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內容簡介

　　The purpose of this book is to provide core material in nonlinear analysis for mathematicians. physicists， engineers， and mathematical biologists. The main goal is to provide a working knowledge of manifolds， dynamical systems， tensors， and differential forms. Some applications to Hamiltonian mechanics， fluid mechanics， electromagnetism. plasma dynamics and control theory are given in Chapter 8， using both invariant and index notation. The current edition of the book does not deal with Riemannian geometry in much detail， and it does not treat Lie groups， principal bundles， or Morse theory. Some of this is planned for a subsequent edition. Meanwhile， the authors will make available to interested readers supplementary chapters on Lie Groups and Differential Topology and invite comments on the book's contents and development.

內頁插圖

目錄

Preface
Background Notation
CHAPTER 1 Topology
1.1 Topological Spaces
1.2 Metric Spaces
1.3 Continuity
1.4 Subspaces. Products. and Quotients
1.5 Compactness
1.6 Connectedness
1.7 Baire Spaces

CHAPTER 2 Banach Spaces and Differential Calculus
2.1 Banach Spaces
2.2 Linear and Multilinear Mappings
2.3 The Derivativc
2.4 Propcrties of che Dcrivarive
2.5 The Inverse and Implicit Function Theorems

CHAPTER 3 Manifolds and Vector Bundles
3.1 Manifolds
3.2 Submanifolds. Products. and Mappings
3.3 The Tangcnt Bundle
3.4 Veaor Bundles
3.5 Submersions. Immersions and Transversality

CHAPTER 4 Vector Fields and Dynamical Systems
4.1 Vector Fields and Flows
4.2 Vector Fields as Differemial Operators
4.3 An Imroduction to Dynamical Systems
4.4 Frobenius' Theorcm and Foliations

CHAPTER 5 Tensors
5.1 Tensors in Linear Spaces
5.2 Tensor Bundles and Tensor Fields
5.3 The Lie Derivative: Algebraic Approach
5.4 The Lie Derivative: Dynamic Approach
5.5 Partitions of Unity

CHAPTER 6 Differential Forms
6.1 Exterior Algebra
6.2 Determinants. Volumes. and the Hodge Star Operator
6.3 Differential Forms
6.4 The Exterior Derivative. tnterior Produa. and Lie Derivative
6.5 Orientation. Volume Elements, and the Codifferential

CHAPTER 7 Integration on Manifolds
7.1 The Definition of (he Integral
7.2 Stokes' Theorem
7.3 The Classical Theorems of Green. Gauss, and Stokes
7.4 Induced Flows on Function Spaces and Ergodicity
7.5 Introduction to Hodge-deRham Theory and Topological Applicarions of
Differential Forms

CHAPTER 8 Applications
8.1 Hamiltonian Mechanics
8.2 Fluid Mechanics
8.3 Electromagnctism
8.3 The Lie-Poisson Bracket in Continuum Mechanics and Plasma Physics
8.4 Constraints and Control

References
Index
Supplementary Chapters-Available from the authors as they are produced
S-1 Lie Groups
S-2 Introduction to Differential Topology
S-3 Topics in Riemannian Geometry

前言/序言

流形、張量分析和應用（第2版） [Manifollds, Tensor Analysis, and Applications 2nd ed] 下載 mobi epub pdf txt 電子書

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