| 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： |
図書
