The Differentiation Lemma and the Reynolds Transport Theorem for submanifolds with corners

Abstract

The Reynolds Transport Theorem, colloquially known as “differentiation under the integral sign”, is a central tool of applied mathematics, finding application in a variety of disciplines such as fluid dynamics, quantum mechanics, and statistical physics. In this work, we state and prove generalizations thereof to submanifolds with corners evolving in a manifold via the flow of a smooth time-independent or time-dependent vector field. Thereby we close a practically important gap in the mathematical literature, as related works require various “boundedness conditions” on domain or integrand that are cumbersome to satisfy in common modeling situations. By considering manifolds with corners, a generalization of manifolds and manifolds with boundary, this work constitutes a step towards a unified treatment of classical integral theorems for the “unbounded case” for which the boundary of the evolving set can exhibit some irregularity.