I have stumbled upon this in differential geometry dealing with regular surfaces:
We define the following surface (a hyperboloid) as
$ K = \{ (x,y,z) \in R^3 | x^2+y^2-z^2 = 1 \} $
and we define the upper half hyperboloid as:
$ S = \{ (x,y,z) \in K | y>0 \} $
We are to prove that S is a regular surface and we are to obtain a regular parametrization of it.
All I know is that the original hyperboloid is a regular surface (for example as regular values of differentiable function) but what to do about upper half hyperboloid being regular surface? Thanks all helpers
From $z^2=x^2+y^2-1$ the surface K can be recover by the images of these paramétrizations : $f(x,y)=(x,y,\sqrt{x^2+y^2-1})$ and $g=(x,y,-\sqrt{x^2+y^2-1})$. The two functions are defined on the region ${(x,y) \in \mathbb {R}^2} / x^2+y^2-1>=0$ .