The Differentiation Lemma and the Reynolds Transport Theorem for Submanifolds with Corners

Abstract

We state and prove generalizations of the Differentiation Lemma and the Reynolds Transport Theorem in the general setting of smooth manifolds with corners (e.g. cuboids, spheres, R n , simplices). Several examples of manifolds with corners are inspected to demonstrate the applicability of the theorems. We consider both the time-dependent and time-independent generalization of the transport theorem. As the proofs do not require the integrand to have compact support (i.e. we neither employ Stokes' theorem nor any formalism relying on that assumption), they also apply to the 'unbounded' case. As such, they are of use to most cases of practical interest to the applied mathematician and theory-oriented physicist. Though the identities themselves have been known for a while, to our knowledge they have thus far not been considered under these conditions in the literature. This work was motivated by the study of the continuity equation in relativistic quantum theory and the general theory of relativity.

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