Uploads from Mathemaniac

Currently Playing: Generalised Stokes' Theorem | Differential forms #3

Second channel video: https://www.youtube.com/watch?v=xPIOzXpKxNU Previous video: https://youtu.be/j79ihqK0-gE Why does the symbol ∂ denote both derivative and boundary? Is it a coincidence? Well yes and no... 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 Remarks: (a) This is true for *constant* v₁ and v₂, but in general, for manifolds, or when v₁ and v₂ are not constant vectors, but vector *fields*, one needs to add a term involving the commutator: -ω([v₁, v₂]). Since I don’t want to confuse people by using differential geometry jargon, I skipped this part in the video. (b) We’ve changed where the vector emanates from by flipping it, which incurs the same order of error as ε², so we’re not allowed to do this. However, dω is the integral of ω around the parallelogram - we’re “estimating” this integral by a normal Riemann sum, which isn’t accurate enough. One can use a “midpoint rule” or “trapezoidal rule” instead. More details in Needham’s Visual Differential Geometry and Forms. Sources: https://jeff560.tripod.com/calculus.html https://mathoverflow.net/questions/46252/is-the-boundary-partial-s-analogous-to-a-derivative John M. Lee, Introduction to Smooth Manifolds (pp. 384 -386) 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 Video chapters: 00:00 Intro 01:31 FTC (reprise) 03:50 Differential forms recap 06:12 0-form 09:12 1-form 14:12 2-form 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!