Covariant derivative by Ciro Santilli 37 Updated 2025-07-16
A generalized definition of derivative that works on manifolds.
TODO: how does it maintain a single value even across different coordinate charts?
EdX by Ciro Santilli 37 Updated 2025-07-16
As of 2022:
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Fuck that.
Also, they have an ICP.
November 2023 course search:
Udacity by Ciro Santilli 37 Updated 2025-07-16
It is a shame that they refocused to more applied courses. This also highlights their highly "managed" approach to content creation. Their 2022 pitch on front page says it all:
for as few as 10 hours a week, you can get the in-demand skills you need to help land a high-paying tech job
they are focused on the highly paid character of many software engineering jobs.
But one cool point of this website is how they hire tutors to help on the courses. This is a very good thing. It is a fair way of monetizing: e-learning websites must keep content free, only charge for certification.
Linear operator by Ciro Santilli 37 Updated 2025-07-16
We define it as a linear map where the domain is the same as the image, i.e. an endofunction.
Examples:
Solving differential equations was apparently Lie's original motivation for developing Lie groups. It is therefore likely one of the most understandable ways to approach it.
It appears that Lie's goal was to understand when can a differential equation have an explicitly written solution, much like Galois theory had done for algebraic equations. Both approaches use symmetry as the key tool.
Advantages over Riemann:
Video 1.
Riemann integral vs. Lebesgue integral by The Bright Side Of Mathematics (2018)
Source.
youtube.com/watch?v=PGPZ0P1PJfw&t=808 shows how Lebesgue can be visualized as a partition of the function range instead of domain, and then you just have to be able to measure the size of pre-images.
One advantage of that is that the range is always one dimensional.
But the main advantage is that having infinitely many discontinuities does not matter.
Infinitely many discontinuities can make the Riemann partitioning diverge.
But in Lebesgue, you are instead measuring the size of preimage, and to fit infinitely many discontinuities in a finite domain, the size of this preimage is going to be zero.
So then the question becomes more of "how to define the measure of a subset of the domain".
Which is why we then fall into measure theory!

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