Another day to discuss the idea of arclength reparameterizations and do a little geometry.
John presented Shifrin 1.1.6. Mark presented Shifrin 1.1.8. I think this is our first real geometry theorem. Note that the result proves that the constant speed straight line gives the shortest curve between two points. That is a geometry result.
Then I presented on Shifrin 1.1.9. There are several challenging things happening here. First, you had to have a bit of a physics insight to get it started. Then, you had to find and solve a differential equation. I am not surprised you found this challenging. Don’t sweat it. As we practice we will get better at this.
You should know that “taking derivatives to form a differential equation, and then solving that equation” is a pretty common technique in differential geometry. It will come up again.
It is time to move on to the next topic. We will now start to study the Frenet-Serret apparatus. This is a neat construction of some objects related to a curve that will allow us to do a lot of geometry very cheaply. Basically, we will shortcut 200 years worth of ad-hoc work on curves and jump right to the “right system” for proving theorems. The first step is to learn to do the computations and see how they connect to some basic geometry with examples.