= Volume integral of a curl
{title2=$\int_V\nabla\times G\,dV=\int_{\partial V}n\times G\,dS$}
The <divergence theorem> applied to $G\times c$ for every constant vector $c$ gives $\int_V\nabla\times G\,dV=\int_{\partial V}n\times G\,dS$, with outward <unit normal> $n$. Indeed $\nabla\cdot(G\times c)=c\cdot(\nabla\times G)$, while $(G\times c)\cdot n=c\cdot(n\times G)$. Equality of the scalar projections for every $c$ proves the vector identity. Reversing the surface <cross product> reverses the sign.
Back to article page
