Why care about differential forms? | Differential forms #1

Описание: Second channel:    / @2maniac  

dx is not just something you put in integrals - it has a very specific meaning! But the story of how we got to that specific meaning is quite interesting and tells you how differential forms hold the secret to many areas of maths and physics.

This is more to document the history of differential forms and how concepts came to be. If I need to talk about differential forms in isolation of the history, I wouldn't have motivated differential forms like that. But historically, the reasons for studying differential forms have been quite "algebraic" in the sense that it involves many computations, and that's what I'm presenting here.

This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject.

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Sources:
1) Early stages of development: https://tinyurl.com/3x2cxwxw
2) Later stages of development: https://tinyurl.com/4y48vhka
3) Cartan’s moving frame formalism: https://arxiv.org/pdf/1810.03872
4) Joseph Mellor’s blog post:   / an-intro-to-differential-forms  
5) Duarte Maia’s blog post turned pdf: https://math.uchicago.edu/~dmaia/docu...
which is in turn based on Dan Piponi’s pdf: http://yaroslavvb.com/papers/notes/pi...
6) Tristan Needham Visual Differential Geometry and Forms

Especially for the Poincaré lemma part of the series, none of the last three sources really addressed that, even though that was the motivation for studying differential forms in the past. But there is a very satisfying visualisation to that lemma.

Further reading (I came across but didn’t use in this video):
a) History of differential forms by someone else: https://www.math.toronto.edu/mgualt/w...
b) Terry Tao’s differential forms: https://www.math.ucla.edu/~tao/prepri...

Credit:
i) Élie Cartan portrait: Fair use, https://en.wikipedia.org/w/index.php?...

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