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@@romainmorleghem4132 Yea, a tangent vector can be viewed as a functional in certain contexts, specifically when defined as a derivation. In diff geom, a tangent vector at a point can be thought of as a linear map (or functional as you said) acting on the space of smooth functions around the point

@whdaffer1 yesss. If you check our videos on manifolds you will find it there. But it’s basically a local coordinate system that “flattens” the manifold. Let us know how we can help 😎👍🏻

@@VittoriaPasolini-ne4pm Ciao ancora Vittoria hahah allora, ti spiego qua quello che ho risposto nell’altro commento, ma lo faccio in italiano. Dimmi se ti torna adesso: Il motivo per cui mappiamo la varietà nello spazio euclideo è una conseguenza della definizione stessa della derivata. Per calcolare una derivata, abbiamo bisogno di un modo per misurare i cambiamenti lungo segmenti rettilinei (pensa al concetto di limite). Su una varietà, che è curva e non necessariamente incorporata in uno spazio euclideo di dimensione superiore, non esiste un modo intrinseco per definire segmenti rettilinei o distanze tra due punti nel caso generale. Senza questi, la derivata non è ben definita. Mappando la varietà nello spazio euclideo tramite un local chart (phi), "appiattiamo" temporaneamente una piccola regione della varietà. Ciò ci consente di utilizzare gli strumenti familiari del calcolo (limiti, derivate, ecc.) nell'impostazione euclidea. In altre parole, ora abbiamo "segmenti rettilinei" per misurare le distanze tra due punti, il che è necessario nella definizione della derivata (di nuovo, pensa al limite)

@@dibeos grazie per la risposta esaustiva! Io ho fatto ingegneria, quindi la mia elasticità matematica è pari a 0! Credevo che tra le mille diavolerie dei matematici ci fosse anche il modo di definire le derivate in spazi curvi a milledimensioni! Non c'era un corso di differenziale che io ricordi, qualcosa per chi faceva cristalli o materiali, mi pare...l'approssimazione a livello "locale" di spazi curvi a piatti, lo vidi fare solo nel corso di relatività generale, che non seguivo, ovviamente, dove c'era tanta differenziale, tensori di curvatura, di Ricci, ecc ...roba che mi è subito uscita dalla testa, ovviamente!

@ certo, capisco perfettamente… allora, ciò che io e Sofia stiamo cercando di fare in questo canale è “aprire le porte” della matematica pura (e un po’ della fisica matematica) a persone che hanno già una certa base di matematica, ma vogliono approfondire ancora di più. Quindi, ogni volta che spieghiamo qualcosa nei nostri video che non sia abbastanza chiara, dimmi pure! Così possiamo migliorare le nostre spiegazioni nei prossimi video 😎👍🏻

@@dibeos seguiro' sicuramente! Nei video in italiano, c'è un professore di Liceo, Arrigo Amadori, forse piu "pazzo" di voi, che fece 8 sabati pomeriggio a spiegare la geometria di riemann ai "muratori", rendendolo comprensibile per'altro...ve lo lascio qui... th-cam.com/video/7mCHzvE2pJw/w-d-xo.html

@VittoriaPasolini-ne4pm Great question. The reason we map the manifold into a Euclidean space is a consequence of the very definition of the derivative. To compute a derivative, we need a way to measure changes along straight-line segments (think of the concept of a limit). On a manifold, which is curved and not necessarily embedded in a higher-dimensional Euclidean space, there is no guaranteed way to define straight-line segments or distances between two points in the general case. Without these, the derivative is not well-defined. By mapping the manifold to Euclidean space through a local chart (phi), we temporarily “flatten” a small region of the manifold. This allows us to use the familiar tools of calculus (limits, derivatives, etc.) in the Euclidean setting. In other words, now we have “straight-line segments” to measure distances between two points, which is necessary in the definition of the derivative (again, think of a limit). Let me know if this helps.

@@charleshartlen3914 Thank you for the comment! So, the expressions are consistent with the differential framework used in calculus on manifolds. The parameterized curve gamma(t) maps from the real line R to the manifold M, and the local chart phi maps M to R^n. Composing these (x(t) = phi(gamma(t))) lets us work in R^n and compute derivatives using standard calculus. At t_0, V_p = d(phi(gamma(t)))/dt |(t=t_0) represents the instantaneous velocity vector in R^n, written as [dx^1/dt, …, dx^n/dt] |(t=t_0). Both sides represent the same velocity: the left is in terms of the composite function phi(gamma), and the right is its expanded coordinate representation. I think that what is not clear to you is that x(t) = [x^1(t), …, x^n(t)] is not a “family of solutions”. It is a single trajectory in R^n. x(t) describes a single curve, and the derivative dx^i/dt |_(t=t_0) evaluates its instantaneous rate of change at a specific t_0

​@@dibeos if that is true then it implies that the x1(t),...xn(t) portion of the expression represents what we need for the full pathway--and the derivative of this is very much the general form of the tangent space through the entire pathway, but how can this be equated to Vp?

@@PackMowin awesome! Thanks, Zach 😎

@@BabatopeFagbenle-rk6jy we are glad that you liked it! Please, let us know what kind of videos you would like to watch in the channel, so that we can make you “cry for joy” again 😄

@@willy8285 thanks for the encouragement Willy, it really helps us to keep going 💪🏻

@@ayandaripa6192 thanks for the encouraging words, Ayan! It really helps us 😎

@@farrasabdelnour your welcome, Farras. Thanks for the encouragement 😎

@@manfredbogner9799 Danke fürs Erkennen! 😎

@@benjamingoldstein1111 thanks for letting us know, Benjamin!!! We will do it 😎👌🏻

@@dibeos Cool!

@@rathalas_enjoyer thank you so much!!! It really means a lot to us…

@@mouha003 thanks for the encouragement, and we really hope to be very useful!! Let us know how we can help 😎

@@connorcriss thanks, I do my best to seduce people into math 😏

I'll second that! Very nice, clear video by the way, we need more like this, that explain at a really basic level. Well done, subbed.

@@letstree1764 thank you so much for the encouragement!!!! 😎

@@redroach401 yes!!! It will be very fun!!! Stay tuned 😎

@10011011110 thanks for the nice words! Our goal is to slowly build up to more complex concepts (while starting from the most intuitive things) 😎 Glad it helped you!

Hey, so the math might seem really odd and difficult but it’s really just the surface of shape in what ever dimension and that surface is literally all of the space any movement can take place in the “manifold calculus”

@davidake8604 thanks! They’re just images from the video but in black and white

@joelmarques6793 once again, excellent comment! thanks for encouraging us 😎👍🏻

@@plranisch9509 thanks for the nice words!!! 😎👌🏻let us know what kind of content you’d like us to post about. Thanks for the encouragement again

@@RayaneAoussar thanks!!! We just use keynotes

Thanks!!! Your comments are always nice, and they really encourage us to keep going 💪🏻😎

@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)