Reynolds transport theorem for smooth deformations of currents on manifolds

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 S, 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 S. The essence of the resulting computations is that the derivative operator is represented by the dual of the Lie derivative operation on smooth forms.