L Y N N H L 0 MIS and S H L M S T ERN B ERG Department of Mathematics, Harvard University ADVANCED CALCULUS REVISED EDITION JONES AND BARTLETT PUBLISHERS Boston London , ~"\' ~ ", :,i.; J) Editorial, Sales, and Customer Service Offices: Jones and Bartlett Publishers, Inc, One Exeter Plaza Boston, MA 02116 Jones and Bartlett Publishers International POBox 1498 London W6 7RS England Copyright © 1990 by Jones and Bartlett Publishers, Inc Copyright © 1968 by Addison-Wesley Publishing Company, Inc All rights reserved No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owner Printed in the United States of America 10 Library of Congress Cataloging-in-Publication Data Loomis, Lynn H Advanced calculus / Lynn H Loomis and Shlomo Sternberg -Rev ed p cm Originally published: Reading, Mass : Addison-Wesley Pub Co., 1968 ISBN 0-86720-122-3 Calculus I Sternberg, Shlomo II Title QA303.L87 1990 89-15620 515 dc20 CIP 'I ' , PREFACE This book is based on an honors course in advanced calculus that we gave in the 1960's The foundational material, presented in the unstarred sections of Chapters through 11, was normally covered, but different applications of this basic material were stressed from year to year, and the book therefore contains more material than was covered in anyone year It can accordingly be used (with omissions) as a text for a year's course in advanced calculus, or as a text for a three-semester introduction to analysis These prerequisites are a good grounding in the calculus of one variable from a mathematically rigorous point of view, together with some acquaintance with linear algebra The reader should be familiar with limit and continuity type arguments and have a certain amount of mathematical sophistication AB possible introductory texts, we mention Differential and Integral Calculus by R Courant, Calculus by T Apostol, Calculus by M Spivak, and Pure Mathematics by G Hardy The reader should also have some experience with partial derivatives In overall plan the book divides roughly into a first half which develops the calculus (principally the differential calculus) in the setting of normed vector spaces, and a second half which deals with the calculus of differentiable manifolds Vector space calculus is treated in two chapters, the differential calculus in Chapter 3, and the basic theory of ordinary differential equations in Chapter The other early chapters are auxiliary The first two chapters develop the necessary purely algebraic theory of vector spaces, Chapter presents the material on compactness and completeness needed for the more substantive results of the calculus, and Chapter contains a brief account of the extra structure encountered in scalar product spaces Chapter is devoted to multilinear (tensor) algebra and is, in the main, a reference chapter for later use Chapter deals with the theory of (Riemann) integration on Euclidean spaces and includes (in exercise form) the fundamental facts about the Fourier transform Chapters and 10 develop the differential and integral calculus on manifolds, while Chapter 11 treats the exterior calculus of E Cartan The first eleven chapters form a logical unit, each chapter depending on the results of the preceding chapters (Of course, many chapters contain material that can be omitted on first reading; this is generally found in starred sections.) On the other hand, Chapters 12, 13, and the latter parts of Chapters and 11 are independent of each other, and are to be regarded as illustrative applications of the methods developed in the earlier chapters Presented here are elementary Sturm-Liouville theory and Fourier series, elementary differential geometry, potential theory, and classical mechanics We usually covered only one or two of these topics in our one-year course We have not hesitated to present the same material more than once from different points of view For example, although we have selected the contraction mapping fixed-point theorem as our basic approach to the in1plicit-function theorem, we have also outlined a "Newton's method" proof in the text and have sketched still a third proof in the exercises Similarly, the calculus of variations is encountered twice-once in the context of the differential calculus of an infinite-dimensional vector space and later in the context of classical mechanics The notion of a submanifold of a vector space is introduced in the early ohapters, while the invariant definition of a manifold is given later on In the introductory treatment of vector space theory, we are more careful and precise than is customary In fact, this level of precision of language is not maintained in the later chapters Our feeling is that in linear algebra, where the concepts are so clear and the axioms so familiar, it is pedagogically sound to illustrate various subtle points, such as distinguishing between spaces that are normally identified, discussing the naturality of various maps, and so on Later on, when overly precise language would be more cumbersome, the reader should be able to produce for hin1self a more precise version of any assertions that he finds to be formulated too loosely Similarly, the proofs in the first few chapters are presented in more formal detail Again, the philosophy is that once the student has mastered the notion of what constitutes a fonnal mathematical proof, it is safe and more convenient to present arguments in the usual mathematical colloquialisms While the level of formality decreases, the level of mathematical sophistication does not Thus increasingly abstract and sophisticated mathematical objects are introduced It has been our experience that Chapter contains the concepts most difficult for students to absorb, especially the notions of the tangent space to a manifold and the Lie derivative of various objects with respect to a vector field There are exercises of many different kinds spread throughout the book Some are in the nature of routine applications Others ask the r~ader to fill in or extend various proofs of results presented in the text Sometimes whole topics, such as the Fourier transform or the residue calculus, are presented in exercise form Due to the rather abstract nature of the textual material, the student is strongly advised to work out as many of the exercises as he possibly can Any enterprise of this nature owes much to many people besides the authors, but we particularly wish to acknowledge the help of L Ahlfors, A Gleason, R Kulkarni, R Rasala, and G Mackey and the general influence of the book by Dieudonne We also wish to thank the staff of Jones and Bartlett for their invaluable help in preparing this revised edition Cambridge, Massachusetts 1968, 1989 L.H.L S.S CONTENTS Chapter 10 11 12 Chapter 1 Chapter 2 *7 Chapter I n trod uction Logic: quantifiers The logical connectives Negations of quantifiers Sets Restricted variables Ordered pairs and relations Functions and mappings Product sets; index notation Composition Duality The Boolean operations Partitions and equivalence relations 6 10 12 14 15 17 19 Vector Spaces Fundamental notions Vector spaces and geometry Product spaces and Hom(V, TV) Affine subspaces and quotient spaces Direct sums Bilinearity 21 36 43 52 56 67 Finite-Dimensional Vector Spaces Bases Dimension The dual space Matrices Trace and determinant Matrix computations The diagonalization of a quadratic form 71 77 81 88 99 102 111 The Differential Calculus Review in IR Norms Continuity 117 121 126 10 Equivalent norms Infinitesimals The differential Directional derivatives; the mean-value theorem The differential and product spaces The differential and IR n • Elementary applications 11 The implicit-function theorem 12 Sub manifolds and Lagrange multipliers *13 Functional dependence *14 Uniform continuity and function-valued mappings *15 The calculus of variations *16 The second differential and the classification of critical points *17 The Taylor formula Chapter Compactness and Completeness *2 10 Metric spaces; open and closed sets Topology Sequential convergence Sequential compactness Compactness and uniformity Equicontinuity Completeness A first look at Banach algebras The contraction mapping fixed-point theorem The integral of a parametrized arc 11 The complex number system *12 Weak methods Chapter 5 132 136 140 146 152 156 161 164 172 175 179 182 186 191 195 201 202 205 210 215 216 223 228 236 240 245 Scalar Product Spaces Scalar products Orthogonal projection Self-adjoint transformations Orthogonal transformations Compact transformations 248 252 257 262 264 Chapter Differential Equations The fundamental theorem Differentiable dependence on parameters The linear equation The nth-order linear equation Solving the inhomogeneous equation The boundary-value problem Fourier series Chapter 7 Chapter 8 10 266 274 276 281 288 294 301 Multilinear Functionals Bilinear functionals Multilinear functionals Permutations The sign of a permutation The subspace an of alternating tensors The determinant The exterior algebra Exterior powers of scalar product spaces The star operator 305 306 308 309 310 312 316 319 320 Integration Introduction Axioms Rectangles and paved sets The minimal theory The minimal theory (continued) Contented sets When is a set contented? Behavior under linear distortions Axioms for integration Integration of contented functions 11 The change of variables formula 12 Successive integration 13 Absolutely integrable functions 14 Problem set: The Fourier transform 321 322 324 327 328 331 333 335 336 338 342 346 351 355 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Differentiable Manifolds Atlases Functions, convergence Differentiable manifolds The tangent space Flows and vector fields Lie derivatives Linear differential forms Computations with coordinates Riemann metrics 364 367 369 373 376 383 390 393 397 The Integral Calculus on Manifolds Compactness Partitions of unity Densities Volume density of a Riemann metric Pullback and Lie derivatives of densities The divergence theorem More complicated domains 403 405 408 411 416 419 424 Exterior Calculus Exterior differential forms Oriented manifolds and the integration of exterior differential forms The operator d Stokes' theorem Some illustrations of Stokes' theorem The Lie derivative of a differential form Appendix "Vector analysis" Appendix II Elementary differential geometry of surfaces in [3 Potential Theory in 429 433 438 442 449 452 457 459 lEn Solid angle Green's formulas The maximum principle Green's functions 474 476 477 479 566 13.15 CLASSICAL MECHANICS (-1,0) (1, 0) Fig 13.14 (See Fig 13.14.) In the xy-plane define the local coordinates ~ and 1] by setting Thus the curves ~ = const represent ellipses with semimajor axis ~ and foci at the two fixed points, while the curves 1] = const are hyperbolas with semimajor axis 1] and the same foci Note that < 11] I ::;: ::;: ~ < 00 The equations of these two curves are and so that Thus dy y ~ d~ 1] -= F- 1- d1] , 1]2 and therefore If we now rotate about the x-axis to get the analogue of cylindrical coordinates in space, we have the coordinates -- Special Symbols Symbol Page Symbol Page -t (3x) SA 12 & n 12 XA 12 =} II ¢=} E C (\Ix) 11, 117, 202 14,43 14 F(x, 0) 16 17 >25 u,n,u,n A' {} -< >R- AXE 7r e([a, b]) 9 L Xiai L(A) 17 19,53,56 24, 121 27 27 r 10 oj R[A] 10 11 La 32 7rj 33,47 ~ 572 28, 31, 74 NOTATION INDEX Symbol N(T), m(T) R(T), CR(T) Hom(V, W) 8j EEl cl(V) #A V* AO T* t* fl.(T) tr(T) D(t) lub, glb II 1/ 1/ 1, 1/ 1/ 1/2, 1/ 1/", Br(~)