Geometric integration theory:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
2008
|
| Series: | Cornerstones
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 339 S. Ill., graph. Darst. 235 mm x 155 mm |
| ISBN: | 9780817646769 0817646760 9780817646790 |
Staff View
MARC
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| 100 | 1 | |a Krantz, Steven G. |d 1951- |0 (DE-588)130535907 |4 aut | |
| 245 | 1 | 0 | |a Geometric integration theory |c Steven G. Krantz ; Harold R. Parks |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2008 | |
| 300 | |a XIII, 339 S. |b Ill., graph. Darst. |c 235 mm x 155 mm | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 0 | |a Cornerstones | |
| 650 | 4 | |a Currents (Calculus of variations) | |
| 650 | 4 | |a Geometric measure theory | |
| 650 | 0 | 7 | |a Geometrische Maßtheorie |0 (DE-588)4125258-5 |2 gnd |9 rswk-swf |
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Record in the Search Index
| _version_ | 1804137512008941568 |
|---|---|
| adam_text | Contents
Preface
......................................................... xi
1 Basics...................................................... 1
1.1
Smooth Functions
.......................................... 1
1.2
Measures
................................................. 6
1.2.1
Lebesgue Measure
................................... 10
1.3
Integration
................................................ 12
1.3.1
Measurable Functions
................................ 12
1.3.2
The Integral
......................................... 14
1.3.3
Lebesgue Spaces
..................................... 20
1.3.4
Product Measures and the Fubini-Tonelli Theorem
........ 21
1.4
The Exterior Algebra
........................................ 22
1.5
The Generalized Pythagorean Theorem
........................ 25
1.6
The Hausdorff Distance and
Steiner
Symmetrization
............. 33
1.7
Borei
and Suslin Sets
....................................... 42
2
Carathéodory s
Construction and Lower-Dimensional Measures
.... 53
2.1
The Basic Definition
........................................ 53
2.1.1
Hausdorff Measure and Spherical Measure
............... 55
2.1.2
A Measure Based on Parallelepipeds
.................... 56
2.1.3
Projections and Convexity
............................. 57
2.1.4
Other Geometric Measures
............................ 58
2.1.5
Summary
........................................... 59
2.2
The Densities of a Measure
.................................. 61
2.3
A One-Dimensional Example
................................. 63
2.4
Carathéodory s
Construction and Mappings
..................... 64
2.5
The Concept of Hausdorff Dimension
......................... 67
2.6
Some Cantor Set Examples
.................................. 69
2.6.1
Basic Examples
...................................... 70
2.6.2
Some Generalized Cantor Sets
......................... 72
2.6.3
Cantor Sets in Higher Dimensions
...................... 73
Contents
Invariant
Measures and the Construction of
Haar
Measure
......... 77
3.1
The Fundamental Theorem
................................... 78
3.2 Haar
Measure for the Orthogonal Group and the Grassmannian
---- 84
3.2.1
Remarks on the Manifold Structure of G(N, M)
.......... 88
Covering Theorems and the Differentiation of Integrals
............ 91
4.1
Wiener s Covering Lemma and Its Variants
..................... 91
4.2
The Besicovitch Covering Theorem
........................... 99
4.3
Decomposition and Differentiation of Measures
................. 109
4.4
The Riesz Representation Theorem
............................ 115
4.5
Maximal Functions Redux
................................... 122
Analytical Tools: The Area Formula, the
Coarea
Formula, and
Poincaré
Inequalities
......................................... 125
5.1
The Area Formula
.......................................... 125
5.1.1
Linear Maps
........................................ 126
5.1.2
C1 Functions
........................................ 132
5.1.3
Rademacher s Theorem
............................... 134
5.2
The
Coarea
Formula
........................................ 137
5.2.1
Measure Theory of Lipschitz Maps
..................... 140
5.2.2
Proof of the
Coarea
Formula
........................... 142
5.3
The Area and
Coarea
Formulas for
С
Submanifolds
............. 143
5.4
Rectifiable Sets
............................................ 148
5.5
Poincaré
Inequalities
........................................ 151
The Calculus of Differential Forms and Stokes s Theorem
..........159
6.1
Differential Forms and Exterior Differentiation
.................. 159
6.2
Stokes s Theorem
.......................................... 164
Introduction to Currents
...................................... 173
7.1
A Few Words about Distributions
............................. 174
7.2
The Definition of a Current
.................................. 177
7.3
Constructions Using Currents and the Constancy Theorem
........ 183
7.4
Further Constructions with Currents
........................... 189
7.4.1
Products of Currents
.................................. 189
7.4.2
The Pushforward
..................................... 190
7.4.3
The Homotopy Formula
............................... 193
7.4.4
Applications of the Homotopy Formula
.................. 193
7.5
Rectifiable Currents with Integer Multiplicity
................... 195
7.6
Slicing
................................................... 204
7.7
The Deformation Theorem
................................... 211
7.8
Proof of the Unsealed Deformation Theorem
................... 217
7.9
Applications of the Deformation Theorem
...................... 222
Contents ix
8
Currents and the Calculus of Variations
.........................225
8.1
Proof of the Compactness Theorem
........................... 225
8.1.1
Integer-Multiplicity O-Currents
......................... 226
8.1.2
A Rectifiability Criterion for Currents
................... 231
8.1.3
MBVFunctions
..................................... 232
8.1.4
The Slicing Lemma
.................................. 237
8.1.5
The Density Lemma
.................................. 238
8.1.6
Completion of the Proof of the Compactness Theorem
..... 240
8.2
The Flat Metric
............................................ 241
8.3
Existence of Currents Minimizing Variational Integrals
........... 244
8.3.1
Minimizing Mass
.................................... 244
8.3.2
Other Integrands and Integrals
......................... 245
8.4
Density Estimates for Minimizing Currents
..................... 250
9
Regularity of Mass-Minimizing Currents
........................255
9.1
Preliminaries
.............................................. 256
9.2
The Height Bound and Lipschitz Approximation
................ 262
9.3
Currents Defined by Integrating over Graphs
.................... 269
9.4
Estimates for Harmonic Functions
............................ 272
9.5
The Main Estimate
......................................... 286
9.6
The Regularity Theorem
..................................... 303
9.7
Epilogue
.................................................. 308
Appendix
.......................................................311
A.
1
Transfinite
Induction
........................................ 311
A.2 Dual Spaces
............................................... 313
A.3 Line Integrals
.............................................. 316
A.4 Pullbacks and Exterior Derivatives
............................ 319
References
......................................................323
Index of Notation
................................................329
Index
...........................................................335
|
| adam_txt |
Contents
Preface
. xi
1 Basics. 1
1.1
Smooth Functions
. 1
1.2
Measures
. 6
1.2.1
Lebesgue Measure
. 10
1.3
Integration
. 12
1.3.1
Measurable Functions
. 12
1.3.2
The Integral
. 14
1.3.3
Lebesgue Spaces
. 20
1.3.4
Product Measures and the Fubini-Tonelli Theorem
. 21
1.4
The Exterior Algebra
. 22
1.5
The Generalized Pythagorean Theorem
. 25
1.6
The Hausdorff Distance and
Steiner
Symmetrization
. 33
1.7
Borei
and Suslin Sets
. 42
2
Carathéodory's
Construction and Lower-Dimensional Measures
. 53
2.1
The Basic Definition
. 53
2.1.1
Hausdorff Measure and Spherical Measure
. 55
2.1.2
A Measure Based on Parallelepipeds
. 56
2.1.3
Projections and Convexity
. 57
2.1.4
Other Geometric Measures
. 58
2.1.5
Summary
. 59
2.2
The Densities of a Measure
. 61
2.3
A One-Dimensional Example
. 63
2.4
Carathéodory's
Construction and Mappings
. 64
2.5
The Concept of Hausdorff Dimension
. 67
2.6
Some Cantor Set Examples
. 69
2.6.1
Basic Examples
. 70
2.6.2
Some Generalized Cantor Sets
. 72
2.6.3
Cantor Sets in Higher Dimensions
. 73
Contents
Invariant
Measures and the Construction of
Haar
Measure
. 77
3.1
The Fundamental Theorem
. 78
3.2 Haar
Measure for the Orthogonal Group and the Grassmannian
---- 84
3.2.1
Remarks on the Manifold Structure of G(N, M)
. 88
Covering Theorems and the Differentiation of Integrals
. 91
4.1
Wiener's Covering Lemma and Its Variants
. 91
4.2
The Besicovitch Covering Theorem
. 99
4.3
Decomposition and Differentiation of Measures
. 109
4.4
The Riesz Representation Theorem
. 115
4.5
Maximal Functions Redux
. 122
Analytical Tools: The Area Formula, the
Coarea
Formula, and
Poincaré
Inequalities
. 125
5.1
The Area Formula
. 125
5.1.1
Linear Maps
. 126
5.1.2
C1 Functions
. 132
5.1.3
Rademacher's Theorem
. 134
5.2
The
Coarea
Formula
. 137
5.2.1
Measure Theory of Lipschitz Maps
. 140
5.2.2
Proof of the
Coarea
Formula
. 142
5.3
The Area and
Coarea
Formulas for
С
'
Submanifolds
. 143
5.4
Rectifiable Sets
. 148
5.5
Poincaré
Inequalities
. 151
The Calculus of Differential Forms and Stokes's Theorem
.159
6.1
Differential Forms and Exterior Differentiation
. 159
6.2
Stokes's Theorem
. 164
Introduction to Currents
. 173
7.1
A Few Words about Distributions
. 174
7.2
The Definition of a Current
. 177
7.3
Constructions Using Currents and the Constancy Theorem
. 183
7.4
Further Constructions with Currents
. 189
7.4.1
Products of Currents
. 189
7.4.2
The Pushforward
. 190
7.4.3
The Homotopy Formula
. 193
7.4.4
Applications of the Homotopy Formula
. 193
7.5
Rectifiable Currents with Integer Multiplicity
. 195
7.6
Slicing
. 204
7.7
The Deformation Theorem
. 211
7.8
Proof of the Unsealed Deformation Theorem
. 217
7.9
Applications of the Deformation Theorem
. 222
Contents ix
8
Currents and the Calculus of Variations
.225
8.1
Proof of the Compactness Theorem
. 225
8.1.1
Integer-Multiplicity O-Currents
. 226
8.1.2
A Rectifiability Criterion for Currents
. 231
8.1.3
MBVFunctions
. 232
8.1.4
The Slicing Lemma
. 237
8.1.5
The Density Lemma
. 238
8.1.6
Completion of the Proof of the Compactness Theorem
. 240
8.2
The Flat Metric
. 241
8.3
Existence of Currents Minimizing Variational Integrals
. 244
8.3.1
Minimizing Mass
. 244
8.3.2
Other Integrands and Integrals
. 245
8.4
Density Estimates for Minimizing Currents
. 250
9
Regularity of Mass-Minimizing Currents
.255
9.1
Preliminaries
. 256
9.2
The Height Bound and Lipschitz Approximation
. 262
9.3
Currents Defined by Integrating over Graphs
. 269
9.4
Estimates for Harmonic Functions
. 272
9.5
The Main Estimate
. 286
9.6
The Regularity Theorem
. 303
9.7
Epilogue
. 308
Appendix
.311
A.
1
Transfinite
Induction
. 311
A.2 Dual Spaces
. 313
A.3 Line Integrals
. 316
A.4 Pullbacks and Exterior Derivatives
. 319
References
.323
Index of Notation
.329
Index
.335 |
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| id | DE-604.BV023223410 |
| illustrated | Illustrated |
| index_date | 2024-07-02T20:16:59Z |
| indexdate | 2024-07-09T21:13:28Z |
| institution | BVB |
| isbn | 9780817646769 0817646760 9780817646790 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016409267 |
| oclc_num | 166372340 |
| open_access_boolean | |
| owner | DE-824 DE-19 DE-BY-UBM DE-355 DE-BY-UBR DE-20 DE-703 DE-11 DE-188 DE-91G DE-BY-TUM DE-83 DE-29T |
| owner_facet | DE-824 DE-19 DE-BY-UBM DE-355 DE-BY-UBR DE-20 DE-703 DE-11 DE-188 DE-91G DE-BY-TUM DE-83 DE-29T |
| physical | XIII, 339 S. Ill., graph. Darst. 235 mm x 155 mm |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Birkhäuser |
| record_format | marc |
| series2 | Cornerstones |
| spelling | Krantz, Steven G. 1951- (DE-588)130535907 aut Geometric integration theory Steven G. Krantz ; Harold R. Parks Boston [u.a.] Birkhäuser 2008 XIII, 339 S. Ill., graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Cornerstones Currents (Calculus of variations) Geometric measure theory Geometrische Maßtheorie (DE-588)4125258-5 gnd rswk-swf Geometrische Maßtheorie (DE-588)4125258-5 s DE-604 Parks, Harold R. aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016409267&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Krantz, Steven G. 1951- Parks, Harold R. Geometric integration theory Currents (Calculus of variations) Geometric measure theory Geometrische Maßtheorie (DE-588)4125258-5 gnd |
| subject_GND | (DE-588)4125258-5 |
| 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 Steven G. Krantz ; Harold R. Parks |
| title_fullStr | Geometric integration theory Steven G. Krantz ; Harold R. Parks |
| title_full_unstemmed | Geometric integration theory Steven G. Krantz ; Harold R. Parks |
| title_short | Geometric integration theory |
| title_sort | geometric integration theory |
| topic | Currents (Calculus of variations) Geometric measure theory Geometrische Maßtheorie (DE-588)4125258-5 gnd |
| topic_facet | Currents (Calculus of variations) Geometric measure theory Geometrische Maßtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016409267&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT krantzsteveng geometricintegrationtheory AT parksharoldr geometricintegrationtheory |