Consider an arbitrary, continuously differentiable function of two variables, $V(x,y)$, and the coordinate substitution $x=\rho\cos\phi$, $y=\rho\sin\phi$.
Prove that there is no functional relationship between $\rho$ and $\phi$; that is, there is no function g such that $g(\rho,\phi)=0$ for all $\rho$ and $\phi$.
My clue is:
We know $\rho=x^2+y^2$ and $\tan\phi=\frac{y}{x}$
Assume there exists g such that $g(\rho,\phi)=0$, and prove it is conflict with something else.
Can anyone help me the rest? Thank you!