Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics V resit / Question 3 / b

Use the outward orientation on the unit hemisphere and counterclockwise boundary orientation on the unit circle. With $\mathbf{r}(t)=(\cos t,\sin t,0)$ and $f=x=\cos t$, the vector line integral isFor the surface integral, $\nabla f=\mathbf{i}$ and the outward unit normal vector is $\mathbf{n}=(x,y,z)$, since the radius is one. The cross product isThe integral of $y$ is zero by reflection symmetry. Using the same spherical polar coordinates as in ii, with $a=1$, the other component isHenceverifying the vector-valued Stokes identity directly, component by component. The wedge symbol in the paper denotes the cross product here.

Choose the scalar fieldThe allowed vector-valued Stokes identity givesEvery boundary point lies on the unit circle in the $xy$-plane, so $f=1/2$ throughout the boundary. Since the line integral of $d\mathbf{r}$ around a closed curve is the zero displacement,Only the boundary value of this scalar field mattered: the interior of the surface need not be part of a sphere. Either consistent choice of orientation gives the same zero answer.