What I mean by "closed volume integral" is the symbol $\unicode{x2230}$ notated by \oiiint in latex {which does not show up here}.
I did some searching on this topic and found out that there is no such thing as an open or closed volume in 3D which makes no sense to me as the volume of a sphere for example is closed but the volume of a hemisphere can be considered open as it is bounded by the hemisphere's boundary.
I also tried to find some formulas where the closed volume integral would appear and I only found this equation on wikipedia which I have never seen before https://en.m.wikipedia.org/wiki/Template:Oiiint $$\frac{1}{\sqrt{2\pi}} \unicode{x2230}_\Omega \frac{\partial F}{\partial s} d\Sigma$$
Lastly I saw in a reddit post that the symbol can be used for generalising the divergence theorem in 4D.
Can someone explain the use of this symbol to me?