Contents

Preface   page xi

Glossary of notation xv

Introduction 1

1. 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
2. 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
3. Transformations 115
   17. Lie groups 115
   18. Lie derivative 121
   19. Holonomy 132
   20. Contact transformations 136
   21. Symmetries 141
4. 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
5. 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
6. 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
7. 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
8. 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
9. 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