Geometric Hodge Star Operator with Applications to the Theorems of Gauss and Green

TL;DR

This paper introduces a geometric dual to the Hodge star operator called the star operator, extending classical divergence and Green's theorems to a broad class of domains called chainlets, including fractals and soap bubbles.

Contribution

The paper develops the star operator on chainlets, generalizing divergence and curl theorems to irregular and fractal domains with a new proof and extension of classical theorems.

Findings

Defines the star operator as a geometric dual to Hodge star

Proves the star theorem relating integrals over chainlets and their duals

Derives generalized divergence and Green's theorems for chainlet domains

Abstract

The classical divergence theorem for an $n$-dimensional domain $A$ and a smooth vector field $F$ in $n$-space $$\int_{\partial A} F \cdot n = \int_A div F$$ requires that a normal vector field $n(p)$ be defined a.e. $p \in \partial A$. In this paper we give a new proof and extension of this theorem by replacing $n$ with a limit $\star \partial A$ of 1-dimensional polyhedral chains taken with respect to a norm. The operator $\star$ is a geometric dual to the Hodge star operator and is defined on a large class of $k$-dimensional domains of integration $A$ in $n$-space the author calls {\em chainlets}. Chainlets include a broad range of domains, from smooth manifolds to soap bubbles and fractals. We prove as our main result the Star theorem $$\int_{\star A} \omega = (-1)^{k(n-k)}\int_A \star \omega.$$ When combined with the general Stokes' theorem for chainlet domains $$\int_{\partial A} \omega = \int_A d \omega$$ this result yields optimal and concise forms of Gauss' divergence theorem $$\int_{\star \partial A}\omega = (-1)^{(k-1)(n-k+1)} \int_A d\star \omega$$ and Green's curl theorem $$\int_{\partial A} \omega = \int_{\star A} \star d\omega.$$