  We extend the Cauchy residue theorem to a large class of domains including
differential chains that represent, via canonical embedding into a space of
currents, divergence free vector fields and non-Lipschitz curves. That is,
while the classical Cauchy theorems involve integrals over piecewise smooth
parameterized curves, these classical theorems actually hold for far more
general notions of "curve." We also extend the definition of winding number to
these domains and show that it behaves as expected.
