Infinite cylinder covered by one chart

  • Thread starter Thread starter beans73
  • Start date Start date
  • Tags Tags
    Cylinder Infinite
AI Thread Summary
The discussion focuses on understanding the mapping of an infinite cylinder using a single chart in manifold theory. It highlights the parameters θ and z, where θ ranges from 0 to 2π and z spans from negative to positive infinity. Two mapping examples are provided: ρ=tan^{-1}(z) + π and ρ=e^z, each representing different parameterizations of the cylinder. The user seeks clarification on the meaning of ρ and the overall mapping process, particularly how the cylinder is parameterized from an annulus and a plane. The conversation emphasizes the complexities of grasping these concepts in manifold theory.
beans73
Messages
12
Reaction score
0
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?
 
Physics news on Phys.org
What is ρ?
 
the following attachments are the working out I'm talking about...
 

Attachments

  • manifold1.jpg
    manifold1.jpg
    27.5 KB · Views: 1,063
  • manifold2.jpg
    manifold2.jpg
    21 KB · Views: 1,035
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?
 
Thread 'Help with Time-Independent Perturbation Theory "Good" States Proof'
(Disclaimer: this is not a HW question. I am self-studying, and this felt like the type of question I've seen in this forum. If there is somewhere better for me to share this doubt, please let me know and I'll transfer it right away.) I am currently reviewing Chapter 7 of Introduction to QM by Griffiths. I have been stuck for an hour or so trying to understand the last paragraph of this proof (pls check the attached file). It claims that we can express Ψ_{γ}(0) as a linear combination of...
Back
Top