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- W2952356543 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 $$int_{star partial A}omega = (-1)^{(k-1)(n-k+1)} int_A dstar omega$$ and Green's curl theorem $$int_{partial A} omega = int_{star A} star domega.$$" @default.
- W2952356543 created "2019-06-27" @default.
- W2952356543 creator A5078367645 @default.
- W2952356543 date "2004-11-18" @default.
- W2952356543 modified "2023-09-27" @default.
- W2952356543 title "Geometric Hodge Star Operator with Applications to the Theorems of Gauss and Green" @default.
- W2952356543 hasPublicationYear "2004" @default.
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