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

Abstract

The classical divergence theorem for an $n$-dimensional domain $A$ and a smooth vector field $F$ in $n$-space $${\int }_{\partial A}F\cdot n={\int }_{A}divF$$ 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 $${\int }_{\partial A}\omega ={\int }_{\star A}\star d\omega .$$

26 pages, 4 figures