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This is a very known result, but I don't have some proof. Someone known or has some proof of it?

Let be $\omega = P\;dx + Q\;dy$ be a $C^1$ differential form on a domain $D$. If $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} ,$ then $\omega$ is locally exact.

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    Cartan's book *Elementary theory of analytic f$u$nctions ...* gives [a very readable answer](http://books.google.com/books?id=KsGbqTBjyo$U$C&pg=PA49) to yo$u$r question.2011-12-02

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If you have a curl-free field $W = (W_1, W_2, W_3)$ in a neighborhood of the origin, it is the gradient of a function $f$ given by $ f(x,y,z) = \int_0^1 \; \left( \; x W_1(tx, ty,tz) + y W_2(tx, ty,tz) + z W_3(tx, ty,tz) \; \right) dt.$

In your case, take $W_3 = 0$ and drop the dependence on $z$ from $f, \; W_1$ and $W_2.$ Note how this is set up so that $f=0$ at the origin.

There is more information at Anti-curl operator