多維實分析（第2卷）

前言

This book, which is in two parts, provides an introduction to the theory of vector-valued functions on Euclidean space.  We focus on four main objects of studyand in addition consider the interactions between these. Volume I is devoted todifferentiation. Differeutiable functions on Rn come first, in Chapters 1 through 3.Next, differentiable manifolds embedded in Rn are discussed, in Chapters 4 and 5. InVolume 11 we take up integration. Chapter 6 deals with the theory of n-dimensionalintegration over . Finally, in Chapters 7 and 8 lower-dimensional integration oversubmanifolds of R" is developed; particular attention is paid to vector analysis andthe the6ry of differential forms, which are treated independently from each other.Generally speaking, the emphasis is on geometric aspects of analysis rather than onmatters belonging to functional analysis.In presenting the material we have been intentionally concrete, aiming at athorough understanding of Euclidean space. Once this case is properly understood,it becomes easier to move on to abstract metric spaces or manifolds and to infinite-dimensional function spaces. If the general theory is introduced too soon, the readermight get confused about its relevance and lose motivation. Yet we have tried toorganize the book as economically as we could, for instance by making use of linearalgebra whenever possible and minimizing the number of ~ arguments, alwayswithout sacrificing rigor. In many cases, a fresh look at old problems, by ourselvesand others, led to results or proofs in a form not found in current analysis textbooks.Quite often, similar techniques apply in different parts of mathematics; on the otherhand, different techniques may be used to prove the same result. We offer ampleillustration of these two principles, in the theory as well as the exercises.A working knowledge of analysis in one real variable and linear algebra is aprerequisite; furthermore, familiarity with differentiable mappings and submani-folds of Rn, as discussed in volume I, for instance. The main parts of the theorycan be used as a text for an introductory course of one semester, as we have beendoing for second-year students in Utrecht during the last decade. Sections at theend of many chapters usually contain applications that can be omitted in case oftime constraints.This volume contains 234 exercises, out of a total of 568, offering variationsand applications of the main theory, as well as special cases and openings towardapplications beyond the scope of this book.  Next to routine exercises we triedalso to include exercises that represent some mathematical idea. The exercises areindependent from each other unless indicated otherwise.

內容概要

《多維實分析(第2卷)(英文版)》講述了：In presenting the material we have been intentionally concrete, aiming at athorough understanding of Euclidean space. Once this case is properly understood,it becomes easier to move on to abstract metric spaces or manifolds and to infinite-dimensional function spaces. If the general theory is introduced too soon, the readermight get confused about its relevance and lose motivation. Yet we have tried toorganize the book as economically as we could, for instance by making use of linearalgebra whenever possible and minimizing the number of ~ arguments, alwayswithout sacrificing rigor. In many cases, a fresh look at old problems, by ourselvesand others, led to results or proofs in a form not found in current analysis textbooks.Quite often, similar techniques apply in different parts of mathematics; on the otherhand, different techniques may be used to prove the same result. We offer ampleillustration of these two principles, in the theory as well as the exercises.

作者簡介

作者：(荷蘭)杜斯特馬特

書籍目錄

Volume Ⅱ  Preface  Acknowledgments  Introduction  6  Integration    6.1  Rectangles    6.2  Riemann integrability    6.3  Jordan measurability    6.4  Successive integration    6.5  Examples of successive integration    6.6  Change of Variables Theorem: formulation and examples     6.7  Partitions of unity      6.8  Approximation of Riemann integrable functions    6.9  Proof of Change of Variables Theorem    6.10  Absolute Riemann integrability    6.11  Application of integration: Fourier transformation    6.12  Dominated convergence    6.13  Appendix: two other proofs of Change of Variables Theorem　7  Integration over Submanifolds    7.1  Densities and integration with respect to density    7.2  Absolute Riemann integrability with respect to density      7.3  Euclidean d-dimensional density    7.4  Examples of Euclidean densities    7.5  Open sets at one side of their boundary    7.6  Integration of a total derivative    7.7  Generalizations of the preceding theorem    7.8  Gauss' Divergence Theorem    7.9  Applications of Gauss' Divergence Theorem  8  Oriented Integration    8.1  Line integrals and properties of vector fields    8.2  Antidifl'erentiation    8.3  Green's and Cauchy's Integral Theorems    8.4  Stokes' Integral Theorem    8.5  Applications of Stokes' Integral Theorem    8.6  Apotheosis: differential forms and Stokes' Theorem    8.7  Properties of differential forms    8.8  Applications of differential forms    8.9  Homotopy Lemma    8.10  Poincard's Lemma    8.11  Degree of mapping　Exercises    Exercises for Chapter 6    Exercises for Chapter 7    Exercises for Chapter 8　Notation　IndexVolume Ⅰ  Preface  Acknowledgments  Introduction　1  Continuity    1.1  Inner product and norm    1.2  Open and closed sets    1.3  Limits and continuous mappings    1.4  Composition of mappings    1.5  Homeomorphisms    1.6  Completeness    1.7  Contractions    1.8  Compactness and uniform continuity    1.9  Connectedness　2  Differentiation    2.1  Linear mappings    2.2  Differentiahle mappings    2.3  Directional and partial derivatives    2.4  Chain rule    2.5  Mean Value Theorem    2.6  Gradient    2.7  Higher-order derivatives    2.8  Taylor's formula    2.9  Critical points     2.10　Commuting limit operations　3  Inverse Function and Implicit Function Theorems    3.1  Diffeomorphisms    3.2  Inverse Function Theorems    3.3  Applications of Inverse Function Theorems    3.4  Implicitly defined mappings    3.5  Implicit Function Theorem    3.6  Applications of the Implicit Function Theorem    3.7  Implicit and Inverse Function Theorems on C　4  Manifolds    4.1  Introductory remarks    4.2  Manifolds    4.3  Immersion Theorem    4.4  Examples of immersions    4.5  Submersion Theorem    4.6  Examples of submersions    4.7  Equivalent definitions of manifold    4.8  Morse's Lemma　5  Tangent Spaces    5.1  Definition of tangent space    5.2  Tangent mapping    5.3  Examples of tangent spaces    5.4  Method of Lagrange multipliers    5.5  Applications of the method of multipliers    5.6  Closer investigation of critical points    5.7  Gaussian curvature of surface    5.8  Curvature and torsion of curve in R3      5.9  One-parameter groups and infinitesimal generators    5.10  Linear Lie groups and their Lie algebras    5.11  Transversality　Exercises    Review Exercises    Exercises for Chapter 1    Exercises lot Chapter 2    Exercises for Chapter 3    Exercises for Chapter 4    Exercises for Chapter 5　Notation　Index

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