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.

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