多維實分析（第1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 study and in addition consider the interactions between these. Volume I is devoted to differentiation. Differentiable functions on Rn come first, in Chapters 1 through 3. Next, differentiable manifolds embedded in R are discussed, in Chapters 4 and 5. In Volume 11 we take up integration. Chapter 6 deals with the theory of n-dimensional integration over R. Finally, in Chapters 7 and 8 lower-dimensional integration over submanifolds of Rn is developed; particular attention is paid to vector analysis and the theory of differential forms, which are treated independently from each other. Generally speaking, the emphasis is on geometric aspects of analysis rather than on matters belonging to functional analysis.In presenting the material we have been intentionally concrete, aiming at a thorough understanding of Euclidean space. Once this case is properly understood, it becomes easier to move on to abstract metric spaces or manifolds and to infinitedimensional function spaces. If the general theory is introduced too soon, the reader might get con. fused about its relevance and lose motivation. Yet we have tried to organize the book as economically as we could, for instance by making use of linear algebra whenever possible and minimizing the number of arguments, always without sacrificing rigor. In many cases, a fresh look at old problems, by ourselves and 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 other hand, different techniques may be used to prove the same result. We offer ample illustration of these two principles, in the theory as well as the exercises.

內容概要

　　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 study and in addition consider the interactions between these. Volume I is devoted to differentiation. Differentiable functions on Rn come first， in Chapters 1 through 3. Next， differentiable manifolds embedded in R are discussed， in Chapters 4 and 5. In Volume 11 we take up integration. Chapter 6 deals with the theory of n-dimensional integration over R. Finally， in Chapters 7 and 8 lower-dimensional integration over submanifolds of Rn is developed; particular attention is paid to vector analysis and the theory of differential forms， which are treated independently from each other. Generally speaking， the emphasis is on geometric aspects of analysis rather than on matters belonging to functional analysis.

作者簡介

作者：(荷蘭)杜斯特馬特

書籍目錄

Volume 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  Differentiable 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 oflnverse 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 for Chapter 2　　Exercises for Chapter 3　　Exercises for Chapter 4　　Exercises for Chapter 5　Notation　IndexVolume 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  Antidifferentiation　　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 Poincare's Lemma　　8.11 Degree of mapping　Exercises　　Exercises for Chapter 6　　Exercises for Chapter 7　　Exercises for Chapter 8　Notation　Index

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