Past exam of the mathematical methods II course of the University of Oxford / 2017 / 8 / b / Solution

The divergence theorem states ${\iint }_{S}F\cdot ndS={\iiint }_V\nabla \cdot FdV$ for a sufficiently smooth field and a piecewise smooth closed boundary with outward unit normal. Choose $F=r/3$. Since $\nabla \cdot r=3$,The factor $1/3$ comes from the three spatial dimensions; the outward orientation is especially important on the inner wall of the drilled solid.