Geometric measure theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Berlin [u.a.]
Springer
1996
|
| Edition: | Reprint of the 1969 ed. |
| Series: | Grundlehren der mathematischen Wissenschaften
153 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 676 S. |
| ISBN: | 3540606564 9783540606567 |
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| 100 | 1 | |a Federer, Herbert |d 1920-2010 |e Verfasser |0 (DE-588)104821169X |4 aut | |
| 245 | 1 | 0 | |a Geometric measure theory |c Herbert Federer |
| 250 | |a Reprint of the 1969 ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1996 | |
| 300 | |a XIV, 676 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Grundlehren der mathematischen Wissenschaften |v 153 | |
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Record in the Search Index
| _version_ | 1804124978110529537 |
|---|---|
| adam_text | Contents
Introduction
.................
CHAPTER ONE
Grassmann
algebra
1.1.
Tensor products
........................... 8
1.2.
Graded algebras
........................... 11
1.3.
The exterior algebra of a vectorspace
.................. 13
1.4.
Alternating forms and duality
..................... 16
1.5.
Interior multiplications
........................ 21
1.6.
Simple m-vectors
........................... 23
1.7.
Inner products
........................... 27
1.8.
Mass and comass
.......................... 38
1.9.
The symmetric algebra of a vectorspace
................. 41
1.10.
Symmetric forms and polynomial functions
............... 43
CHAPTER TWO
General measure theory
2.1.
Measures and measurable sets
..................... 51
2.1.1.
Numerical summation
.................. 51
2.1.2. —
3.
Measurable sets
..................... 53
2.1.4.-5.
Measure hulls
...................... 56
2.1.6.
Ulam
numbers
...................... 58
2.2.
Borei
and Suslin sets
......................... 59
2.2.1.
Borei
families
...................... 59
2.2.2. — 3.
Approximation by closed subsets
.............. 60
2.2.4.
Nonmeasurable sets
................... 62
2.2.5.
Radon measures
..................... 62
2.2.6.
The space of sequences of positive integers
.......... 63
2.2.7. — 9.
Lipschitzian maps
.................... 63
2.2.10.-13.
Suslinsets
........................ 65
2.2.14.-15.
Borei
and Baire functions
................. 70
2.2.16.
Separability of supports
.................. 71
2.2.17.
Images of Radon measures
................ 72
2.3.
Measurable functions
......................... 72
2.3.1.-2.
Basic properties
..................... 72
2.3.3. — 7.
Approximation theorems
................. 76
2.3.8. —10.
Spaces of measurable functions
............... 78
2.4.
Lebesgue integration
......................... 80
2.4.1.-5.
Basic properties
..................... 80
2.4.6.-9.
Limit theorems
..................... 84
2.4.10.—11.
Integrals over subsets
................... 85
2.4.12.-17.
Lebesgue spaces
..................... 86
2.4.18.
Compositions and image measures
............. 90
2.4.19.
Jensen s inequality
.................... 91
2.5.
Linear functionals
.......................... 91
2.5.1.
Lattices of functions
................... 91
2.5.2.-6.
Danieli
integrals
..................... 92
2.5.7.—12.
Linear functionals on Lebesgue spaces
........... 98
2.5.13,—15.
Riesz s representation theorem
............... 106
2.5.16.
Curve length
...................... 109
2.5.17.-18.
Riemann-Stieltjes integration
............... 110
2.5.19.
Spaces of
Danieli
integrals
................. 113
2.5.20.
Decomposition of
Danieli
integrals
............. 114
2.6.
Product measures
.......................... 114
2.6.1.-4.
Fubini s theorem
..................... 114
2.6.5.
Lebesgue measure
.................... 119
2.6.6.
Infinite cartesian products
................. 120
2.6.7.
Integration by parts
................... 121
2.7.
Invariant measures
.......................... 121
2.7.1.-3.
Definitions
....................... 121
2.7.4.— 13.
Existence and uniqueness of invariant integrals
........ 123
2.7.14.—15.
Covariant measures are Radon measures
.......... 131
2.7.16.
Examples
........................ 133
2.7.17.
Nonmeasurable sets
................... 141
2.7.18.
Ł,
continuity of group actions
............... 141
2.8.
Covering theorems
.......................... 141
2.8.1.-3.
Adequate families
.................... 141
2.8.4.-8.
Coverings with enlargement
................ 143
2.8.9.-15.
Centered ball coverings
.................. 145
2.8.16.-20.
Vitali
relations
...................... 151
2.9. Derivates.............................. 152
2.9.1.-5.
Existence of
dérivâtes
................... 152
2.9.6.-10.
Indefinite integrals
.................... 155
2.9.11.—13.
Density and approximate continuity
............ 158
2.9.14.—18.
Additional results on derivation using centered balls
..... 159
2.9.19. —25.
Derivatives of curves with finite length
........... 163
2.10.
Carathéodory s
construction
...................... 169
2.10.1.
The general construction
................. 169
2.10.2.-6.
The measures
ЈГ1,
<T,
¿T1 , á? 1, V , J?,
Ж?
......... 171
2.10.7.
Relation to Riemann-Stieltjes integration
.......... 174
2.10.8.-11.
Partitions and multiplicity integrals
............. 175
2.10.12.-14.
Curvelength
...................... 176
2.10.15.—16.
Integralgeometric measures
................ 178
2.10.17.-19.
Densities
........................ 179
2.10.20.
Remarks on approximating measures
............ 181
2.10.21. Spaces
of Lipschitzian
functions and closed subsets
...... 182
2.10.22, —23.
Approximating measures of increasing sequences
....... 184
2.10.24.
Direct construction of the upper integral
........... 186
2.10.25. —27.
Integrals of measures of
counterimages
........... 188
2.10.28.-29.
Sets of Cantor type
.................... 191
2.10.30.-31. Steiner symmetrization................... 195
2.10.32.-42.
Inequalities between basic measures
............. 196
2.10.43.-44.
Lipschitzian extension of functions
............. 201
2.10.45.-46.
Cartesian products
.................... 202
2.10.47.-48.
Subsets of finite Hausdorff measure
............. 204
CHAPTER THREE
Rectifiability
3.1.
Differentials and tangents
....................... 209
3.1.1.—10.
Differentiation and approximate differentiation
........ 209
3.1.11.
Higher differentials
.................... 218
3.1.12.-13.
Partitions of unity
.................... 223
3.1.14.-17.
Differentiable extension of functions
............ 225
3.1.18.
Factorization of maps near generic points
.......... 229
3.1.19.-20.
Submanifolds of Euclidean space
.............. 231
3.1.21.
Tangent vectors
..................... 233
3.1.22.
Relative differentiation
.................. 235
3.1.23.
Local flattening of a submanifold
.............. 236
3.1.24.
Analytic functions
.................... 237
3.2.
Area and
coarea
of Lipschitzian maps
.................. 241
3.2.1.
Jacobians
........................ 241
3.2.2. — 7.
Area of maps of Euclidean spaces
.............. 242
3.2.8.— 12.
Coarea
of maps of Euclidean spaces
............. 247
3.2.13.
Applications; Euier s function
Γ
.............. 250
3.2.14.-15.
Rectifiable sets
...................... 251
3.2.16.— 19.
Approximate tangent vectors and differentials
........ 252
3.2.20. —22.
Area and coatea of maps of rectifiable sets
.......... 256
3.2.23. —24.
Cartesian products
.................... 260
3.2.25. —26.
Equality of measures of rectifiable sets
............ 261
3.2.27.
Areas of projections of rectifiable sets
............ 262
3.2.28.
Examples
........................ 263
3.2.29.
Rectifiable sets and manifolds of class
1........... 267
3.2.30. — 33.
Further results on
coarea
................. 268
3.2.34. —40.
Steineťs
formula and Minkowski content
.......... 271
3.2.41.-44.
Brunn^Minkowski theorem
................ 277
3.2.45.
Relations between the measures
Ά™
............. 279
3.2.46.
Hausdorff measures in Riemannian manifolds
........ 280
3.2.47. —49.
Integralgeometry on spheres
................ 282
3.3.
Structure theory
........................... 287
3.3.1. —4.
Tangential properties of arbitrary Suslin sets
......... 287
3.3.5.-18.
Rectifiability and projections
............... 292
3.3.19.-21.
Examples of unrectifiable sets
............... 302
3.3.22.
Rectifiability and density
................. 309
3.4.
Some properties of highly differentiable functions
............. 310
3.4.1.-4.
Measures of/{x: dim
im
0/(x)áv} ............
310
3.4.5.-12.
Analytic varieties
.................... 318
CHAPTER FOUR
Homological integration theory
4.1.
Differential forms and currents
..................... 343
4.1.1.
Distributions
...................... 343
4.1.2. —
4.
Regularization
...................... 346
4.1.5.
Distributions representable by integration
.......... 349
4.1.6.
Differential forms and m-vectorfields
............ 351
4.1.7.
Currents
........................ 355
4.1.8.
Cartesian products
.................... 360
4.1.9.-10.
Homotopies
....................... 363
4.1.11.
Joins, oriented simplexes
................. 364
.12.-19.
Flat chains
....................... 367
.20.-21.
Relation to integralgeometry measure
............ 378
.22. - 23.
Polyhedral chains and flat approximation
.......... 379
.24.-28.
Rectifiable currents
.................... 380
.29.
Lipschitz neighborhood retracts
.............. 386
.30.
Transformation formula
.................. 387
.31.
Oriented submanifolds
.................. 389
.32.
Projective
maps and polyhedral chains
........... 392
4.1.33.
Duality formulae
..................... 394
4.1.34.
Lie product of vectorfields
................. 394
4.2.
Deformations and compactness
.................... 395
4.2.1.
Slicing normal currents by real valued functions
....... 395
4.2.2.
Maps with singularities
.................. 396
4.2.3.-6.
Cubical subdivisions
................... 398
4.2.7.-9.
Deformation theorem
................... 404
4.2.10.
Isoperimetric inequality
.................. 408
4.2.11,—14.
Flat chains and integralgeometric measure
.......... 408
4.2.15.-16.
Closure theorem
..................... 411
4.2.17.-18.
Compactness theorem
.................. 414
4.2.19.— 24.
Approximation by polyhedral chains
............ 415
4.2.25.
Indecomposable integral currents
.............. 420
4.2.26.
Flat chains modulo
v
................... 423
4.2.27.
Locally rectifiable currents
................ 432
4.2.28.-29.
Analytic chains
..................... 433
4.3.
Slicing
............................... 435
4.3.1.— 8.
Slicing flat chains by maps into R
............. 435
4.3.9. —12.
Homotopies, continuity of slices
.............. 445
4.3.13.
Slicing by maps into manifolds
............... 451
4.3.14.
Oriented cones
...................... 452
4.3.15.
Oriented cylinders
.................... 455
4.3.16.-19.
Oriented tangent cones
.................. 455
4.3.20.
Intersections of flat chains
................. 460
4.4.
Homology groups
........................... 463
4.4.1.
Homology theory with coefficient group
Z
.......... 463
4.4.2. — 3.
Isoperimetric inequalities
................. 466
4.4.4.
Compactness properties of homology classes
.........470
4.4.5.-6.
Homology theories with coefficient groups
R
and Zv
.....472
4.4.7.
Two simple examples
...................473
4.4.8.
Homotopy groups of cycle groups
.............474
4.4.9.
Cohomology groups
...................474
4.5.
Normal currents of dimension
π
in R
..................474
4.5.1.—4.
Sets with locally finite perimeter
..............474
4.5.5.
Exterior normals
.....................477
4.5.6.
Gauss-Green theorem
..................478
4.5.7.-10.
Functions corresponding to locally normal currents
......480
4.5.11.—12.
Densities and locally finite perimeter
............506
4.5.13.-17.
Examples and applications
................509
CHAPTER FIVE
Applications to the calculus of variations
5.1.
Integrands and minimizing currents
.........-......... 515
5.1.1.
Parametric integrands and integrals
............. 515
5.1.2.
Ellipticity of parametric integrands
............. 517
5.1.3.
Convexity, parametric Legendre condition
.......... 518
5.1.4.
Diffeomorphic
invariance
of ellipticity
............ 519
5.1.5.
Lowersemicontinuity of the integral
............. 519
5.1.6.
Minimizing currents
................... 521
5.1.7.-8.
Isotopie
deformations, variations
.............. 524
5.1.9.
Nonparatnetric integrands
................. 527
5.1.10.
Nonparametric Legendre condition
............. 529
5.1.11.
Euler-Lagrange formulae
................. 530
5.2.
Regularity of solutions of certain differential equations
.......... 532
5.2.1.-2.
L2 and Holder conditions
................. 532
5.2.3.
Strongly elliptic systems
.................. 534
5.2.4.
Sobolev s inequality
................... 537
5.2.5.-6.
Generalized harmonic functions
.............. 538
5.2.7.-10.
Convolutions with essentially homogeneous functions
..... 541
5.2.11.-13.
Elementary solutions
................... 547
5.2.14.
Holder estimate for linear systems
............. 552
5.2.15.-18.
Nonparametric variational problems
............ 554
5.2.19.
Maxima of real valued solutions
.............. 560
5.2.20.
One dimensional variational problems
........... 564
5.3.
Excess and smoothness
........................ 565
5.3.1. —6.
Estimates involving excess
................. 565
5.3.7.
A limiting process
.................... 581
5.3.8.-13.
The decrease of excess
.................. 585
5.3.14. —17.
Regularity of minimizing currents
............. 606
5.3.18.— 19.
Minimizing currents of dimension
m
in
R™*1
......... 613
5.3.20.
Minimizing currents of dimension
1
in R
.......... 617
5.3.21.
Minimizing flat chains modulo
ν
.............. 619
5.4.
Further results on area minimizing currents
.............. . 619
5.4.1.
Terminology
...................... 619
5.4.2.
Weak convergence of variation measures
.......... 620
5.4.3.-5.
Density
ratios and tangent
cones
.............. 621
5.4.6.-7.
Regularity of area minimizing currents
........... 628
5.4.8.-9.
Cartesian products
.................... 631
5.4.10.—14.
Study of cones by differential geometry
........... 632
5.4.15.-16.
Currents of dimension
m
in
R 1*1
.............. 644
5.4.17.
Lack of uniqueness and symmetry
............. 647
5.4.18.
Nonparametric surfaces, Bernstein s theorem
......... 649
5.4.19.
Holomorphic varieties
.................. 651
5.4.20.
Boundary regularity
................... 654
Bibliography
............................... 655
Glossary of some standard notations
.................... 669
List of basic notations defined in the text
................... 670
Index
.................................. 672
|
| any_adam_object | 1 |
| author | Federer, Herbert 1920-2010 |
| author_GND | (DE-588)104821169X |
| author_facet | Federer, Herbert 1920-2010 |
| author_role | aut |
| author_sort | Federer, Herbert 1920-2010 |
| author_variant | h f hf |
| building | Verbundindex |
| bvnumber | BV010511101 |
| classification_rvk | QH 150 SK 430 |
| classification_tum | MAT 280f MAT 580f |
| ctrlnum | (OCoLC)473420321 (DE-599)BVBBV010511101 |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | Reprint of the 1969 ed. |
| format | Book |
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| id | DE-604.BV010511101 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:54:15Z |
| institution | BVB |
| isbn | 3540606564 9783540606567 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007006333 |
| oclc_num | 473420321 |
| open_access_boolean | |
| owner | DE-91 DE-BY-TUM DE-20 DE-355 DE-BY-UBR DE-29T DE-739 DE-703 |
| owner_facet | DE-91 DE-BY-TUM DE-20 DE-355 DE-BY-UBR DE-29T DE-739 DE-703 |
| physical | XIV, 676 S. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | Springer |
| record_format | marc |
| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften Classics in mathematics |
| spelling | Federer, Herbert 1920-2010 Verfasser (DE-588)104821169X aut Geometric measure theory Herbert Federer Reprint of the 1969 ed. Berlin [u.a.] Springer 1996 XIV, 676 S. txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 153 Classics in mathematics Geometrie (DE-588)4020236-7 gnd rswk-swf Maßtheorie (DE-588)4074626-4 gnd rswk-swf Geometrische Maßtheorie (DE-588)4125258-5 gnd rswk-swf Geometrische Maßtheorie (DE-588)4125258-5 s Maßtheorie (DE-588)4074626-4 s Geometrie (DE-588)4020236-7 s 1\p DE-604 Erscheint auch als Online-Ausgabe 978-3-642-62010-2 Grundlehren der mathematischen Wissenschaften 153 (DE-604)BV000000395 153 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007006333&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 |
| spellingShingle | Federer, Herbert 1920-2010 Geometric measure theory Grundlehren der mathematischen Wissenschaften Geometrie (DE-588)4020236-7 gnd Maßtheorie (DE-588)4074626-4 gnd Geometrische Maßtheorie (DE-588)4125258-5 gnd |
| subject_GND | (DE-588)4020236-7 (DE-588)4074626-4 (DE-588)4125258-5 |
| title | Geometric measure theory |
| title_auth | Geometric measure theory |
| title_exact_search | Geometric measure theory |
| title_full | Geometric measure theory Herbert Federer |
| title_fullStr | Geometric measure theory Herbert Federer |
| title_full_unstemmed | Geometric measure theory Herbert Federer |
| title_short | Geometric measure theory |
| title_sort | geometric measure theory |
| topic | Geometrie (DE-588)4020236-7 gnd Maßtheorie (DE-588)4074626-4 gnd Geometrische Maßtheorie (DE-588)4125258-5 gnd |
| topic_facet | Geometrie Maßtheorie Geometrische Maßtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007006333&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT federerherbert geometricmeasuretheory |