Let \(\mathbf F=\langle x+z^2,\,y+x^2,\,z+y^2\rangle\). Compute the outward flux through the boundary of a cube of side \(2\) and choose the correct conclusion.
\(\nabla\cdot\mathbf F=1+1+1=3\) is constant, so the flux is \(3\cdot\text{vol}=3\cdot8=24\) wherever the cube sits. The terms \(z^2,x^2,y^2\) contribute no divergence.
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.