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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!

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  • $\begingroup$ 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... $\endgroup$
    – messenger
    Aug 5 at 22:14

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