Past exam of the mathematical methods II course of the University of Oxford / 2019 / 8 / c / i / Solution

Write $r^2=x^2+y^2$ and work away from the $z$ axis, where $r>0$. The vector field is independent of $z$ and has zero vertical component, so only the vertical curl can be nonzero:ThereforeThis local spin in an azimuthal power-law flow depends on the radial variation of the tangential speed, not merely on the sign of that speed.