Suppose I have a function $F: R^3 \to R^3$ which satisfies:
1) There exists $\Psi: R^3 \to R$ such that $F = \nabla \Psi$ and
2) $F(x)$ depends only on $\|x\|$
Can I conclude that $\|F(x)\| = C\frac{1}{\|x\|^2}$ for some constant $C$? Or maybe at least $\|F(x)\| \le C\frac{1}{\|x\|^2}$? Maybe there is another condition to add that would let me make the conclusion?
Thanks in advance.