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 / i / Solution

The logarithm is defined away from the $z$ axis, where $x^2+y^2>0$. For the vector potential $\psi {\hat{\mathbf{e}}}_z=(0,0,\psi )$, its curl is $({\psi }_y,-{\psi }_x,0)$. Sincewe obtainFor $a>0$ this is clockwise circulation about the axis, the velocity structure of a point vortex in each transverse plane.