The cylinder is given by $C=\{(x,y,z):x^2+y^2=1\}$. I want to prove that $C$ iis diffeomorphic to $\mathbb{R}^2-{(0,0)}$.
I have come up with a possible function, but I've been unable to prove it is surjective. The function I defined is
$$f(x,y)=\left(\frac{x}{\sqrt{x^2+y^2}},\frac{y}{\sqrt{x^2+y^2}},xy\right)$$.
By definition of $f=(f_1,f_2,f_3)$ is clear that $f_1^2+f_2^2=1$. The function is also injective because if $f(x,y)=f(w,z)$ then $$ \frac{x}{\sqrt{x^2+y^2}}=\frac{w}{\sqrt{w^2+z^2}} \\ \frac{y}{\sqrt{x^2+y^2}}=\frac{z}{\sqrt{w^2+z^2}}\\ xy=wz $$
The product of the first two imply $\frac{xy}{x^2+y^2}=\frac{wz}{w^2+z^2}$, since $xy=wz$ both denominators are the same, then applying this two the first two equation follows $x=w,y=z$.
Surjectivity is, as I said before, a problem. To find coordinates for $(a,b,c)$ I need to set $c=xy$, which doesn't seem to have enough freedom to allow me vary $x$ and $y$ as much as I would like to generate all the possible $a,b$.
On the differentiability, each component function is a composition of continuous differentiable functions in $\mathbb{R}-\{(0,0)\}$, so $f$ is differentiable.