Computing parametrizations for a differentiable $2$-manifold with boundary

Question

Consider the following subset of $\mathbb{R}^{3}$ \begin{equation} C=\{(x,y,z)\in\mathbb{R}^{3}\:|\:0\leq x\leq 1,\:0\leq y\leq 1,\:z=x^{2}+y^{2} \}. \end{equation} Intuitively, this looks like a differentiable $2$-manifold with boundary, the boundary being the points at $z=1$.

I am looking to parametrize $C$ using the least possible charts. I think that I'll need two.

I have tried over and over to find such charts, but since I have to get the boundary, the problem got a little hard. I could come up with something like (for most of the manifold, missing a point) \begin{equation} f_{1}\colon[0,1)\times[0,1)\to\mathbb{R}^{3}\qquad\qquad f_{1}(u,v)=(1-u,\:1-v,\:1-u^{2}-v^{2}) \end{equation}

But this one has the issue that $[0,1)\times[0,1)$ is not an open set of $\mathbb{H}^{2}=\{(x,y)\in\mathbb{R}^{2}\:y\geq 0\}.$

Any tips? Thank you all.

EDIT

I feel like my failure could be in assuming that I could do that with only two charts. Since the boundary is kind of circular, it doesn't really look like $\mathbb{H}^{2}$.

Answer 1

One chart is enough. Take $I=[0,1]$ and $$f:I^2\rightarrow\mathbb{R}^3:(u,v)\mapsto (u,v,u^2+v^2)$$.