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Currently Playing: What are differential forms? | Differential forms #2

Second channel video on hairy ball theorem: https://youtu.be/TLHbOMNKtzw Second channel: https://www.youtube.com/@2maniac Next video: https://youtu.be/6B9eWS1GCR0 Previous video: https://youtu.be/dNT8OLNMsQ0 What are differential forms? How do we visualise them? There are two perspectives: "how it looks" and "what it does", to be explained in this video. 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: howitlooksvswhatitdoes Comment on orientation: you might’ve heard of Fubini’s theorem, which roughly states that you can integrate in whichever variable first. First off, the kind of integrals in Fubini are slightly different and are insensitive to orientations. In some vague sense, those integrals always go from left to right (x direction), and bottom to top (y direction). When you integrate with respect to x first, then y, i.e. dx dy, you integrate in the usual orientation, and the differential form has the usual orientation too. But when you integrate with respect to y first, then x, i.e. dy dx, then you integrate in the opposite orientation, but the differential form also has the opposite orientation, and the minus signs cancel out. Sources: 1) DFormPy library: https://arxiv.org/pdf/2201.10517 2) Joseph Mellor’s blog post: https://joseph-mellor1999.medium.com/an-intro-to-differential-forms-b3ec00d43d31 3) Duarte Maia’s blog post turned pdf: https://math.uchicago.edu/~dmaia/documents/visualizing_diff_forms.pdf which is in turn based on Dan Piponi’s pdf: http://yaroslavvb.com/papers/notes/piponi-on.pdf 4) Tristan Needham Visual Differential Geometry and Forms 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: https://www.facebook.com/mathemaniacyt Instagram: https://www.instagram.com/_mathemaniac_/ Twitter: https://twitter.com/mathemaniacyt Patreon: https://www.patreon.com/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!