Let $u=(x,y,z)\in \mathbb{R}^3 , r(u)=r(x,y,z)=\sqrt{x^2+y^2+z^2}$.
Denote $f: (0,\infty) \to \mathbb{R},f\in C^1$.
Let $F=f(r(u))u$ be a vector field in $\mathbb{R}^3\setminus \{{(0,0,0)}\}$.
I have to find out if $F$ must be a conservative vector field in $\mathbb{R}^3\setminus \{{(0,0,0)}\}$.
My solution :
I consider to prove that $F$ is not necessarily a conservative vector field by proving that that curl is not zero,
using a spherical coordinate substitution $x=\rho \cos \theta \sin\phi, y=\rho \sin\theta \sin\phi, z=\rho \cos\phi , 0\leq \rho < \infty , 0\leq \theta < 2 \pi , 0\leq \phi \leq \pi , J=\rho^2 sin \phi$
Now , $r(x,y,z)=\rho\cdot|\rho^2 sin \phi|=\rho^3sin \phi$
$F(u)=f(\rho^3sin \phi)\cdot(\rho \cos \theta \sin\phi,\rho \sin\theta \sin\phi,\rho \cos\phi)$
Now I want to find the curl of $F$ but I think i very complex hence I am wondering if my approach is correct ?
Any help is welcome.