beans73
- 12
- 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 [itex]\in[/itex] (0,2pi] and z [itex]\in[/itex] (-[itex]\infty[/itex], +[itex]\infty[/itex]). It goes on to give two examples of maps that can be used:
1. ρ=tan[itex]^{-1}[/itex](z) + pi [itex]\in[/itex] (pi/2, 3*pi/2)
2. ρ=e[itex]^{z}[/itex] [itex]\in[/itex] (0,[itex]\infty[/itex])
I was wondering if anyone could help me understand a little more of what's going on here?
1. ρ=tan[itex]^{-1}[/itex](z) + pi [itex]\in[/itex] (pi/2, 3*pi/2)
2. ρ=e[itex]^{z}[/itex] [itex]\in[/itex] (0,[itex]\infty[/itex])
I was wondering if anyone could help me understand a little more of what's going on here?
![Handwritten notes with a small sketch of a cylinder's z axis and angle theta, followed by text defining theta in (0, 2π] and z in (-∞, +∞), and a mapping of theta to phi and z to a radial coordinate rho = tan^-1(z) + π in (π/2, 3π/2).](/data/attachments/43/43605-1b74218b85b4a14d7d3391be35612697.jpg?hash=G3Qhi4W0oU)
![Handwritten notes on a topology argument: a small sketch of concentric circles forming an open annulus, a polar-coordinate diagram with angle phi marked, and the line Another simple possibility is to take Lambda = e^z, z in (0, infinity), and phi = theta in (0, 2pi], which maps to an open annulus with zero inner radius and infinite outer radius.](/data/attachments/43/43606-f8037a0d3a54f7a869fee637e30b6cc8.jpg?hash=-AN6DTpU96)