Paraboloid and parametrizations

Question

After having studied some concepts and definitions regarding manifolds (at a basic level), I wondered if a paraboloid with equation $P = \{ x^2 + y^2 = z\}$ was a manifold. The answer was affirmative since if one considers the function $F(x,y) = x^2 + y^2 - z$, it is clear that $P = F^{-1}(0), F \in C^\infty$ and $DF(x,y) = (2x,2y,-1)$, which has range greater than $0$. I find no difficulties when giving a parametrization locally on every point of this manifold, with the exception to $(0,0,0)$. I seems clear to me that $\phi(r,\theta) = (r\cos \theta, r \sin \theta, r^2)$ is a parametrization, but I cannot find an open set/neighbourhood in which that function is a parametrization around $(0,0,0)$. I tried to use the set $U = \{ 0\} \times (0,2\pi)$, but I found that this is not an open set.
Which is the correct local parametrization for that point? Is there any? If not, isn't this a contradiction to the definition of manifold?

Answer 1

As mentioned in the comments, the projection on the xy plane gives you a global parametrization for the paraboloid, which is also a local parametrization around every point. The open $U$ is the whole plane, and the map is just $\phi:\mathbb R^2\to\mathbb R^3$, $\phi(x,y)=(x,y,x^2+y^2)$. Note that the same argument applies by replacing the paraboloid for the graph of any smooth functions. Something very interesting here is that a partial converse holds: in light of the implicit function theorem, any submanifold $S\subset\mathbb R^n$ admits around a given point $p\in S$ a local graph-like chart $\phi:U\subset\mathbb R^{n-1}\to \mathbb R^n$, where one of the coordinates is written as a function of the others. Hope that helps.