Manifolds, tensor analysis, and applications

Cover......Page 1
Preface......Page 2
1.1 Topological Spaces......Page 5
1.2 Metric Spaces......Page 12
1.3 Continuity......Page 16
1.4 Subspaces, Products, and Quotients......Page 19
1.5 Compactness......Page 24
1.6 Connectedness......Page 30
1.7 Baire Spaces......Page 34
2.1 Banach Spaces......Page 38
2.2 Linear and Multilinear Mappings......Page 52
2.3 The Derivative......Page 69
2.4 Properties of the Derivative......Page 75
2.5 The Inverse and Implicit Function Theorems......Page 104
3.1 Manifolds......Page 128
3.2 Submanifolds, Products, and Mappings......Page 136
3.3 The Tangent Bundle......Page 142
3.4 Vector Bundles......Page 151
3.5 Submersions, Immersions, and Transversality......Page 175
3.6 The Sard and Smale Theorems......Page 195
4.1 Vector Fields and Flows......Page 211
4.2 Vector Fields as Differential Operators......Page 232
4.3 An Introduction to Dynamical Systems......Page 259
4.4 Frobenius’ Theorem and Foliations......Page 281
5.1 Tensors on Linear Spaces......Page 292
5.2 Tensor Bundles and Tensor Fields......Page 301
5.3 The Lie Derivative: Algebraic Approach......Page 309
5.4 The Lie Derivative: Dynamic Approach......Page 318
5.5 Partitions of Unity......Page 324
6.1 Exterior Algebra......Page 337
6.2 Determinants, Volumes, and the Hodge Star Operator......Page 345
6.3 Differential Forms......Page 357
6.4 The Exterior Derivative, Interior Product, and Lie Derivative......Page 362
6.5 Orientation, Volume Elements and the Codifferential......Page 386
7.1 The Definition of the Integral......Page 399
7.2 Stokes’ Theorem......Page 410
7.3 The Classical Theorems of Green, Gauss, and Stokes......Page 434
7.4 Induced Flows on Function Spaces and Ergodicity......Page 442
7.5 Introduction to Hodge–deRham Theory......Page 463
8.1 Hamiltonian Mechanics......Page 482
8.2 Fluid Mechanics......Page 502
8.3 Electromagnetism......Page 514
8.4 The Lie–Poisson Bracket in Continuum Mechanics and Plasma Physics......Page 522
8.5 Constraints and Control......Page 535
References......Page 542
9 An Introduction to Lie Groups......Page 553
9.1 Basic Definitions and Properties......Page 555
9.2 Some Classical Lie Groups......Page 571
9.3 Actions ofLie Groups......Page 597