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

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

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

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

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

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

Dibujo A

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

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

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

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

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

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

Dibujo B