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 $$\int_{\partial A} F \cdot n = \int_A div F$$ 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 $$\int_{\partial A} \omega = \int_{\star A} \star d\omega.$$