Integrating unit vectors

Question

If I want to find a function $V(r)$ such that ${\bf F} = -\nabla V$, where ${\bf F} = f(r)\hat{\bf r}$, how do I integrate ${\bf F}$ along the radial direction?

Answer 1

Let for $\vec{x}\in\mathbb{R}^d$ let $r=|\vec{x}|$. Note that if $g:[0,\infty)\rightarrow \mathbb{R}$, then the gradient of the function $V(\vec{x})= g(|\vec{x}|)$ is given by $$ \displaystyle \nabla V = \frac{\vec{r}}{r}g'. $$ Thus to solve your problem we need to find $g$ such that $$ -\frac{\vec{r}}{r}g'(r)=f(r)\vec{r}, $$ which leads to $$ g(r)=-\int_0^r f(t)\,t\,dt + C. $$