Manifolds, tensors, and forms: an introduction for mathematicians and physicists
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge, UK
Cambridge University Press
2014
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | xii, 329 Seiten Diagramme |
| ISBN: | 9781107042193 |
Internformat
MARC
| LEADER | 00000nam a2200000 c 4500 | ||
|---|---|---|---|
| 001 | BV041592707 | ||
| 003 | DE-604 | ||
| 005 | 20180507 | ||
| 007 | t | ||
| 008 | 140128s2014 |||| |||| 00||| eng d | ||
| 010 | |a 2013036056 | ||
| 020 | |a 9781107042193 |9 978-1-10-704219-3 | ||
| 035 | |a (OCoLC)869889631 | ||
| 035 | |a (DE-599)GBV76849253X | ||
| 040 | |a DE-604 |b ger |e rda | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-19 |a DE-11 |a DE-20 |a DE-91G |a DE-898 |a DE-188 |a DE-824 | ||
| 084 | |a SK 370 |0 (DE-625)143234: |2 rvk | ||
| 084 | |a MAT 530f |2 stub | ||
| 100 | 1 | |a Renteln, Paul |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Manifolds, tensors, and forms |b an introduction for mathematicians and physicists |c Paul Renteln, California State University San Bernardino and California Institute of Technology |
| 264 | 1 | |a Cambridge, UK |b Cambridge University Press |c 2014 | |
| 300 | |a xii, 329 Seiten |b Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 0 | 7 | |a Tensorrechnung |0 (DE-588)4192487-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Homologietheorie |0 (DE-588)4141714-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Differentialgeometrie |0 (DE-588)4012248-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Differentialgeometrie |0 (DE-588)4012248-7 |D s |
| 689 | 0 | 1 | |a Tensorrechnung |0 (DE-588)4192487-3 |D s |
| 689 | 0 | 2 | |a Homologietheorie |0 (DE-588)4141714-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027037739&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-027037739 | ||
Datensatz im Suchindex
| _version_ | 1804151786628448256 |
|---|---|
| adam_text | Titel: Manifolds, tensors, and forms
Autor: Renteln, Paul
Jahr: 2014
Contents
Preface page ix
1 Linear algebra 1
1.1 Vector spaces 1
1.2 Linear maps 3
1.3 Exact sequences 4
1.4 Quotient spaces 6
1.5 Matrix representations 7
1.6 The dual space 8
1.7 Change of basis 9
1.8 Upstairs or downstairs? 11
1.9 Inner product spaces 14
1.10 The Riesz lemma 19
1.11 Adjoint maps, transpose maps, and duality 20
Additional exercises 21
2 Multilinear algebra 30
2.1 The tensor product 30
2.2 General tensors 33
2.3 Change of basis 34
2.4 Tensors as multilinear maps 34
2.5 Symmetry types of tensors 35
2.6 Alternating tensors and the space / p V of p-vectors 38
2.7 The exterior algebra 41
2.8 The induced linear transformation / T 42
2.9 The Hodge dual 44
Additional exercises 49
vi Contents
3 Differentiation on manifolds 54
3.1 Basic topology* 54
3.2 Multivariable calculus facts 59
3.3 Coordinates 60
3.4 Differentiable manifolds 62
3.5 Smooth maps on manifolds 68
3.6 Immersions and embeddings 70
3.7 The tangent space 73
3.8 The cotangent space T*M 79
3.9 The cotangent space as jet space* 81
3.10 Tensor fields 83
3.11 Differential forms 87
3.12 The exterior derivative 89
3.13 The interior product 93
3.14 Pullback 95
3.15 Pushforward 97
3.16 Integral curves and the Lie derivative 100
Additional exercises 104
4 Homotopy and de Rham cohomology 116
4.1 Homotopy 117
4.2 The Poincaré lemma 120
4.3 de Rham cohomology 122
4.4 Diagram chasing* 125
4.5 The Mayer-Vietoris sequence* 128
Additional exercises 134
5 Elementary homology theory 139
5.1 Simplicial complexes 139
5.2 Homology 144
5.3 The Euler characteristic 149
Additional exercises 151
6 Integration on manifolds 158
6.1 Smooth singular homology 158
6.2 Integration on chains 159
6.3 Change of variables 160
6.4 Stokes theorem 163
6.5 de Rham s theorem 169
Additional exercises 174
Contents vii
7 Vector bundles 176
7.1 The definitions 176
7.2 Connections 181
7.3 Cartan s moving frames and connection forms 183
7.4 Curvature forms and the Bianchi identity 184
7.5 Change of basis 185
7.6 The curvature matrix and the curvature operator 186
Additional exercises 188
8 Geometric manifolds 193
8.1 Index gymnastics 194
8.2 The Levi-Civita connection 199
8.3 The Riemann curvature tensor 204
8.4 More curvature tensors 206
8.5 Flat manifolds 208
8.6 Parallel transport and geodesies 212
8.7 Jacobi fields and geodesic deviation 215
8.8 Holonomy 216
8.9 Hodge theory 221
Additional exercises 225
9 The degree of a smooth map 249
9.1 The hairy ball theorem and the Hopf fibration 252
9.2 Linking numbers and magnetostatics 255
9.3 The Poincaré-Hopf index theorem and the Gauss-Bonnet
theorem 259
Appendix A Mathematical background 263
Appendix B The spectral theorem 271
Appendix C Orientations and top-dimensional forms 21A
Appendix D Riemann normal coordinates 276
Appendix E Holonomy of an infinitesimal loop 281
Appendix F Frobenius theorem 284
Appendix G The topology of electrical circuits 296
Appendix H Intrinsic and extrinsic curvature 308
References 317
Index 321
|
| any_adam_object | 1 |
| author | Renteln, Paul |
| author_facet | Renteln, Paul |
| author_role | aut |
| author_sort | Renteln, Paul |
| author_variant | p r pr |
| building | Verbundindex |
| bvnumber | BV041592707 |
| classification_rvk | SK 370 |
| classification_tum | MAT 530f |
| ctrlnum | (OCoLC)869889631 (DE-599)GBV76849253X |
| discipline | Mathematik |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01701nam a2200397 c 4500</leader><controlfield tag="001">BV041592707</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20180507 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">140128s2014 |||| |||| 00||| eng d</controlfield><datafield tag="010" ind1=" " ind2=" "><subfield code="a">2013036056</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9781107042193</subfield><subfield code="9">978-1-10-704219-3</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)869889631</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)GBV76849253X</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rda</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-19</subfield><subfield code="a">DE-11</subfield><subfield code="a">DE-20</subfield><subfield code="a">DE-91G</subfield><subfield code="a">DE-898</subfield><subfield code="a">DE-188</subfield><subfield code="a">DE-824</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 370</subfield><subfield code="0">(DE-625)143234:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">MAT 530f</subfield><subfield code="2">stub</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Renteln, Paul</subfield><subfield code="e">Verfasser</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Manifolds, tensors, and forms</subfield><subfield code="b">an introduction for mathematicians and physicists</subfield><subfield code="c">Paul Renteln, California State University San Bernardino and California Institute of Technology</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Cambridge, UK</subfield><subfield code="b">Cambridge University Press</subfield><subfield code="c">2014</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">xii, 329 Seiten</subfield><subfield code="b">Diagramme</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">Includes bibliographical references and index</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Tensorrechnung</subfield><subfield code="0">(DE-588)4192487-3</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Homologietheorie</subfield><subfield code="0">(DE-588)4141714-8</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Differentialgeometrie</subfield><subfield code="0">(DE-588)4012248-7</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Differentialgeometrie</subfield><subfield code="0">(DE-588)4012248-7</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2="1"><subfield code="a">Tensorrechnung</subfield><subfield code="0">(DE-588)4192487-3</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2="2"><subfield code="a">Homologietheorie</subfield><subfield code="0">(DE-588)4141714-8</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">HBZ Datenaustausch</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027037739&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-027037739</subfield></datafield></record></collection> |
| id | DE-604.BV041592707 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:00:21Z |
| institution | BVB |
| isbn | 9781107042193 |
| language | English |
| lccn | 2013036056 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027037739 |
| oclc_num | 869889631 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-11 DE-20 DE-91G DE-BY-TUM DE-898 DE-BY-UBR DE-188 DE-824 |
| owner_facet | DE-19 DE-BY-UBM DE-11 DE-20 DE-91G DE-BY-TUM DE-898 DE-BY-UBR DE-188 DE-824 |
| physical | xii, 329 Seiten Diagramme |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | Cambridge University Press |
| record_format | marc |
| spelling | Renteln, Paul Verfasser aut Manifolds, tensors, and forms an introduction for mathematicians and physicists Paul Renteln, California State University San Bernardino and California Institute of Technology Cambridge, UK Cambridge University Press 2014 xii, 329 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Tensorrechnung (DE-588)4192487-3 gnd rswk-swf Homologietheorie (DE-588)4141714-8 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 s Tensorrechnung (DE-588)4192487-3 s Homologietheorie (DE-588)4141714-8 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027037739&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Renteln, Paul Manifolds, tensors, and forms an introduction for mathematicians and physicists Tensorrechnung (DE-588)4192487-3 gnd Homologietheorie (DE-588)4141714-8 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4192487-3 (DE-588)4141714-8 (DE-588)4012248-7 |
| title | Manifolds, tensors, and forms an introduction for mathematicians and physicists |
| title_auth | Manifolds, tensors, and forms an introduction for mathematicians and physicists |
| title_exact_search | Manifolds, tensors, and forms an introduction for mathematicians and physicists |
| title_full | Manifolds, tensors, and forms an introduction for mathematicians and physicists Paul Renteln, California State University San Bernardino and California Institute of Technology |
| title_fullStr | Manifolds, tensors, and forms an introduction for mathematicians and physicists Paul Renteln, California State University San Bernardino and California Institute of Technology |
| title_full_unstemmed | Manifolds, tensors, and forms an introduction for mathematicians and physicists Paul Renteln, California State University San Bernardino and California Institute of Technology |
| title_short | Manifolds, tensors, and forms |
| title_sort | manifolds tensors and forms an introduction for mathematicians and physicists |
| title_sub | an introduction for mathematicians and physicists |
| topic | Tensorrechnung (DE-588)4192487-3 gnd Homologietheorie (DE-588)4141714-8 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Tensorrechnung Homologietheorie Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027037739&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT rentelnpaul manifoldstensorsandformsanintroductionformathematiciansandphysicists |