Infinite cylinder covered by one chart

[beans73]

Hi there. I have just been doing some reading on manifolds, and I'm finding some of it hard to grasp. An example we were given was of the infinite cylinder and how to construct its map, and that it would be with one chart. the reading initially says that we can use θ \in (0,2pi] and z \in (-\infty, +\infty). It goes on to give two examples of maps that can be used:

1. ρ=tan^{-1}(z) + pi \in (pi/2, 3*pi/2)
2. ρ=e^{z} \in (0,\infty)

I was wondering if anyone could help me understand a little more of what's going on here?

 

[voko]

What is ρ?

 

[beans73]

the following attachments are the working out I'm talking about...

 

[voko]

I could not understand some words (I am very bad at reading anything hand written - even if written by myself!) but I think I understand that in the first case, the cylinder is paramaterized via a map from an annulus (ring), in another from a plane without the unit circle at the origin. What difficulty are you having with that?