Differential forms and connections:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
1999
|
| Ausgabe: | Reprint. |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008999740&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | X, 256 S. graph. Darst. |
| ISBN: | 0521462592 0521468000 |
Internformat
MARC
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| 100 | 1 | |a Darling, R. W. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Differential forms and connections |c R. W. R. Darling |
| 250 | |a Reprint. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 1999 | |
| 300 | |a X, 256 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Differentialgeometrie - Differentialform - Konnexion | |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-008999740 | |
Datensatz im Suchindex
| DE-BY-TUM_call_number | 0202 MAT 530f 2002 A 689 |
|---|---|
| DE-BY-TUM_katkey | 1207659 |
| DE-BY-TUM_location | 02 |
| DE-BY-TUM_media_number | 040020605318 |
| _version_ | 1821931778594570240 |
| adam_text | Contents
Preface ix
1 Exterior Algebra 1
1.1 Exterior Powers of a Vector Space 1
1.2 Multilinear Alternating Maps and Exterior Products 5
1.3 Exercises 7
1.4 Exterior Powers of a Linear Transformation 8
1.5 Exercises 12
1.6 Inner Products 13
1.7 The Hodge Star Operator 17
1.8 Exercises 20
1.9 Some Formal Algebraic Constructions 21
1.10 History and Bibliography 23
2 Exterior Calculus on Euclidean Space 24
2.1 Tangent Spaces - the Euclidean Case 24
2.2 Differential Forms on a Euclidean Space 28
2.3 Operations on Differential Forms 31
2.4 Exercises 33
2.5 Exterior Derivative 35
2.6 Exercises 39
2.7 The Differential of a Map 41
2.8 The Pullback of a Differential Form 43
vi Contents
2.9 Exercises 47
2.10 History and Bibliography 49
2.11 Appendix: Maxwell s Equations 50
3 Submanifolds of Euclidean Spaces 53
3.1 Immersions and Submersions 53
3.2 Definition and Examples of Submanifolds 55
3.3 Exercises 60
3.4 Parametrizations 61
3.5 Using the Implicit Function Theorem to Parametrize a Submanifold 64
3.6 Matrix Groups as Submanifolds 69
3.7 Groups of Complex Matrices 71
3.8 Exercises 72
3.9 Bibliography 75
4 Surface Theory Using Moving Frames 76
4.1 Moving Orthonormal Frames on Euclidean Space 76
4.2 The Structure Equations 78
4.3 Exercises 79
4.4 An Adapted Moving Orthonormal Frame on a Surface 81
4.5 The Area Form 85
4.6 Exercises 87
4.7 Curvature of a Surface 88
4.8 Explicit Calculation of Curvatures 91
4.9 Exercises 94
4.10 The Fundamental Forms: Exercises 95
4.11 History and Bibliography 97
5 Differential Manifolds 98
5.1 Definition of a Differential Manifold 98
5.2 Basic Topological Vocabulary 100
5.3 Differentiable Mappings between Manifolds 102
5.4 Exercises 104
5.5 Submanifolds 105
5.6 Embeddings 107
Contents vji
5.7 Constructing Submanifolds without Using Charts 110
5.8 Submanifolds-with-Boundary HI
5.9 Exercises 114
5.10 Appendix: Open Sets of a Submanifold 116
5.11 Appendix: Partitions ofUnity 117
5.12 History and B ibliography 119
6 Vector Bundles 120
6.1 Local Vector Bundles 120
6.2 Constructions with Local Vector Bundles 122
6.3 General Vector Bundles 125
6.4 Constructing a Vector Bundle from Transition Functions 130
6.5 Exercises 132
6.6 The Tangent Bundle of a Manifold 134
6.7 Exercises 139
6.8 History and Bibliography 141
6.9 Appendix: Constructing Vector Bundles 141
7 Frame Fields, Forms, and Metrics 144
7.1 Frame Fields for Vector Bundles 144
7.2 Tangent Vectors as Equivalence Classes of Curves 147
7.3 Exterior Calculus on Manifolds 148
7.4 Exercises 151
7.5 Indefinite Riemannian Metrics 152
7.6 Examples of Riemannian Manifolds 153
7.7 Orthonormal Frame Fields 156
7.8 An Isomorphism between the Tangent and Cotangent Bundles 160
7.9 Exercises 161
7.10 History and Bibliography 163
8 Integration on Oriented Manifolds 164
8.1 Volume Forms and Orientation 164
8.2 Criterion for Orientability in Terms of an Atlas 167
8.3 Orientation of Boundaries 169
8.4 Exercises 172
viii Contents
8.5 Integration of an n-Form over a Single Chart 174
8.6 Global Integration of n-Forms 178
8.7 The Canonical Volume Form for a Metric 181
8.8 Stokes s Theorem 183
8.9 The Exterior Derivative Stands Revealed 184
8.10 Exercises 187
8.11 History and Bibliography 189
8.12 Appendix: Proof of Stokes s Theorem 189
9 Connections on Vector Bundles 194
9.1 Koszul Connections 194
9.2 Connections via Vector-Bundle-valued Forms 197
9.3 Curvature of a Connection 202
9.4 Exercises 206
9.5 Torsion-free Connections 212
9.6 Metric Connections 216
9.7 Exercises 219
9.8 History and Bibliography 222
10 Applications to Gauge Field Theory 223
10.1 The Role of Connections in Field Theory 223
10.2 Geometric Formulation of Gauge Field Theory 225
10.3 Special Unitary Groups and Quaternions 231
10.4 Quaternion Line Bundles 233
10.5 Exercises 238
10.6 The Yang-Mills Equations 242
10.7 Self-duality 244
10.8 Instantons 247
10.9 Exercises 249
10.10 History and Bibliography 250
Bibliography 251
Index 253
|
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| id | DE-604.BV013209164 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T10:42:59Z |
| institution | BVB |
| isbn | 0521462592 0521468000 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008999740 |
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| physical | X, 256 S. graph. Darst. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| spellingShingle | Darling, R. W. Differential forms and connections Differentialgeometrie - Differentialform - Konnexion Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4012248-7 |
| title | Differential forms and connections |
| title_auth | Differential forms and connections |
| title_exact_search | Differential forms and connections |
| title_full | Differential forms and connections R. W. R. Darling |
| title_fullStr | Differential forms and connections R. W. R. Darling |
| title_full_unstemmed | Differential forms and connections R. W. R. Darling |
| title_short | Differential forms and connections |
| title_sort | differential forms and connections |
| topic | Differentialgeometrie - Differentialform - Konnexion Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Differentialgeometrie - Differentialform - Konnexion Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008999740&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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Teilbibliothek Physik
| Signatur: |
0202 MAT 530f 2002 A 689 Lageplan |
|---|---|
| Exemplar 1 | Ausleihbar Am Standort |