Stokes’ theorem for nonsmooth chains

Abstract:

Much of the vast literature on the integral during the last two centuries concerns extending the class of integrable functions. In contrast, our viewpoint is akin to that taken by Hassler Whitney [Geometric integration theory, Princeton Univ. Press, Princeton, NJ, 1957] and by geometric measure theorists because we extend the class of integrable domains. Let $\omega$ be an n-form defined on ${\mathbb {R}^m}$. We show that if $\omega$ is sufficiently smooth, it may be integrated over sufficiently controlled, but nonsmooth, domains $\gamma$. The smoother is $\omega$, the rougher may be $\gamma$. Allowable domains include a large class of nonsmooth chains and topological n-manifolds immersed in ${\mathbb {R}^m}$. We show that our integral extends the Lebesgue integral and satisfies a generalized Stokes’ theorem.