1210.4528 (Jenny Harrison)

Operator Calculus of Differential Chains and Differential Forms    [PDF]

Jenny Harrison

We describe a topological predual 'B to the Fr\'echet space of differential forms B defined in an open subset U of \R^n. This proper subspace of currents B' has useful properties: Subspaces of finitely supported Dirac chains and polyhedral chains are dense, offering a unification of discrete and continuum viewpoints. Operators can be defined constructively and geometrically on Dirac chains of arbitrary dimension and dipole order. The operators are continuous, and are thus defined on limits of Dirac chains, including polyhedral chains, submanifolds, stratified sets, and fractals. The operator algebra contains operators predual to exterior derivative, Hodge star, Lie derivative, wedge and interior product on differential forms, yielding simplifications and extensions of the classical integral theorems of calculus including the theorems of Stokes, Gauss-Green, and Kelvin-Stokes to arbitrary dimension and codimension. The limit chains, called "differential chains" may be highly irregular, and the differential forms may be discontinuous across the frontier of U. We announce new fundamental theorems for nonsmooth domains and their boundaries evolving in a flow. We close with broad generalizations of the Leibniz integral rule and Reynolds' transport theorem.

View original: http://arxiv.org/abs/1210.4528