Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 10B / ii / Solution

Close the surface with the unit disk $D$ in the plane $z=0$. Its outward normal is $-e_z$, but $F\cdot (-e_z)=-z^4=0$ there. AlsoThe $x$ and $y$ terms integrate to zero by symmetry. The divergence theorem therefore gives