Operator calculus - the exterior differential complex

We describe a topological predual to differential forms constructed as an inductive limit of a sequence of Banach spaces. This subspace of currents has nice properties, in that Dirac chains and polyhedral chains are dense, and its operator algebra contains operators predual to exterior derivative, Hodge star, Lie derivative, and interior product. Using these operators, we establish higher order divergence theorems for net flux of k-vector fields across nonsmooth boundaries, Stokes' theorem for domains in open sets which are not necessarily regular, and a new fundamental theorem for nonsmooth domains and their boundaries moving in a smooth flow. We close with broad generalizations of the Leibniz integral rule and Reynold's transport theorem.