Length, Angles, and Area on Surfaces Differential Geometry Reading Stream (with Proofs!) Episode 15

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Welcome back to all my new subscribers, thanks again for liking my videos!

That includes everyone who just listens to these, even passively in the background or while sleeping, people who just watch the intros, or people who just listen to the reading and skip all my rants. I appreciate all of you!

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Back with a long boi edit of almost an hour and a half of quite the differential geometry related content as shown by the timestamps below.

This is what a little over 2 hours of footage edited down looks like lol.

I want to keep getting practice from making more reading videos/have more followers before I start doing these live. Imagine watching 30+ extra minutes of mess ups, "um"s, "uhh"s, pauses, cat interruptions, drinking water, etc. I think these are better without all that jazz, but it would still be fun to mix live math streams with these more tightly edited ones.

I said last week I was going to do a casual/"just chatting" comment reading video and I didn't 😥
So I was going to catch up on reading comments and award the weekly viewer comment of the week at the end of this one but after a long filming sesh (and I hadn't edited it yet so I didn't know at the time if the footage I just shot was gonna edit down to a final cut that's more like 1.1 or 1.8 hours long) that's gonna be the bonus video later this week.

After the early poll results on my community tab, I decided to try splitting the longer videos like this into multiple vids and upload throughout the week. I'll keep doing more ~30 minute videos but I also like these long bois though because I can fit more sections in one video. 2 sections per video makes a more digestible 30 minute video but I can get through this book by the end of the year reading 3+ sections per video.

I hope you guys enjoy this one as much as I enjoyed making it!

Get the Kreyszig Differential Geometry book I've been reading in these videos: https://amzn.to/3U6OnbH

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TIMESTAMPS
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0:13 Rap intro (shoutout to @TravTheWhite)
1:00 Why bother learning differential geometry?
1:27 I mention this    • What's a Tensor?  
2:21 Ptolemy, Copernican Revolution
5:53 Newtonian Mechanics, Absolute Space and Absolute Time
7:28 Immanuel Kant has entered the chat; Space Absolutism (Newton) vs. Space Relationalism (Leibniz)
8:45 An attempt at a modern interpretation of Leibniz Space Relationalism
10:34 Kant’s take on Space per Transcendental Idealism in the Critique of Pure Reason
10:53 Kant’s 4 Antinomies of Reason
12:14 Kant’s 1st Antinomy of Space & Time
12:44 Back to Newtonian physics but non-inertial reference frames
15:19 Grad school flashback
16:35 General Relativity
18:55 3 Tests of GR
20:46 Beyond GR (?)
22:31 Appreciate the philosophical and mathematical situation modern physics is currently in
22:40 Differential Geometry and Tensor Calculus
27:13 My name is Gregorio Ricci-Curbastro but everybody calls me Ricci.
28:26 You chose to learn differential geometry the hard way (POV)
28:47 Starting to read § 33. Special Tensors
35:00 Problem 33.1
35:16 Solution to Problem 33.1
36:26 Problem 33.2
36:33 Solution to Problem 33.2
37:35 Problem 33.3
37:41 Solution to Problem 33.3
37:56 Problem 33.4
38:21 Solution to Problem 33.4
38:29 § 34. Normal to a surface.
42:31 Surface orientability
43:26 How to make a paper Möbius Strip
45:16 What was once the back & front of a piece of paper are now continuous on the Möbius Strip as shown by the different colored lines, hence the Möbius Strip is not orientable!
46:05 Back to reading about surface orientation in § 34
46:22 § 35. Measurement of lengths and angles in a surface
56:08 Theorem 35.1 Necessary & Sufficient condition for a coordinate system on a surface S to be orthogonal
59:08 Problem 35.1
59:20 Solution to Problem 35.1
1:02:42 § 36. Area
1:10:56 PROOF that the double integral formulation for area in the limiting condition of a portion of a plane is equivalent to the definition for plane areas.
1:13:20 Definition 36.1 Formula for the area A(H) of a portion H of a surface S as a double integral of a function in terms of the first fundamental form
1:16:34 § 37. Remarks on the definition of area.
1:21:34 Problem 37.1
1:21:59 Solution to Problem 37.1
1:22:36 Problem 37.2
1:22:47 Solution to Problem 37.2
1:23:36 Problem 37.3
1:23:44 Solution to Problem 37.3

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