Geometric integration theory:
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Prinhceton, NJ
Princeton University Press
1971
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| Ausgabe: | Forth printing |
| Schriftenreihe: | Princeton mathematical series
21 |
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| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 387 Seiten |
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Datensatz im Suchindex
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|---|---|
| adam_text | Table of Contents
Preface v
Introduction 3
A. The general problem of integration
1. The integral as a function of the domain 3
2. Polyhedral chains 5
3. Two continuity hypotheses 5
4. A further continuity hypothesis 6
5. Some examples 7
6. The case r = n 7
7. The r vector of an oriented r cell 8
8. On r vectors and boundaries of (r + l) cells .... 9
9. Grassmann algebra 10
10. The dual algebra 11
11. Integration of differential forms 13
B. Some classical topics
12. Grassmann algebra in metric oriented w space .... 13
13. The same, n = 3 14
14. The differential of a mapping 15
15. Jacobians 16
16. Transformation of the integral 17
17. Smooth manifolds 18
18. Particular forms of integrals in 3 space 19
19. The Theorem of Stokes 21
20. The exterior differential 21
21. Some special formulas in metric oriented E3 .... 23
22. An existence theorem 24
23. De Rham s Theorem 25
C. Indications of general theory
24. Normed spaces of chains and cochains 27
25. Continuous chains 28
26. On 0 dimensional integration 30
ix
x TABLE OF CONTENTS
Part I: Classical theory
Chapter I. Grassmann algebra
1. Multivectors 35
2. Multicovectors 37
3. Properties of V[r] and Fw 38
4. Alternating r linear functions 39
5. Use of coordinate systems 40
6. Exterior products 41
7. Interior products 42
8. n vectors in w space 44
9. Simple multivectors 44
10. Linear mappings of vector spaces 46
11. Duality 47
12. Euclidean vector spaces 48
13. Mass and comass 52
14. Mass and comass of products 55
15. On projections 56
Chapter II. Differential forms
1. The differential of a smooth mapping 58
2. Some properties of differentials 60
3. Differential forms 61
4. Smooth mappings 62
5. Use of coordinate systems 63
6. Jacobians 66
7. The inverse and implicit function theorems 68
8. The exterior differential 70
9. A representation of vectors and covectors 74
10. Smooth manifolds 75
11. The tangent space of a smooth manifold 75
12. Differential forms in smooth manifolds 76
13. A characterization of the exterior differential .... 77
Chapter III. Riemann integration theory
1. The r vector of an oriented r simplex 80
2. The r vector of an r chain 81
3. Integration over cellular chains 82
4. Some properties of integrals 83
5. Relation to the Riemann integral 84
6. Integration over open sets 85
TABLE OF CONTENTS xi
7. The transformation formula 87
8. Proof of the transformation formula 89
9. Transformation of the Riemann integral 92
10. Integration in manifolds 92
11. Stokes Theorem for a parallelepiped 94
12. A special case of Stokes Theorem 96
13. Sets of zero s extent 97
14. Stokes Theorem for standard domains 99
15. Proof of the theorem 101
16. Regular forms in Euclidean space 103
17. Regular forms in smooth manifolds 106
18. Stokes Theorem for standard manifolds 108
19. The iterated integral in Euclidean space 110
Chapter IV. Smooth manifolds
A. Manifolds in Euclidean space
1. The imbedding theorem 113
2. The compact case 113
3. Separation of subsets of Em 114
4. Regular approximations 115
5. Proof of Theorem 1A, M compact 115
6. Admissible coordinate systems in M 116
7. Proof of Theorem 1 A, M not compact 117
8. Local properties of M in Em 117
9. On w directions in Em 119
10. The neighborhood of M in Em 120
11. Projection along a plane 123
B. Triangulation of manifolds
12. The triangulation theorem 124
13. Outline of the proof 124
14. Fullness 125
15. Linear combinations of edge vectors of simplexes . . . 127
16. The quantities used in the proof 128
17. The complex L 128
18. The complex L* 129
19. The intersections of M with L* 130
20. The complex A 131
21. Imbedding of simplexes in M 132
22. The complexes Kp 133
23. Proof of the theorem 134
xii TABLE OF CONTENTS
C. Cohomology in manifolds
24. ^ regular forms 135
25. Closed forms in star shaped sets 136
26. Extensions of forms 137
27. Elementary forms 138
28. Certain closed forms are derived 141
29. Isomorphism of cohomology rings 142
30. Periods of forms 143
31. The Hopf invariant 143
32. On smooth mappings of manifolds 145
33. Other expressions for the Hopf invariant 147
Part II: General theory
Chapter V. Abstract integration theory
1. Polyhedral chains 152
2. Mass of polyhedral chains 153
3. The flat norm 154
4. Flat cochains 156
5. Examples 158
6. The sharp norm 159
7. Sharp cochains 160
8. Characterization of the norms 163
9. An algebraic criterion for a multicovector 165
10. Sharp r forms 167
11. Examples 171
12. The semi norms A ^ j^4]* are norms 173
13. Weak convergence 175
14. Some relations between the spaces of chains and cochains 177
15. The p norms 178
16. The mass of chains 179
17. Separability of spaces of chains 181
18. Non separability of spaces of cochains 182
Chapter VI. Some relations between chains and functions
1. Continuous chains on the real line 187
2. 0 chains in E1 defined by functions of bounded variation . 190
3. Sharp functions times 0 chains 193
4. The part T of a chain of finite mass 194
5. Functions of bounded variation in El defined by 0 chains 196
6. Some related analytical theorems 198
TABLE OF CONTENTS xiii
7. Continuous r chains in E 109
8. On compact cochains 202
9. The boundary of a smooth chain 204
10. Continuous chains in smooth manifolds 205
Chapter VII. General properties of chains and cochains
1. Sharp functions times chains 208
2. Sharp functions times cochains 212
3. Supports of chains and cochains 213
4. On non compact chains 217
5. On polyhedral approximations 219
6. The r vector of an r chain 220
7. Sharp chains at a point 221
8. Molecular chains are dense 223
9. Flat r chains in Er~k are zero 224
10. Flat cochains in complexes 225
11. Elementary flat cochains in a complex 226
12. The isomorphism theorem 229
Chapter VIII. Chains and cochains in open sets
1. Chains and cochains in open sets, elementary properties . 232
2. Chains and cochains in open sets, further properties . . 236
3. Properties of mass 241
4. On the open sets to which a chain belongs 243
5. An expression for flat chains 246
6. An expression for sharp chains 248
Part III: Lebesgue theory
Chapter IX. Flat cochains and differential forms
1. n cochains in E 255
2. Some properties of fullness 256
3. Properties of projections 257
4. Elementary properties of Dx(p, a) 258
5. The r form defined by a flat r cochain 260
6. Flat r forms 262
7. Flat r forms and flat r cochains 263
8. Flat r direction functions 266
9. Flat forms defined by components 268
10. Approximation to Dx(p) by X a/ or j 270
11. Total differentiability of Lipschitz functions . . . .271
12. On the exterior differential of r forms 272
xiv TABLE OF CONTENTS
13. On averages of r forms 275
14. Products of cochains 276
15. Lebesgue chains 280
16. Products of cochains and chains 281
17. Products and weak limits 284
18. Characterization of the products 286
Chapter X. Lipschitz mappings
1. Affine approximations to Lipschitz mappings .... 289
2. The approximation on the edges of a simplex .... 290
3. Approximation to the Jacobian 292
4. The volume of affine approximations 293
5. A continuity lemma 295
6. Lipschitz chains 296
7. Lipschitz mappings of open sets 298
8. Lipschitz mappings and flat cochains 302
9. Lipschitz mappings and flat forms 302
10. Lipschitz mappings and sharp functions 305
11. Lipschitz mappings and products 306
12. On the flat norm of Lipschitz chains 307
13. Deformations of chains 308
Chapter XL Chains and additive set functions
1. On finite dimensional Banach spaces 311
2. Vector valued additive set functions 312
3. Vector valued integration 314
4. Point functions times set functions 316
5. Relations between a set function and its variation . . . 318
6. On positive linear functionals 320
7. On bounded linear functionals 322
8. Linear functions of sharp r forms 323
9. The sharp norm of r vector valued set functions . . . 325
10. Molecular set functions 325
11. Sharp chains and set functions 326
12. Bounded Borel functions times chains 328
13. The part of a chain in a Borel set 329
14. Chains and point functions 330
15. Characterization of the sharp norm 331
16. Expression for the sharp norm 333
17. Other expressions for the norm 335
Appendix I. Vector and linear spaces
1. Vector spaces 342
TABLE OF CONTENTS xv
2. Linear transformations 343
3. Conjugate spaces 343
4. Direct sums, complements 344
5. Quotient spaces 345
6. Pairing of linear spaces 345
7. Abstract homology 346
8. Normed linear spaces 346
9. Euclidean linear spaces 348
10. Affine spaces 349
11. Barycentric coordinates 351
12. Affine mappings 352
13. Euclidean spaces 353
14. Banach spaces 353
15. Semi conjugate spaces 354
Appendix II. Geometric and topological preliminaries
1. Cells, simplexes 356
2. Polyhedra, complexes 357
3. Subdivisions 357
4. Standard subdivisions 358
5. Orientation 360
6. Chains and cochains 361
7. Boundary and coboundary 362
8. Homology and cohomology 362
9. Products in a complex 363
10. Joins 364
11. Subdivisions of chains 364
12. Cartesian products of cells 365
13. Mappings of complexes 366
14. Some properties of planes 367
15. Mappings of w pseudomanifolds into w space .... 368
16. Distortion of triangulations of Em 370
Appendix III. Analytical preliminaries
1. Existence of certain functions 372
2. Partitions of unity 373
3. Smoothing functions by taking averages 373
4. The Weierstrass approximation theorem 375
5. Lebesgue theory 376
6. The space L1 378
Index of symbols 379
Index of terms 383
|
| adam_txt |
Table of Contents
Preface v
Introduction 3
A. The general problem of integration
1. The integral as a function of the domain 3
2. Polyhedral chains 5
3. Two continuity hypotheses 5
4. A further continuity hypothesis 6
5. Some examples 7
6. The case r = n 7
7. The r vector of an oriented r cell 8
8. On r vectors and boundaries of (r + l) cells . 9
9. Grassmann algebra 10
10. The dual algebra 11
11. Integration of differential forms 13
B. Some classical topics
12. Grassmann algebra in metric oriented w space . 13
13. The same, n = 3 14
14. The differential of a mapping 15
15. Jacobians 16
16. Transformation of the integral 17
17. Smooth manifolds 18
18. Particular forms of integrals in 3 space 19
19. The Theorem of Stokes 21
20. The exterior differential 21
21. Some special formulas in metric oriented E3 . 23
22. An existence theorem 24
23. De Rham's Theorem 25
C. Indications of general theory
24. Normed spaces of chains and cochains 27
25. Continuous chains 28
26. On 0 dimensional integration 30
ix
x TABLE OF CONTENTS
Part I: Classical theory
Chapter I. Grassmann algebra
1. Multivectors 35
2. Multicovectors 37
3. Properties of V[r] and Fw 38
4. Alternating r linear functions 39
5. Use of coordinate systems 40
6. Exterior products 41
7. Interior products 42
8. n vectors in w space 44
9. Simple multivectors 44
10. Linear mappings of vector spaces 46
11. Duality 47
12. Euclidean vector spaces 48
13. Mass and comass 52
14. Mass and comass of products 55
15. On projections 56
Chapter II. Differential forms
1. The differential of a smooth mapping 58
2. Some properties of differentials 60
3. Differential forms 61
4. Smooth mappings 62
5. Use of coordinate systems 63
6. Jacobians 66
7. The inverse and implicit function theorems 68
8. The exterior differential 70
9. A representation of vectors and covectors 74
10. Smooth manifolds 75
11. The tangent space of a smooth manifold 75
12. Differential forms in smooth manifolds 76
13. A characterization of the exterior differential . 77
Chapter III. Riemann integration theory
1. The r vector of an oriented r simplex 80
2. The r vector of an r chain 81
3. Integration over cellular chains 82
4. Some properties of integrals 83
5. Relation to the Riemann integral 84
6. Integration over open sets 85
TABLE OF CONTENTS xi
7. The transformation formula 87
8. Proof of the transformation formula 89
9. Transformation of the Riemann integral 92
10. Integration in manifolds 92
11. Stokes' Theorem for a parallelepiped 94
12. A special case of Stokes' Theorem 96
13. Sets of zero s extent 97
14. Stokes' Theorem for standard domains 99
15. Proof of the theorem 101
16. Regular forms in Euclidean space 103
17. Regular forms in smooth manifolds 106
18. Stokes' Theorem for standard manifolds 108
19. The iterated integral in Euclidean space 110
Chapter IV. Smooth manifolds
A. Manifolds in Euclidean space
1. The imbedding theorem 113
2. The compact case 113
3. Separation of subsets of Em 114
4. Regular approximations 115
5. Proof of Theorem 1A, M compact 115
6. Admissible coordinate systems in M 116
7. Proof of Theorem 1 A, M not compact 117
8. Local properties of M in Em 117
9. On w directions in Em 119
10. The neighborhood of M in Em 120
11. Projection along a plane 123
B. Triangulation of manifolds
12. The triangulation theorem 124
13. Outline of the proof 124
14. Fullness 125
15. Linear combinations of edge vectors of simplexes . . . 127
16. The quantities used in the proof 128
17. The complex L 128
18. The complex L* 129
19. The intersections of M with L* 130
20. The complex A' 131
21. Imbedding of simplexes in M 132
22. The complexes Kp 133
23. Proof of the theorem 134
xii TABLE OF CONTENTS
C. Cohomology in manifolds
24. ^ regular forms 135
25. Closed forms in star shaped sets 136
26. Extensions of forms 137
27. Elementary forms 138
28. Certain closed forms are derived 141
29. Isomorphism of cohomology rings 142
30. Periods of forms 143
31. The Hopf invariant 143
32. On smooth mappings of manifolds 145
33. Other expressions for the Hopf invariant 147
Part II: General theory
Chapter V. Abstract integration theory
1. Polyhedral chains 152
2. Mass of polyhedral chains 153
3. The flat norm 154
4. Flat cochains 156
5. Examples 158
6. The sharp norm 159
7. Sharp cochains 160
8. Characterization of the norms 163
9. An algebraic criterion for a multicovector 165
10. Sharp r forms 167
11. Examples 171
12. The semi norms A ^ j^4]* are norms 173
13. Weak convergence 175
14. Some relations between the spaces of chains and cochains 177
15. The p norms 178
16. The mass of chains 179
17. Separability of spaces of chains 181
18. Non separability of spaces of cochains 182
Chapter VI. Some relations between chains and functions
1. Continuous chains on the real line 187
2. 0 chains in E1 defined by functions of bounded variation . 190
3. Sharp functions times 0 chains 193
4. The part T of a chain of finite mass 194
5. Functions of bounded variation in El defined by 0 chains 196
6. Some related analytical theorems 198
TABLE OF CONTENTS xiii
7. Continuous r chains in E" 109
8. On compact cochains 202
9. The boundary of a smooth chain 204
10. Continuous chains in smooth manifolds 205
Chapter VII. General properties of chains and cochains
1. Sharp functions times chains 208
2. Sharp functions times cochains 212
3. Supports of chains and cochains 213
4. On non compact chains 217
5. On polyhedral approximations 219
6. The r vector of an r chain 220
7. Sharp chains at a point 221
8. Molecular chains are dense 223
9. Flat r chains in Er~k are zero 224
10. Flat cochains in complexes 225
11. Elementary flat cochains in a complex 226
12. The isomorphism theorem 229
Chapter VIII. Chains and cochains in open sets
1. Chains and cochains in open sets, elementary properties . 232
2. Chains and cochains in open sets, further properties . . 236
3. Properties of mass 241
4. On the open sets to which a chain belongs 243
5. An expression for flat chains 246
6. An expression for sharp chains 248
Part III: Lebesgue theory
Chapter IX. Flat cochains and differential forms
1. n cochains in E" 255
2. Some properties of fullness 256
3. Properties of projections 257
4. Elementary properties of Dx(p, a) 258
5. The r form defined by a flat r cochain 260
6. Flat r forms 262
7. Flat r forms and flat r cochains 263
8. Flat r direction functions 266
9. Flat forms defined by components 268
10. Approximation to Dx(p) by X a/\ or j 270
11. Total differentiability of Lipschitz functions . . . .271
12. On the exterior differential of r forms 272
xiv TABLE OF CONTENTS
13. On averages of r forms 275
14. Products of cochains 276
15. Lebesgue chains 280
16. Products of cochains and chains 281
17. Products and weak limits 284
18. Characterization of the products 286
Chapter X. Lipschitz mappings
1. Affine approximations to Lipschitz mappings . 289
2. The approximation on the edges of a simplex . 290
3. Approximation to the Jacobian 292
4. The volume of affine approximations 293
5. A continuity lemma 295
6. Lipschitz chains 296
7. Lipschitz mappings of open sets 298
8. Lipschitz mappings and flat cochains 302
9. Lipschitz mappings and flat forms 302
10. Lipschitz mappings and sharp functions 305
11. Lipschitz mappings and products 306
12. On the flat norm of Lipschitz chains 307
13. Deformations of chains 308
Chapter XL Chains and additive set functions
1. On finite dimensional Banach spaces 311
2. Vector valued additive set functions 312
3. Vector valued integration 314
4. Point functions times set functions 316
5. Relations between a set function and its variation . . . 318
6. On positive linear functionals 320
7. On bounded linear functionals 322
8. Linear functions of sharp r forms 323
9. The sharp norm of r vector valued set functions . . . 325
10. Molecular set functions 325
11. Sharp chains and set functions 326
12. Bounded Borel functions times chains 328
13. The part of a chain in a Borel set 329
14. Chains and point functions 330
15. Characterization of the sharp norm 331
16. Expression for the sharp norm 333
17. Other expressions for the norm 335
Appendix I. Vector and linear spaces
1. Vector spaces 342
TABLE OF CONTENTS xv
2. Linear transformations 343
3. Conjugate spaces 343
4. Direct sums, complements 344
5. Quotient spaces 345
6. Pairing of linear spaces 345
7. Abstract homology 346
8. Normed linear spaces 346
9. Euclidean linear spaces 348
10. Affine spaces 349
11. Barycentric coordinates 351
12. Affine mappings 352
13. Euclidean spaces 353
14. Banach spaces 353
15. Semi conjugate spaces 354
Appendix II. Geometric and topological preliminaries
1. Cells, simplexes 356
2. Polyhedra, complexes 357
3. Subdivisions 357
4. Standard subdivisions 358
5. Orientation 360
6. Chains and cochains 361
7. Boundary and coboundary 362
8. Homology and cohomology 362
9. Products in a complex 363
10. Joins 364
11. Subdivisions of chains 364
12. Cartesian products of cells 365
13. Mappings of complexes 366
14. Some properties of planes 367
15. Mappings of w pseudomanifolds into w space . 368
16. Distortion of triangulations of Em 370
Appendix III. Analytical preliminaries
1. Existence of certain functions 372
2. Partitions of unity 373
3. Smoothing functions by taking averages 373
4. The Weierstrass approximation theorem 375
5. Lebesgue theory 376
6. The space L1 378
Index of symbols 379
Index of terms 383 |
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| author | Whitney, Hassler 1907-1989 |
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| author_facet | Whitney, Hassler 1907-1989 |
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| bvnumber | BV022179895 |
| classification_rvk | SK 430 |
| ctrlnum | (OCoLC)632015181 (DE-599)BVBBV022179895 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | Forth printing |
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| id | DE-604.BV022179895 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:20:27Z |
| indexdate | 2024-07-09T20:51:54Z |
| institution | BVB |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015394642 |
| oclc_num | 632015181 |
| open_access_boolean | |
| owner | DE-706 |
| owner_facet | DE-706 |
| physical | XV, 387 Seiten |
| publishDate | 1971 |
| publishDateSearch | 1971 |
| publishDateSort | 1971 |
| publisher | Princeton University Press |
| record_format | marc |
| series | Princeton mathematical series |
| series2 | Princeton mathematical series |
| spelling | Whitney, Hassler 1907-1989 (DE-588)11904045X aut Geometric integration theory Hassler Whitney Forth printing Prinhceton, NJ Princeton University Press 1971 XV, 387 Seiten txt rdacontent n rdamedia nc rdacarrier Princeton mathematical series 21 Integrales Medida, Teoría de la Integration Mathematik (DE-588)4072852-3 gnd rswk-swf Integrationstheorie (DE-588)4138369-2 gnd rswk-swf Algebraische Topologie (DE-588)4120861-4 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Integrationstheorie (DE-588)4138369-2 s Geometrie (DE-588)4020236-7 s 1\p DE-604 Algebraische Topologie (DE-588)4120861-4 s Integration Mathematik (DE-588)4072852-3 s 2\p DE-604 Princeton mathematical series 21 (DE-604)BV000019035 21 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015394642&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 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 | Whitney, Hassler 1907-1989 Geometric integration theory Princeton mathematical series Integrales Medida, Teoría de la Integration Mathematik (DE-588)4072852-3 gnd Integrationstheorie (DE-588)4138369-2 gnd Algebraische Topologie (DE-588)4120861-4 gnd Geometrie (DE-588)4020236-7 gnd |
| subject_GND | (DE-588)4072852-3 (DE-588)4138369-2 (DE-588)4120861-4 (DE-588)4020236-7 |
| title | Geometric integration theory |
| title_auth | Geometric integration theory |
| title_exact_search | Geometric integration theory |
| title_exact_search_txtP | Geometric integration theory |
| title_full | Geometric integration theory Hassler Whitney |
| title_fullStr | Geometric integration theory Hassler Whitney |
| title_full_unstemmed | Geometric integration theory Hassler Whitney |
| title_short | Geometric integration theory |
| title_sort | geometric integration theory |
| topic | Integrales Medida, Teoría de la Integration Mathematik (DE-588)4072852-3 gnd Integrationstheorie (DE-588)4138369-2 gnd Algebraische Topologie (DE-588)4120861-4 gnd Geometrie (DE-588)4020236-7 gnd |
| topic_facet | Integrales Medida, Teoría de la Integration Mathematik Integrationstheorie Algebraische Topologie Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015394642&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000019035 |
| work_keys_str_mv | AT whitneyhassler geometricintegrationtheory |