This post covers two class meetings.
Today I handed out project descriptions for surfaces. Also, we started talking about the tasks in Shifrin 2.4 about the covariant derivative and geodesics.
Our discussion started with student presentations today: - Jesse M did Shifrin 2.3 #6 - Corbyn did #7a - Christine did #7b - Emily did #7c…sorta.
Sladana presented Shifrin 2.3# 2b, Emily presented 2.3#12. This gave me the opportunity to talk about Gauss’ Theorema Egregium. This is a central result of classical differential geometry, and one of the motivating results for modern differential (Reimannian) geometry.
Today Mark did Shifrin 2.3#5.
Sladana presented her project on the Bishop Framing and parallel curves. Christine finished her discussion of Shifrin 2.3#1 and Emily finished 2.3 #2c.
Mark shared with us his work on a project to describe curves which live on quadric surfaces by a differential equation using only curvature and torsion.
Emily presented about the tangent spherical image of a curve. This is a bit like a curves version of the Gauss map.
Christine presented on Bertrand mates.
Corbyn gave the presentation on his project for curves. This bit about treating the framing as a rigid body and studying the rotational motion as a physicist does is neat. So, we learned about the concept of areal velocity and saw the Darboux vector. By the way, Gaston Darboux was an important French geometer.
Today, Jesse M presented his project on the theorems of Fenchel and Fary-Milnor.
Today we had four presentations from Shifrin 2.2.
We got down into the details! Presentations from Shifrin section 2.2:
Back from break, and back to work.
As the week ended, we lost all momentum and didn’t get a lot done. But most of you turned in project papers, and that is a good thing.
Jesse Started work on Shifrin 2.1 #14. (I think he finished (a).) We started work in class on exercise 12.
Today we talked more about basic geometry on surfaces. We had lots of opportunity to talk about maps and the way that they distort things. An important distinction that came up is the difference between bending a surface around in space (which won’t change the inner geometry) and the actual curvature of a surface.
We discussed more of the exercises from Shifrin on the basics of parametrized surfaces.
More work on basics of surfaces. Emily did Shfrin # 4a; Corbyn did # 7. We talked for a bit about the details of that last one. Some of the arguments are subtle.
More time to discuss the basics of surfaces. From Shifrin: Corbyn did # 2, Sladana did # 3a,b; Corbyn did # 4b.
Today we began our study of surfaces. We talked about the basic picture of a parametrized surace: with \(u\)-curves, \(v\)-curves, tangent vectors, tangent plane, normal vectors, and the first fundamental form.
Today was our last day of talking about only curves as a class. We discussed how integration (like when computing arclength) really is a difficult problem, and we have made up names of functions to describe some integrals that we understand. Names like \(\ln x\), \(\sin x\), and \(e^x\). Many of you ran across “elliptic integrals” and these are similar, but of the next level of complextity.
Short discussion of Involutes and Evolutes today. I hope we can finish this up and move on to surfaces next week.
Today we took time to talk a bit more about the detail for curves. Emily finished Struik \(S\)1-6 #11 and we learned the new vocabulary word vertex. A vertex is a point where \(\kappa’=0\), and this connects with the task Emily did by seeing the radius of the osculating circle seems to be not changing at that point.
Today was a bit slow. So, I talked a little bit about the Fundamental Theorem of Curve theory, which is really a corollary to the Existence and Uniqueness theorem for ordinary differential equations. The idea is that the curvature and torsion functions are all the information we need to write down the Frenet-Serret equations. And those are a coupled system of ordinary differential equations! In the end, that means that knowledge of \(\kappa\) and \(\tau\) is enough to completely determine the curve, up to some rigid motion of Euclidean space.
Exam today on the basic computational aspects of curve theory.
I spent some time discussing the intricacies of the argument that justifies calling the osculating plane the “plane through three consecutive points.”
Today we solved many problems by using the Frenet-Serret apparatus and equations. The fun part is to see how quickly we can get real geometry theorems using this tool.
Today we discussed two exercises from Shifrin. Corbyn did 1.2.11, which was a patient and careful computation. Mark did 1.2.8, which showed that a curve which has all of its normal lines through a fixed point must be a planar circle. This used all three of the “big tricks” from classical differential geometry. In order.
Today Emily asked me to show how I start to think about one of these tasks. There is value in watching an “expert” figure stuff out, so I did it. The task in question was exercise 1.2.10 from Shifrin. This ended up taking the whole hour, so we didn’t do anything else. I’ll put a copy of my “solution” below, but it doesn’t have the thinking part in it.
We spent some time in discussion today where the students checked their work on lots of example calculuations together.
We only had two presentations today. Jesse T did 1.2.2c from Shifrin and Corbyn did 1.2.3a.
We started working through computations of the Frenet framing of a curve. This is a set of vectors \(T, N, B\) defined at each point of the curve. The best way to think of them is as a moving frame. The basic geometry of a curve will be encoded in the way this frame moves around.
Another day to discuss the idea of arclength reparameterizations and do a little geometry.
We made some more progress today on parameterized curves.
We first talked a little bit about “leveling up,” and about how to read mathematics effectively (use a pen; work along; do extra examples by yourself; reread multiple times).
For our first meeting, we considered two problems about parameterized curves.
This is the class blog. I’ll have more to say later.