  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.
