I'm having problems trying to proof Poincaré's lemma for Star-Shaped sets
Let $F:U\to \mathbb R^{2}$ be $C^{1}$ functions,where $U\subset \mathbb R^{2}$ is a Star-shaped set. If $F_x:U\to \mathbb R$ and $F_y:U\to \mathbb R$ are its components such that it satisfies $$ \frac{dF_x}{dx}=\frac{dF_y}{dy} \space $$ then F is conservative.
I have been trying to build such function, but I can't think of a integral where I could use the Fundamental Theorem of Calculus (since the $U$ isn't neither an open rectangle nor an open ball)
Any suggestions about how I could build such Function?