La función $f:{\mathbb{R}}^2\setminus \{(x,y)|x\ne 0\}⟶\mathbb{R}$

Dada por $f(x,y)=\frac{y}{x}$

$$f\circ T(r,\theta )=f(T(r,\theta ))$$

$$f\circ T(r,\theta )=f(r\cos \theta ,r\sin \theta )$$

$$f\circ T(r,\theta )=\frac{r\sin \theta }{r\cos \theta }$$

$$f\circ T(r,\theta )=\tan \theta$$

https://www.geogebra.org/classic/exswze2h

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En el otro ejemplo $g:{\mathbb{R}}^2\setminus \{(0,0)\}⟶\mathbb{R}$ $$g(x,y)=\frac{2xy}{x^2+y^2}$$

$$g\circ T(r,\theta )=g(T(r,\theta ))$$

$$g\circ T(r,\theta )=g(r\cos \theta ,r\sin \theta )$$

$$g\circ T(r,\theta )=\frac{2r\cos \theta r\sin \theta }{(r\cos \theta {)}^2+(r\sin \theta {)}^2}$$

$$g\circ T(r,\theta )=\frac{2r^2\cos \theta \sin \theta }{r^2}$$

$$g\circ T(r,\theta )=\sin (2\theta )$$

En el siguiente enlace puedes observar una animación del cambio de coordenadas.

https://www.geogebra.org/classic/cvn5wpwz