Advanced Calculus A Differential Forms Approach

To my students-and especially to those who never stopped asking questions.

Harold M. Ed\Nards

Advanced Calculus A Differential Forms Approach

i Springer Science+Business Media, LLC

Harold M. Edwards Courant Institute New York University New York, NY 10012

Library of Congress Cataloging In-Publication Data

Edwards, Harold M. Advanced calculus : a differential fonns approach I Harold M.

Edwards. -- [3rd ed.] p. em.

Includes index. ISBN 978-1-4612-6688-4 (alk. paper) I. Calculus. I. Title.

QA303.E24 1993 515--dc20

Printed on acid-free paper © 1994 Harold M. Edwards Originally published by Birkhäuser Boston in 1994 Reprinted 1994 with corrections from the original Houghton Mifflin edition.

93-20657 CIP

m® Birkhiiuser I®>

Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any fonn or by any means, electronic, mechanical, photo­copying, recording, or otherwise, without prior pennission of the copyright owner.

Permission to photocopy for internal or personal use of specific clients is granted by Springer Science+Business Media, LLC for libraries and other users registered with the Copyright Clear­ance Center (CCC), provided that the base fee of$6.00 per copy, plus $0.20 per page is paid di­rectly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should be ad­dressed directly to Birkhiiuser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A.

ISBN 978-1-4612-6688-4 ISBN 978-1-4612-0271-4 (eBook) DOI 10.1007/978-1-4612-0271-4

9 8 7 6 5

Preface to the 1994 Edition

My first book had a perilous childhood. With this new edition, I hope it has reached a secure middle age.

The book was born in 1969 as an "innovative text­book"-a breed everyone claims to want but which usu­ally goes straight to the orphanage. My original plan had been to write a small supplementary textbook on differen­tial forms, but overly optimistic publishers talked me out of this modest intention and into the wholly unrealistic ob­jective (especially unrealistic for an unknown 30-year-old author) of writing a full-scale advanced calculus course that would revolutionize the way advanced calculus was taught and sell lots of books in the process.

I have never regretted the effort that I expended in the pursuit of this hopeless dream-{}nly that the book was published as a textbook and marketed as a textbook, with the result that the case for differential forms that it tried to make was hardly heard. It received a favorable tele­graphic review of a few lines in the American Mathematical Monthly, and that was it. The only other way a potential reader could learn of the book's existence was to read an advertisement or to encounter one of the publisher's sales­men. Ironically, my subsequent books-Riemann :S Zeta Function, Fermat:S Last Theorem and Galois Theory-sold many more copies than the original edition of Advanced Calculus, even though they were written with no commer­cial motive at all and were directed to a narrower group of readers.

When the original publisher gave up on the book, it was republished, with corrections, by the Krieger Publish­ing Company. This edition enjoyed modest but steady sales for over a decade. With that edition exhausted and with Krieger having decided not to do a new printing, I am enor­mously gratified by Birkhauser Boston's decision to add this title to their fine list, to restore it to its original, easy-to­read size, and to direct it to an appropriate audience. It is at their suggestion that the subtitle "A Differential Forms Approach" has been added.

I wrote the book because I believed that differential forms provided the most natural and enlightening approach

vi Preface

to the calculus of several variables. With the exception of Chapter 9, which is a bow to the topics in the calculus of one variable that are traditionally covered in advanced calculus courses, the book is permeated with the use of differential forms.

Colleagues have sometimes expressed the opinion that the book is too difficult for the average student of advanced calculus, and is suited only to honors students. I disagree. I believe these colleagues think the book is difficult be­cause it requires that they, as teachers, rethink the material and accustom themselves to a new point of view. For stu­dents, who have no prejudices to overcome, I can see no way in which the book is more difficult than others. On the contrary, my intention was to create a course in which the students would learn some useful methods that would stand them in good stead, even if the subtleties of uniform convergence or the rigorous definitions of surface integrals in 3-space eluded them. Differential forms are extremely useful and calculation with them is easy. In linear alge­bra, in implicit differentiation, in applying the method of Lagrange multipliers, and above all in applying the gen­eralized Stokes theorem fas w = fs dw (also known as the fundamental theorem of calculus) the use of differential forms provides the student with a tool of undeniable useful­ness. To learn it requires a fraction of the work needed to learn the notation of div, grad, and curl that is often taught, and it applies in any number of dimensions, whereas div, grad, and curl apply only in three dimensions.

Admittedly, the book contains far too much material for a one year course, and if a teacher feels obliged to cover everything, this book will be seen as too hard. Some topics, like the derivation of the famous equation E = mc2 or the rigorous development of the theory of Lebesgue integration as a limiting case of Riemann integration, were included because I felt I had something to say about them which would be of interest to a serious student or to an honors class that wanted to attack them. Teachers and students alike should regard them as extras, not requirements.

My thanks to Professor Creighton Buck for allowing us to reuse his kind introduction to the 1980 edition, to Sheldon Axler for his very flattering review of the book in the American Mathematical Monthly of December 1982, and to Birkhauser for producing this third edition.

Harold Edwards New York 1993

Introduction

It is always exciting to teach the first quarter (or first semester) of a calculus course-especially to students who have not become blase from previous exposure in high school. The sheer power of the new tool, supported by the philosophical impact of Newton's vision, and spiced by the magic of notational abracadabra, carries the course for you. The tedium begins when one must supply all the details of rigor and technique. At this point, one becomes envious of the physicist who frosts his elementary course by refer­ences to quarks, gluons and black holes, thereby giving his students the illusion of contact with the frontiers of research in physics.

In mathematics, it is much harder to bring recent research into an intro­ductory course. This is the achievement of the author of the text before you. He has taken one of the jewels of modern mathematics---, the theory of differential forms-and made this far reaching generalization of the fundamental theorem of calculus the basis for a second course in calculus. Moreover, he has made it pedagogically accessible by basing his approach on physical intuition and applications. Nor are conventional topics omitted, as with several texts that have attempted the same task. (A quick glance at the Index will make this more evident than the Table of Contents.)

Of course this is an unorthodox text! However, it is a far more honest attempt to present the essence of modern calculus than many texts that emphasize mathematical abstraction and the formalism of rigor and logic. It starts from the calculus of Leibniz and the Bernouillis, and moves smoothly to that of Cartan.

This is an exciting and challenging text for students (and a teacher) who are willing to follow Frost's advice and "take the road less travelled by."

R. Creighton Buck January 1980

Preface

There is a widespread misconception that math books must be read from beginning to end and that no chapter can be read until the preceding chapter has been thor­oughly understood. This book is not meant to be read in such a constricted way. On the contrary, I would like to encourage as much browsing, skipping, and back­tracking as possible. For this reason I have included a synopsis, I have tried to keep the cross-references to a minimum, and I have avoided highly specialized notation and terminology. Of course the various subjects covered are closely interrelated, and a full appreciation of one section often depends on an understanding of some other section. Nonetheless, I would hope that any chapter of the book could be read with some understanding and profit independently of the others. If you come to a state­ment which you don't understand in the middle of a passage which makes relatively good sense, I would urge you to push right on. The point should clarify itself in due time, and, in any case, it is best to read the whole section first before trying to fill in the details. That is the most important thing I have to say in this preface. The rest of what I have to say is said, as clearly as I could say it, in the book itself. If you learn anywhere near as much from reading it as I have learned from writing it, then we will both be very pleased.

New York 1969

Contents

Chapter 1 Constant Forms

1.1 One-Forms 1 1.2 Two-Forms 5 1.3 The Evaluation of Two-Forms, Pullbacks 8 1.4 Three-Forms 15 1.5 Summary 19

Chapter 2 Integrals

2.1 Non-Constant Forms 22 2.2 Integration 24 2.3 Definition of Certain Simple Integrals.

Convergence and the Cauchy Criterion 29 2.4 Integrals and Pullbacks 38 2.5 Independence of Parameter 44 2.6 Summary. Basic Properties of Integrals 49

Chapter 3 Integration and Differentiation

3.1 The Fundamental Theorem of Calculus 52 3.2 The Fundamental Theorem in Two Dimensions 58 3.3 The Fundamental Theorem in Three Dimensions 65 3.4 Summary. Stokes Theorem 72

Chapter 4 Linear Algebra

4.1 Introduction 4.2 Constant k-Forms on n-Space

76 86

Contents x

4.3 Matrix Notation. Jacobians 94

4.4 The Implicit Function Theorem for Affine Maps 105

4.5 Abstract Vector Spaces 113

4.6 Summary. Affine Manifolds 127

Chapter 5 Differential Calculus

5.1 The Implicit Function Theorem for Differentiable Maps 132

5.2 k-Forms on n-Space. Differentiable Maps 142 5.3 Proofs 151 5.4 Application: Lagrange Multipliers 160 5.5 Summary. Differentiable Manifolds 190

Chapter 6 Integral Calculus

6.1 Summary 196 6.2 k-Dimensional Volume 197 6.3 Independence of Parameter and the

Definition of fs w 200 6.4 Manifolds-with-Boundary and Stokes' Theorem 214

6.5 General Properties of Integrals 219

6.6 Integrals as Functions of S 224

Chapter 7 Practical Methods of Solution

7.1 Successive Approximation 226

7.2 Solution of Linear Equations 235

7.3 Newton's Method 242

7.4 Solution of Ordinary Differential Equations 245

7.5 Three Global Problems 256

Chapter8 Applications

8.1 Vector Calculus 265

8.2 Elementary Differential Equations 270

8.3 Harmonic Functions and Conformal Coordinates 278

8.4 Functions of a Complex Variable 289

8.5 Integrability Conditions 313

Contents xi

8.6 Introduction to Homology Theory 8.7 Flows 8.8 Applications to Mathematical Physics

Chapter 9 Further Study of Limits

9.1 The Real Number System 9.2 Real Functions of Real Variables 9.3 Uniform Continuity and Differentiability 9.4 Compactness 9.5 Other Types of Limits 9.6 Interchange of Limits 9.7 Lebesgue Integration 9.8 Banach Spaces

Appendices

Answers to Exercises

Index

320 328 333

357 381 387 392 399 407 426 447

456

468

504

Synopsis

There are four major topics covered in this book: con­vergence, the algebra of forms, the implicit function theorem, and the fundamental theorem of calculus.

Of the four, convergence is the most important, as well as the most difficult. It is first considered in §2.3 where it arises in connection with the definition of definite integrals as limits of sums. Here, and throughout the book, con­vergence is defined in terms of the Cauchy Criterion (see Appendix 1). The convergence of definite integrals is considered again in §6.2 and §6.3. In Chapter 7 the idea of convergence occurs in connection with processes of successive approximation; this is a particularly simple type of convergence and Chapter 7 is a good introduction to the general idea of convergence. Of course any limit involves convergence, so in this sense the idea of con­vergence is also encountered in connection with the defini­tion of partial derivatives (§2.4 and §5.2); in these sections, however, stress is not laid on the limit concept per se. It is in Chapter 9 that the notions of limit and con­vergence are treated in earnest. This entire chapter is de­voted to these subjects, beginning with real numbers and proceeding to more subtle topics such as uniform con­tinuity, interchange of limits, and Lebesgue integration.

The algebra of forms is the most elementary of the four topics listed above, but it is the one with which the reader is least likely to have some previous acquaintance. For this reason Chapter 1 is devoted to an introduction of the notation, the elementary operations, and, most impor­tant, the motivating ideas of the algebra of forms. This introductory chapter should be covered as quickly as possible; all the important ideas it contains are repeated in more detail later in the book. In Chapter 2 the algebra of forms is extended to non-constant forms (§2.1, §2.4). In Chapter 4 the algebra of constant forms in considered

Synopsis xiv

again, from the beginning, defining terms and avoiding appeals to geometrical intuition in proofs. In the same way Chapter 5 develops the algebra of (non-constant) forms from the beginning. Finally, it is shown in Chap­ter 6 (especially §6.2) that the algebra of forms corre­sponds exactly to the geometrical ideas which originally motivated it in Chapter 1. Several important applications of the algebra of forms are given; these include the theory of determinants and Cramer's rule (§4.3, §4.4), the theory of maxima and minima with the method of Lagrange multipliers (§5.4), and integrability conditions for differ­ential equations (§8.6).

The third of the topics listed above, the implicit func­tion theorem, is a topic whose importance is too fre­quently overlooked in calculus courses. Not only is it the theorem on which the use of calculus to find maxima and minima is based (§5.4), but it is also the essential in­gredient in the definition of surface integrals. More generally, the implicit function theorem is essential to the definition of any definite integral in which the domain of integration is a k-dimensional manifold contained in a space of more than k dimensions (see §2.4, §2.5, §6.3, §6.4, §6.5). The implicit function theorem is first stated (§4.1) for affine functions, in which case it is little more than the solution of m equations in n unknowns by the techniques of high school algebra. The general (non­affine) theorem is almost as simple to state and apply (§5.1 ), but it is considerably more difficult to prove. The proof, which is by the method of successive approxima­tions, is given in §7.1. Other more practical methods of solving m equations in n unknowns are discussed later in Chapter 7, including practical methods of solving affine equations (§7.2).

The last of the four topics, the fundamental theorem of calculus, is the subject of Chapter 3. Included under the heading of the "fundamental theorem" is its generaliza­tion to higher dimensions

fs dw = fas w

which is known as Stokes' theorem. The complete state­ment and proof of Stokes' theorem (§6.5) requires most of the theory of the first six chapters and can be regarded as one of the primary motivations for this theory.

In broad outline, the first three chapters are almost entirely introductory. The next three chapters are the core

Synopsis xv

of the calculus of several variables, covering linear alge­bra, differential calculus, and integral calculus in that order. For the most part Chapters 4-6 do not rely on Chapters 1-3 except to provide motivation for the ab­stract theory. Chapter 7 is almost entirely independent of the other chapters and can be read either before or after them. Chapter 8 is an assortment of applications; most of these applications can be understood on the basis of the informal introduction of Chapters 1-3 and do not require the more rigorous abstract theory of Chapters 4-6. Finally, Chapter 9 is almost entirely inde­pendent of the others. Only a small amount of adjustment would be required if this chapter were studied first, and many teachers may prefer to order the topics in this way.