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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

$$\iint_S F\, d\vec S \stackrel{?}{=} \iiint_V \nabla F \, dS$$?

user256872
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    $F\vec{dS}$ has three components, each of which can be written as something like $(F,0,0)\cdot \vec{dS}$, etc. Why not try taking this approach and proving/disproving your guess with a formula you can derive? – Ninad Munshi Sep 26 '23 at 04:36
  • @NinadMunshi i did not think of that... thank you! I tried it, and seemed to have gotten to that answer. Am I correct? – user256872 Sep 26 '23 at 06:17

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