Basic concepts of synthetic differential geometry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer Acad. Publ.
1996
|
| Series: | Kluwer texts in the mathematical sciences
13 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 320 S. |
| ISBN: | 079233941X |
Staff View
MARC
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| 245 | 1 | 0 | |a Basic concepts of synthetic differential geometry |c by René Lavendhomme |
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Record in the Search Index
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|---|---|
| adam_text | Table of contents
Table of contents v
Introduction xi
1 Differential calculus and integrals 1
1.1 Introduction : Kock Lawvere Axiom 1
1.1.1 Introduction 1
1.1.2 Notes on logic 2
1.1.3 A few remarks on infinitesimals 4
1.1.4 Euclidean R modules 5
1.2 The rudimentary differential calculus 6
1.2.1 The notion of derivative 6
1.2.2 A Taylor formula 8
1.2.3 Functions of several variables 11
1.2.4 Application: a study of homogeneity 15
1.3 The rudimentary integral calculus 17
1.3.1 Preorder 17
1.3.2 Integral on the interval [0, 1] 19
1.3.3 Integral on an interval [a, b] 21
1.3.4 A few applications of the axiom of integration . . 24
1.4 Commented bibliography 30
2 Weil algebras and infinitesimal linearity 33
2.1 Weil algebras 33
2.1.1 Introduction 33
2.1.2 Weil algebras 34
2.1.3 The general Kock axiom 41
v
vi
2.2 The general microlinearity of R 43
2.2.1 The wonderful myopia of R 43
2.2.2 Quasi colimits of small objects 50
2.2.3 Perception by R of quasi colimits of small objects 55
2.3 General microlinearity 57
2.3.1 The notion of general microlinearity 57
2.3.2 A few elementary consequences of general micro¬
linearity 57
2.4 Commented bibliography 59
3 Tangency 61
3.1 Tangent bundle 61
3.1.1 The tangent module at a point 61
3.1.2 Vector bundle and tangent bundle 66
3.2 Vector fields 68
3.2.1 The fl modules of vector fields 68
3.2.2 Lie algebra of vector fields 71
3.3 Derivations 76
3.3.1 Directional derivatives 76
3.3.2 Reflexive objects 80
3.4 Micro squares 87
3.5 Commented bibliography 100
4 Differential forms 101
4.1 Differential forms with values in an i2 module 101
4.1.1 Singular definition 101
4.1.2 Classical differential forms 104
4.2 The exterior differential 108
4.2.1 Infinitesimal n chains 108
4.2.2 Integral of a n form on an infinitesimal n chain .110
4.2.3 The exterior differential 112
4.3 Integral of differential forms 117
4.3.1 Integral of a n form on an n interval 117
4.3.2 Stokes formula 120
4.4 The exterior algebra of differential forms 123
4.4.1 Multilinear forms 123
4.4.2 The exterior product 124
vii
4.4.3 Interior products. Lie derivatives 126
4.5 De Rham s Theorem 131
4.5.1 Integration as a cochain morphism 131
4.5.2 The differentiation morphism D 132
4.5.3 Complements 136
4.6 Commented bibliography 139
5 Connections 141
5.1 Connection, covariant derivative and spray 141
5.1.1 Introduction and definition 141
5.1.2 Covariant derivative 147
5.1.3 Sprays 151
5.2 Vertical and horizontal microsquares 158
5.2.1 Vertical horizontal decomposition 158
5.2.2 Connecting mappings or connection forms . . . .161
5.2.3 Parallel transport 164
5.3 Torsion and curvature 166
5.3.1 Differential forms with value in a tangent fibre
bundle 166
5.3.2 Torsion 169
5.3.3 Curvature 173
5.4 Commented bibliography 180
6 Global actions 181
6.1 Lie objects 181
6.1.1 Definition of Lie objects 181
6.1.2 Differential forms 183
6.1.3 Connections 191
6.1.4 Extensions of connections and actions 193
6.2 Curvature of a connection on E and torsion of a connec¬
tion on L 196
6.2.1 Curvature of a connection 196
6.2.2 Bianchi identity 200
6.2.3 Torsion of a connection on L 201
6.3 Weil s characteristic homomorphism 204
6.3.1 Algebraic preliminaries 204
6.3.2 A derivation formula 205
viii
6.3.3 The Weil homomorphism 209
6.4 Commented bibliography 210
7 On the algebra of the geometry of mechanics 211
7.1 Structured Lie objects 211
7.1.1 The Riemannian case 211
7.1.2 Pre symplectic and symplectic structures 216
7.1.3 Complex situations 226
7.1.4 Hermitian and Kaelherian objects 230
7.2 Lie algebras of Lie groups 233
7.2.1 Lie groups and Lie algebras 239
7.2.2 Elementar linear examples 241
7.2.3 Global examples of Lie groups and Lie algebras . 243
7.2.4 Actions of Lie groups 248
7.3 The cotangent bundle 254
7.3.1 Vector fields on a vector bundle 254
7.3.2 Symplectic structure on cotangent bundle .... 259
7.3.3 The case of FT 265
7.4 Commented bibliography 267
8 Note on toposes and models of S.D.G. 269
8.1 Multisorted language of higher order 269
8.1.1 The language 269
8.1.2 Terms and formulas 271
8.1.3 Prepositional calculus 272
8.1.4 Predicate calculus 276
8.1.5 Higher order theory 278
8.2 The concept of topos 279
8.2.1 (Insufficient) introduction to categories 279
8.2.2 Definition and examples of toposes 285
8.2.3 Toposes and set theoretical notions 294
8.2.4 Language and theory of a topos 297
8.3 Some models of S.D.G 302
8.3.1 The algebraic model 302
8.3.2 Varying sets on C^ algebras 304
8.3.3 A good model of S.D.G 308
8.4 Commented bibliography 310
ix
Bibliography 313
Index 317
|
| any_adam_object | 1 |
| author | Lavendhomme, René |
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| discipline | Mathematik |
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| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:58:53Z |
| institution | BVB |
| isbn | 079233941X |
| language | English |
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| physical | XV, 320 S. |
| publishDate | 1996 |
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| publisher | Kluwer Acad. Publ. |
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| series | Kluwer texts in the mathematical sciences |
| series2 | Kluwer texts in the mathematical sciences |
| spelling | Lavendhomme, René Verfasser aut Basic concepts of synthetic differential geometry by René Lavendhomme Dordrecht [u.a.] Kluwer Acad. Publ. 1996 XV, 320 S. txt rdacontent n rdamedia nc rdacarrier Kluwer texts in the mathematical sciences 13 Geometry, Differential Synthetische Differentialgeometrie (DE-588)4462361-6 gnd rswk-swf Synthetische Differentialgeometrie (DE-588)4462361-6 s DE-604 Kluwer texts in the mathematical sciences 13 (DE-604)BV005450041 13 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007205185&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lavendhomme, René Basic concepts of synthetic differential geometry Kluwer texts in the mathematical sciences Geometry, Differential Synthetische Differentialgeometrie (DE-588)4462361-6 gnd |
| subject_GND | (DE-588)4462361-6 |
| title | Basic concepts of synthetic differential geometry |
| title_auth | Basic concepts of synthetic differential geometry |
| title_exact_search | Basic concepts of synthetic differential geometry |
| title_full | Basic concepts of synthetic differential geometry by René Lavendhomme |
| title_fullStr | Basic concepts of synthetic differential geometry by René Lavendhomme |
| title_full_unstemmed | Basic concepts of synthetic differential geometry by René Lavendhomme |
| title_short | Basic concepts of synthetic differential geometry |
| title_sort | basic concepts of synthetic differential geometry |
| topic | Geometry, Differential Synthetische Differentialgeometrie (DE-588)4462361-6 gnd |
| topic_facet | Geometry, Differential Synthetische Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007205185&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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