Differential Forms in Mathematical Physics (eBook)

Front Cover 1
Differential Forms in Mathematical Physics 4
Copyright Page 5
Contents 10
Preface 8
PART I BASIC CONCEPTS 18
Chapter 1 Topological Preliminaries 20
Summary 20
1. Introduction 20
2. Topological spaces 21
3. Closure. Interior. Boundary 23
4. Continuous maps. Homeomorphisms 25
5. Properties of topological spaces 27
6. Special topologies 30
Chapter 2 Differential Calculus on Rn 36
Summary 36
1. Introduction 37
2. Differentiation of maps of severable variables 38
3. Properties of differentiable maps 39
4. The directional derivative 41
5. Partial derivatives 42
6. The matrix of a differential. Jacobians 44
7. C1-maps and diffeomorphisms 49
8. The mean value theorem 49
9. Higher order differentials and Taylor's formula 50
10. The notion of Ck-diffeomorphism 53
11. The inverse function theorem 54
12. The implicit function theorem 55
PART II MANIFOLDS 60
Chapter 3 Differentiable manifolds 62
Summary 62
1. Introduction 62
2. Charts and atlases 63
3. Definition of differentiable manifolds 65
4. Properties of differentiable manifolds 66
5. Examples of differentiable manifolds 70
Chapter 4 Differential calculus on manifolds 76
1. Differentiable maps 76
2. Tangent vectors and tangent spaces 82
3. The differential of a map 88
4. The rank of a map 93
5. Submersion. Immersion. Submanifolds 95
6. Applications to mathematical physics 99
Chapter 5 Lie groups 101
Summary 101
1. Introduction and historical background 101
2. Definition of a Lie group. Examples 103
3. Left-invariant vector fields 105
4. The Lie algebra of a Lie group 107
5. Lie group homomorphisms 110
6. Lie subgroups of a Lie group 112
7. One-parameter subgroups 113
8. The exponential map 115
9. The canonical coordinate system 121
10. The adjoint representation 122
11. Lie transformation groups 125
12. Homogeneous spaces of Lie groups 129
13. Application: F. Klein's "Erlanger Program" 131
Chapter 6 Fiber bundles 133
1. Introduction 133
2. Definition of a fiber bundle. Examples 135
3. Tangent and cotangent bundles 139
4. Tensor bundles 143
5. Vector bundles 147
6. Principal fiber bundles 148
7. Associated bundles 152
PART III DIFFERENTIAL FORMS 156
Chapter 7 Basic concepts of differential forms 158
Summary 158
1. Definition of a differential form 159
2. Operations on differential forms 166
3. Invariant differential forms on Lie groups 191
4. Applications to mathematical physics 196
Chapter 8 The Frobenius Theory 225
1. Introduction 225
2. The condition of Frobenius 229
3. The Frobenius integrability theorem 233
4. Applications to mathematical physics 248
PART IV INTEGRATION THEORY ON MANIFOLDS 275
Chapter 9 Integration of differential forms 276
1. Integration in the Euclidean space Rn 276
2. Orientation 279
3. Integration of n-forms on Rn 284
4. Integration over chains 286
5. Integration on oriented manifolds 294
Chapter 10 The de Rham cohomology 310
Summary 310
1. De Rham cohomology 310
2. Differentiable singular homology 324
3. The de Rham theorems 327
4. The Hodge theorem 331
5. Poincaré duality 341
6. Applications to mathematical physics 344
PART V THEORY OF CONNECTIONS 350
Chapter 11 Connections on fibre bundles 352
Summary 352
1. The affine connection in classical differential geometry 353
2. Affine connections in the sense of Koszul 357
3. Cartan connections 375
4. Ehresmann connections 380
5. Linear connections 401
PART VI INTRINSIC MATHEMATICAL PHYSICS 404
Chapter 12 Hamiltonian mechanics and geometry 406
Summary 406
1. Introduction 406
2. Conservative mechanics, Symplectic manifolds 409
3. Time dependent mechanics 438
4. A geometrical approach to Hamilton's principle 441
5. Geometric interpretation of the Hamilton–Jacobi equation 453
Chapter 13 General Theory of Relativity 457
Summary 457
1. Introduction (Historical development of the theory of gravitation) 457
2. Symmetries in general relativity 463
3. The five-dimensional theory of Kaluza–Klein 468
4. Relativistic fluid mechanics 473
Bibliography 500
Subject Index 502