 Solution

Write $r=\sqrt{x^2+y^2}>0$. The cylindrical coordinates give ${\mathbf{e}}_{\theta }=(-y/r,x/r,0)$ and ${\mathbf{e}}_r=(x/r,y/r,0)$, so the vector components of the vector field areThe first two components of the curl, calculated in Cartesian coordinates, are $-x{f}^{\prime }(z)/r$ and $-y{f}^{\prime }(z)/r$. For the third component, the chain rule givesand henceCombining these vector components in the cylindrical basis yieldsThe restriction $r>0$ matters because the cylindrical basis is undefined on the axis.