Class summary: 15 Sept.
Today in class I talked about the homework mechanics:
1. Please start each problem on a new page -- this will make it easier to add and respond to
comments.
2. You will have at least 2 weeks to work on a problem and receive full credit. Also, any
problem with MORE than 3/4 circle when we finish it in class will also receive full credit if you
eventually complete it.
3. Complete all the problems and you will have earned an A for the course.
4. If you turn in work by Wednesday evening, then (most times) you will receive it back with
comments in class Thursday.
3.1. Remember that angle is a local notion – it only pertains to what is happening near the vertex
of the angle. Thus “rays” (extending infinitely in one direction) are and not appropriate
(especially on the non-planar surfaces). It is better to think in terms of “directed line segments”
or short “vectors” (length and direction). We can talk about intrinsic vector on spheres, cones,
and cylinders; but they do not form a vector space on these surfaces. Also, remember that a
(small) local part of a cylinder is locally isometric to the plane – that is, has the same metric or
geometry as a small patch on the plane. The same applies to cones except at the cone point.
5.1. We are considering the annular hyperbolic planes to extend indefinitely in all directions. To
get more of an idea of this looks like, visit the Tompkins County Library (at the Green Street bus
stop) to see an exhibit of large (and symmetric) hyperbolic planes by Daina Taimina.
5.3. The pseudosphere is to the hyperbolic plane as a cone or cylinder is to the Euclidean plan.
The pseudosphere is a surface of revolution formed by rotating the graph of a function r=f(r)
about the z-axis. You can find an equation for z as a function of r but for r as a function of z.
But, you can still show that f(r) is a continuously differentiable function using results you first
learned in 1st semester calculus or the Inverse Function Theorem.
5.4c. We discussed in small groups the definition of “rotation” – we all know what it is but it is
difficult to define simply.