Prove the injectivity and differentiability of
$$x(u,v) = (u \cos(v), u \sin(v), \frac{u^2}{2})$$
where $u \in (0,\infty),$ $v \in (0,2\pi).$
What I really want to prove is that (U,x) is a simple surface.
where U=(0,inf)x(0,2$\pi$)
What I have: For the injectivity Let (u1,v1),(u2,v2) $\epsilon$ U
Got to prove x(u1,v1)=x(u2,v2) $\Rightarrow$ (u1,v1)=(u2,v2)
$\frac{u1^2}{2}$ = $\frac{u2^2}{2}$ $\Rightarrow$ u1=$\pm$u2
u1*cosv1 = u2*cosv2 $\Rightarrow$ ?
u1*senv1 = u2*senv2 $\Rightarrow$ ?
And I don't have any clue about how to prove differentiability