流形、张量分析和应用 第2版

PrefaceBackground NotationCHAPTER 1  Topology1.1 Topological Spaces1.2 Metric Spaces1.3 Continuity1.4 Subspaces, Products, and Quotients1.5 Compactness1.6 Connectedness1.7 Baire SpacesCHAPTER 2  Banach Spaces and Differential Calculus2.1 Banach Spaces2.2 Linear and Multilinear Mappings2.3 The Derivative2.4 Properties of the Derivative2.5 The Inverse and Implicit Function TheoremsCHAPTER 3  Manifolds and Vector Bundles3.1 Manifolds3.2 Submanifolds, Products, and Mappings3.3 The Tangent Bundle3.4 Vector Bundles3.5 Submersions, Immersions and TransversalityCHAPTER 4  Vector Fields and Dynamical Systems4.1 Vector Fields and Rows4.2 Vector Fields as Differential Operators4.3 An Introduction to Dynamical Systems4.4 Frobenius' Theorem and FoliationsCHAPTER 5  Tensors5.1 Tensors in Linear Spaces5.2 Tensor Bundles and Tensor Fields5.3 The Lie Derivative: Algebraic Approach5.4 The Lie Derivative: Dynamic Approach5.5 Partitions of UnityCHAPTER 6  Differential Forms6.1 Exterior Algebra6.2 Determinants, Volumes. and the Hodge Star Operator6.3 Differential Forms6.4 The Exterior Derivative, Interior Product. and Lie Derivative6.5 Orientation, Volume Elements, and the CodifferentialCHAPTER 7  Integration on Manifolds7.1 The Definition of the Integral7.2 Stokes Theorem7.3 The Classical Theorems of Green, Gauss, and Stokes7.4 Induced Flows on Function Spaces and Ergodicity7.5 Introduction to Hodge-deRham Theory and Topological Applications of Differential FormsCHAPTER 8  Applications8.1 Hamiltonian Mechanics8.2 Fluid Mechanics8.3 Electromagnetism8.3 The Lie-Poisson Bracket in Continuum Mechanics and Plasma Physics8.4 Constraints and ControlReferencesIndexSupplementary Chapters--Available from the authors as they are producedS-1 Lie GroupsS-2 Introduction to Differential TopologyS-3 Topics in Riemannian Geometry