Curl-free field that is not conservative

Remarks

The third component of $\nabla \times F$ is $\partial_x \big( x / (x^2 + y^2) \big) - \partial_y \big( -y / (x^2 + y^2) \big) = \frac{y^2 - x^2}{(x^2 + y^2)^2} - \frac{y^2 - x^2}{(x^2 + y^2)^2} = 0$, and the other two components are zero. On the unit circle $\gamma(t) = (\cos t, \sin t, 0)$, $t \in [0, 2\pi]$, we have $F(\gamma(t)) \cdot \gamma'(t) = 1$. So the circulation is $2\pi \ne 0$, and path independence fails. The domain $U$ is not simply connected, so this does not contradict the theorem for convex or simply connected sets. Locally, $F = \nabla \theta$, where $\theta$ is the polar angle of $(x, y)$. But no continuous choice of $\theta$ exists on all of $U$.