Let \(\mathbf F=\dfrac{\langle x,y,z\rangle}{(x^2+y^2+z^2)^{3/2}}\), with \(\nabla\cdot\mathbf F=0\) away from the origin. Compute the outward flux through the unit sphere.
The theorem does not apply: \(\mathbf F\) is not smooth at the origin, which lies inside the ball. Directly: on the sphere \(\mathbf F=\mathbf n\), so \(\mathbf F\cdot\mathbf n=1\) and the flux is the area, \(4\pi\). Answering \(0\) is the typical mistake.
The divergence theorem relates the flux of a field \(\mathbf F\) through a closed surface \(S\), with outward normal, to the triple integral of its divergence over the solid \(E\) that \(S\) encloses.