Let $n\in\mathbb{N}$ and let $F_i:\mathbb{R}^n\rightarrow\mathbb{R}\,\forall i\in\{1,\dots,n\}$ be some functions which obey certain conditions (smooth, for example), such that $\forall \{i,j\}\subset\{1,\dots,n\},\,\partial_iF_j=\partial_jF_i$.
How do you then prove that the system of partial differential equations $F_i=\partial_if\,\forall i\in\{1,\dots,n\}$ has a solution (for some unknown $f:\mathbb{R}^n\rightarrow\mathbb{R}$), what are the conditions that $f$ fulfills with relation to the conditions on $F_i$, and how do you explicitly write down $f$ then?
