The better proof of hairy ball theorem (and generalisations) скачать в хорошем качестве

The better proof of hairy ball theorem (and generalisations)

Main channel:    / @mathemaniac   This second channel is mainly for more novel topics that are not usually covered in a curriculum. Frankly, even though I still like making videos of the type currently covered in the main channel, I like the type of videos on this second channel much more, so I would love to grow the second channel to find the right audience this time around. You might have seen a proof of the hairy ball theorem, but do you know there is a very elegant proof of the more general Poincaré-Hopf theorem by Hopf himself? This proof is more natural and elegant, and gives a lot more information on vector fields on a sphere, while not requiring much more effort (arguably less!). This is why I love this genius proof! Files for download: https://www.mathemaniac.co.uk/download Remarks for the video: (a) 2 things: i) to define an index, the zero has to be “isolated”, i.e. for sufficiently small region around the zero, we can encircle only 1 zero. ii) I haven’t been very specific about the path to take to encircle the zero, or the vector field that indicates the reference direction; yet the definition of index relies on them! In fact, it should depend on these things continuously. However, α can only change by a multiple of 2π, so the index has to be an integer. When you alter the path and the reference direction, if the index jumps, then it doesn’t change continuously, so the index has to remain the same value. As long as the path only encircles 1 zero, and the reference direction doesn’t have a zero inside the path, the index doesn’t depend on the details of them. (b) You might be wondering why you can’t use this argument to prove that all index sums are 0. The reason is that the reference direction might be useful (i.e. has no zeros) in a particular region, but not everywhere on the sphere. With the reference direction potentially changing across regions, the argument doesn’t work for the reference vector field. (c) This argument works for surfaces that are compact (so that there are only finitely many isolated zeros), orientable (all edges can cancel out), and have no boundary (all edges can cancel out). (d) This is not the original proof of the Poincaré-Hopf theorem, which was back in 1926. Hopf was giving this proof specific to 2 dimensions during a lecture series in 1946. In fact, I spent too long trying to think about how this argument can generalise. Sources: 1) Original proof in Heinz Hopf (1956), Differential Geometry in the Large (the part I am animating appears in the first half of the book, which was in a 1946 lecture series) 2) I read about this in Tristan Needham’s Visual Differential Geometry and Forms 3) I used this for double torus: https://math.stackexchange.com/questi... Further reading (I came across but didn’t use in this video): (1) Another proof from a Mostly Mental video:    • Hairy Ball Theorem - Combing a Coconut   (2) 3Blue1Brown video:    • The Hairy Ball Theorem   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!