If $U$ is open in $\mathbb{R}^2$ and $f: U\to \mathbb{R}$ is $C^{\infty}$, then $S=\{(x,y,z)\in\mathbb{R}^3:z=f(x,y)\}$ is a surface with a single atlas$(\mathbb{R}^2, Id_{\mathbb{R}^2})$.
I am beginning to study and surfaces and I am trying to understand many things that I had not seen before. So I would like someone to explain to me why what I put up is a surface with all the details, thank you very much.