This question arose to me when studying the method of characteristics, but it is really independent (there is no need to know about the method in order to answer my question).
Suppose I have a surface $\mathcal{S}$ in $\mathbb{R}^3$ of which I know an explicit parametrization of the form
$$\Big\{ \big( x, y, z(x,y) \big) \, : x,y \in \mathbb{R} \Big\} = \mathcal{S} \, $$
(I know the function $z(x,y)$ ). Suppose I also know there exists another parametrization of the same surface, this time an implicit one, of the form
$$\Big\{ \big( t, \xi ( t,s) , \eta(t,s) \big) \, : t,s \in \mathbb{R} \Big\} = \mathcal{S} \, . $$
(I don't know the functions $\xi(t,s)$, $\eta(t,s)$ ). Am I allowed to identify these two parametrizations point by point? i.e. am I allowed to write
$$\big( x, y, z(x,y) \big) = \big( t, \xi ( t,s) , \eta(t,s) \big) \, ,$$
implying for instace that $x=t$? Or the only thing I know is that they both sweep the surface $\mathcal{S}$?
The relation is
$z(x,y)=\eta(t,s)$ and $x=t$,$y=\xi(t,s)$
So
$z(t,\xi(t,s))=\eta(t,s)$