Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 16 / a / iii / Solution

In cylindrical coordinates, $x^2+y^2=r^2$ and $r>0$, soThe potential vector has components $(A_r,A_{\theta },A_z)=(0,0,a\ln r)$. Expanding the prescribed cylindrical curl formula givesThereforeSince ${\hat{\mathbf{e}}}_{\theta }=(-\sin \theta ,\cos \theta ,0)$, this agrees with the Cartesian result and its clockwise orientation.