Burke W.L. Applied differential geometry. - Cambridge: Cambridge University Press, 1985. - xvii, 414 p.: ill. Bibliogr.: p.409-410. - Ind.: p.411-414. - ISBN 0-521-26317-4  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Оглавление / Contents

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