From taking and working in physics, I know that, given some uniform vector field $\vec{V}$, the flux of $\vec{v}$ through any closed surface is necessarily zero. I do not, however, understand, why this is, either intuitively or in a mathematically rigorous way.
For simple surfaces, it can be seen easily. Take, for example, a cube "parallel" to some vector field $\vec{V}$. By quick geometric examination, or argument by symmetry, we can see that only two sides "contribute" to the total flux, and their area are opposite in direction, thus the total flux is $0$. Similar approaches can be taken for spheres, etc.
For more complex surfaces, however, this is both bizzare and amazing. Why, intuitively and rigorously, is this true?
Because the flux is proportional to the projected area, not the area itself (there is a cosine factor), and the surface can in a way be seen as flat. As a closed surface has as many entering as exiting faces, the whole contribution is zero.