Geometric Integration Theory

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ISBN: 9780691079721 | 0691079722
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Copyright: 12/8/2015

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Geared toward upper-level undergraduates and graduate students, this treatment of geometric integration theory consists of three parts: an introduction to classical theory, a postulational approach to general theory, and a final section that continues the general study with Lebesgue theory.

Preface	p. iii
Introduction	p. 3
The general problem of integration
The integral as a function of the domain	p. 3
Polyhedral chains	p. 5
Two continuity hypotheses	p. 5
A further continuity hypothesis	p. 6
Some examples	p. 7
The case r = n	p. 7
The r-vector of an oriented r-cell	p. 8
On r-vectors and boundaries of (r + 1)-cells	p. 9
Grassmann algebra	p. 10
The dual algebra	p. 11
Integration of differential forms	p. 13
Some classical topics
Grassmann algebra in metric oriented n-space	p. 13
The same, n = 3	p. 14
The differential of a mapping	p. 15
Jacobians	p. 16
Transformation of the integral	p. 17
Smooth manifolds	p. 18
Particular forms of integrals in 3-space	p. 19
The Theorem of Stokes	p. 21
The exterior differential	p. 21
Some special formulas in metric oriented E[superscript 3]	p. 23
An existence theorem	p. 24
De Rham's Theorem	p. 25
Indications of general theory
Normed spaces of chains and cochains	p. 27
Continuous chains	p. 28
On 0-dimensional integration	p. 30
Classical theory
Grassmann algebra
Multivectors	p. 35
Multicovectors	p. 37
Properties of V[subscript r] and V[superscript r]	p. 38
Alternating r-linear functions	p. 39
Use of coordinate systems	p. 40
Exterior products	p. 41
Interior products	p. 42
n-vectors in n-space	p. 44
Simple multivectors	p. 44
Linear mappings of vector spaces	p. 46
Duality	p. 47
Euclidean vector spaces	p. 48
Mass and comass	p. 52
Mass and comass of products	p. 55
On projections	p. 56
Differential forms
The differential of a smooth mapping	p. 58
Some properties of differentials	p. 60
Differential forms	p. 61
Smooth mappings	p. 62
Use of coordinate systems	p. 63
Jacobians	p. 66
The inverse and implicit function theorems	p. 68
The exterior differential	p. 70
A representation of vectors and covectors	p. 74
Smooth manifolds	p. 75
The tangent space of a smooth manifold	p. 75
Differential forms in smooth manifolds	p. 76
A characterization of the exterior differential	p. 77
Riemann integration theory
The r-vector of an oriented r-simplex	p. 80
The r-vector of an r-chain	p. 81
Integration over cellular chains	p. 82
Some properties of integrals	p. 83
Relation to the Riemann integral	p. 84
Integration over open sets	p. 85
The transformation formula	p. 87
Proof of the transformation formula	p. 89
Transformation of the Riemann integral	p. 92
Integration in manifolds	p. 92
Stokes' Theorem for a parallelepiped	p. 94
A special case of Stokes' Theorem	p. 96
Sets of zero s-extent	p. 97
Stokes' Theorem for standard domains	p. 99
Proof of the theorem	p. 101
Regular forms in Euclidean space	p. 103
Regular forms in smooth manifolds	p. 106
Stokes' Theorem for standard manifolds	p. 108
The iterated integral in Euclidean space	p. 110
Smooth manifolds
Manifolds in Euclidean space
The imbedding theorem	p. 113
The compact case	p. 113
Separation of subsets of E[superscript m]	p. 114
Regular approximations	p. 115
Proof of Theorem 1A, M compact	p. 115
Admissible coordinate systems in M	p. 116
Proof of Theorem 1A, M not compact	p. 117
Local properties of M in E[superscript m]	p. 117
On n-directions in E[superscript m]	p. 119
The neighborhood of M in E[superscript m]	p. 120
Projection along a plane	p. 123
Triangulation of manifolds
The triangulation theorem	p. 124
Outline of the proof	p. 124
Fullness	p. 125
Linear combinations of edge vectors of simplexes	p. 127
The quantities used in the proof	p. 128
The complex L	p. 128
The complex L*	p. 129
The intersections of M with L*	p. 130
The complex K	p. 131
Imbedding of simplexes in M	p. 132
The complexes K[subscript p]	p. 133
Proof of the theorem	p. 134
Cohomology in manifolds
[mu]-regular forms	p. 135
Closed forms in star shaped sets	p. 136
Extensions of forms	p. 137
Elementary forms	p. 138
Certain closed forms are derived	p. 141
Isomorphism of cohomology rings	p. 142
Periods of forms	p. 143
The Hopf invariant	p. 143
On smooth mappings of manifolds	p. 145
Other expressions for the Hopf invariant	p. 147
General theory
Abstract integration theory
Polyhedral chains	p. 152
Mass of polyhedral chains	p. 153
The flat norm	p. 154
Flat cochains	p. 156
Examples	p. 158
The sharp norm	p. 159
Sharp cochains	p. 160
Characterization of the norms	p. 163
An algebraic criterion for a multicovector	p. 165
Sharp r-forms	p. 167
Examples	p. 171
The semi-norms [vertical bar A vertical bar superscript flat], [vertical bar A vertical bar superscript sharp] are norms	p. 173
Weak convergence	p. 175
Some relations between the spaces of chains and cochains	p. 177
The [rho]-norms	p. 178
The mass of chains	p. 179
Separability of spaces of chains	p. 181
Non-separability of spaces of cochains	p. 182
Some relations between chains and functions
Continuous chains on the real line	p. 187
0-chains in E[superscript 1] defined by functions of bounded variation	p. 190
Sharp functions times 0-chains	p. 193
The part	p. 194
Functions of bounded variation in E[superscript 1] defined by 0-chains	p. 196
Some related analytical theorems	p. 198
Continuous r-chains in E[superscript n]	p. 199
On compact cochains	p. 202
The boundary of a smooth chain	p. 204
Continuous chains in smooth manifolds	p. 205
General properties of chains and cochains
Sharp functions times chains	p. 208
Sharp functions times cochains	p. 212
Supports of chains and cochains	p. 213
On non-compact chains	p. 217
On polyhedral approximations	p. 219
The r-vector of an r-chain	p. 220
Sharp chains at a point	p. 221
Molecular chains are dense	p. 223
Flat r-chains in E[superscript r-k] are zero	p. 224
Flat cochains in complexes	p. 225
Elementary flat cochains in a complex	p. 226
The isomorphism theorem	p. 229
Chains and cochains in open sets
Chains and cochains in open sets, elementary properties	p. 232
Chains and cochains in open sets, further properties	p. 236
Properties of mass	p. 241
On the open sets to which a chain belongs	p. 243
An expression for flat chains	p. 246
An expression for sharp chains	p. 248
Lebesgue theory
Flat cochains and differential forms
n-cochains in E[superscript n]	p. 255
Some properties of fullness	p. 256
Properties of projections	p. 257
Elementary properties of D[subscript X](p, [alpha])	p. 258
The r-form defined by a flat r-cochain	p. 260
Flat r-forms	p. 262
Flat r-forms and flat r-cochains	p. 263
Flat r-direction functions	p. 266
Flat forms defined by components	p. 268
Approximation to D[subscript X](p) by [characters not reproducible]	p. 270
Total differentiability of Lipschitz functions	p. 271
On the exterior differential of r-forms	p. 272
On averages of r-forms	p. 275
Products of cochains	p. 276
Lebesgue chains	p. 280
Products of cochains and chains	p. 281
Products and weak limits	p. 284
Characterization of the procducts	p. 286
Lipschitz mappings
Affine approximations to Lipschitz mappings	p. 289
The approximation on the edges of a simplex	p. 290
Approximation to the Jacobian	p. 292
The volume of affine approximations	p. 293
A continuity lemma	p. 295
Lipschitz chains	p. 296
Lipschitz mappings of open sets	p. 298
Lipschitz mappings and flat cochains	p. 302
Lipschitz mappings and flat forms	p. 302
Lipschitz mappings and sharp functions	p. 305
Lipschitz mappings and products	p. 306
On the flat norm of Lipschitz chains	p. 307
Deformations of chains	p. 308
Chains and additive set functions
On finite dimensional Banach spaces	p. 311
Vector valued additive set functions	p. 312
Vector valued integration	p. 314
Point functions times set functions	p. 316
Relations between a set function and its variation	p. 318
On positive linear functionals	p. 320
On bounded linear functionals	p. 322
Linear functions of sharp r-forms	p. 323
The sharp norm of r-vector valued set functions	p. 325
Molecular set functions	p. 325
Sharp chains and set functions	p. 326
Bounded Borel functions times chains	p. 328
The part of a chain in a Borel set	p. 329
Chains and point functions	p. 330
Characterization of the sharp norm	p. 331
Expression for the sharp norm	p. 333
Other expressions for the norm	p. 335
Vector and linear spaces
Vector spaces	p. 342
Linear transformations	p. 343
Conjugate spaces	p. 343
Direct sums, complements	p. 344
Quotient spaces	p. 345
Pairing of linear spaces	p. 345
Abstract homology	p. 346
Normed linear spaces	p. 346
Euclidean linear spaces	p. 348
Affine spaces	p. 349
Barycentric coordinates	p. 351
Affine mappings	p. 352
Euclidean spaces	p. 353
Banach spaces	p. 353
Semi-conjugate spaces	p. 354
Geometric and topological preliminaries
Cells, simplexes	p. 356
Polyhedra, complexes	p. 357
Su bdivisions	p. 357
Standard subdivisions	p. 358
Orientation	p. 360
Chains and cochains	p. 361
Boundary and coboundary	p. 362
Homology and cohomology	p. 362
Products in a complex	p. 363
Joins	p. 364
Subdivisions of chains	p. 364
Cartesian products of cells	p. 365
Mappings of complexes	p. 366
Some properties of planes	p. 367
Mappings of n-pseudomanifolds into n-space	p. 368
Distortion of triangulations of E[superscript m]	p. 370
Analytical preliminaries
Existence of certain functions	p. 372
Partitions of unity	p. 373
Smoothing functions by taking averages	p. 373
The Weierstrass approximation theorem	p. 375
Lebesgue theory	p. 376
The space L[superscript 1]	p. 378
Index of symbols	p. 379
Index of terms	p. 382
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