Is this field conservative?

Question

I'm looking for a potential of this field:

$ F=\frac{f(r)}{r} (x,y,z) $ where $ r=\sqrt{x^2+y^2+z^2} $ and $ f $ is a $ C^1 $ function.

Any help is appreciated.

Thanks a lot!

Answer 1

Let $g$ be an anti-derivative of $f$, i.e., $g'=f$, then it can be easily show that $$ \nabla g(r)=f(r) \frac{(x,y,z)}{r}, $$ since $$ \frac{\partial r}{\partial x}=\frac{x}{r},\,\,\frac{\partial r}{\partial y}=\frac{y}{r} \,\,\,\text{and} \,\,\,\frac{\partial r}{\partial z}=\frac{z}{r}. $$ This shows that the field is conservative, as $$ \int_a^b F\big(\gamma(s)\big)\cdot \gamma'(s)\,ds=g\big(\gamma(b)\big)-g\big(\gamma(a)\big), $$ i.e., the integral is independent of the route we followed - it only depends on the endpoints.