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 called 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 \[ \displaystyle\int_{\star A} \omega = (-1)^{k(n-k)}\int_A \star \omega \] where $\omega$ is a $k$-form and $A$ is an $(n-k)$-chainlet. When combined with the general Stokes' theorem ([H1, H2]) \[\int_{\partial A} \omega = \int_A d \omega\] this result yields optimal and concise forms of Gauss' divergence theorem \[\int_{\star \partial A}\omega = (-1)^{(k)(n-k)} \int_A d\star \omega\] and Green's curl theorem \[\int_{\partial A} \omega = \int_{\star A} \star d\omega.\]