Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2018 / Paper 2 / 8

The vector field is radial, while the differential displacement along a centred circle is tangential. These directions are orthogonal, so the dot product vanishes at every point. Hence

Use the outward unit normal on the unit sphere and the convention that $\theta$ is measured from the positive vertical axis. Then $\mathbf{F}\cdot \mathbf{n}=\sin \theta$, while the spherical coordinates area element is $dS=\sin \theta d\theta d\varphi$. The flux is therefore