Compute the flux of \(\mathbf F=\langle x,y,z\rangle\) through the upper hemisphere \(z=\sqrt{1-x^2-y^2}\) with outward normal. (Hint: close it with the disk \(z=0\).)
The total flux through hemisphere plus disk is \(3\cdot\tfrac{2\pi}{3}=2\pi\). On the disk the normal is \(-\mathbf k\) and \(\mathbf F\cdot(-\mathbf k)=-z=0\), so all the flux crosses the hemisphere: \(2\pi\).
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.