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