# 河南快赢481软件下载: ACourseinAnalysisVol. ii differentiation and integration of functions of several variables, vector calculus

A Course in Analysis Vol. II Differentiation and integration of Functions of Several Variables, Vector Calculus A Course in Analysis Vol. I Part 1 Introductory Calculus Part 2 Analysis of Functions of One Real Variable Vol. II Part 3 Differentiation of Functions of Several Variables Part 4 Integration of Functions of Several Variables Part 5 Vector Calculus Vol. III Part 6 Measure and Integration Theory Part 7 Complex-valued Functions of a Complex Variable Part 8 Fourier Analysis Vol. IV Part 9 Ordinary Differential Equations Part 10 Partial Differential Equations Part 11 Calculus of Variations Vol. V Part 12 Functional Analysis Part 13 Operator Theory Part 14 Theory of Distributions Vol. VI Part 15 Differential Geometry of Curves and Surfaces Part 16 Differentiable Manifolds and Riemannian Geometry Part 17 Lie Groups Vol. VII Part 18 History of Analysis A Course in Analysis Vol. II Differentiation and Integration of Functions of Several Variables, Vector Calculus Niels Jacob Kristian P Evans Swansea University, UK Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Jacob, Niels. A course in analysis / by Niels Jacob (Swansea University, UK), Kristian P. Evans (Swansea University, UK). volumes cm Includes bibliographical references and index. Contents: volume 1. Introductory calculus, analysis of functions of one real variable Identifiers: ISBN 978-9814689083 (hardcover : alk. paper) -- ISBN 978-9814689090 (pbk : alk. paper) 1. Mathematical analysis--Textbooks. 2. Mathematics--Study and teaching (Higher) 3. Calculus--Textbooks. I. Evans, Kristian P. II. Title. QA300.J27 2015 515--dc23 2015029065 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright ? 2016 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-3140-95-0 ISBN 978-981-3140-96-7 (pbk) Printed in Singapore Preface A detailed description of the content of Volume II of our Course in Analysis will be provided in the introduction. Here we would like to take the opportunity to thank those who have supported us in writing this volume. We owe a debt of gratitude to James Harris who has typewritten a major part of the manuscript. Thanks for typewriting further parts are expressed to Huw Fry and Yelena Liskevich who also provided a lot of additional support. Lewis Bray, James Harris and Elian Rhind did a lot of proofreading for which we are grateful. We also want to thank the Department of Mathematics, the College of Science and Swansea University for providing us with funding for typewriting. Finally we want to thank our publisher, in particular Tan Rok Ting and Ng Qi Wen, for a pleasant collaboration. Niels Jacob Kristian P. Evans Swansea, January 2016 Introduction This is the second volume of our Course in Analysis and it is designed for second year students or above. In the first volume, in particular in Part 1, the transition from school to university determined the style and approach to the material, for example by introducing abstract concepts slowly or by giving very detailed (elementary) calculations. Now we use an approach that is suitable for students who are more used to the university style of doing mathematics. As we go through the volumes our intention is to guide and develop students to nurture a more professional and rigorous approach to mathematics. We will still take care with motivations (some lengthy) when introducing new concepts, exploring new notions by examples and their limitations by counter examples. However some routine calculations are taken for granted as is the willingness to “fight” through more abstract concepts. In addition we start to change the way we use references and students should pick up on this. Calculus and analysis in one dimension is so widely taught that it is difficult to trace back in textbooks the origin of how we present and prove results nowadays. Some comments about this were made in Volume I. The more advanced the material becomes, the more appropriate it becomes to point out in more detail the existing literature and our own sources. Still we are in a territory where a lot of material is customary and covered by “standard approaches”. However in some cases authors may claim more originality and students should know about the existing literature and give it fair credit - as authors of books are obliged to do. We hope that the more experienced reader will consider our referencing as being fair, please see further details below. The goal of this volume is to extend analysis for real-valued functions of one real variable to mappings from domains in m to n, i.e to vector-valued mappings of several variables. At a first glance we need to address three wider fields: convergence and continuity; linear approximation and differentiability; and integration. As it turns out, to follow this programme, we need to learn in addition much more about geometry. Some of the geometry is related to topological notions, e.g. does a set have several components? Does a set have “holes”? What is the boundary of a set? Other geometric notions will be related to the vector space structure of n, e.g. quadratic forms, orthogonality, convexity, some types of symmetries, say rotationally invariant functions. But we also need a proper understanding of elementary differential geometric concepts such as parametric curves and surfaces (and later on manifolds and sub-manifolds). Therefore we have included a fair amount of geometry in our treatise starting with this volume. The situation where we introduce integrals is more difficult. The problem is to define a volume or an area for certain subsets in G ? n. Once this is done for a reasonably large class of subsets a construction of the integral along the lines of the one-dimensional case is possible. It turns out that Lebesgue’s theory of measure and integration is much better suited to this and we will develop this theory in the next volume. Our approach to volume (and surface) integrals following Riemann’s ideas (as transformed by Darboux) is only a first, incomplete attempt to solve the integration problem. However it is essentially sufficient to solve concrete problems in analysis, geometry as well as in mathematical physics or mechanics. Let us now discuss the content of this volume in more detail. In the first four chapters we cover convergence and continuity. Although our main interest is in handling mappings f : G → n, G ? m, in order to be prepared for dealing with convergence of sequences of functions, continuity of linear operators, etc., we discuss convergence and continuity in metric spaces as we introduce the basic concepts of point set topology. However we also spend time on normed spaces. We then turn to continuous mappings and study their basic properties in the context of metric and topological spaces. Eventually we consider mappings f : G → n and for this we investigate some topological properties of subsets of n. In particular we discuss the notion of compactness and its consequences. The main theoretical concepts are developed along the lines of N. Bourbaki, i.e. J. Dieudonné [10], however when working in the more concrete Euclidean context we used several different sources, in particular for dealing with connectivity we preferred the approach of M. Heins [24]. A general reference is also [9] Differentiability is the topic in Chapter 5 and Chapter 6. First we discuss partial derivatives of functions f : G → , G ? n, and then the differential of mappings f : G → n, G ? m. These chapters must be viewed as “standard” and our approach does not differ from any of the approaches known to us. Once differentiability is established we turn to applications and here geometry is needed. An appropriate name for Chapter 7 and Chapter 8 would be G. Monge’s classical one: “Applications of Analysis to Geometry”. We deal with parametric curves in n, with some more details in the case n = 3, and we have a first look at parametric surfaces in 3. In addition to having interesting and important applications of differential calculus we prepare our discussion of the integral theorems of vector calculus where we have to consider boundaries of sets either as parametric curves or as parametric surfaces. These two chapters benefit greatly from M. DoCarmo’s textbook [11]. In Chapter 9 to Chapter 11 we extend the differential calculus for functions of several variables as we give more applications, many of them going back to the times of J. d’Alembert, L. Euler and the Bernoulli family. Key phrases are Taylor formula, local extreme values under constraints (Lagrange multipliers) or envelopes. However note that Taylor series or more generally power series for functions of several variables are much more difficult to handle due to the structure of convergence domains and we postpone this until we discuss complex-valued functions of complex variables (in Volume III). Of a more theoretical nature, but with a lot of applications, is the implicit function theorem and its consequence, the inverse mapping theorem. In general in these chapters we follow different sources and merge them together. In particular, since some of the classical books include nice applications, however in their theoretical parts they are now outdated, some effort is needed to obtain a coherent view. The book of O. Forster [19] was quite helpful in treating the implicit function theorem. In Chapter 12 curvilinear coordinates are addressed - a topic which is all too often neglected nowadays. However, when dealing with problems that have symmetry, for example in mathematical physics, curvilinear coordinates are essential. In our understanding, they also form a part of classical differential geometry, as can be already learned from G. Lamé’s classical treatise. Almost every book about differential calculus in several variables discusses convexity, e.g. convex sets which are useful when dealing with the mean value theorem, or convex functions when handling local extreme values. We have decided to treat convex sets and functions in much more detail than what other authors do by including more on the geometry of convex sets (for example separating hyperplanes) where S. Hildebrandt’s treatise [26] was quite helpful. We further do this by discussing extreme points (Minkowski’s theorem) and its applications to extreme values of convex functions on compact convex sets. We also look at a characterisation of differentiable convex functions by variational inequalities as do we discuss the Legendre transform (conjugate functions) of convex functions and metric projections onto convex sets. All of this can be done in n at this stage of our Course and it will be helpful in other parts such as the calculus of variations, functional analysis and differential geometry. Most of all, these are beautiful results. After introducing continuity or differentiability (or integrability) we can consider vector spaces of functions having these (and some additional) properties. For example we may look at the space C(K) of all continuous functions defined on a compact set K ? n, which is equipped with the norm ∥g∥∞ := supx∈K|g(x)| a Banach space. Already in classical analysis the question whether an arbitrary continuous function can be approximated by simpler functions such as polynomials or trigonometrical functions was discussed. It was also discussed whether a uniformly bounded sequence of continuous functions always has a uniformly convergent subsequence. We can now interpret the first question as the problem to find a “nice” dense subset in the Banach space C(K) whereas the second problem can be seen as to find or characterise (pre-) compact sets in C(K). We deliberately put these problems into the context of Banach spaces, i.e. we treat the problems as problems in functional analysis. We prove the general Stone-Weierstrass theorem, partly by a detour, by first proving Korovkin’s approximatiuon theorem, and we prove the Arzela-Ascoli theorem. We strongly believe at this stage of the Course that students should start to understand the benefit of reformulating concrete classical problems as problems of functional analysis. The final chapter of Part 3 deals with line integrals. We locate line integrals in Part 3 and not Part 4 since eventually they are reduced to integrals of functions defined on an interval and not on a domain in n, n 1. We discuss the basic definition, the problem of rectifying curves and we start to examine the integrability conditions. As already indicated, defining an integral in the sense of Riemann for a bounded function f : G → , G ? compact, is not as straightforward as it seems. In Chapter 16 we give more details about the problems and we indicate our strategy to overcome these difficulties. A first step is to look at iterated integrals for functions defined on a hyper-rectangle (which we assume to be axes parallel) and this is done in the natural frame of parameter dependent integrals. In the following chapter we introduce and investigate Riemann integrals (volume integrals) for functions defined on hyper- rectangles. This can be done rather closely along the lines we followed in the one-dimensional case. Identifying volume integrals with iterated integrals allows us to reduce the actual integration problem to one-dimensional integrals. Integrating functions on sets G other than hyper-rectangles is much more difficult. The main point is that we do not know what the volume of a set in n is, hence Riemann sums are difficult to introduce. Even the definition of an integral for step functions causes a problem. It turns out that the boundary ?G determines whether we can define, say for a bounded continuous function f : G → an integral. This leads to a rather detailed study of boundaries and their “content” or “measure”. Basically it is the intertwining of the topological notion “boundary” with the (hidden) measure theoretical notion “set of measure zero” which causes difficulties. We devote Chapter 19 to these problems and once we end up with the concept of (bounded) Jordan measurable sets, we can construct integrals for bounded (continuous) functions defined on bounded Jordan measurable sets. In our presentation of this part we combine parts of the approaches of [20], [25] and [26]. In order to evaluate volume integrals we need further tools, in particular the transformation theorem. Within the Riemann context this theorem is notoriously difficult and lengthy to prove which is essentially due to the problems mentioned above, i.e. the mixture of topological and measure theoretical notions. In the context of Lebesgue’s theory of integration the transformation theorem admits a much more transparent proof, we also refer to our remarks in Chapter 21. For this reason we do not provide a proof here but we clearly state the result and give many applications. Eventually we return to improper and parameter dependent integrals, but now in the context of volume integrals. Many of these considerations will become of central importance when investigating partial differential equations. The final part of this volume is devoted to vector calculus in 2, but most of all in 3. A pure mathematician’s point of view could be to first introduce manifolds including E. Cartan’s exterior calculus, then to introduce integrals for k-forms over m-dimensional sub-manifolds of n-dimensional manifolds, and then to eventually prove the general Stokes’ theorem. By specialising we can now derive the classical theorems of Gauss, Green and Stokes. This programme neither takes the historical development into account nor is it suitable for second year students. Thus we decided on a more classical approach. Chapter 23 gives in some sense a separate introduction to Part 5, hence we can be more brief here. In Chapter 24 we discuss the problem of how to define the area of a parametric surface and then we turn to surface integrals for scalar-valued functions as well as for vector fields. With line and surface integrals at our disposal we can prove Gauss’ theorem (in 3 and later on in n), Stokes’ theorem in 3 and Green’s theorem in the plane. One part of our investigations is devoted to the question of in what type of domain can we prove these theorems? Another part deals with applications. Our aim is to give students who are interested in applied mathematics, mathematical physics or mechanics the tools (and the ideas of the mathematical background) needed to solve such problems. Only in Volume VI will we provide a rigorous proof of the general Stokes’ theorem. As in Volume I we have provided solutions to all problems (ca. 275) and since we depend on a lot of results from linear algebra we have collected these results in an appendix. Since many of our considerations are geometry related, the text contains a substantial number of figures (ca. 150). All of these figures were done by the second named author using LaTex. Finally a remark about referring to Volume I. When referring to a theorem, a lemma, a definition, etc. in Volume I we write, for example, Theorem I.25.9 etc., and when referring to a formula we write, for example, (I.25.10) etc. As in Volume I, problems marked with a * are more challenging. Contents Preface Introduction List of Symbols Part 3:Differentiation of Functions of Several Variables 1Metric Spaces 2Convergence and Continuity in Metric Spaces 3More on Metric Spaces and Continuous Functions 4Continuous Mappings Between Subsets of Euclidean Spaces 5Partial Derivatives 6The Differential of a Mapping 7Curves in ?n 8Surfaces in ?3. A First Encounter 9Taylor Formula and Local Extreme Values 10 Implicit Functions and the Inverse Mapping Theorem 11 Further Applications of the Derivatives 12 Curvilinear Coordinates 13 Convex Sets and Convex Functions in ?n 14 Spaces of Continuous Functions as Banach Spaces 15 Line Integrals Part 4:Integration of Functions of Several Variables 16 Towards Volume Integrals in the Sense of Riemann 17 Parameter Dependent and Iterated Integrals 18 Volume Integrals on Hyper-Rectangles 19 Boundaries in ?n and Jordan Measurable Sets 20 Volume Integrals on Bounded Jordan Measurable Sets 21 The Transformation Theorem: Result and Applications 22 Improper Integrals and Parameter Dependent Integrals Part 5:Vector Calculus 23 The Scope of Vector Calculus 24 The Area of a Surface in ?3 and Surface Integrals 25 Gauss’ Theorem in ?3 26 Stokes’ Theorem in ?2 and R3 27 Gauss’ Theorem for ?n Appendices Appendix I: Vector Spaces and Linear Mappings Appendix II: Two Postponed Proofs of Part 3 Solutions to Problems of Part 3 Solutions to Problems of Part 4 Solutions to Problems of Part 5 References Mathematicians Contributing to Analysis (Continued) Subject Index List of Symbols In general, symbols already introduced in Volume I are not listed here and we refer to the List of Symbols in Volume I. the set of all multi-indices α! = α1! · … · αn! for α = (α1,., αn) for α = (α1,.,αn) α ≤ β αj ≤ βj, α, α + β = (α1 + β1,., αn + βn), α, for α = (α1,., αn) and x ∈ n (X) power set of X fj the jth component of f prj projection on the jth factor or component f = (f1,., fn) vector-valued f with components R(f ) = ran(f) range of f Γ(f) graph of f f|K restriction of f to K (f ∨