Operator Calculus of Differential Chains and Differential Forms

TL;DR

This paper develops an operator calculus framework for differential chains and forms, unifying discrete and smooth structures, and extends classical theorems to rough domains using a new topological approach.

Contribution

It introduces a calculus for differential chains with a strong topology, enabling continuous boundary and dual operators, and generalizes Reynolds' Transport Theorem to rough domains.

Findings

Boundary operator is continuous in the differential chain setting.

Partitions of unity and Cartesian wedge product exist within this framework.

A generalized Reynolds' Transport Theorem applies to rough domains.

Abstract

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.