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

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dc.creatorHarrison, Jenny
dc.date2004-11-18
dc.date.accessioned2026-07-07T04:31:38Z
dc.date.available2026-07-07T04:31:38Z
dc.descriptionThe 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 $$\int_{\star A} ω= (-1)^{k(n-k)}\int_A \star ω.$$ When combined with the general Stokes' theorem for chainlet domains $$\int_{\partial A} ω= \int_A d ω$$ this result yields optimal and concise forms of Gauss' divergence theorem $$\int_{\star \partial A}ω= (-1)^{(k-1)(n-k+1)} \int_A d\star ω$$ and Green's curl theorem $$\int_{\partial A} ω= \int_{\star A} \star dω.$$
dc.description26 pages, 4 figures
dc.identifierhttps://arxiv.org/abs/math-ph/0411063
dc.identifierhttp://arxiv.org/abs/math-ph/0411063
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/57892
dc.subjectMathematical Physics
dc.subject49Q15
dc.titleGeometric Hodge Star Operator with Applications to the Theorems of Gauss and Green
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