Example of a map

The map $f: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ with values

$f(x)=\begin{pmatrix} \sqrt{x_{1}^2+x_{2}^2}\cos(x_{3}) \\ \sqrt{x_{1}^2+x_{2}^2}\sin(x_{3})\end{pmatrix}.$

has components the functions $f_{i}: \mathbb{R}^{2} \rightarrow \mathbb{R}, i=1,2$ , , with values

$f_{1}(x)=\sqrt{x_{1}^2+x_{2}^2}\cos(x_{3}), f_{2}(x)=\sqrt{x_{1}^2+x_{2}^2}\sin(x_{3}).$

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