Let there be vector field $F$ and some surface $S$ which is without holes and closed.
We may want to calculate: $$ \iint\operatorname{curl}F\cdot n\cdot ds $$ where $n$ is the outer normal vector to the surface and $ds$ is the area element.
But $\operatorname{curl}F$ is a vector field in itself so let $G=\operatorname{curl}F$.
So according to Gauss theorem $\iint G\cdot n\cdot ds=\iiint_v\operatorname{div}G\,dv$ which is actually $\iiint \operatorname{div}(\operatorname{curl}F)\,dv$.
But we also know that $\operatorname{div}(\operatorname{curl}F)=0$.
So is $\iiint \operatorname{div}(\operatorname{curl}F)\,dv$ always $0$?