Differential functions Probem Let $f:\mathbb{R}^n\to\mathbb{R}$ and $h:\mathbb{R}^n\to\mathbb{R}$ to be differential functions such that
$$\text{grad }f(X)=h(X)X.$$
Show that $f$ is constant over r-sphere centered at the origin.
I have no good idea to show that.

If $\|x_0\| = \|x_1\|>0$ then there is a differentiable path $t \mapsto x(t)$ such that $x(0) = x_0$, $x(1) = x_1$ and $\|x(t)\| = \|x_0\|$ for all $t$.
In particular, we have $\langle Dx(t), x(t) \rangle = 0$ for all $t \in (0,1)$.
Let $\phi(t) = f(x(t))$, then $\phi'(t) = \langle \nabla f(x(t)), Dx(t) \rangle
= h(x(t)) \langle x(t), Dx(t) \rangle = 0$, and so $\phi(0) = \phi(1)$.
