Let $r = ||X||$. Let $g$ be a differentiable function of one variable. Now to show that the vector field defined by $$F(X) = \frac{g'(r)}{r} X$$ in the domain $X \neq 0$ always admits a potential function. What is the potential function?
I learnt that if the domain of the vector field is the whole space or a rectangle or a open ball i.e a connected set than $D_if_j = D_jf_i$ for all indices $i,j$ implies that $F$ has a potential function, where $f_i$'s are the component functions of $F$. But in this question the domain is punctured at $X=0$ so we can't conclude that $F$ has a potential function by previous statement.
So how can I prove that $F$ admits a potential function in this case.