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Modern Birkhäuser Classics

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Calculus

A Differential Forms Approach

Harold M. Edwards

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A dvanced Calculus

A D ifferential Forms A pproach

Harold M. Edwards

Reprint of the 1994 Edition

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

ISSN 2197- ISBN 978-0-8176-8411- DOI 10.1007/978-0-8176-8412- Springer New York Heidelberg Dordrecht London

Library of Congress Control Number:

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.

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Springer is part of Springer Science+Business Media (www.birkhauser-science.com)

ISSN 2197-1811 (electronic) ISBN 978-0-8176-8412-9 (eBook)

2013953495

© Harold M. Edwards 2014

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 forms approach I Harold M. Edwards.-- [3rd ed.) p. em. Includes index. ISBN 0-8176-3707-9 (alk. paper)

1. Calculus. I. Title. QA303.E24 1993 515--dc

Printed on acid-free paper © 1994 Harold M. Edwards Reprinted 1994 with corrections from the original Houghton Mifflin edition.

93- CIP

~® Birkhiiuser HWJ

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 form or by any means, electronic, mechanical, photo- copying, recording, or otherwise, without prior permission of the copyright owner.

Permission to photocopy for internal or personal use of specific clients is granted by Birkhliuser Boston 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 Birkhauser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A.

ISBN 0-8176-3707- ISBN 3-7643-3707-

Printed and bound by Quinn-Woodbine, Woodbine, NJ Printed in the USA

9 8 7 6 5

To my students-and especially to those who never

stopped asking questions.

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 text- book"-a breed everyone claims to want but which usu- ally goes straight to the orphanage. My original plan had been to write a small supplementary textbook on differen- tial forms, but overly optimistic publishers talked me out of this modest intention and into the wholly unrealistic ob- jective (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-only 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 tele- graphic 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 sales- men. 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 commer- cial 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 Publish- ing 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 enor- mously gratified by Birkhauser Boston's decision to add this title to their fine list, to restore it to its original, easy-to- read 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

x Prefa ce

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 thai 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 be-

cause it requires that they, as teachers, rethink the material

and accustom themselves to a new point of view. For stu- dents, 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 alge- bra, in implicit differentiation, in applying the method of Lagrange multipliers, and above all in applying the gen- eralized Stokes theorem las W = Isdw (also known as the fundamental theorem of calculus) the use of differential forms provides the student with a tool of undeniable useful- ness. To learn it requires a fraction of the work needed 10 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 everythi ng, this book will be seen as too hard. Some topics,

like the derivation of the famous equation E = mc 2 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 10 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

NeH! York / 993

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 thor- oughly 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 back- tracking 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 state- ment 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 .2 Two-Forms 1 .3 The Evaluation of Two-Forms. Pullbacks 1.4 Three-Forms 1.5 Summary

Chapter 2 Integrals

2.1 Non-Constant Forms 2.2 Integration 2.3 Definition of Certain Simple Integrals. Convergence and the Cauchy Criterion 2.4 Integrals and Pullbacks 2.5 Independence of Parameter 2.6 Summary. Basic Properties of Integrals

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

Contents xv

8.6 Introduc tion to Homology Theory

8.7 Flows

8.8 App lications to Mathematical Physics

Ch apt er 9 Further Study of Limits

9.1 Th e Real Number System 9.2 Real Functi ons of Real Variables

9 .3 Uniform Continuity and Differentiability

9 .4 Compactness

9.5 Oth er Types of Limits 9.6 Interchange of Limits

9.7 Lebesgue Integration

9.8 Banach Spaces

Appendices

Answers to Exercises

Index

Synopsis xviii

again, fro m the beginnin g, defining terms and a voiding appeals to geometrical intuiti on in proofs. In the same way Chapter 5 develops the algebra of (non-consta nt) form s from the beginnin g. Finally, it is shown in Chap- tcr 6 (especially §6.2) that the algebra of form s corre- spo nds exactly to the geometrical ideas which origina lly moti vated it in Chap ter I. Several import ant app lications of the algebra of forms are given ; these include the theory of determin ant s a nd Cramer's rule (§4.3, §4.4), the theory of maxima and minima with the meth od of Lagrange multi pliers (§5.4), and integrability con ditions for d iffer- ential equa tions (§8.6). Th e thir d of the top ics listed above, th e implicit func- tion theorem, is a topic whose import ance is too fre- quently overloo ked in calculus courses. No t 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 in- gredient 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 integrat ion is a k-d imensional manifold contained in a space of more than k dimensions (see §2.4, §2.5, §6.3, §6.4, §6.5). Th e implicit function theorem is first sta ted (§4.1) for affine functions, in which case it is little more

than the solution of m equations in /I un knowns by the

techniques of high schoo l algebra. The genera l (non - a ffi ne) theorem is almost as simple to state and a pply (§5.1), but it is considerab ly more di ffic ult to prove. T he proof, which is by the method of successive approxima- tions, is given in §7.1. Other more practical meth ods of

solving m equations in n unkn owns are discussed later in

Chap ter 7, including practical meth ods of solving a ffi ne equations (§7.2). T he last of the four topics, the funda mental theorem of calculus, is the subject of Chap ter 3. Included under the headin g of the " funda menta l theorem" is its genera liza- tion to higher dimensions

Is dw = Ias w

which is known as Sto kes' theorem. Th e complete state- ment a nd proof of Sto kes' theorem (§6.5) requires most of the theory of the first six cha pters and can be regard ed as one of the primary mot ivat ions for this theory. In broad outline, the first thr ee cha pters are almost entirely introductory. T he next thr ee chapters are the core

Synopsis xix

of th e calculus of severa l variables, covering linear alge- bra, differential calculus, and int egral ca lculus in th at

order. Fo r th e most part Chapters 4- 6 do not rely on

Chapters 1- 3 excep t to provide moti vat ion for the a b-

strac t th eory. C hap ter 7 is almost entirely independ ent of th e other cha pters and can be read either before or after th em. Chap ter 8 is an assortment of applicatio ns ; most of th ese appli cati on s ca n be und erstood on th e

basis of th e inform al int rodu cti on of Chap ters 1- 3 and

do not require th e more rigor ou s abstract th eor y of

Chapters 4 -6. F inally, Chapter 9 is almost enti rely inde-

pendent of th e ot hers. Only a sma ll amo unt of adj ustment wou ld be req uired if th is c hap ter were studied first, a nd ma ny teach ers may prefer to order the topics in this way.