Consider a force of form $\vec F=N\hat v$, where $\hat v$ is a unit vector in direction of velocity or equivalently, in direction of change of displacement and N is a constant. To prove that this is a path-dependent force we will consider the integral $$\oint\vec F\cdot d\vec r$$ Where $d\vec r$ is displacement in any general direction. Since the direction of force and displacement are the same we get that $$\oint\vert\vec F\vert\vert d\vec r\vert$$ Making it clear that this indeed is path-dependent.
The question is can we prove this using the curl form of the integral, that is in general $$\vec\nabla\times\vec F\neq0?$$
Also, can we generalize to this where N is some scalar function instead of a constant?