  Differential chains are a proper subspace of de Rham currents given as an
inductive limit of Banach spaces endowed with a geometrically defined strong
topology. Boundary is a continuous operator, as are operators that dualize to
Hodge star, Lie derivative, pullback and interior product. Partitions of unity
exist in this setting, as does Cartesian wedge product. Subspaces of finitely
supported Dirac chains and polyhedral chains are both dense, leading to a
unification of the discrete with the smooth continuum. We conclude with an
application generalizing a simple version of the Reynolds' Transport Theorem to
rough domains.
