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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?

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    $\begingroup$ Doesn’t $(x\rightarrow x, y\rightarrow y\sin{x})$ take $f$ to $g$? $\endgroup$ – G. Smith May 16 at 1:51
  • $\begingroup$ @G.Smith I think that the domain and range of function would somewhat change when you will switch from y to ysinx. So we must check the range of each function. $\endgroup$ – Unique May 16 at 5:44
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    $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$ – Qmechanic May 16 at 8:05
  • $\begingroup$ @Unique: good point but in the case the domain and range of both $f$ and $g$ are unbounded. $\endgroup$ – Cinaed Simson May 17 at 7:44
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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 $\Omega$.

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