Flux across nonsmooth boundaries and fractal Gauss/Green/Stokes' theorems

Abstract

By replacing the parametrization of a domain with polyhedral approximations we give optimal extensions of theorems of Gauss, Green and Stokes'. Permitted domains of integration range from smooth submanifolds to structures that may not be locally Euclidean and have no tangent vectors defined anywhere. One may still calculate divergence and curl over a domain, and flux across its boundary which itself may have no normal vectors defined anywhere.