Past exam of the mathematical methods II course of the University of Oxford / 2021 / 10 / ii / Solution

Assume $a>0$ and choose the outward orientation. On the sphere, $\mathbf{n}=(x,y,z)/a$, so $\mathbf{F}\cdot \mathbf{n}=(x^2+y^2+2z^2)/a=(a^2+z^2)/a$. In spherical coordinates, $z=a\cos \theta$ and $dS=a^2\sin \theta d\theta d\varphi$. The direct surface integral isIndependently, $\nabla \cdot \mathbf{F}=1+1+2=4$. Its volume integral over the ball isThe two calculations agree exactly with the divergence theorem. The expression requested is a scalar flux of a vector field, namely the dot product with the directed surface area element.