 Volume integral of a curl

The divergence theorem applied to $G\times c$ for every constant vector $c$ gives ${\int }_V\nabla \times GdV={\int }_{\partial V}n\times GdS$, 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.