Divergence theorem for a non-closed surface?

[NicolasM]

Is there some way we can apply divergence (Gauss') theorem for an open surface, with boundaries? Like a paraboloid that ends at some point, but isn't closed with a plane on the top.

I found this at Wikipedia:
It can not directly be used to calculate the flux through surfaces with boundaries

but I couldn't find some further explanation, like under what conditions it can be applied, or in what way we can "indirectly" use it.

 

[Blazejr]

Well you can use divergence theorem to calculate flux through your paraboloid WITH plane attached. Then if you can calculate flux through the plane you can substract it from what you got; what is left is obviously flux through paraboloid.

 

[davidbenari]

IMHO, it makes no sense to speak of the divergence theorem for a non-closed surface because of the way the theorem is derived. I don't know if you already know this (or maybe I'm the one who lacks knowledge) but in the derivation of this theorem one makes reference to the addition of the flux in all infinitesimally small cubes that make up a surface and the only flux that "survives" is the one from the outward facing-sides on the most exterior cubes of your closed surface.

 

[DivergentSpectrum]

No, but there is a curl theorem for open surfaces (stokes)