Is it possible for a vector field in $\mathbb{R}^2$ to have nonzero scalar curl
$\bullet$ at a point
$\bullet$ along a line
$\bullet$ along a curve (which is not a line)
$\bullet$ in a finite region
yet have zero scalar curl everywhere else?
I am not asking for vector fields such as $\begin{bmatrix}-\frac{y}{x^2+y^2}\\\frac{x}{x^2+y^2}\end{bmatrix}$, which are either undefined or have zero scalar curl everywhere, but for fields with defined, nonzero sections.