I've been reading Manfredo Do Carmo's Differential Geometry of Curves and Surfaces and was wondering what are the conditions that need to hold for a surface parameterization as this is not defined in the book. For example, one of the exercises asks whether the parameterization:
$\textbf{x}(u,v)=(u+v,u+v,uv)$ defined on the set $U=\{(u,v)\in R^2|u>v\}$ is a parameterization of the plane $P=\{(x,y,z)\in R^3|x=y\}$. Apparently, this is not a parameterization since $\textbf{x}$ is not one-to-one. I don't see why this should be a condition (i.e. being one-to-one) in order to be a parameterization. Is this a condition because the plane is a regular surface?