Gauss divergence theorem says from https://en.wikipedia.org/wiki/Divergence_theorem
that
$$ \iiint_{V}(\nabla\cdot F)\,dV=\iint_{S}(F\cdot n)\,dS. $$
Now, let $G=\nabla\times F$ for a nice vector field $F$, then from the above result, we get $$ \iint_{S}(G\cdot n)\,dS=\iiint_{V}(\nabla\cdot G)\,dV=0, $$ since $\nabla\cdot G=\nabla\cdot (\nabla\times F)=0$. But, it seems it should not be true and I could not understand what I am missing. Indeed, Stokes theorem (from https://en.wikipedia.org/wiki/Stokes%27_theorem#Theorem) does not give this value to be zero always.
Can someone please explain what is going on here?
Thanks a lot.