Integration of Forms in Affine Spaces

As a first step towards a theory of integration of differential forms on manifolds, we will present the particular case of integration on certain subsets of an affine space (without necessarily having an inner-product structure). In Section 2.8.3 we introduced the rigorous concept of an affine simplex and, later, in Section 3.4, we developed the idea of the multivector uniquely associated to an oriented affine simplex. Moreover, we have already advanced, on physical grounds, the notion that the evaluation of an r-form on an r-vector conveys the meaning of the calculation of the physical content of the quantity represented by the form within the volume represented by the multivector. For this idea to be of any practical use, we should be able to pursue it to the infinitesimal limit. Namely, given an r-dimensional domain D in an affine space A and, given at each point of this domain a (continuously varying, say) r-form ω, we would like to be able to subdivide the domain into small r-simplexes and define the total content as the limit of the sum of the evaluations of the r-forms on a point of each of the simplexes. In this way, we would have a generalization of the concept of Riemann integral.

Simplicial Complexes

A domain of integration within an n-dimensional affine space A may be a rather general set.