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亚伯拉罕编著的《流形张量分析和应用(第2版)

》旨在为数学家、物理学家、工程和数学生物专业的全面介绍非线性分析的知识。书中介绍了流形、动力系统、张量和微分形式的背景知识和哈密顿力学、流体力学、电磁学、等离子动力学和控制理论等的应用。这本书起点低，读者只要了解本科生线性代数知识和高等微积分即可。

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 Derivative

2.4 Properties of the Derivative

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 Tangent Bundle

3.4 Vector Bundles

3.5 Submersions, Immersions and Transversality

CHAPTER 4 Vector Fields and Dynamical Systems

4.1 Vector Fields and Rows

4.2 Vector Fields as Differential Operators

4.3 An Introduction to Dynamical Systems

4.4 Frobenius' Theorem 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, Interior Product. and Lie Derivative

6.5 Orientation, Volume Elements, and the Codifferential

CHAPTER 7 Integration on Manifolds

7.1 The Definition of the 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 Applications of Differential Forms

CHAPTER 8 Applications

8.1 Hamiltonian Mechanics

8.2 Fluid Mechanics

8.3 Electromagnetism

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

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书籍介绍

《流形，张量分析和应用（第2版）》旨在为数学家、物理学家、工程和数学生物专业的全面介绍非线性分析的知识。书中介绍了流形、动力系统、张量和微分形式的背景知识和哈密顿力学、流体力学、电磁学、等离子动力学和控制理论等的应用。这本书起点低，读者只要了解本科生线性代数知识和高等微积分即可。

目次：拓扑；巴拿赫空间和微积分；流形和向量丛；向量域和动力系统；张量；微分形式；流形上的积分；应用。

读者对象：数学专业的高年级本科生和研究生。