Integral Calculus

Summary The integral ∫ s ω was defined in Chapter 2 for a continuous k-form on n-space and for S a k-dimensional domain in n-space, parameterized on a bounded domain D of k-space. (In Chapter 2 only the cases n ≤ 3 were considered, but the same definitions apply, with only minor modifications, to cases where n > 3.) The definition had very serious defects: it was necessary to parameterize a domain in order to define integrals over it, and it was never proved that such a parameterization can be given nor that the resulting number ∫ s ω is independent of the choice of the parameterization. This chapter is devoted to the rigorous definition of ∫ s ω for a continuous k-form ω on n-space and for a suitably general class of k-dimensional domains S in n-spac—enamely, the class of compact, oriented, differentiable, k-dimensional manifolds-with-boundary. The subject of this chapter is merely the definition of ∫ s ω, an intuitive definition of which is very easily given in simple cases, and the proof that this definition has all the properties which would have been expected on the basis of the intuitive definition. Such definitions and proofs are of great theoretical importance, but have little practical significance. In practice, integrals ∫ s ω are rarely evaluated. The integrand (the field) and the concept of integration (which gives the field its meaning) are the important ideas, and the actual number ∫ s ω is usually of no interest. When it is actually necessary to perform an integration, the domain of integration S (as well as the integrand ω) must be relatively simple for there to be any hope of success—simple enough that the parameterization of S, and hence a definition of ∫ s ω, would present no problem.