Reynolds transport theorem for smooth deformations of currents on manifolds

Abstract

The Reynolds transport theorem for the rate of change of an integral over an evolving domain is generalized. For a manifold B, a differentiable motion m of B in the manifold [Formula: see text], an r-current T in B, and the sequence of images m( t) ♯ T of the current under the motion, we consider the rate of change of the action of the images on a smooth r-form in [Formula: see text]. The essence of the resulting computations is that the derivative operator is represented by the dual of the Lie derivative operation on smooth forms.