Flux of vector field proportional to 1/r^3 through sphere

[jameson2]

Homework Statement

Consider the vector field $$\vec F=\frac{\vec r}{r^3}$$ with $$\vec r=x\hat{i}+y\hat{j}+z\hat{k}$$. Compute the flux of $$\vec F$$ out of a sphere of radius "a" centred
at the origin.

Homework Equations

The Gauss Divergence Theorem $$\int_D dV \nabla \bullet F=\int_S F\bullet dA$$

The Attempt at a Solution

I think I'm either missing or not understanding something in this question. When I compute $$\nabla \bullet F$$, I get zero, which means the flux is zero, I think. But this doesn't seem right at all. What am I missing?

 

[ystael]

The vector field $$\vec{F}$$ has a singularity at the origin, so you can't use the divergence theorem (at least not in its most common form). Compute the flux directly from the definition instead.