 | Sussman G.J. Functional differential geometry / G.J.Sussman, J.Wisdom, W.Farr. - London: MIT press, 2013. - xx, 228 p.: ill. - Bibliogr.: p.217-218. - Ind.: p.219-228. - ISBN 978-2-262-01934-7
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Preface ..................................................... xi
Prologue .................................................... xv
1 Introduction ................................................. 1
2 Manifolds ................................................... 11
2.1 Coordinate Functions ................................... 12
2.2 Manifold Functions ..................................... 14
3 Vector Fields and One-Form Fields ........................... 21
3.1 Vector Fields .......................................... 21
3.2 Coordinate-Basis Vector Fields ......................... 26
3.3 Integral Curves ........................................ 29
3.4 One-Form Fields ........................................ 32
3.5 Coordinate-Basis One-Form Fields ....................... 34
4 Basis Fields ................................................ 41
4.1 Change of Basis ........................................ 44
4.2 Rotation Basis ......................................... 47
4.3 Commutators ............................................ 48
5 Integration ................................................. 55
5.1 Higher Dimensions ...................................... 57
5.2 Exterior Derivative .................................... 62
5.3 Stokes's Theorem ....................................... 65
5.4 Vector Integral Theorems ............................... 67
6 Over a Map .................................................. 71
6.1 Vector Fields Over a Map ............................... 71
6.2 One-Form Fields Over a Map ............................. 73
6.3 Basis Fields Over a Map ................................ 74
6.4 Pullbacks and Pushforwards ............................. 76
7 Directional Derivatives ..................................... 83
7.1 Lie Derivative ......................................... 85
7.2 Covariant Derivative ................................... 93
7.3 Parallel Transport .................................... 104
7.4 Geodesic Motion ....................................... 111
8 Curvature .................................................. 115
8.1 Explicit Transport .................................... 116
8.2 Torsion ............................................... 124
8.3 Geodesic Deviation .................................... 125
8.4 Bianchi Identities .................................... 129
9 Metrics .................................................... 133
9.1 Metric Compatibility .................................. 135
9.2 Metrics and Lagrange Equations ........................ 137
9.3 General Relativity .................................... 144
10 Hodge Star and Electrodynamics ............................. 153
10.1 The Wave Equation ..................................... 159
10.2 Electrodynamics ....................................... 160
11 Special Relativity ......................................... 167
11.1 Lorentz Transformations ............................... 172
11.2 Special Relativity Frames ............................. 179
11.3 Twin Paradox .......................................... 181
A Scheme ..................................................... 185
В Our Notation ............................................... 195
С Tensors .................................................... 211
References ................................................. 217
Index ...................................................... 219
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