Coordinate transformations

Question

I have two scalar functions of $x$ and $y$ that I can define:

$$f(x,y)=x^2+y^2\qquad \text{and}\qquad g(x,y)=x^2 + \sin^2(x) y^2.$$

Is it true that there is literally no coordinate change that will take one to the other?

Answer 1

FWIW, note that $$\mathrm{d}f\wedge\mathrm{d}g ~=~ h(x,y)\mathrm{d}x\wedge\mathrm{d}y,$$ where $$h(x,y)~:=~ 2y\{2x(\sin^2(x)-1)-y^2\sin(2x) \} ~=~ -4y\cos(x)\{x \cos(x)+y^2\sin(x) \}.$$ The pair $(f,g)$ are by definition functionally independent within the set $$\Omega~:=~\{(x,y)\in \mathbb{R}^2| h(x,y)\neq 0\}.$$ By the inverse function theorem the pair $(f,g)$ constitutes local coordinates in sufficiently small neighborhoods of $\Omega$.