@@dukenukem9770 well, we are planning a series of videos on diff geom and tensor calc in which we will give concrete examples of how to measure distances. 😎
@willyh.r.1216 we really appreciate your support, Willy! Just because you asked us we will keep going… 😉 The next video will be about the Core of Linear Algebra. But we added your suggestions to our list of ideas 😎
You could make a video on elementary estimates of prime counting function due to Erdos and on Schnirelmann’s theorem proof(which is again elementary). Also u could introduce polynomial methods in combi (especially stuff like Alon’s combinatorial nullstellensatz and its beautiful applications)
@Barry-p8u wow these are cool ideas. We do need to research these topics in order to learn more, but thanks for letting us know! We will include them in our list of ideas 😎
i personally find the pdfs a bit easier to read first . i don't know if i'm not used to accent or the pace or just too much new things, but thank you for providing them! I feel like you are gradually taking an ml guy who's trying to learn about dynamical systems to differential geometry and beyond!
It’s actually a misconception that the manifold isn’t embedded in an external space, i.e. that the external space “isn’t there.” The manifold IS embedded in an external Euclidean space, so long as one is invoking the Levi-Civita connection. (Different connections imply embeddings in more abstract geometries). It’s simply that one doesn’t need to explicitly reference that external space - the reference remains there, implicitly, through the connection.
@@pseudolullus but that’s EXACTLY one of the next immediate steps in the explanation! You are absolutely right. Variational calculus is the key to find the geodesics, especially when the manifold is too wired (too warped and in very higher dimensions)
@@joeeeee8738 yeah, with the Manhattan distance geodesics aren’t smooth curves like in the Euclidean case. In this case, they are piecewise linear paths, following only horizontal and vertical segments. The shortest path minimizes the sum of absolute differences along each axis, rather than the straight-line distance. It’s really cool to see how different metrics reshape our view of “natural” paths (geodesics or not)
@@ЗакировМарат-в5щ thanks for the feedback. Could you please let us know exactly what was too complex in the example? Because this way we can improve for the next one 😎
Notice you said 'spit'... If I ranked this video over all of your videos, some of which are truly great, this video would always be at the bottom. Buried the lead, unintuitive tangent space with unembedded angle left without explanation until way later, lots of hand waving, and references dropped but no real intuitive connection or explanation made. Please try again, but ask what are you trying to explain here, and stay focused on that without claiming the proofs won't fit in the margin.
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I’d love to learn how the metric is applied to the tangent space, thank you!
@@raajchatterjee3901 thanks for letting us know!!! We will publish a video about Riemannian metric
I am interested in seeing your take on how the Riemannian metric provides a measuring stick for each tangent space via the inner product.
@@dukenukem9770 well, we are planning a series of videos on diff geom and tensor calc in which we will give concrete examples of how to measure distances. 😎
Please keep up with producing more visualized high level math topics.
@willyh.r.1216 we really appreciate your support, Willy! Just because you asked us we will keep going… 😉 The next video will be about the Core of Linear Algebra. But we added your suggestions to our list of ideas 😎
You could make a video on elementary estimates of prime counting function due to Erdos and on Schnirelmann’s theorem proof(which is again elementary). Also u could introduce polynomial methods in combi (especially stuff like Alon’s combinatorial nullstellensatz and its beautiful applications)
@Barry-p8u wow these are cool ideas. We do need to research these topics in order to learn more, but thanks for letting us know! We will include them in our list of ideas 😎
i personally find the pdfs a bit easier to read first . i don't know if i'm not used to accent or the pace or just too much new things, but thank you for providing them! I feel like you are gradually taking an ml guy who's trying to learn about dynamical systems to differential geometry and beyond!
It’s actually a misconception that the manifold isn’t embedded in an external space, i.e. that the external space “isn’t there.” The manifold IS embedded in an external Euclidean space, so long as one is invoking the Levi-Civita connection. (Different connections imply embeddings in more abstract geometries).
It’s simply that one doesn’t need to explicitly reference that external space - the reference remains there, implicitly, through the connection.
I feel like it's a philosophical debate more than mathematical. One could always embed a manifold if that helps.
your videos are great. please do a video about integration on manifolds
@@6388-s2n thanks! Yeah, it will be a pleasure 😎
0:24 We can always try to minimize the path length with variational calculus even if the surface is curved :P (I know, that's not the point here)
@@pseudolullus but that’s EXACTLY one of the next immediate steps in the explanation! You are absolutely right. Variational calculus is the key to find the geodesics, especially when the manifold is too wired (too warped and in very higher dimensions)
@dibeos crap, I had to stop the vid and I thought you were going to go on a slightly different way
8:44 dt under the root is wrong.
Beautiful!
What if the distance is not quadratic? I.e. the Manhattan distance?
@@joeeeee8738 yeah, with the Manhattan distance geodesics aren’t smooth curves like in the Euclidean case. In this case, they are piecewise linear paths, following only horizontal and vertical segments. The shortest path minimizes the sum of absolute differences along each axis, rather than the straight-line distance. It’s really cool to see how different metrics reshape our view of “natural” paths (geodesics or not)
Please add actual examples if possible!
More simple but practical examples would be awesome.
@@ЗакировМарат-в5щ thanks for the feedback. Could you please let us know exactly what was too complex in the example? Because this way we can improve for the next one 😎
Another coincidence; was finding ways to explain geodesics and I guess this adds to the list 😅
Sir from Which country You people are originally ??
@TopeshMitter I (Luca) am originally from Brazil and Sofia was born in Ukraine. We are married and have been living in Italy for a few years now 😎
Please make a detailed Research Video on Grothendieck it will be a nice Video and also helpful for your channel .
Oh, her accent is straight from Ukraine 🤣
@ actually, she grew up in LA, California 😅 that’s why
don't be so formal. It is not good idea for teaching beginners. Use simple methods. Your method is not good for beginners.
@@specialrelativity8222 thanks for the advice! Let us know if we fixed that in future videos 😉
Notice you said 'spit'... If I ranked this video over all of your videos, some of which are truly great, this video would always be at the bottom. Buried the lead, unintuitive tangent space with unembedded angle left without explanation until way later, lots of hand waving, and references dropped but no real intuitive connection or explanation made. Please try again, but ask what are you trying to explain here, and stay focused on that without claiming the proofs won't fit in the margin.
I feel like you just spit on the comment section for some reason 😒