Mathematics for physics: an illustrated handbook
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo
World Scientific
[2018]
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| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | xviii, 282 Seiten Illustrationen, Diagramme |
| ISBN: | 9789813233911 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804178384155049984 |
|---|---|
| adam_text | Contents
/I
Preface v
Notation ix
1. Mathematical structures 1
1.1 Classifying mathematical concepts.......................... 1
1.2 Defining mathematical structures and mappings.............. 2
2. Abstract algebra 5
2.1 Generalizing numbers....................................... 5
2.1.1 Groups ............................................ 6
2.1.2 Rings.............................................. 8
2.2 Generalizing vectors....................................... 8
2.2.1 Inner products of vectors......................... 10
2.2.2 Norms of vectors ................................. 11
2.2.3 Multilinear forms on vectors...................... 12
2.2.4 Orthogonality of vectors ......................... 14
2.2.5 Algebras: multiplication of vectors .............. 15
2.2.6 Division algebras ................................ 16
2.3 Combining algebraic objects .............................. 17
2.3.1 The direct product and direct sum ................ 18
2.3.2 The free product ................................. 19
2.3.3 The tensor product................................ 20
2.4 Dividing algebraic objects................................ 22
2.4.1 Quotient groups................................... 22
2.4.2 Semidirect products............................... 23
2.4.3 Quotient rings.................................... 24
xiii
xiv Contents
2.4.4 Related constructions and facts ................ 25
2.5 Summary................................................. 25
3. Vector algebras 27
3.1 Constructing algebras from a vector space................ 27
3.1.1 The tensor algebra ............................... 27
3.1.2 The exterior algebra ............................. 28
3.1.3 Combinatorial notations .......................... 30
3.1.4 The Hodge star ................................... 32
3.1.5 Graded algebras................................. 34
3.1.6 Clifford algebras............................... 34
3.1.7 Geometric algebra .............................. 36
3.2 Tensor algebras on the dual space ...................... 38
3.2.1 The structure of the dual space .............. 38
3.2.2 Tensors........................................... 40
3.2.3 Tensors as multilinear mappings................... 40
3.2.4 Abstract index notation .......................... 41
3.2.5 Tensors as multi-dimensional arrays............... 43
3.3 Exterior forms ........................................... 44
3.3.1 Exterior forms as multilinear mappings............ 44
3.3.2 Exterior forms as completely anti-symmetric ten-
sors ..................................................... 45
3.3.3 Exterior forms as anti-symmetric arrays .......... 46
3.3.4 The Clifford algebra of the dual space ........... 46
3.3.5 Algebra-valued exterior forms .............. 47
3.3.6 Related constructions and facts .............. 49
4. Topological spaces 51
4.1 Generalizing surfaces .................................... 51
4.1.1 Spaces ......................................... 52
4.1.2 Generalizing dimension............................ 52
4.1.3 Generalizing tangent vectors...................... 53
4.1.4 Existence and uniqueness of additional structure 53
4.1.5 Summary .......................................... 54
4.2 Generalizing shapes ...................................... 55
4.2.1 Defining spaces .................................. 56
4.2.2 Mapping spaces.................................... 57
4.3 Constructing spaces ...................................... 60
Contents
xv
4.3.1 Cell complexes ................................... 60
4.3.2 Projective spaces ............................. 61
4.3.3 Combining spaces............................... 62
4.3.4 Classifying spaces............................. 64
5. Algebraic topology 67
5.1 Constructing surfaces within a space................... 68
5.1.1 Simplices...................................... 68
5.1.2 Triangulations................................. 69
5.1.3 Orientability.................................. 70
5.1.4 Chain complexes ............................... 71
5.2 Counting holes that aren’t boundaries..................... 71
5.2.1 The homology groups............................ 71
5.2.2 Examples ......................................... 73
5.2.3 Calculating homology groups ...................... 75
5.2.4 Related constructions and facts .............. 75
5.3 Counting the ways a sphere maps to a space ............ 76
5.3.1 The fundamental group ............................ 77
5.3.2 The higher homotopy groups ....................... 79
5.3.3 Calculating the fundamental group ................ 80
5.3.4 Calculating the higher homotopy groups ........... 80
5.3.5 Related constructions and facts .............. 80
6. Manifolds 83
6.1 Defining coordinates and tangents ........................ 84
6.1.1 Coordinates ...................................... 84
6.1.2 Tangent vectors and differential forms ........... 85
6.1.3 Frames............................................ 89
6.1.4 Tangent vectors in terms of frames ............... 91
6.2 Mapping manifolds......................................... 92
6.2.1 Diffeomorphisms .................................. 92
6.2.2 The differential and pullback .................... 92
6.2.3 Immersions and embeddings......................... 94
6.2.4 Critical points................................... 95
6.3 Derivatives on manifolds ................................. 96
6.3.1 Derivations ..................................... 96
6.3.2 The Lie derivative of a vector field ............. 97
6.3.3 The Lie derivative of an exterior form ........... 99
XVI
Contents
6.3.4 The exterior derivative of a 1-form ............. 101
6.3.5 The exterior derivative of a k-form ............. 104
6.3.6 Relationships between derivations................ 106
6.4 Homology on manifolds ..................................... 107
6.4.1 The Poincare lemma .............................. 107
6.4.2 de Rham cohomology............................... 108
6.4.3 Poincare duality................................. 109
7. Lie groups 111
7.1 Combining algebra and geometry............................ Ill
7.1.1 Spaces with multiplication of points............. Ill
7.1.2 Vector spaces with topology ..................... 112
7.2 Lie groups and Lie algebras............................... 113
7.2.1 The Lie algebra of a Lie group................... 114
7.2.2 The Lie groups of a Lie algebra ................. 115
7.2.3 Relationships between Lie groups and Lie algebras 116
7.2.4 The universal cover of a Lie group .............. 117
7.3 Matrix groups ............................................ 119
7.3.1 Lie algebras of matrix groups ................... 119
7.3.2 Linear algebra................................... 120
7.3.3 Matrix groups with real entries ................. 122
7.3.4 Other matrix groups ............................. 123
7.3.5 Manifold properties of matrix groups........... 124
7.3.6 Matrix group terminology in physics ............. 126
7.4 Representations ..................................... 127
7.4.1 Group actions.................................. 128
7.4.2 Group and algebra representations ........ 130
7.4.3 Lie group and Lie algebra representations .... 131
7.4.4 Combining and decomposing representations . . . 132
7.4.5 Other representations.......................... 134
7.5 Classification of Lie groups .......................... 135
7.5.1 Compact Lie groups .............................. 136
7.5.2 Simple Lie algebras............................ 138
7.5.3 Classifying representations.................... 140
8. Clifford groups 141
8.1 Classification of Clifford algebras....................... 141
8.1.1 Isomorphisms .................................... 141
Contents xvii
8.1.2 Representations and spinors ...................... 143
8.1.3 Pauli and Dirac matrices.......................... 145
8.1.4 Chiral decomposition.............................. 148
8.2 Clifford groups and representations ...................... 149
8.2.1 Reflections....................................... 149
8.2.2 Rotations ........................................ 150
8.2.3 Lie group properties ............................. 152
8.2.4 Lorentz transformations .......................... 153
8.2.5 Representations in spacetime...................... 156
8.2.6 Spacetime and spinors in geometric algebra . . . 159
9. Riemannian manifolds 161
9.1 Introducing parallel transport of vectors............... 161
9.1.1 Change of frame .................................. 161
9.1.2 The parallel transporter ......................... 162
9.1.3 The covariant derivative.......................... 163
9.1.4 The connection ................................... 165
9.1.5 The covariant derivative in terms of the connection 166
9.1.6 The parallel transporter in terms of the connection 169
9.1.7 Geodesics and normal coordinates.................. 170
9.1.8 Summary .......................................... 172
9.2 Manifolds with connection ................................ 175
9.2.1 The covariant derivative on the tensor algebra . . 175
9.2.2 The exterior covariant derivative of vector-valued
forms............................................. 177
9.2.3 The exterior covariant derivative of algebra-valued
forms............................................. 179
9.2.4 Torsion........................................... 181
9.2.5 Curvature ........................................ 184
9.2.6 First Bianchi identity............................ 187
9.2.7 Second Bianchi identity .......................... 190
9.2.8 The holonomy group ............................... 193
9.3 Introducing lengths and angles............................ 194
9.3.1 The Riemannian metric............................. 194
9.3.2 The Levi-Civita connection ....................... 196
9.3.3 Independent quantities and dependencies........... 198
9.3.4 The divergence and conserved quantities........... 199
9.3.5 Ricci and sectional curvature..................... 203
9.3.6 Curvature and geodesics .......................... 206
xviii
Contents
9.3.7 Jacobi fields and volumes....................... 209
9.3.8 Summary.........................................212
9.3.9 Related constructions and facts ................ 215
10. Fiber bundles 217
10.1 Gauge theory...................................................... 217
10.1.1 Matter fields and gauges ....................... 217
10.1.2 The gauge potential and field strength.......... 218
10.1.3 Spinor fields................................... 219
10.2 Defining bundles.................................................. 222
10.2.1 Fiber bundles .................................. 222
10.2.2 G-bundles .................................................225
10.2.3 Principal bundles ........................................ 226
10.3 Generalizing tangent spaces....................................... 229
10.3.1 Associated bundles ....................................... 229
10.3.2 Vector bundles ............................................230
10.3.3 Frame bundles............................................. 233
10.3.4 Gauge transformations on frame bundles........... 237
10.3.5 Smooth bundles and jets................................... 241
10.3.6 Vertical tangents and horizontal equivariant forms 242
10.4 Generalizing connections.......................................... 246
10.4.1 Connections on bundles ................................... 246
10.4.2 Parallel transport on the frame bundle ................... 247
10.4.3 The exterior covariant derivative on bundles . . . 250
10.4.4 Curvature on principal bundles ............... 251
10.4.5 The tangent bundle and solder form........................ 252
10.4.6 Torsion on the tangent frame bundle....................... 256
10.4.7 Spinor bundles............................................ 257
10.5 Characterizing bundles............................................ 259
10.5.1 Universal bundles......................................... 259
10.5.2 Characteristic classes.................................... 262
10.5.3 Related constructions and facts ............... 263
Appendix A Categories and functors 265
A.l Generalizing sets and mappings ................................... 265
A.2 Mapping mappings.................................................. 266
Bibliography 269
Index 271
Mathematics for Physics
An Illustrated Handbook
This unique book complements traditional textbooks by providing a visual yet rigorous
survey of the mathematics used in theoretical physics beyond that typically covered in
undergraduate math and physics courses. The exposition is pedagogical but compact, and
the emphasis is on defining and visualizing concepts and relationships between them, as
well as listing common confusions, alternative notations and jargon, and relevant facts and
theorems. Special attention is given to detailed figures and geometric viewpoints, while
topics are avoided which are well covered in textbooks, such as historical motivations, proofs
and derivations, and tools for practical calculations. The primary physical models targeted
are general relativity, spinors, and gauge theories, with notable chapters on Riemannian
geometry, Clifford algebras, and fiber bundles.
World Scientific
www.worldscientific.com
10816 he
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| spelling | Marsh, Adam Verfasser aut Mathematics for physics an illustrated handbook Adam Marsh New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo World Scientific [2018] © 2018 xviii, 282 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Mathematik (DE-588)4037944-9 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Mathematik (DE-588)4037944-9 s Mathematische Physik (DE-588)4037952-8 s 2\p DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030255302&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030255302&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Marsh, Adam Mathematics for physics an illustrated handbook Mathematik (DE-588)4037944-9 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| subject_GND | (DE-588)4037944-9 (DE-588)4037952-8 (DE-588)4123623-3 |
| title | Mathematics for physics an illustrated handbook |
| title_auth | Mathematics for physics an illustrated handbook |
| title_exact_search | Mathematics for physics an illustrated handbook |
| title_full | Mathematics for physics an illustrated handbook Adam Marsh |
| title_fullStr | Mathematics for physics an illustrated handbook Adam Marsh |
| title_full_unstemmed | Mathematics for physics an illustrated handbook Adam Marsh |
| title_short | Mathematics for physics |
| title_sort | mathematics for physics an illustrated handbook |
| title_sub | an illustrated handbook |
| topic | Mathematik (DE-588)4037944-9 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| topic_facet | Mathematik Mathematische Physik Lehrbuch |
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