I know the divergence theorem can be applied to closed flux integrals $$\iint_S \vec F \cdot d\vec S = \iiint_V \nabla \cdot \vec F \, dV$$ but what if I have an integral of the form $$\iint_S F \, d\vec S$$
where $S$ is some closed surface? can Stokes's Theorem be neatly applied for it?
i.e. something like for a closed surface S enclosing a volume V where $d\vec S$ is an outward-oriented normal
$$\iint_S F\, d\vec S \stackrel{?}{=} \iiint_V \nabla F \, dS$$?