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I Real functions - FUnctions of several variables - Integral formulas Stokes, Gauss, Green, etc... Further results on harmonic functions Trang 9 This text was produced for the second p
Introduction to Analysis in Several Variables Advanced Calculus Michael E Taylor Introduction to Analysis in Several Variables Advanced Calculus PIift/d�nAL/ UNDERGRADUA��' 46 Introduction to Analysis in Several Variables Advanced Calculus Michael E Taylor • •• AMERICAN MATHEMATICAL SOCIETY �;r AMS Providence, Rhode Island USA .:;., " EDITORIAL COMMITTEE Gerald B Folland (Chair) Steven J Miller Jamie Pommersheim Serge Tabachnikov 2010 Mathematics Subject Classification Primary 26B05, 26BlO, 26B12, 26B15, 26B20 For additional information and updates on this book, visit www.amsoorg/bookpages/amstext-46 Library of Congress Cataloging-in-Publication Data Names: Taylor, Michael E., 1946- author Title: Introduction to analysis in several variables : advanced calculus / Michael E Taylor Description: Providence, Rhode Island American Mathematical Society, [2020] I Series: Pure and applied undergraduate texts, 1943-9334 ; volume 46 I Includes bibliographical references and index Identifiers: LCCN 20200097351 ISBN 9781470456696 (paperback) I ISBN 9781470460167 (ebook) Subjects: LCSH: Calculus I FUnctions of several real variables I Functions of several complex variables I AMS: Real functions - FUnctions of several variables - Continuity and differentia tion questions I Real functions - FUnctions of several variables - Implicit function theorems, Jacobians, transformations with several variables I Real functions - FUnctions of several variables - Calculus of vector functions I Real functions - FUnctions of several variables - Integration: length, area, volume I Real functions - FUnctions of several variables - Integral formulas (Stokes, Gauss, Green, etc.) Classification: LCC QA303.2 T38 2020 I DDC 515-dc23 LC record available at https://lccn.loc.gov/2020009735 Copying and reprinting Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center For more information, please visit www.ams.org/publications/pubpermissions Send requests for translation rights and licensed reprints to reprint-permission@ams.org © 2020 by the American Mathematical Society All rights reserved The American Mathematical Society retains all rights except those granted to the United States Government Printed in the United States of America o The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 25 24 23 22 21 20 Contents Preface vii Some basic notation xi Chapter 1 Background 1 1.1 One-variable calculus 2 1.2 Euclidean spaces 17 1.3 Vector spaces and linear transformations 22 1.4 Determinants 31 Chapter 2 Multivariable differential calculus 39 2.1 The derivative 39 2.2 Inverse function and implicit function theorems 56 2.3 Systems of differential equations and vector fields 68 Chapter 3 Multivariable integral calculus and calculus on surfaces 87 3.1 The Riemann integral in n variables 88 3.2 Surfaces and surface integrals 117 3.3 Partitions of unity 145 3.4 Sard's theorem 146 3.5 Morse functions 147 3.6 The tangent space to a manifold 148 Chapter 4 Differential forms and the Gauss-Green-Stokes formula 153 4.1 Differential forms 154 4.2 Products and exterior derivatives of forms 160 4.3 The general Stokes formula 164 - v vi Contents 4.4 The classical Gauss, Green, and Stokes formulas 169 4.5 Differential forms and the change ofvariable formula 179 Chapter 5 Applications of the Gauss-Green-Stokes formula 185 5.1 Holomorphic functions and harmonic functions 186 5.2 Differential forms, homotopy, and the Lie derivative 200 5.3 Differential forms and degree theory 205 Chapter 6 Differential geometry of surfaces 221 6.1 Geometry ofsurfaces I: geodesics 225 6.2 Geometry of surfaces II: curvature 238 6.3 Geometry ofsurfaces III: the Gauss-Bonnet theorem 252 6.4 Smooth matrix groups 265 6.5 The derivative of the exponential map 283 6.6 A spectral mapping theorem 288 Chapter 7 Fourier analysis 291 7.1 Fourier series 294 7.2 The Fourier transform 310 7.3 Poisson summation formulas 330 7.4 Spherical harmonics 332 7.5 Fourier series on compact matrix groups 372 7.6 Isoperimetric inequality 378 Appendix A Complementary material 381 A.1 Metric spaces, convergence, and compactness 382 A.2 Inner product spaces 393 A.3 Eigenvalues and eigenvectors 398 A.4 Complements on power series 402 A.5 The Weierstrass theorem and the Stone-Weierstrass theorem 408 A.6 Further results on harmonic functions 410 A.7 Beyond degree theory-introduction to de Rham theory 416 Bibliography 437 Index 441 Preface This text was produced for the second part of a two-part sequence on advanced cal culus, whose aim is to provide a firm logical foundation for analysis, for students who have had threesemesters ofcalculus and a course in linear algebra The first part treats analysis in one variable, and the text [49] was written to cover that material The text at hand treats analysis in several variables These two texts can be used as companions, but they are written so that they can be used independently, if desired Chapter 1 treats background needed for multivariable analysis The first section gives a brief treatment of one-variable calculus, including the Riemann integral and the fundamental theorem ofcalculus This section distills material developed in more detail in the companion text [49] We have included it here to facilitate the indepen dent use of this text Subsequent sections in Chapter 1 present the basic linear algebra background of use for the rest of this text They include rnaterial on n-dimensional Euclidean spaces and other vector spaces, on linear transformations on such spaces, and on determinants of such linear transformations Chapter 2 develops multidimensional differential calculus on domains in n-dimen sional Euclidean space Itn The first section defines the derivative of a differentiable map F : (') Itm, at a point x E ('), for (') open in Itn, as a linear map from Itn to Itm, and establishes basic properties, such as the chain rule The next section deals with the inverse function theorem, giving a condition for such a map to have a differentiable inverse, when n = m The third section treats n X n systems of differential equations, bringing in the concepts ofvector fields and flows on an open set (') E Itn While the emphasis here is on differential calculus, we do make use of integral calculus in one variable, as exposed in Chapter l Chapter 3 treats multidimensional integral calculus We define the Riemann in tegral for a class of functions on Itn and establish basic properties, including a change ofvariable formula We then study smooth m-dimensional surfaces in Itn, and extend - vii viii Preface the Riemann integral to a class of functions on such surfaces Going further, we ab stract the notion of surface to that of a manifold, and study a class of manifolds known as Riemannian manifolds These possess an object known as a metric tensor We also define the Riemann integral for a class of functions on such manifolds The change of variable formula is instrumental in this extension of the integral In Chapter4 we introduce a further class ofobjects that can be defined on surfaces, differentialforms A k-form can be integrated over a k-dimensional surface, endowed with an extra structure, an orientation Again the change of variable formula plays a role in establishing this Important operations on differential forms include products and the exterior derivative A key result of Chapter 4 is a general Stokes formula, an important integral identity that can be seen as a multidimensional version of the fun damental theorem of calculus In §4.4 we specialize this general Stokes formula to classical cases, known as theorems of Gauss, Green, and Stokes A concluding section of Chapter 4 makes use of material on differential forms to give another proof of the change of variable formula for the integral, much different from the proof given in Chapter 3 Chapter 5 is devoted to several applications of the material on the Gauss- Green Stokes theoremsfrom Chapter 4 In §5.1 we use Green's theorem to derive fundamental properties of holomorphic functions of a complex variable Sprinkled throughout ear lier sections are some allusions to functions of complex variables, particularly in some of the exercises in §§2.1-2.2 Readers with no previous exposure to complex variables might wish to return to these exercises after getting through §5.1 In this section, we also discuss some results on the closelyrelated study of harmonic functions One result is Liouville's theorem, stating that a bounded harmonic function on all of Rn must be constant When specialized to holomorphic functions on C = R', this yields a proof of the fundamental theorem of algebra In §5.2 we define the notion of smoothly homotopic maps and consider the be havior of closed differential forms under pullback by smoothly homotopic maps This material is then applied in §5.3,whichintroducesdegree theoryand derives some inter esting consequences Key results include the Brouwer fixed point theorem, the Jordan Brouwer separation theorem (in the smooth case), and the study of critical points of a vector field tangent to a compact surface, and connections with the Euler characteris tic We also show how degree theoryyields another proof of the fundamental theorem of algebra Chapter 6 applies results of Chapters 2-5 to the study of the geometry of surfaces (and more generally of Riemannian manifolds) Section 6.1 studies geodesics, which are locally length-minimizing curves Section 6.2 studies curvature Several varieties of curvature arise, including Gauss curvature and Riemann curvature, and it is of great interest to understand the relations between them Section 6.3 ties the curvature study of §6.2 to material on degree theory from §5.3, in a result known as the Gauss-Bonnet theorem Section 6.4 studies smooth matrix groups, which are smooth surfaces in M(n, F) that are also groups These carry left and right invariant metric tensors, with important