I honestly don't understand manifolds and only studied Physics and math until Differential Calculus in uni undergrad, but you guys made it super simple to understand this level of math at MY level. Now I feel like studying more of this after watching. Thanks to both of you!
@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”
@BCarli1395 thanks for the nice words! We really appreciate it. We are working on a longer video about the core of Linear Algebra. Hopefully we will post it this weekend - stay tuned 😎
Both of you are very brilliant people who can explain various issues very simply and this shows the depth of your insight. I am sure that your future works will bring various fields under the microscope and light!
Very cool video! I was surprised when I saw how few subscribers you have, this is very well done! Keep up the good work, I hope you get big because you deserve it
I'd be interested in a video zooming in on that leap to the chain rool. Somebody's gotta type it. So I do. Great job, guys! Nice visuals, great explanations!
Yup Tangent spaces literally put the “differential” (irrespective of d^n or position dependability) into geometry! Any of you studied (NCG) C* by any chance?
I can't understand why I need to map into the Euclidean space the "manifold".. If the manifold represents the space itself, why the derivate are not defined, and I need of an Euclidean space?
@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.
Thanks, really nice video. I like the animations. So you chose the 'differential approach' like Do Carmo. I also know the 'geometric approach' with equivalence classes of curves and the 'physician's approach' with vectors transforming according to transition maps. Jänich describes all three approaches in his book 'vector analysis'. Do you know other approaches?
@@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 😎👍🏻
Anzi, lo chiedo in italiano... Perché bisogna fare il passaggio da quello spazio multidimensionale al classico tre dimensioni? Non sono "definite" le derivate e tutte le proprietà dell analisi, nel multispqzio?
@@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... ruclips.net/video/7mCHzvE2pJw/видео.html
@@pahandulanga1039 well, it depends. If the manifold is 2D (so, a surface) then indeed its tangent space at each point will be a tangent plane. But if the manifold is a 3D space, for example, then the tangent space at each point is a Euclidean (flat) 3D space
I honestly don't understand manifolds and only studied Physics and math until Differential Calculus in uni undergrad, but you guys made it super simple to understand this level of math at MY level. Now I feel like studying more of this after watching. Thanks to both of you!
@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”
Much appreciated. Would love more visualized video on Differential Geometry, Topological Manifold, Algebraic Topology.
I have seen this explained in many ways, but this concise, visually appealing video is about the best introduction I have seen. Thanks.
@BCarli1395 thanks for the nice words! We really appreciate it. We are working on a longer video about the core of Linear Algebra. Hopefully we will post it this weekend - stay tuned 😎
Both of you are very brilliant people who can explain various issues very simply and this shows the depth of your insight. I am sure that your future works will bring various fields under the microscope and light!
@@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
Very cool video! I was surprised when I saw how few subscribers you have, this is very well done! Keep up the good work, I hope you get big because you deserve it
@@rathalas_enjoyer thank you so much!!! It really means a lot to us…
I really like Differential Geometry and think this is a really good explanation. Thank You!
@@letstree1764 thank you so much for the encouragement!!!! 😎
Please make a video on fiber bundle 🙏
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.
i'm exited to study this at college then saying that i know everything because of you, thank for both of you
@@mouha003 thanks for the encouragement, and we really hope to be very useful!! Let us know how we can help 😎
I'd be interested in a video zooming in on that leap to the chain rool. Somebody's gotta type it. So I do.
Great job, guys! Nice visuals, great explanations!
@@benjamingoldstein1111 thanks for letting us know, Benjamin!!! We will do it 😎👌🏻
@@dibeos Cool!
Once again... Excellent work!
@joelmarques6793 once again, excellent comment! thanks for encouraging us 😎👍🏻
The set of tangent vectors to à point of a line/curve/surface, collectively it is a vector space.
Great topic, thanks 👍
@@frankdearr2772 thanks for the recognition
Yup Tangent spaces literally put the “differential” (irrespective of d^n or position dependability) into geometry!
Any of you studied (NCG) C* by any chance?
I can't understand why I need to map into the Euclidean space the "manifold".. If the manifold represents the space itself, why the derivate are not defined, and I need of an Euclidean space?
@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.
amazing video!! which software u use to make those animation please?
I'm currently working on a math project for uni
@@RayaneAoussar thanks!!! We just use keynotes
Thanks, really nice video. I like the animations.
So you chose the 'differential approach' like Do Carmo. I also know the 'geometric approach' with equivalence classes of curves and the 'physician's approach' with vectors transforming according to transition maps. Jänich describes all three approaches in his book 'vector analysis'.
Do you know other approaches?
Awesome.
I have a question. How did you do the images for the pdf file?
@davidake8604 thanks! They’re just images from the video but in black and white
Can we say that a tangent vector is a functionnal ?
@@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
I'm assuming that you defined the concept of a "chart" in some previous video?
@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 😎👍🏻
Anzi, lo chiedo in italiano... Perché bisogna fare il passaggio da quello spazio multidimensionale al classico tre dimensioni? Non sono "definite" le derivate e tutte le proprietà dell analisi, nel multispqzio?
@@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...
ruclips.net/video/7mCHzvE2pJw/видео.html
just started differential geometry course , how to get good material for begginers
Top!
Is this different from a tangent plane?
@@pahandulanga1039 well, it depends. If the manifold is 2D (so, a surface) then indeed its tangent space at each point will be a tangent plane. But if the manifold is a 3D space, for example, then the tangent space at each point is a Euclidean (flat) 3D space
Cool
Subbed
@@PackMowin awesome! Thanks, Zach 😎
Dude you are absolutely mogging in the thumbnail
@@connorcriss thanks, I do my best to seduce people into math 😏