Let $f:\Bbb R^3\to \Bbb R$ such that $∂f/∂x$ + $∂f/∂y$ + $∂f/∂z$ $= 0$ for any point of $\Bbb R^3$. Proof that there exists a diffeomorphism $g:\Bbb R^3\to \Bbb R^3$, such that $∂(f∘g)/∂x=0$ for any point of $\Bbb R^3$.
So, I was thinking to use $g$ linear. I tried to use $g(x,y,z)=(2x,2y,2z)$ because is one-to-one, differentiable and $g^{-1}$ is differentiable, but that doesn't work. Anybody can help me?
If $f$ is a map to value in $\mathbb{R}$ than
$\frac{\partial (f\circ g)}{\partial x}= \frac{\partial f}{\partial x} \frac{\partial g_1}{\partial x} + \frac{\partial f}{\partial y} \frac{\partial g_2}{\partial x} + \frac{\partial f}{\partial z} \frac{\partial g_3}{\partial x} $
And so you can choose for example the linear map $g(x,y,z)=(x,x+y,x+z)$