Past exam of the mathematical methods II course of the University of Oxford / 2021 resit / 8 / c / Solution

Away from the origin, write $\mathbf{F}=r^{-\beta -1}\mathbf{r}$. The curl product rule for a scalar field and the gradient of a radial function giveConsequently the surface integral of curl over the upper hemisphere is zero. Its outward unit normal vector induces a counterclockwise equator when viewed from above. Parametrize that boundary byThe radial vector field is perpendicular to this tangent, soThis verifies Stokes theorem directly without passing through the singular origin.