Let be the “top” face of the cube, which can be parametrized by for We leave it to the reader to confirm that which points outside of the cube.
The flux across this face is:
This double integral is not trivial to compute, requiring multiple trigonometric substitutions. This example is not meant to stress integration techniques, so we leave it to the reader to confirm the result is
Note how the result is independent of no matter the size of the cube, the flux through the top surface is always
An argument of symmetry shows that the flux through each of the six faces is thus the total flux through the faces of the cube is
It takes a bit of algebra, but we can show that Thus the Divergence Theorem would seem to imply that the total flux through the faces of the cube should be
but clearly this does not match the result from above. What went wrong?
Revisit the statement of the Divergence Theorem. One of the conditions is that the components of must be differentiable on the domain enclosed by the surface. In our case, is not differentiable at the origin — it is not even defined! As does not satisfy the conditions of the Divergence Theorem, it does not apply, and we cannot expect
Since is differentiable everywhere except the origin, the Divergence Theorem does apply over any domain that does not include the origin. Let be any surface that encloses the cube used before, and let be the domain between the cube and note how does not include the origin and so the Divergence Theorem does apply over this domain. The total outward flux over is thus which means the amount of flux coming out of is the same as the amount of flux coming out of the cube. The conclusion: the flux across any surface enclosing the origin will be
This has an important consequence in electrodynamics. Let be a point charge at the origin. The electric field generated by this point charge is
i.e., it is with where is a physical constant (the “permittivity of free space”). Gauss’s Law states that the outward flux of across any surface enclosing the origin is