| Introduction A.The general problem of integration 1.The integral as a function of the domain 2.Polyhedral chains 3.Two continuity hypotheses 4.A further continuity hypothesis 5.Some examples 6.The case r=n 7.The r-vector of an oriented r-cell 8.On r-vectors and boundaries of (r+1)-cells 9.Grassmann algebra 10.The dual algebra 11.Integration of differential forms B.Some classical topics 12.Grassmann algebra in metric oriented n-space 13.The same, n-3 14.The differential of a mapping 15.Jacobians 16.Transformation of the integral 17.Smooth manifolds 18.Particular forms of integrals in 3-space 19.The Theorem of Stokes 20.The exterior differential 21.Some special formulas in metric oriented Ea 22.An existence theorem 23, De Rham's Theorem C.Indications of general theory 24.Normed spaces of chains and cochains 25.Continuous chains 26.On 0-dimensional integration PartⅠ:Classical theory Chapter Ⅰ.Grassmann algebra 1.Multivectors 2.Multicovectors 3.Properties of V[ir] and V[r] 4.Alternating r-linear functions 5.Use of coordinate systems 6.Exterior products 7.Interior products 8.n-vectors in n-space 9.Simple multiveetors 10.Linear mappings of vector spaces 11.Duality 12.Euclidean vector spaces 13.Mass and eomass 14.Mass and comass of products 15.On projections Chapter Ⅱ.Differential forms 1.The differential of a smooth mapping 2.Some properties of differentials 3.Differential forms 4.Smooth mappings 5.Use of coordinate systems 6.Jacobians 7.The inverse and implicit function theorems 8.The exterior differential 9.A representation of vectors and covectors 10.Smooth manifolds 11.The tangent space of a smooth manifold 12.Differential forms in smooth manifolds 13.A characterization of the exterior differential Chapter Ⅲ.Riemann integration theory 1.The r-vector of an oriented r-simplex 2.The r-vector of an r-chain 3.Integration over cellular chains 4.Some properties of integrals 5.Relation to the Riemann integral 6.Integration over open sets …… PartⅡ:General theory PartⅢ:Lebesgue theory Appendix Index of symbols Index of terms |
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