Is it possible to have three smooth functions of two variables $(x,y) \in \mathbb{R}^2$
$$\begin{array}{rcl} u &=& f_1(x,y)\\ v &=& f_2(x,y)\\ w &=& f_3(x,y)\\ \end{array} $$
such that the image $(u,v,w)$ is equal to $\mathbb{R}^3$
I was thinking about this in relation to parametric families of member curves/functions where the curves transition smoothly when changing the parameters. Then such a family where the member curves are parameterized by three variables $u,v,w$ could be changed into a family where the member curves are parameterized by two variables $x,y$, while still keeping smooth transitions.
I tried something like a tangens function e.g. $u = tan(x)$ and $v = \lfloor \frac{1}{2}+\frac{x}{\pi} \rfloor$ but this is not smooth (at every $x = \frac{1+k}{2} \pi$ ) and $v$ is only in $\mathbb{N}$.
No, such a surjective smooth map does not exist:
Else each point of $\mathbb R^2$ would be a critical point , because there is no surjective linear map $\mathbb R^2\to \mathbb R^3$.
But then each point of $\mathbb R^3$ would be a critical value, which contradicts Sard's theorem according to which the set of critical values in the codomain $\mathbb R^3$ of $f$ must have measure zero.
Remark
In contrast to the above non-existence result beware that there do exist for any $n,p\gt0$ continuous (non smooth) surjective maps $\mathbb R^n\to \mathbb R^p$ .
This most unexpected phenomenon made quite a stir when Giuseppe Peano discovered it in 1890 .