Table of Contents
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
B Our Notation 195
C Tensors 211
References 217
Index 219
