A first divergence theorem

Our task in this appendix is to extend the fundamental theorem of the calculus to functions defined on subsets of ℝN. If the result is to have a modest generality and to be of some use, then this task cannot be short and easy, for several reasons. First, it is not obvious how to pass from the merely local description of ∂Ω in Chapter 0, (viii), to the evaluation of integrals over ∂Ω as a whole; Definition D.I and Lemma D.2 are preparations for this step. Second, we must attend to the smoothness both of the function being integrated and of the boundary ∂Ω. Third, conditions that allow a straightforward proof of the divergence theorem (such as those in Theorem D.3) are often too restrictive for applications. Although we take only two primitive steps towards relaxing the conditions in Theorem D.3, those steps require a certain length if they are to be elementary and transparent.

Notation Throughout this appendix, ℝN is to have dimension N ≥ 2. The notation of Cartesian products will be taken beyond the definition of A × B in Chapter 0, (i); for example, (–β, β)N–1 denotes the cube Q′(0,β) described rather fully in Definition D.I, and {a} × [O,a]N–1 denotes a face of the cube (0, a)N, more precisely, the intersection of the hyperplane {x ∈ ℝN | x1 = a} and the closed cube [0,a]N.