几何Hodge星算子及其对Gauss定理和Green定理的应用
Geometric Hodge Star Operator with Applications to the Theorems of Gauss and Green
摘要：
$n$维域$A$和$n$维空间$$\int_{\partial A} F \cdot n = \int_A div F$$中光滑向量场$F$的经典发散定理要求定义一个法向量场$n(p)$ a.e. $p \in \partial A$。本文用一维多面体链关于范数的极限$\star \partial A$代替$n$，给出了这个定理的一个新的证明和推广.算子$\star$是霍奇星算子的几何对偶，定义在$n$ -空间中的一大类$k$维积分域$A$上，作者称之为{ \em chainlets}。Chainlets包括广泛的域，从光滑流形到肥皂泡和分形。我们证明了星定理$$\int_{\star A} ω= (-1)^{k(n-k)}\int_A \star ω.$$。当与小链域的一般Stokes定理$$\int_{\partial A} ω= \int_A d ω$$相结合时，这个结果得到了高斯散度定理$$\int_{\star \partial A}ω= (-1)^{(k-1)(n-k+1)} \int_A d\star ω$$和格林旋度定理$$\int_{\partial A} ω= \int_{\star A} \star dω.$$的最佳和简洁形式
英文摘要：
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 $$\int_{\star A} ω= (-1)^{k(n-k)}\int_A \star ω.$$ 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 $$\int_{\partial A} ω= \int_{\star A} \star dω.$$
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