I am reading a book by Moskowitz and Paliogiannis where I miss a point when trying to complete a proof of the Poincare lemma (which is left to the reader).
The following facts are proved clearly in the book.
Fact 1: Let $F:\Omega\to\mathbb{R}^{n}$ be a $C^{1}$-mapping (vector field), $\Omega$ an open ball in $\mathbb{R}^{n}$, and $$ \forall i,j\in\{1,\dots,n\}\; \forall x\in\Omega: \quad \frac{\partial F_{i}}{\partial x_{j}}(x)=\frac{\partial F_{j}}{\partial x_{i}}(x). $$ Then $F$ is conservative, i.e., there is a $C^{1}$-funcion $f:\Omega\to\mathbb{R}$ such that $\mbox{grad} f =F$.
Fact 2: Let $F:\Omega\to\mathbb{R}^{n}$ be a $C^{1}$-mapping, $\Omega$ a region (=open and connected) in $\mathbb{R}^{n}$, and for any closed $C^{1}$-curve $\gamma$ in $\Omega$, it holds that $$ \int_{\gamma}F\cdot ds=0. $$ Then $F$ is conservative.
The goal is to prove the following:
Theorem (Poincare lemma): Fact 1 remains true if $\Omega$ is any simply connected subset of $\mathbb{R}^{n}$.
I miss a point when trying to pass from balls to simply connected domains. I suppose that one should elaborate on Fact 2 and use the simple connectedness of $\Omega$.
Question: How should I construct the potential $f$ on a simply connected domain $\Omega$ provided that I am aware of Facts 1 and 2?