Multivariable Analysis

G. Baley Price

Multivariable Analysis

With 93 Illustrations

Springer-Verlag New York Berlin Heidelberg Tokyo

G. Baley Price Department of Mathematics University of Kansas Lawrence, Kansas 66045 U.S.A.

AMS Classification: 26-0 I

Library of Congress Cataloging in Publication Data Price, G. Baley (Griffith Baley)

Multivariable analysis Bibliography: p. Includes index. I. Calculus. 2. Functions of several real variables.

I. Title. QA303.P917 1984 515 83-20328

© 1984 by Springer-Verlag New York Inc. Softcover reprint of the hardcover 1st edition 1984 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A.

Typeset by Asco Trade Typesetting Ltd., Hong Kong

9 8 7 6 5 4 3 2 1

ISBN-13: 978-1-4612-9747-5 DOl: 10.1007/978-1-4612-5228-3

e-ISBN-13: 978-1-4612-5228-3

To

C. L. B. P.

"Nullum quod tetigit non ornavit."

A mathematician, like a painter or a poet, is a maker of patterns .... The mathematician's patterns, like the painter's or the poet's, must be

beautiful; the ideas, like the colours or the words, must fit together in a harmonious way.

-G. H. Hardy, A Mathematician's Apology

It is undeniable that some of the best inspirations in mathematics-in those parts of it which are as pure mathematics as one can imagine-have come from the natural sciences. We will mention the two most monumental facts.

The first example is, as it should be, geometry .... The second example is calculus-or rather all of analysis, which sprang

from it. The calculus was the first achievement of modern mathematics, and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technicaladvance in exact thinking.

-John von Neumann, "The Mathematician"

A movement for the reform of the teaching of mathematics, which some decades ago made quite a stir in Germany under the leadership of the great mathematician Felix Klein, adopted the slogan "functional thinking." The important thing with the average educated man should have learned in his mathematics classes, so the reformers claimed, is thinking in terms of variables and functions. A function describes how one variable y depends on another x; or more generally, it maps one variety, the range of a variable element x, upon another (or the same) variety. This idea of function or mapping is certainly one of the most fundamental concepts, with accompanies mathematics at every step in theory and application .

. . . But I should have completely failed if you had not realized at least this much, that mathematics, in spite of its age, is not doomed to progres­sive sclerosis by its growing complexity, but is still intensely alive, drawing nourishment from its deep roots in mind and nature.

-Hermann Weyl, "The Mathematical Way of Thinking"

Preface

This book contains an introduction to the theory of functions, with emphasis on functions of several variables. The central topics are the differentiation and integration of such functions. Although many of the topics are familiar, the treatment is new; the book developed from a new approach to the theory of differentiation. Iff is a function of two real variables x and y, its deriva­tives at a point Po can be approximated and found as follows. Let PI' P2 be two points near Po such that Po, PI, P2 are not on a straight line. The linear function of x and y whose values at Po, PI' P2 are equal to those off at these points approximates f near Po; determinants can be used to find an explicit representation of this linear function (think of the equation of the plane through three points in three-dimensional space). The (partial) derivatives of this linear function are approximations to the derivatives of f at Po ; each of these (partial) derivatives of the linear function is the ratio of two determinants. The derivatives off at Po are defined to be the limits of these ratios as PI and P2 approach Po (subject to an important regularity condition). This simple example is only the beginning, but it hints at a theory of differentiation for functions which map sets in IRn into IRm which is both general and powerful, and which reduces to the standard theory of differentiation in the one-dimensional case.

This book develops general theories in which both the methods and the results for functions of several variables are similar to those for functions of a single variable. Although general methods are always employed rather than ad hoc methods, the results (theorems) are similar to the standard one­dimensional theorems and are sometimes better than the traditional theo­rems for functions of several variables. The approach and the general methods employed succeed in unifying many aspects of the theory. The

Vlll Preface

book is elementary in the sense that it does not employ Lebesgue measure or Lebesgue integration.

The treatment is geometric in nature, and the principal geometric tool is the simplex (in the example above, Po, Pi' P2 are the vertices of a simplex). Often the simplex occurs as an element in an oriented Euclidean complex. Chapter 3 is an introduction to the geometry of n-dimensional Euclidean space; it treats convex sets, simplexes, the orientation of simplexes, com­plexes and chains, boundaries of simplexes and chains, the volumes of simplexes, and simplicial subdivisions of cubes and simplexes. Because of the geometric nature of the treatment, numerous figures have been included to make the reading of the text easier. The principal analytic tools are the determinant and the Stolz condition. With each n-simplex in ~n there is associated an (n + 1) by (n + 1) matrix whose determinant is proportional to the volume of the simplex. Appendix 1 contains a complete treatment of determinants, including proofs of all of the theorems used in this book. Some of these theorems are not at all well known, but they find natural and important applications in the theories developed in this book. The Stolz condition, introduced by Otto Stolz in 1893, states that, in the approxi­mation of the increment of a function by a linear function, the remainder term has a certain specified form. This book introduces a Stolz condition for the increment of a function which maps a set in ~n into ~m.

The subject matter of this book includes the theory of differentiation and (Riemann) integration, and a number of related topics in analysis. Chapter 4 treats Sperner's lemma by novel methods which fit easily and naturally into this book's general methods based on oriented simplicial complexes and determinants. Sperner's lemma is used to prove a very general form of the intermediate-value theorem; it is applied in Chapter 5 to prove a very general inverse-function theorem. The most important theorem in the theory of integration is the fundamental theorem of the integral calculus. By defining both derivatives and integrals by means of simplexes, it becomes easy to establish a connection between differentiation and integration. The funda­mental theorem results from properties of determinants and from properties of the boundary of a chain in an oriented simplicial complex. As is well known, Stokes' theorem is a corollary of the fundamental theorem. This book shows that the evaluation of integrals by iterated integrals and Cauchy'S integral theorem are also corollaries of the fundamental theorem of the integral calculus (see Chapters 8 and lO). Chapter 9 contains a treatment of Kronecker's integral; the Kronecker integral formula is closely related to the fundamental theorem. Chapter lO is an introduction to the differentia­tion and integration of functions of a single complex variable and of several complex variables by the methods developed in earlier chapters of the book. As stated above, the fundamental theorem of the integral calculus becomes Cauchy'S integral theorem for functions of one complex variable and also of several complex variables.

The prerequisites for the study of this book are two: a first course in

Preface lX

calculus and the ability to read and understand mathematical definitions, theorems, and proofs. For the student who knows elementary calculus, the book contains everything needed to read and understand the book. Appendix I presents a treatment of determinants (including several relatively unknown theorems) and several topics in linear algebra; Appendix 2 contains the basic theorems on numbers, sets, and functions. Although the book treats only elementary mathematics, it is not always easy. As a result, some readers may desire a more extensive background and more maturity than they have acquired from an elementary course in calculus.

The book consists of a Table of Contents, ten chapters, two appendices, References and Notes, an Index of Symbols, and an Index. The ten chapters and two appendices are divided into 97 sections, numbered in order from I for the first section in Chapter I to 97 for the last section in Appendix 2. There is a set of exercises at the end of each section in the ten chapters; these exercises are designed to illustrate and to supplement the material in the text. For easy reference throughout the book, the important definitions, theorems, corollaries, lemmas, and examples in each section are numbered with boldface numbers containing a decimal point; the digits before the decimal point indicate the number of the section, and the digits after the decimal point are the number of the item in the section. For example, Theorem 20.5 is the fifth numbered item in Section 20. In each section, the equations, formulas, and other special items to which reference is made mostly within the section are numbered (1) to (n) on the right margin.

The relationship of the chapters in this book is indicated by the following diagram.

Theorem 62.7 at the end of Chapter 9 requires Chapter 4, but except for this one theorem, Chapter 9 is independent of Chapter 4 as indicated by the diagram. The diagram shows that a minimum course on the differentiation and integration of functions of several real variables can be taught from Chapters I, 2, 3, and 6. More extensive courses, corresponding to the needs of students and the interests of the instructor, can be obtained by adding chapters to this minimum course in accordance with the diagram above. As the diagram suggests, many different courses are possible.

I am pleased to take this opportunity to acknowledge with appreciation and thanks the assistance that I have received in the preparation and publi­cation of this book. This assistance includes a Guggenheim Fellowship in 1946-1947 and a sabbatical leave from The University of Kansas in 1972-1973. Also, I am indebted to the editorial staff of Springer-Verlag for sugges­tions that have led to many improvements and for their help in preparing

x Preface

the manuscript and publishing the book. Finally, I gratefully acknowledge the assistance of my wife, Cora Lee Beers Price, without whose help and support this book would not have been written. To all of those who have assisted in the writing and publication of this book, I extend my hearty thanks.

Lawrence, Kansas February 28, 1983

G. BALEY PRICE

Contents

CHAPTER 1

Differentiable Functions and Their Derivatives ................. .

1. Introduction ............................................... . 2. Definitions and Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3. Elementary Properties of Differentiable Functions. . . . . . . . . . . . . . . . 23 4. Derivatives of Composite Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5. Compositions with Linear Functions ........................... 48 6. Classes of Differentiable Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 7. The Derivative as an Operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

CHAPTER 2

Uniform Differentiability and Approximations; Mappings....... 68

8. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 9. The Mean-Value Theorem: A Generalization.................... 70

10. Uniform Differentiability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 II. Approximation of Increments of Functions. . . . . . . . . . . . . . . . . . . . . . 85 12. Applications: Theorems on Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 97

CHAPTER 3

Simplexes, Orientations, Boundaries, and Simplicial Subdivisions 102

13. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 14. Barycentric Coordinates, Convex Sets, and Simplexes. . . . . . . . . . . . . 105 15. Orientation of Simplexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 16. Complexes and Chains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 17. Boundaries of Simplexes and Chains. . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 18. Boundaries in a Euclidean Complex. . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 19. Affine and Barycentric Transformations. . . . . . . . . . . . . . . . . . . . . . . . 155

xii

20. Three Theorems on Determinants ............................. . 21. Simplicial Subdivisions ...................................... .

CHAPTER 4

Contents

164 176

Sperner's Lemma and the Intermediate-Value Theorem. . . . . . . . . . 195

22. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 23. Spemer Functions; Spemer's Lemma ....................... , . . . 197 24. A Special Class of Sperner Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 25. Properties of the Degree of a Function. . . . . . . . . . . . . . . . . . . . . . . . . . 208 26. The Degree ofa Curve........................................ 212 27. The Intermediate-Value Theorem .......................... , . . . 217 28. Spemer's Lemn;ta Generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 29. Generalizations to Higher Dimensions. .. . . .. . . . . . . . . . . . . . . .. . . . 229

CHAPTERS

The Inverse-Funcdon Theorem................................. 237

30. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 31. The One-Dimensional Case. .... . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . 240 32. The First Step: A Neighborhood is Covered . . . . . . . . . . . . . . . . . . . . . 243 33. The Inverse-Function Theorem.. . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . 250

CHAPTER 6

Integrals and the Fundamental Theorem of the Integral Calculus 263

34. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 35. The Riemann Integral in IR" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 36. Surface Integrals in IR" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 37. Integrals on an m-Simplex in IR". . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 38. The Fundamental Theorem of the Integral Calculus. . . . . . .. . . . . . . 312 39. The Fundamental Theorem of the Integral Calculus for Surfaces. . . . 333 40. The Fundamental Theorem on Chains. . . . . . . . . . . . . . . . . . . . . . . . . . 342 41. Stokes' Theorem and Related Results. . . . . . . . . . . . . . . . . . . . . . . . . . . 346 42. The Mean-Value Theorem.................................... 350 43. An Addition Theorem for Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 44. Integrals Which Are Independent of the Path . . . . . . . . . . . . . . . . . . . . 358 45. The Area of a Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 46. Integrals of Uniformly Convergent Sequences of Functions . . . . . . . . 365

CHAPTER 7

Zero Integrals, Equal Integrals, and the Transformadon of Integrals. . . . . .. . ... . ... . . . . .. .. . . .. . . . . . . . . . . ... . .. . . . . .. . . .. . 368

47. Introduction................................................ 368 48. Some Integrals Which Have the Value Zero.. . . . .. . . . . . . .. . . . . . . 372 49. Integrals Over Surfaces with the Same Boundary. . . . . . . . . .. . . . .. . 381 50. Integrals on Affine Surfaces with the Same Boundary . . . . . . . . . . . . . 388 51. The Change-of-Variable Theorem. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 396

Contents xiii

CHAPTERS

The Evaluation of Integrals 407

52. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 53. Definitions ................................................. 410 54. Functions and Primitives ..................................... 412 55. Integrals and Evaluations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 56. The Existence of Primitives: Derivatives of a Single Function ...... 422 57. The Existence of Primitives: The General Case. . . . . . . . . . . . . . . . . . . 429 58: Iterated Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

CHAPTER 9

The Kronecker Integral and the Sperner Degree. . . . . . . . . . . . . . . . . 443

59. Preliminaries................................................ 443 60. The Area and the Volume ofa Sphere.......................... 448 61. The Kronecker Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 62. The Kronecker Integral and the Spemer Degree. . . . . . . . . . . . . . . . . . 487

CHAPTER 10

Differentiable Functions of Complex Variables. . . . . . . . . . . . . . . . . . 494

63. Introduction ............................................... . 494

Part I: Functions of a Single Complex Variable. . . . . . . . . . . . . . . . . 496

64. Differentiable Functions; The Cauchy-Riemann Equations. . . . . . . . 496 65. The Stolz Condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 66. Integrals..... ............................................... 501 67. A Special Case of Cauchy's Integral Theorem.................... 514 68. Cauchy's Integral Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 69. Taylor Series for a Differentiable Function . . . . . . . . . . . . . . . . . . . . . . 527 70. Complex-Valued Functions of Real Variables. . . . . . . . . . . . . . . . . . . . 530 71. Cauchy's Integral Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

Part II: Functions of Several Complex Variables. . . . . . . . . . . . . . . . 548 72. Derivatives ................................................. 548 73. The Cauchy-Riemann Equations and Differentiability. . . . . . . . . . . . 555 74. Cauchy's Integral Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

APPENDIX 1

Determinants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

75. Introduction to Determinants ................................. 573 76. Definition of the Determinant of a Matrix. . . . . . . . . . . . . . . . . . . . . . . 575 77. Elementary Properties of Determinants. . . . . . . . . . . . . . . . . . . . . . . . . 577 78. Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 79. Expansions of Determinants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 80. The Multiplication Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 81. Sylvester's Theorem of 1839 and 1851. . . . . . . . . . . . . . . . . . . . . . . . . . . 593

xiv

82. The Sylvester-Franke Theorem ............................... . 83. The Bazin-Reiss-Picquet Theorem ............................ . 84. Inner Products ............................................. . 85. Linearly Independent and Dependent Vectors; Rank ofa Matrix .. . 86. Schwarz's Inequality ........................................ . 87. Hadamard's Determinant Theorem ............................ .

APPENDIX 2

Real Numbers, Euclidean Spaces, and Functions

Contents

594 597 599 600 605 607

611

88. Some Properties of the Real Numbers .......................... 611 89. Introduction to 1R3 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 613 90. Introduction to IR" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 91. The Norm, Distance, and Triangle Inequality in IR" . . . . . . . . . . . . . . . 623 92. Open and Closed Sets and Related Matters in IR" . . . . . . . . . . . . . . . . . 624 93. The Nested Interval Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 94. The Bolzano-Weierstrass Theorem. . . . .. . . .. . . . . . . . . . . . . . . . . . . . 628 95. The Heine-Borel Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 96. Functions .............. .-................................... 630 97. Cauchy Sequences. ... . ... . ... . . . . . . .. . . .. . . . . . . . . . . . . . . . . .. . 641

References and Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645

Index of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649

Index......................................................... 651