Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 19 / d

On the sphere of radius $a$, the outward unit normal is $\mathbf{n}=(x,y,z)/a$. Thus $\mathbf{F}\cdot \mathbf{n}=a$ everywhere. Multiplying by the surface area $4\pi a^2$ givesThe same surface integral follows in spherical coordinates from $dS=a^2\sin \vartheta d\vartheta d\phi$.