Past exam of the mathematics course of the University of Cambridge / 2020 / ia / Paper 2 / 9B / b / Solution

Stokes theorem givesFor the upward orientation, the outer circle $r=2,z=1$ is counterclockwise from above and the inner circle $r=1,z=4$ is clockwise. On a counterclockwise circle of radius $R$ and height $z$,The outer circle contributes $-16\pi$. The inner circle would contribute $-7\pi$ counterclockwise and hence contributes $+7\pi$ with its induced clockwise orientation. Thusagreeing with the direct calculation.