Operator calculus - the exterior differential complex

TL;DR

This paper introduces a topological predual to differential forms as an inductive limit of Banach spaces, enabling advanced theorems for nonsmooth domains and their boundaries using an operator algebra that includes key differential operators.

Contribution

It constructs a new topological predual to differential forms with dense chains and an operator algebra that supports generalized divergence and Stokes' theorems for nonsmooth domains.

Findings

Established higher order divergence theorems for nonsmooth boundaries

Proved Stokes' theorem for irregular domains in open sets

Generalized Leibniz rule and Reynolds' transport theorem

Abstract

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.