Two properties of Bochner integrals

Abstract

Two theorems for Lebesgue integrals, namely the Gauss-Green Theorem relating surface and volume integrals, and the integration-by-parts formula, are shown to possess generalizations where the integrands take values in a Banach space, the integrals are Bochner integrals, and derivatives are Fréchet derivatives. For integration-by-parts, the integrand consists of a continuous linear map applied to a vector-valued function. These results were required for a generalization of the calculus of variations, given in another paper.