I have to proof that the circular cylinder $M=\{(x,y,z)\in\mathbb{R}^3\mid x^2 + y^2 = r^2\}$ is a regular surface, where $r$ is a constant, $r>0$.
Then I have to see also that $\mathrm x\colon \mathbb{R}^2 \to \mathbb{R}^3, \mathrm x(u,v)= (r \cos u, r\sin u, v)$ is a parameterization of $M$ when we restrict the domain of definition to a suitable open $U$ in $\mathbb{R}^2$ . What part of $M$ covers? Find another parameterization that, along with it, cover to all $M$.
With parameterization I mean:
$(i)\,U$ in $\mathbb{R}^2$ is open and $\mathrm x(U)$ in $M$ is open.
$(ii)\,\mathrm x\colon U \subseteq\mathbb{R}^2 \to \mathbb{R}^3$ is differentiable.
$(iii)\,\mathrm x\colon U\subseteq \mathbb{R}^2 \to x(U)$ in $M$ is an homeomorphism.
$(iv)\,\mathrm x$ is regular.
The hardest part for me is the first one, to see if is a regular surface. And in order to see if is a parameterization the part (ii) and (iv) I know how to do.
Thank you.
I think you can calculate the Jaccobi matrix of $r$: $D(r)$. And it is easy to see that the rank of Jaccobi $\geq{2}$. That is to say, $r$ is an immersion. Then I think you can use the property of immersion to prove this.