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!