Preface ..................................................... xi
Glossary of notation ........................................ xv
Introduction ................................................. 1
I Tensors in linear spaces ..................................... 11
1 Linear and affine spaces .................................... 12
2 Differential calculus ....................................... 21
3 Tensor algebra .............................................. 27
4 Alternating products ........................................ 31
5 Special relativity .......................................... 37
6 The uses of covariance ...................................... 44
II Manifolds ................................................... 51
7 Manifolds ................................................... 52
8 Tangent vectors and 1-forms ................................. 59
9 Lie bracket ................................................. 68
10 Tensors on manifolds ........................................ 72
11 Mappings .................................................... 77
12 Cotangent bundle ............................................ 84
13 Tangent bundle .............................................. 90
14 Vector fields and dynamical systems ......................... 94
15 Contact bundles ............................................. 99
16 The geometry of thermodynamics ............................. 108
III Transformations ........................................... 115
17 Lie groups ................................................. 115
18 Lie derivative ............................................. 121
19 Holonomy ................................................... 132
20 Contact transformations .................................... 136
21 Symmetries ................................................. 141
IV The calculus of differential forms ......................... 147
22 Differential forms ......................................... 147
23 Exterior calculus .......................................... 153
24 The operator ............................................... 159
25 Metric symmetries .......................................... 169
26 Normal forms ............................................... 173
27 Index notation ............................................. 176
28 Twisted differential forms ................................. 183
29 Integration ................................................ 194
30 Cohomology ................................................. 202
V Applications of the exterior calculus ....................... 207
31 Diffusion equations ........................................ 207
32 First-order partial differential equations ................. 213
33 Conservation laws .......................................... 219
34 Calculus of variations ..................................... 225
35 Constrained variations ..................................... 233
36 Variations of multiple integrals ........................... 239
37 Holonomy and thermodynamics ................................ 245
38 Exterior differential systems .............................. 248
39 Symmetries and similarity solutions ........................ 258
40 Variational principles and conservation laws ............... 264
41 When not to use forms ...................................... 268
VI Classical electrodynamics .................................. 271
42 Electrodynamics and differential forms ..................... 272
43 Electrodynamics in spacetime ............................... 282
44 Laws of conservation and balance ........................... 285
45 Macroscopic electrodynamics ................................ 293
46 Electrodynamics of moving bodies ........................... 298
VII Dynamics of particles and fields .......................... 305
47 Lagrangian mechanics of conservative systems ............... 306
48 Lagrange's equations for general systems ................... 311
49 Lagrangian field theory .................................... 314
50 Hamiltonian systems ........................................ 320
51 Symplectic geometry ........................................ 325
52 Hamiltonian optics ......................................... 333
53 Dynamics of wave packets ................................... 338
VIII Calculus on fiber bundles ................................ 347
54 Connections ................................................ 349
55 Parallel transport ......................................... 354
56 Curvature and torsion ...................................... 358
57 Covariant differentiation .................................. 365
58 Metric connections ......................................... 367
IX Gravitation ................................................ 371
59 General relativity ......................................... 372
60 Geodesies .................................................. 374
61 Geodesic deviation ......................................... 377
62 Symmetries and conserved quantities ........................ 382
63 Schwarzschild orbit problem ................................ 387
64 Light deflection ........................................... 393
65 Gravitational lenses ....................................... 395
66 Moving frames .............................................. 402
Bibliography ............................................... 409
Index ...................................................... 411
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