Geometric integration theory

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Geometric integration theory / by Hassler Whitney

資料種別:: 図書
出版情報:: Princeton, N.J. : Princeton University Press, 1957
形態:: xv, 387 p ; 24 cm
シリーズ名:: Princeton mathematical series ; 21 <BA00109433>
著者名:: Whitney, Hassler <DA01209699>
ISBN:: 9780691079721 [0691079722]
書誌ID:: BA00654929

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書誌詳細

主題:

数値積分学; 積分学

言語:

英語

目次情報:

Preface

Introduction

The general problem of integration / A：

The integral as a function of the domain / 1：

Polyhedral chains / 2：

Two continuity hypotheses / 3：

A further continuity hypothesis / 4：

Some examples / 5：

The case r = n / 6：

The r-vector of an oriented r-cell / 7：

On r-vectors and boundaries of (r + 1)-cells / 8：

Grassmann algebra / 9：

The dual algebra / 10：

Integration of differential forms / 11：

Some classical topics / B：

Grassmann algebra in metric oriented n-space / 12：

The same, n = 3 / 13：

The differential of a mapping / 14：

Jacobians / 15：

Transformation of the integral / 16：

Smooth manifolds / 17：

Particular forms of integrals in 3-space / 18：

The Theorem of Stokes / 19：

The exterior differential / 20：

Some special formulas in metric oriented E[superscript 3] / 21：

An existence theorem / 22：

De Rham's Theorem / 23：

Indications of general theory / C：

Normed spaces of chains and cochains / 24：

Continuous chains / 25：

On 0-dimensional integration / 26：

Classical theory / Part I：

Multivectors / Chapter I：

Multicovectors

Properties of V[subscript r] and V[superscript r]

Alternating r-linear functions

Use of coordinate systems

Exterior products

Interior products

n-vectors in n-space

Simple multivectors

Linear mappings of vector spaces

Duality

Euclidean vector spaces

Mass and comass

Mass and comass of products

On projections

Differential forms / Chapter II：

The differential of a smooth mapping

Some properties of differentials

Smooth mappings

The inverse and implicit function theorems

A representation of vectors and covectors

The tangent space of a smooth manifold

Differential forms in smooth manifolds

A characterization of the exterior differential

Riemann integration theory / Chapter III：

The r-vector of an oriented r-simplex

The r-vector of an r-chain

Integration over cellular chains

Some properties of integrals

Relation to the Riemann integral

Integration over open sets

The transformation formula

Proof of the transformation formula

Transformation of the Riemann integral

Integration in manifolds

Stokes' Theorem for a parallelepiped

A special case of Stokes' Theorem

Sets of zero s-extent

Stokes' Theorem for standard domains

Proof of the theorem

Regular forms in Euclidean space

Regular forms in smooth manifolds

Stokes' Theorem for standard manifolds

The iterated integral in Euclidean space

Manifolds in Euclidean space / Chapter IV：

The imbedding theorem

The compact case

Separation of subsets of E[superscript m]

Regular approximations

Proof of Theorem 1A, M compact

Admissible coordinate systems in M

Proof of Theorem 1A, M not compact

Local properties of M in E[superscript m]

On n-directions in E[superscript m]

The neighborhood of M in E[superscript m]

Projection along a plane

Triangulation of manifolds

The triangulation theorem

Outline of the proof

Fullness

Linear combinations of edge vectors of simplexes

The quantities used in the proof

The complex L

The complex L*

The intersections of M with L*

The complex K

Imbedding of simplexes in M

The complexes K[subscript p]

Cohomology in manifolds

[mu]-regular forms

Closed forms in star shaped sets

Extensions of forms

Elementary forms / 27：

Certain closed forms are derived / 28：

Isomorphism of cohomology rings / 29：

Periods of forms / 30：

The Hopf invariant / 31：

On smooth mappings of manifolds / 32：

Other expressions for the Hopf invariant / 33：

General theory / Part II：

Abstract integration theory / Chapter V：

Mass of polyhedral chains

The flat norm

Flat cochains

Examples

The sharp norm

Sharp cochains

Characterization of the norms

An algebraic criterion for a multicovector

Sharp r-forms

The semi-norms [vertical bar A vertical bar superscript flat], [vertical bar A vertical bar superscript sharp] are norms

Weak convergence

Some relations between the spaces of chains and cochains

The [rho]-norms

The mass of chains

Separability of spaces of chains

Non-separability of spaces of cochains

Some relations between chains and functions / Chapter VI：

Continuous chains on the real line

0-chains in E[superscript 1] defined by functions of bounded variation

Sharp functions times 0-chains

The part

Functions of bounded variation in E[superscript 1] defined by 0-chains

Some related analytical theorems

Continuous r-chains in E[superscript n]

On compact cochains

The boundary of a smooth chain

Continuous chains in smooth manifolds

General properties of chains and cochains / Chapter VII：

Sharp functions times chains

Sharp functions times cochains

Supports of chains and cochains

On non-compact chains

On polyhedral approximations

Sharp chains at a point

Molecular chains are dense

Flat r-chains in E[superscript r-k] are zero

Flat cochains in complexes

Elementary flat cochains in a complex

The isomorphism theorem

Chains and cochains in open sets / Chapter VIII：

Chains and cochains in open sets, elementary properties

Chains and cochains in open sets, further properties

Properties of mass

On the open sets to which a chain belongs

An expression for flat chains

An expression for sharp chains

Lebesgue theory / Part III：

Flat cochains and differential forms / Chapter IX：

n-cochains in E[superscript n]

Some properties of fullness

Properties of projections

Elementary properties of D[subscript X](p, [alpha])

The r-form defined by a flat r-cochain

Flat r-forms

Flat r-forms and flat r-cochains

Flat r-direction functions

Flat forms defined by components

Approximation to D[subscript X](p) by [characters not reproducible]

Total differentiability of Lipschitz functions

On the exterior differential of r-forms

On averages of r-forms

Products of cochains

Lebesgue chains

Products of cochains and chains

Products and weak limits

Characterization of the procducts

Lipschitz mappings / Chapter X：

Affine approximations to Lipschitz mappings

The approximation on the edges of a simplex

Approximation to the Jacobian

The volume of affine approximations

A continuity lemma

Lipschitz chains

Lipschitz mappings of open sets

Lipschitz mappings and flat cochains

Lipschitz mappings and flat forms

Lipschitz mappings and sharp functions

Lipschitz mappings and products

On the flat norm of Lipschitz chains

Deformations of chains

Chains and additive set functions / Chapter XI：

On finite dimensional Banach spaces

Vector valued additive set functions

Vector valued integration

Point functions times set functions

Relations between a set function and its variation

On positive linear functionals

On bounded linear functionals

Linear functions of sharp r-forms

The sharp norm of r-vector valued set functions

Molecular set functions

Sharp chains and set functions

Bounded Borel functions times chains

The part of a chain in a Borel set

Chains and point functions

Characterization of the sharp norm

Expression for the sharp norm

Other expressions for the norm

Vector and linear spaces / Appendix I：

Vector spaces

Linear transformations

Conjugate spaces

Direct sums, complements

Quotient spaces

Pairing of linear spaces

Abstract homology

Normed linear spaces

Euclidean linear spaces

Affine spaces

Barycentric coordinates

Affine mappings

Euclidean spaces

Banach spaces

Semi-conjugate spaces

Geometric and topological preliminaries / Appendix II：

Cells, simplexes

Polyhedra, complexes

Su bdivisions

Standard subdivisions

Orientation

Chains and cochains

Boundary and coboundary

Homology and cohomology

Products in a complex

Joins

Subdivisions of chains

Cartesian products of cells

Mappings of complexes

Some properties of planes

Mappings of n-pseudomanifolds into n-space

Distortion of triangulations of E[superscript m]

Analytical preliminaries / Appendix III：

Existence of certain functions

Partitions of unity

Smoothing functions by taking averages

The Weierstrass approximation theorem

The space L[superscript 1]

Index of symbols

Index of terms

Preface

Introduction

The general problem of integration / A：

The integral as a function of the domain / 1：

Polyhedral chains / 2：

Two continuity hypotheses / 3：

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