Lecture summary (roughly this time, but will get more detailed in later lectures):

- Waffle about metamathematical content
- Rough overview of the course
- Group and field actions
- A field is something with 2 substructures: an additive abelian group and a multiplicative abelian monoid
- Tangent spaces
- Affine spaces

I am starting a series of notes based on MATH4033: Calculus on Manifolds, given by Prof. Guowu Meng at HKUST during Spring 2023-2024. This series will be mostly metamathematical, and will not contain substantial mathematical content, because I don’t understand 80% of the course well enough to write about it. Still, I go to the lectures because it’s fun, I actually do learn something, and because I can have a six-hour-long back-to-back lecture session twice per week, so I can fast during that period to reduce weight.

This series is titled ‘Calculus III’ because Prof. Meng calls this course that.

This will be written as a pseudo-socratic dialogue, except that it is one-sided. Meaning that I react to Prof Meng’s words, but not vice versa. The lines below ‘spoken’ by Prof. Meng (M) below are not exact transcriptions, and are in fact heavily paraphrased. The lines below ‘spoken’ by me (B) are mere thoughts.

M: Most agree that Einstein’s gravity is the best so far.

B: Bro said einstein like 愛因斯坦 in mandarin.

M: It is much easier to be a mathematician than a physicist. That is why I escaped from physics to maths. Problems in physics are too difficult, while those in maths is easier. When Poincare defined the Poincare conjecture, it was only in 3D. Stephen Smale was the first person to make significant progress. He proved it for dimensions >= 5. Physicists don’t care, but mathematicians will like anything beautiful.

(waffled for 30 min, say not waffle anymore and continued waffling for 7 min)

M: algebra is a secondary thing is maths, the core of math is physics, geometry and number theory

B: wow exactly the same as what I thought a few months ago, now that i think about it again maybe combinatorics should also be included

M: most multivariable calculus courses miss one thing about tangent space. On a curve, you have a tangent line, but not just a line, it is a line with a distinguished point. This gives us a vector space by point-vector correspondence.

M: reference point O is for observer

B: obviously physicist lmao a mathematician would say origin (right? idk)

M: between topological spaces there are, well not topological maps, continuous maps (chuckles)