∯${\bf A}×d{\bf S}$类似于散度定理的公式

hbghlyj

维基百科上看到的一个公式,不知如何推导的:
${\lower{20px}{_{∂V}}\displaystyle \mathbf {A} \times d\mathbf {S} \ =\ -\iiint _{V}\nabla \times \mathbf {A} \,dV}$

hbghlyj

Let $\bf c$ be an arbitrary constant vector,
${\Large∯}{\bf A}×{\bf c}·d{\bf S}=\iiint∇·({\bf A}×{\bf c})dV=\iiint[(∇×{\bf A})·{\bf c}-\cancelto0{(∇×{\bf c})}·{\bf A}]dV=\iiint(∇×{\bf A})·{\bf c}dV$
⇒$-{\bf c}·{\Large∯}{\bf A}×d{\bf S}={\bf c}·\iiint∇×{\bf A}\,dV$
Since $\bf c$ is arbitrary, the result follows.