Osculating Sphere Differential Geometry Reading Stream Episode 5

Описание: Thanks to everyone who subscribed from one of my recent uploads!!

Big proof energy this time around on the differential geometry stream.

I start off defining two math terms brought up by Eric Weinstein on a podcast and responding to some of your comments from the last video.

Also 1:11 EDIT: I found it hard to believe Riemannian Manifolds wouldn't be in the diff geo book I'm reading at all, but it turns out two-dimensional Riemannian Manifolds ("surfaces"/"spaces" in the book) are touched on in upcoming §27.

§19 like I said previously is one of/the most substantial section in this chapter as is the next section, §20.

Each is about 4-5 pages, so plus the introduction §19 gets a video of its own.

I hope you enjoy whether you watch or listen.

If you leave a comment I might react to it next time!

The book is Differential Geometry by Erwin Kreyszig, Dover Ed. and this video starts in §19 in Ch. 2 on pg. 49.
Get the book: https://amzn.to/3W2a5QT

When I get to 400 subscribers I'm gonna get a wireless rode mic because my microphone handling skills are whack!

If you guys get me to 10k subscribers, I'll start to do some differential geometry streams live 😎

TIMESTAMPS
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0:00 Eric Weinstein gatekeeping lol
0:11 Defining Riemannian Manifold, Line Bundle, Fiber Bundle, and Pencil (Geometric Bundle)
05:45 Reacting to a few comments on the previous video
09:09 You guys keep me in check 🤝🫡
11:33 Motivation for complex numbers, rant about the history of cubic roots during the early renaissance
(I mistakenly refer to a greek letter xi as “omicron” here…)
30:00 Starting to read §19 Contact, osculating sphere
36:11 Definition 19.1 Contact of order m of curves C, C*
43:35 Rant about Taylor Series, Taylor Formula, and Taylor Theorem
49:43 Back to reading §19
52:23 Definition 19.2 Contact of order m of curve C and surface S
53:13 Things return full-circle to the term “pencil” mentioned in the beginning lol
53:55 Theorem 19.1 with proof!
55:39 Lemma 19.2 with proof!
1:01:05 The trickiest math to read out loud in this section imo
1:11:54 Theorem 19.3 with proof!
1:27:04 Theorem 19.4