多维实分析

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

Index

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 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

2.10Commuting 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

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.