Geometrical Methods in Mathematical Physics

✍ Scribed by Vitali D. Milman, Gideon Schechtman

• Publisher: Cambridge University Press
• Year: 1980
• Tongue: English
• Leaves: 260
• Series: Lecture Notes in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

In recent years the methods of modern differential geometry have become of considerable importance in theoretical physics and have found application in relativity and cosmology, high-energy physics and field theory, thermodynamics, fluid dynamics and mechanics. This textbook provides an introduction to these methods - in particular Lie derivatives, Lie groups and differential forms - and covers their extensive applications to theoretical physics. The reader is assumed to have some familiarity with advanced calculus, linear algebra and a little elementary operator theory. The advanced physics undergraduate should therefore find the presentation quite accessible. This account will prove valuable for those with backgrounds in physics and applied mathematics who desire an introduction to the subject. Having studied the book, the reader will be able to comprehend research papers that use this mathematics and follow more advanced pure-mathematical expositions.

✦ Table of Contents

Geometrical methods of mathematical physics......Page 1
CONTENTS......Page 3
Why study geometry?......Page 7
Aims of this book......Page 8
How to use this book......Page 9
Background assumed of the reader......Page 10
1.1 The space Rn and its topology......Page 11
1.2 Mappings......Page 15
1.3 Weal analysis......Page 19
1.4 Group theory......Page 21
1.5 Linear algebra......Page 23
1.6 The algebra of square matrices......Page 26
1.7 Bibliography......Page 30
2.1 Definition of a manifold......Page 33
2.2 The sphere as a manifold......Page 36
2.3 Other examples of manifolds......Page 38
2.4 Global considerations......Page 39
2.6 Functions on M......Page 40
2.7 Vectors and vector fields......Page 41
2.8 Basis vectors and basis vector fields......Page 44
2.9 Fiber bundles......Page 45
2.10 Examples of fiber bundles......Page 47
2.11 A deeper look at fiber bundles......Page 48
2.12 Vector fields and integral curves......Page 52
2.14 Lie brackets and noncoordinate bases......Page 53
2.15 When is a basis a coordinate basis?......Page 57
2.16 One-forms......Page 59
2.17 Examples of one-forms......Page 60
2.18 The Dirac delta function......Page 61
2.19 The gradient and the pictorial representation of a one-form......Page 62
2.20 Basis one-forms and components of one-forms......Page 65
2.21 Index notation......Page 66
2.22 Tensors and tensor fields......Page 67
2.23 Examples of tensors......Page 68
2.25 Contraction......Page 69
2.26 Basis transformations......Page 70
2.27 Tensor operations on components......Page 73
2.29 The metric tensor on a vector space......Page 74
2.30 The metric tensor field on a manifold......Page 78
2.31 Special relativity......Page 80
2.32 Bibliography......Page 81
3.1 Introduction : how a vector field maps a manifold into itself......Page 83
3.3 Lie dragging a vector field......Page 84
3.4 Lie derivatives......Page 86
3.5 Lie derivative of a one-form......Page 88
3.6 Submanifolds......Page 89
3.7 Frobenius' theorem (vector field version)......Page 91
3.8 Proof of Frobenius' theorem......Page 93
3.9 An example: the generators of S2......Page 95
3.10 Invariance......Page 96
3.11 Killing vector fields......Page 98
3.13 Axial symmetry......Page 99
3.14 Abstract Lie groups......Page 102
3.15 Examples of Lie groups......Page 105
3.16 Lie algebras and their groups......Page 111
3.17 Realizations and representations......Page 115
3.18 Spherical symmetry, spherical harmonics and representations of the rotation group......Page 118
3.19 Bibliography......Page 122
4.1 Definition of volume - the geometrical role of differential forms......Page 123
4.2 Notation and definitions for antisymmetric tensors......Page 125
4.3 Differential forms......Page 127
4.4 Manipulating differential forms......Page 129
4.6 Fields of forms......Page 130
4.8 Volumes and integration on oriented manifolds......Page 131
4.9 N-vectors, duals, and the symbol eij...k......Page 135
4.10 Tensor densities......Page 138
4.11 Generalized Kronecker deltas......Page 140
4.12 Determinants and eij...k......Page 141
4.13 Metric volume elements......Page 142
4.14 The exterior derivative......Page 144
4.15 Notation for derivatives......Page 145
4.16 Familiar examples of exterior differentiation......Page 146
4.17 Integrability conditions for partial differential equations......Page 147
4.18 Exact forms......Page 148
4.19 Proof of the local exactness of closed forms......Page 150
4.20 Lie derivatives of forms......Page 152
4.21 Lie derivatives and exterior derivatives commute......Page 153
4.22 Stokes' theorem......Page 154
4.23 Gauss' theorem and the definition of divergence......Page 157
4.24 A glance at cohomology theory......Page 160
4.25 Differential forms and differential equations......Page 162
4.26 Frobenius' theorem (differential forms version)......Page 164
4.27 Proof of the equivalence of the two versions of Frobenius ' theorem......Page 167
4.28 Conservation laws......Page 168
4.29 Vector spherical harmonics......Page 170
4.30 Bibliography......Page 171
5.1 Simple systems......Page 173
5.2 Maxwell and other mathematical identities......Page 174
5.3 Composite thermodynamic systems: Caratheodory's theorem......Page 175
5.4 Hamiltonian vector fields......Page 177
5.5 Canonical transformation......Page 178
5.6 Map between vectors and one-forms provided by ω......Page 179
5.8 Many-particle systems: symplectic forms......Page 180
5.9 Linear dynamical systems: the symplectic inner product and conservedquantities......Page 181
5.10 Fiber bundle structure of the Hamiltonian equations......Page 184
5.11 Rewriting Maxwell's equations using differential forms......Page 185
5.12 Charge and topology......Page 189
5.13 The vector potential......Page 190
5.15 Role of Lie derivatives......Page 191
5.16 The comoving timederivative......Page 192
5.17 Equation of motion......Page 193
5.18 Conservation of vorticity......Page 194
5.19 The cosmological principle......Page 196
5.20 Lie algebra of maximal symmetry......Page 200
5.21 The metric of a spherically symmetric three-space......Page 202
5.22 Construction of the six Killing vectors......Page 205
5.23 Open, closed, and flat universes......Page 207
5.24 Bibliography......Page 209
6.2 Parallelism on curved surfaces......Page 211
6.3 The co variun t derivative......Page 213
6.4 Components: covariant derivatives of the basis......Page 215
6.5 Torsion......Page 217
6.6 Geodesics......Page 218
6.8 Riemann tensor......Page 220
6.9 Geometric interpretation of the Riemann tensor......Page 222
6.10 Flat spaces......Page 224
6.11 Compatibility of the connection with the volume-measure or the metric......Page 225
6.12 Metric connections......Page 226
6.13 The affine connection and the equivalence principle......Page 228
6.14 Connections and gauge theories: the example of elec tromagnetism......Page 229
6.15 Bibliography......Page 232
APPENDIX: SOLUTIONS AND HINTS FOR SELECTED EXERCISES......Page 234
NOTATION......Page 254
INDEX......Page 256