Відео

@harshavardhan9399 thanks for letting us know, Harsha. We will fix it for the next video (not the one of tomorrow, but next week). Let us know in future comments if we really fixed it 😎

@@christressler3857 exactly! And then we do it with all the local charts (and their intersections), also called local coordinates.

@@abhishekgy38 Your idea is partially correct… The metric tensor operates on vectors in the tangent space, but it doen’t directly “give” the infinitesimal distance. Instead, it “gives” the structure needed to calculate lengths of vectors and angles between them. Infinitesimal distance is derived using the metric tensor, but the tensor itself is a bilinear map that outputs a scalar when applied to two tangent vectors (like the inner product). So, it’s more accurate to say that the metric tensor encodes the geometric information required to measure distance and angles on the manifold

@@Systematizer we are glad that you like it. Please, let us know how to make it even better 😎

@@voyager8958 yessss we will do it ;)

@@advaithnair8152 Manifolds are typically studied over the field of real numbers because they are modeled on Euclidean spaces. But it’s also possible to define analogous structures over arbitrary fields, especially in the context of algebraic geometry. In this case, varieties and schemes generalize the concept of manifolds to work over fields like the complex numbers or finite fields. If you’re interested, we could make a video about it 😎

@@dibeos please do so.

@@MichelPham-z2x hi Michel, thanks for such a nice comment. It really encourages us to keep going! Please, let us know what kind of content you’d like to watch in our channel 😄

@@dibeos I'd like to see your approach about Real Analysis and Differential Geometry. If it is possible. Thanks

@you just named 2 of my top 3 areas in math. So, it will be a pleasure 😎 stay tuned

@@ValidatingUsername A manifold is not just the “surface” but an n-dimensional space that can locally resemble R^n. It can be embedded in higher-dimensional spaces, but “in” does not always imply embedding-it refers to the abstract space itself. For example, a 2D sphere is a 2-manifold, not its “surface”, and it can be considered in its own (without defining a higher dimensional space around it)

@@zubairkhan-en6ze thanks for the nice comment. Please, tell us what kind of content you’d like to see in the channel 😎

@@dibeos thanks for asking I like many things but ll list few of them. 1. How a differential geometry became a core subject for physicists? 2. Hyperbolic geometry and the structure of universe. 3. How algebra is linked with group theory? Especially in the context of continuous groups. 4. How we add structures to manifold and why we do so? 5. And many more ...

@@macroxela That’s right, cyclotomic fields are number systems based on roots of unity

@@paperclips1306 please let us know how we can make it easier to follow next time. We tried to simplify as much as possible, without sacrificing the “juicy” parts. What do you think we can do better? 😄

​​@@dibeos I think a great start is by knowing when you introduce a term that might be unfamiliar e.g euclidean space, parameterize etc so that you can clarify it.This could help reduce confusion and ambiguity

Animation is good too

Likee, hmm maybe like make a mission in start. Give a question. Then like explore ideas. Then like go on blowing minds.. etc etc.. Those are quite engaging....

@@ishannepal3146 thanks for the tips! We are slowly getting better 😄 we will fix what you told us in the next videos 😎

@@drybowser1519 thanks!!! Let us know what kind of content you are interested in 😎

It can be. Any manifold can be embedded into Euclidean space (Whitney embedding theorem)

​@@pavlenikacevic4976why do we need a theorem. The question is obvious anything that has (x,y,z) is embedded in 3D right?

@@paperclips1306 not anything. If you don't specify what you mean by (x,y,z) those can be three "any thing". (Alice, Bob, Mark).

@@sanjeevsoni4962 that’s the whole point of Differential Geometry. A curved manifold is not necessarily embedded in 3D to exist. Of course, it is reaaally hard to imagine it, but a manifold can be a space on its own. That’s the same reason (just as an example) why it is hard to understand the Big Bang in physics. People often ask: “but if the universe (4D manifold) expanded from a “point”, where did it expanded in? What happened around this point?” This question doesn’t make sense because the universe is a manifold on its own, and as far as we know this 4D manifold is not embedded in another higher dimensional one. In Diff Geom we can describe the curvature of a space without relying on an external Euclidean space (x,y,z,…). Another example that might convince you is the fact that a plane (2D) can be completely described without relying on the definition of an external 3D space around it. In other words, manifolds are spaces in their own.

Not having to depend on the coordinates of an external/extrinsic space is very useful, in physics as well as when using manifolds for computation for instance.

It was a pain having to go through all of this when I started studying Physics way back in the day. I was just a first semeter student trying to get my bachelor and went to the uni library to read the books on topology and differential geometry. This stuff kept my up at night.

differential geometry in your first year at uni?

@@imPyroHD because I wanted to study it. I thought it would be great to be ahead in the course.

@@imPyroHD I too studied these by myself first as I was too eager to learn this. Uni introduced this a bit later.

@@stefan-danielwagner6597 thanks for the nice words!!! They mean a lot to us!! 😎

A small suggestion: Consider placing the mic 🎙 at a fixed spot, thus freeing up both your arms, and nobody will be distracted by the bobbing mic throughout the video. 👍🏻

@ thanks for the suggestion, we really appreciate it. We will think of something. Let us know what you think in the next video (this Saturday)

@@joelmarques6793 thanks for the awesome comment! As I said in another comment, these are one of our favorite subjects. So we will post more videos about them!!

@@dibeos I am looking foward to seeing it!

@@GaelSune thank you. Let us know what else you’d like us to post about 😎

@@dibeos Sure! would like to know more about Topology in general and maybe something about calculus of diferences. Thank you!!!

@@user-wr4yl7tx3w thanks!! Differential Geometry is one our favorite areas in math. Let us know what else you’d like us to post about :)

@alexanderbalka1074 well please let us know what was not clear. We want to improve our explanations for future videos

@@bradzoltick6465 hi Brad, thanks for letting us know!! We will make and post it soon 😎👌🏻

@@tinkeringtim7999 yeah, some practical applications would be nice. Well, thanks for letting us know. We will fix that in future videos 😎

@dibeos I'm subscribed so hope to catch them! I like your focus on clarity/simplicity. I'm still undecided if groups are a great way to access the fundamental essence of mathamatics, or have been a great way to obscure and cover that essence.

Thanks Ashutosh

@polyhistorphilomath hahaha yeah, after rewatching the video I also noticed that. Weird, but… I guess you can find a beautiful metaphorical meaning for it

Thanks Davide, let us know how what else can we post about in order to save your algebra exam 😎

@@OpPhilo03 that’s exactly what Sofia and I are doing! Almost all of the videos in our channel related to Group Theory are chapters of the book 😎

@@graf_paper thanks for the awesome words!! There is more coming. We will build a huge library of math videos and pdf summaries so that people can learn math subjects much faster and with less “suffering”. Thanks again 😎

@@al-carissimi hi Alexandre, thanks for letting us know! We will definitely post more videos about abstract algebra! 😎

@@plranisch9509 thanks for letting us know! We will definitely do it!!

@@sabelojupiter6081 thanks for the nice comment. Yeah, it is a solid concept that is flexible enough to appear in many different forms

@brian.westersauce thanks for the clarification, Brian! I (Luca) am from Brazil and Sofia is from Ukraine (she grew up in the US though). We’ve been living in Italy for a few years now.

@@dicipuluscaptiosus thanks for the nice words!!! Slowly but surely we will get there.

Wow, thanks for the nice comment. Yeah, it would be a pleasure. I actually had some classes about the mathematics of the Calabi-Yau manifold in my masters. It is really fascinating! We will definetely do it!

@@dibeos Thank you! I'm not professional, just trying to understanding the basics and the intuition behind them!

Also thumbnail is a good cheat sheet for fast remembering the topic❤

@@SobTim-eu3xu thanks for the nice comment!!! More amazing videos on primes (and number theory in general) are about to come

@dibeos awh, such a miracle and great wish)

@GuillermoSV thanks for the nice comment! We will post another one soon ;) But you can check out this one, it’s technically the next part ua-cam.com/video/pNlcQ0Tx4qs/v-deo.html