Bonus Problem 2

This is a bonus problem for extra credit. A well written, correct solution submitted to me will earn the submitter 0.5 bonus percentage points added to their final course grade.

Problem: Find a vector field$\vec{F}$, well defined on 

$\mathbb{R}^3 - C$

where $C$ is the circle given by $\left\{x^2+y^2=1,z=0\right\}$satisfying:

$\operatorname{curl}\vec{F}=\vec{0}$

but $\vec{F}$ is NOT conservative. Your solution should be an explicitly defined vector field whose formula is given in terms of the usual kinds of functions we encounter in class. Your write up should clearly demonstrate that your vector field has zero curl, is not conservative, and has the correct domain.