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

The closed boundary includes the curved unit hemisphere and the disk at $y=0$. On the disk, $F=(0,y,0)=0$, so that contribution vanishes. On the curved part the outward normal is $n=(x,y,z)$, giving $F\times n=(yz,0,-xy)$.

Use spherical coordinates with the polar axis along positive $y$:The area element is $\sin \theta d\theta d\varphi$. ConsequentlyThis directly performs the surface integral and includes the planar cap, rather than applying the closed-surface theorem to only an open hemisphere.