Given: $\mathbf{F}=F(r)\ \mathbf{\hat{r}}$ is a spherically symmetric radial vector field which is continuous everywhere except at origin.
To prove: $\mathbf{F}=-\nabla \psi$
Proof:
\begin{align} \mathbf{F} \cdot \mathbf{\hat{r}}&=F(r)\ \mathbf{\hat{r}} \cdot \mathbf{\hat{r}}=F(r)=\dfrac{d(\int F\ dr)}{dr}\\ &=\dfrac{d\psi}{dr}=\nabla \psi \cdot \mathbf{\hat{r}} \end{align}
From here we cannot straight away deduce that: $\mathbf{F}=\nabla\psi$
UNKNOWN STEPS ???
$$\mathbf{F}=-\nabla\psi$$
$\mathbf{F}$ is conservative.
Question: From what steps (unknown to me) can we get $\mathbf{F}=-\nabla\psi\ $?