the equation I'm dealing with is motivated by physics, but my problem is purely mathematical, so I think this is a good place to ask. The equation is the following
$$ \nabla h =\frac{\nabla P}{\rho} $$
where $h$, $P$ and $\rho$ are differentiable scalar functions, and $\nabla$ denotes the gradient operator.
I know that:
$\rho$ is a function of spatial coordinates, $\rho=\rho(x,y,z)$.
The function $P$ depends only on $\rho$, $P=P(\rho)$.
I need to show that $h$ depends only on $P$, $h=h(P)$, with this information and the equation with gradients above. How could I do that?
I guess it should be an easy task taking into account the geometrical interpretation of a gradient, but I have forgotten so many things from my calculus course...