A parametrized surface If I am given a map $f:U\subset \mathbb R^2 \to \mathbb R^3$ where $(x,y)\to (f(x,y),g(x,y),h(x,y))$. Is this necessarily a "parametrized surface"? 
Am I right in thinking that any map of the above form satisfies the "parametrized" bit. Is it necessary a surface though?
Please correct me if I am wrong. Thank you.
 A: If you are interested in the viewpoint of the classical differential geometry of surfaces, then we say that your map $f$ is a local parametrization of a surface in $\mathbb R^3$ if and only if $f$ is three-times continuously differentiable (i.e. $f \in C^3$) and the cross product $f_x \times f_y$ is nonzero for each $x, y \in U$. Here $f_x$ and $f_y$ represent the partial derivatives of $f$ w.r.t. $x$ and $y$, respectively. Of course there are other contexts as well.
A: It depends on your definition of "parametrized surface", but, without any more requirements, your maps include cases like
$$
(x,y) \mapsto (x,x,x)  \ ,
$$
which you would hardly call a "surface", would you? Or take simply
$$
(x,y) \mapsto (0,0,0)  \ .
$$
If you think that my examples are tricky because the maps don't "truly" depend on two variables, then think of
$$
(x,y) \mapsto (x+y, x+y, x+y ) \ .
$$
A: You'll want to assume your map is locally one-to-one, I think.  This is true if your map is continuously differentiable and the null space of the Jacobian matrix is trivial.
