin

k=l

(Yxk)

acx:i)JJ ~ax

ax

ax

· (x1) + cos (x1)).!_1+ [sin (x1) + sin (x2)).!_ ax ax 2

Section 15 15.1 (3 v3;2 + i)e 3 • 15.2 -12(dz 1)P + (dz 2)P + 4(dz3)p. 15.3 (a) w 1 = dx 1 + dx 2 - dx 3 ; w 2 3 dx 2 + 3 dx 3 •

=

dx 1

+ 3 dx

2 -

dx 3 ; w3 = -dx 1

-

Section 16

16.2 4. 16.4 (ai)* = X; follows from the relations ((ai)*, ak) = (ai, ak) = (Xh ak) all k, and then (ai, ak) = (X;, ak) = (X;, [akJ#) = (X;, Xk) = o;k· 16.5 tt»*(a;)(Y;) = a 1(ti»•[Y;]) = (ti»•[Y;], X;) = (X;, X;) = ~!.,_ 1 (Xm, X;) wm( Y ;) for each i.

Section 18

Section 20 20.2 20.3 20.4

5ea. 3e 3 1\ e5 e 1 1\ e4•

-

e1 1\ e3 •

Section 22

Section 24 24.1 11/6. 24.2 c o ui: t --> (1, 1 Integral = 0.

+ 2t); c o u~: t--> (1 + t, 1); c o u~: t--> (1 + t, 3 -

2t).

AnBWers to Selected Exercises

24.3 24.4

-6. (a) 0; (b)

/

163

pba.

Section 26 26.1

[( ah)' + (iJh)' + (ah.)2]-1/2 axt

axt

X

axa

[ axt

~ - dx 2 1\ dx 3

~ + -axt dx 3

~ + -axa dx 1

1\ dx 1

26.2 N = [(2xt + x2)2 + (xl - 6x2)2 + 1]-t/2 X [ (2x 1 + x2) - a + (x 1 1 ax

-

6x 2) - a

ax 2

-

a ax•

J •

Section 29 29.1

z' = j.(x 1

(a) z1 = r, z2 = 8; r, 8 polar coordinates. (b) z1 = j.(x 1 + x2), xt).

Section 32 32.2 By Theorem (32.4) w = dft on Ut and w = dft on U2. d(ft - /2) = w - w = 0 on Ut f"\ U,, and since Ut f"\ U2 is connected, this meansft - /t = c a constant on Ut f"\ U2. Definef =/ton Ut, f =It+ con Ut. 32.3 Outline of solution: Let C,.: [0, 1]--+ U be given by C.(t) = (a cos 2rt a sin 2'11't), 1 < a < 2. These 1-simplices are all homotopic, so that

x=

r

]c.

w

is a constant. The 1-form (3 = w - Xa is closed, d(3 = 0, and

r

]c.

f3 =

o

for all a.

Now the region V = U'""" f (a, O): 1 < a < 2} can be deformed to a point in U; so there is a function f on V with df = (3. Then the condition

r f3 = o

]c.

can be used to show that f extends uniquely to a differentiable function on U with df = (3 there.

Index of Symbols

liP II B.(q)

f2f F• B:(q)

Ga pq D;

xp C'(O)

(Xp, Yp)

[X, Y] r;(Rn)

(df)p

q,* X*p

the length of the vector p, 1 closed ball with center q and radius e, 1 the empty set, 2 the complement of the set F, 2 the open ball with center q and radius e, 3 G-delta, a countable intersection of open sets, 3 line segment with endpoints p and q, 13 an operator indicating partial differentiation with respect to the jth cartesian coordinate, 13 possessing continuous partial derivatives up to and through those of the kth order, 18 having all partial derivatives of all orders continuous, 18 a vector at the point p, 47 the tangent vector to the curve C at the point C(O), 48 the space of vectors which are tangent to Rn at p, 48 the linear map of TP(Rn) into T>(Rn) induced by q,, 49 the scalar product of the vectors XP and Yp, 50 the bracket product of two vector fields, 52 the space of covectors on Rn at p, 53 the gradient of f at p, the exterior derivative of f at p, 53 the linear map of T!cp>(Rn) into r;(Rn) induced by q,, 54 the covector determined by the vector X P and the scalar product, 54 165

166

I

INDEX OF SYMBOLS

w*p

V* sign (u) A"V* al

1\ .

. 1\ a"

® (F, w) ...J E(w) S.J. dw

*W aXfJ d"'

fa .. f3

f. w

uk'

a c"""" b

pXq, Sdm Sq, mesh (c) 0 (qt, fJx

CTp(Rn)

cr;(Rn)

a a -·az" az.

Cone (q,)

. ' qk)

the vector determined by the covector wp and the scalar product, 54 the dual of the vector space V, 58 the sign of the permutation u, 59 the space of k-covectors on V, 60 the exterior product of a 1, • • • , a", 61 tensor product, 66 the value of the k-vector F on the k-covector w, 70 inner product, 72 the space Ia E V*: a 1\ w = 0 I determined by the k-covector w, 73 Iv E V: (v, S) = 0 I, the annihilator of the set S, 73 the exterior derivative of w, 75 the dual of w, 80 the cross product or vector product of a and fJ, 82 the standard m-simplex, 84 integral of fJ over dm, 84 the integral of w over q,, 84 a specific (m - I)-simplex on the boundary of d"', 85 the boundary operator for simplices and chains, 88 also an operator associated with complex coordinates, 148 the equivalence of two chains b and c, 91 the cone with base q, and vertex p, 100 first simplicial subdivision of dm, 100 first simplicial subdivision of q,, 100 the mesh of the chain c, 101 the generalized parallelogram with sides parallel to Qt, . . . ' Qk, 30, 107 Lie derivative along the integral curves of the vector field X, 136 space of complex tangent vectors to Rn at p, 144 space of complex covectors on Rn at p, 144 special vector fields associated with analytic coordinates, 146 the cone associated with a simplex q, and a homotopy contracting q, to a point, 151, 152

Index

Accumulation point, 5 Affine chains, 101 Alternating multilinear map, 59 Analytic coordinates, 147 Analytic functions, 147 of one variable, 149 Angle between two vectors, 50 Antisymmetrization operator, 66 Arc, 16 Ball, e-ball centered at q, 1 Boundary of an m-chain, 88 Boundary operator, 88 Bounded set, 7 Bracket of two vector fields, 52 Cauchy sequence, 4 Cauchy-Riemann equations, 150 Cauchy's integral formula, 152 Cauchy's theorem, 151 Chain rule, 14 Chains: affine, 101 equivalence of, 91 maps of, 88 singular Ck m-chains, 87 support of, 91 Closed forms, 151 Closed sets, 2 Compact sets, 6 Complex structure, 143 Composition of two maps, 9 Cone construction, 100 Conformal map, 151 Conjugate: of a form, 144 of a function, 143

Conjugate (cont.) of a vector, 143 Continuity, 9 of a map, 9 uniform, 12 Continuously differentiable of class k, 18, 143 Convergence of a sequence, 4 Convex set, 16 Coordinate systems: adapted to a vector field, 123 analytic, 147 Ck local, 35 complex analytic, 147 cylindrical, 38 simultaneously adapted to x., ... , xk, 121 Cotangent space, r;(R"), 53 Covector, 53 image under a map, 54 k-covectors, 59 scalar product for, 55 0-covectors, 71 Covering, 4 Cross product, 82 Curl, 82 Curves in R 3, 45 Cylindrical coordinates, 38 Decomposable k-covectors, 69 Decreasing sequence of sets, 2 Degree: of a differential form, 74 of a k-vector, 68 Determinants, multiplication rule for, 64 Diameter of a set, 8 167

168

I

INDEX

Differential forms, 74 Divergence, 82 Dot product, 50 Dual: of a k-form, 80 of a vector space, 58 Duality: in polar coordinates, 84 in spherical coordinates, 83 Elementary linear transformations, 30, 31 Equation of continuity, 138 Equivalent m-chains, 91 Even permutations, 59 Exterior derivative, 75 Exterior product: of differential forms, 74 of k-covectors, 68 Fields, of vectors, 52 Finite intersection property, 8 Flow, 120 stationary, 121 Flux of a vector field, 110 Frobenius' theorem, 129 G-delta, 3 Gradient of a function, 53 Gram-Schmidt orthonormalization process, 51 Heine Borel property, 8 Homotopy, 139 Imbedded m-simplex, 85 Implicit function theorem, 33, 37 Inner product, 72 Integral curve of a vector field, 129 Interior: point, 1 of a set, 1-2 Invariant k-forms, 138 Inverse function theorem, 20, 24 Involutive system of vector fields, 129 Jacobian, use of in multiple integrals, 27 Leibnitz's formula, 138 Lesbesgue number, 8 Limit of a sequence, 4 Local coordinate system, 35 cylindrical coordinates, 38

Local coordinate system (cont.) polar coordinates, 38 spherical coordinates, 38 Local coordinates, 35 Manifold, 42 regular k-dimensional CN submanifold of Rn, 42, 43 Maps: alternating multilinear, 59 conformal, 151 continuous, 9 linear, 12 multilinear, 59, 66 open, 12 oscillation of, 12 skew symmetric multilinear, 59 Mesh of an m-chain, 101 Monotone decreasing sequence of sets, 2 Morera's theorem, 151 Multilinear maps, 59, 66 Neighborhood of a point, 2 Nowhere dense set, 46 Odd permutations, 59 Open sets, 2 Operators: antisymmetrization, 66 boundary, 88 Orientation of a singular simplex, 87 Orienting symbols, 94-98 Orthonormalization process, GramSchmidt, 51 Partial derivatives, 13 Perfect sets, 6 Permutations, 59 Poincar~'s lemma, 141 Polar coordinates, 38 Power series, convergence of, 157 Rank, 39 of a k-covector, 70 Rational point, 3 Regular k-dimensional CN submanifold of Rn, 42 Relatively closed sets, 2 Relatively open sets, 2 Scalar product: of covectors, 55 in cylindrical coordinates, 52 of k-covectors, 79

INDEX

Scalar product (cont.) of k-vectors, 79 in spherical coordinates, 52, 57, 83 of vectors, 50 Sequences, 4 Cauchy sequence, 4 eventually in the set J, 5 subsequence, 5 ultimately in the set J, 4 Sets: bounded, 7 closed, 2 compact, 6 convex, 16 nowhere dense, 46 open, 2 perfect, 6 relatively closed, 2 relatively open, 2 Sign of a permutation, 59 Simplex: imbedded singular simplices, 85 maps of singular simplices, 87 singular Ck m-simplex, 84 standard m-simplex, 84 Simplicial subdivision, 100 Singular Ck m-chains, 87 Singular Ck m-simplex, 84 Skew symmetric multilinear map, 59 Space of cotangent vectors, 53 Space of tangent vectors, 48 Spherical coordinates, 38 Stationary flow, 121 Stokes' theorem, 102

I

169

Subspace determined by a k-covector, 69 Support of a chain, 91 Surfaces in R 3, 45 area of, 108 vectors tangent to, 50 Tangent space Tp(Rn), 48 Tangent vector to a curve, 48 Tangent vector to a surface, 50 Taylor series, 17, 156 Taylor's finite development, 156 Tensor product, 66 Vector, 47 conjugate of, 143 field, 52 flux of, 110 image under a map, 49 k-vector, 70 length of, 50 normal to a hypersurface, 57 real, 143 tangent to a surface, 50, 56 0-vector, 71 Vector field: coordinate system adapted to, 123 integral curve of, 123 Vector fields, involutive system of, 129 bracket of, 52 Vector product, 82 Velocity field of a flow, 121 Volume, 108 k-volume, 108

(continued from front flap)

mental existence and uniqueness theorem for ordinary differential equations, the results used in this book, are proved in the body of the text.

Professor Woll's treatment of the theory of functions of several variables can be considered in three general sections. In the first two chapters a theoretical foundation is laid for the material developed in the later chapters. Chapters Three, Four, and Five are basically manipulative, covering such notions as a tangent vector at pERn, covectors at p, multilinear algebra, and differential forms on Rn. The last section, Chapters Six and Seven, is again somewhat theoretical. Chapter Six treats the concept of a flow with velocity field X and the related derivations on vector fields and differential forms. In Chapter Seven Professor Woll shows how the notation and ideas developed in the preceding chapters can be used in the theory of functions of a complex variable.

THE AUTHOR

JOHN W. WOLL, JR. (Ph.D., Princeton University) is Assistant Professor of Mathematics at the University of Washington. Professor Woll has also taught at Lehigh University and the University of California, Berkeley, and has done research in the fields of functional analysis and stochastic processes.

14 /

CHAPTER TWO

Differentiation on R"

The proof is based on (6.2). In fact (6.5) for some qt E Pt-IPt• and (6.4) is obtained by summing (6.5) from k k = n.

=

1 to

Formula (6.4) shows that if the first partial derivatives off exist and are bounded in a neighborhood of the point p, then f is continuous at p. In general the mere existence of first partial derivatives in a neighborhood of p does not imply continuity at p. (6.6)

Example.

The function f defined by

/(z, y)

=

r z' _ yt cos--2z1+y'

when z 1

+ y 1 > 0,

/(z, y)

=

0

when z 1

+y

1

=

0

has first partial derivatives at every point of R 1, but it is not continuous at (0, 0). In fact the sequence Pt = (1/k, 1/k) converges to (0, 0) while 0

= J(O, 0)

'# limt/ >

(!k .!)k

=

1.

Contrast this with the one-dimensional case where the mere existence of a derivative at p is sufficient to insure continuity at p.

A major consequence of equation (6.4) is the chain rule for partial derivatives. The hypothesis of this theorem concerns a map tf>: W--+ U of an open set W of Rm into the open ball U of R" and a function/ defined on U (see Figure 2). For convenience let x 1, • • • , x" denote cartesian coordinate functions on R" and yl, . . . , ym cartesian coordinate functions on Rm. (6.7) Theorem (The chain rule). Suppose that (1) the partial derivatives Dd, . . . , D,.f exist on U and are continuous at tf>(q); (2) the functions x 1 o tf>, •.. , x" o tf> have partial derivatives D;(xk o tf>)(q) at