Let $F$ be a $C^1$ vector field in $\mathbb{R}^3$. I want to prove that the following are equivalent:
I) There exists a function $g$ such that $F=\nabla g$, and
II) $\nabla \times F = (0,0,0)$
For I $\implies$ II, can I use that the mixed partial derivates $f_{xz}$ and $f_{zx}$ are equals? Why?