Why care about differential forms? | Differential forms #1

Second channel:    / @2maniac   Next video:    • Alternative view of integration | Differen...   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. Files for download: Go to https://www.mathemaniac.co.uk/download and enter the following password: tensorsdonottransform 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?... Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels: https://forms.gle/QJ29hocF9uQAyZyH6 If you want to know more interesting Mathematics, stay tuned for the next video! SUBSCRIBE and see you in the next video! If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos. Social media: Facebook:   / mathemaniacyt   Instagram:   / _mathemaniac_   Twitter:   / mathemaniacyt   Patreon:   / mathemaniac   (support if you want to and can afford to!) Merch: https://mathemaniac.myspreadshop.co.uk Ko-fi: https://ko-fi.com/mathemaniac [for one-time support] For my contact email, check my About page on a PC. See you next time!