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The classical divergence theorem for an $n$-dimensional domain $A$ and asmooth 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. $p \in \partialA$. In this paper we give a new proof and extension of this theorem byreplacing $n$ with a limit $\star \partial A$ of 1-dimensional polyhedralchains taken with respect to a norm. The operator $\star$ is a geometric dualto the Hodge star operator and is defined on a large class of $k$-dimensionaldomains of integration $A$ in $n$-space the author calls {\em chainlets}.Chainlets include a broad range of domains, from smooth manifolds to soapbubbles 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 andconcise 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$$\int_{\partial A} \omega = \int_{\star A} \star d\omega.$$

Geometric Hodge star operator with applications to the theorems of Gauss and Green