Operator Calculus of Differential Chains and Differential Forms
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.
Cite
@article{arxiv.1210.4528,
title = {Operator Calculus of Differential Chains and Differential Forms},
author = {Jenny Harrison},
journal= {arXiv preprint arXiv:1210.4528},
year = {2015}
}
Comments
68 pages, 14 figures. First posted on the arxiv in 2012, published in Springer's First online 05 September 2013 and published in The Journal of Geometric Analysis, January 2015. Since this paper was posted, two unrelated applications have been published: arXiv:1310.0508 and Seguin and Fried, Math. Models Methods Appl. Sci. 24, 1729 (2014). arXiv admin note: text overlap with arXiv:1101.0979