# Show that a system with 0 curl in a simply connected region has no closed paths in said region.

I am studying for a nonlinear ODE exam and am somewhat stumped on this question from a practice exam:

Given the system $$x' = X(x,y), y' = Y(x,y)$$ with $$curl(X,Y) = 0$$ in a simply connected region D, show that the system has no closed paths in D.

My approach to this problem was to first note that $$\bigtriangledown \times F = 0$$, thus for some surface S we can apply stoke's theorem as follows, $$\iint_S \bigtriangledown \times F = 0 = \oint F \cdot dl$$

Is this a correct approach? If so is it enough to show that there are no closed paths? I feel like I'm missing an important step or two. Thanks in advance!

• Maybe write $F\cdot dl$ in terms of $x'$ and $y'$, note that it is a nonnegative quantity and what must hold for the integral to be zero... Aug 5 at 22:14