Imagine you have the following closed surface : paraboloid $\displaystyle -x^2-y^2+16 $ closed by the $\displaystyle xy$ plane ($\displaystyle z=0$).

Now imagine the divergence of the given field is worth $\displaystyle 0$. For example the field $\displaystyle F(x,y,z)=y \vec i + x \vec j + x \vec k$.

Does that mean that there's no flux passing through the surface of the closed paraboloid? Or it means that the flux entering the paraboloid is worth the same amount than the flux leaving the paraboloid via the $\displaystyle z=0$ plane?

Or does it mean that the total flux is $\displaystyle 0$ and I can't know how much flux pass through the open paraboloid and the $\displaystyle z=0$ plane?

This is a very important question to me! Thanks in advance.