Past exam of the mathematics course of the University of Cambridge / 2016 / ia / Paper 3 / 12C / d / Solution

Set $s=x^2+y^2>0$. The first two components of the curl vanish since the horizontal components are independent of $z$ and the last component depends only on $z$. For the third,so this is a curl-free vector field everywhere in its domain:Choose the orientation in which the projection onto the $xy$-plane runs counterclockwise, and parametrizeOn this curve the horizontal contribution is $(-\sin t,\cos t)\cdot (-\sin t,\cos t)dt=dt$, while the vertical contribution is $zdz=d(z^2/2)$. Its integral over a closed curve is zero. Thereforewith $-2\pi$ for the opposite orientation. This does not contradict Stokes theorem: the planar disk spanning this curve meets the excluded $z$-axis at $(0,0,200)$, where the vector field is undefined. The horizontal one-form is the angular differential and records one winding about that axis; thus this curl-free vector field has no single-valued global potential of a conservative vector field on its domain.