  Most transport theorems---that is, a formula for the rate of change of an
integral in which both the integrand and domain of integration depend on
time---involve domains that evolve according to a flow map. Such domains are
said to be convecting. Here a transport theorem for nonconvecting domains
evolving on an embedded manifold is established. While the domain is not
convecting, it is assumed that the boundary of the domain does evolve according
to a flow map is some generalized sense. The proof relies on considering the
evolving set as a fixed set in one higher dimension and then using the
divergence theorem. The domains considered can be irregular in the sense that
their boundaries need only be Lipschitz. Tools from geometric measure theory
are used to deal with this irregularity.
