Corollary of Poincaré-Lemma

Question

In the script of my professor, there is the following corollary of the Poincaré Lemma:

If $\mathcal F$ is a vector field with rot$(\mathcal F)=0$ and $U\subset\mathbb R^n$ is an open ball, then there is a function $f:U\to\mathbb R$ with grad$(f)=\mathcal F$.

Could anyone make clear how this follows directly from the Poincaré Lemma? I don't really see it. Any help is greatly appreciated!

Answer 1

The Poincaré lemma say us that in a contractible domain (as it is an open ball) a closed form is exact. This means that if rot$(f)=0$ (the form is closed) there is a (scalar) field $F$ such that $f$ is its external derivative (it is exact), i.e.: $ f=$grad$(F)$, so that rot$($grad$(F))=0$

Answer 2

Poincare lemma says that all closed forms are exact forms. So if $\mbox{rot}(F)=0$ what can you say about the exactness of $F$?