I haven't seen such a PERFECT explanation of this topic as I saw in this video! Differential Geometry and Riemannian Geometry are very difficult topics and require a huge amount of time and dedication to learn. Thank you so much, and please continue making this excellent content!
@@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!!
Love that you're making differential geometry more accessible! Hopefully you will clarify confusing points or mistakes along the way. For example, at 5:53 it seems you have conflated the local [composite] path with the local coordinate axes.
@logansimon1272 Thanks for the nice comment Logan! Let us know what parts of math you are interested in so that Sofia and I can post videos about them. 😎
@@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 😄
These are the only ones bringing the purity of mathematics and tools of physicists to the world out there through animation *and “speed of a curve” is a forgotten notion in most texts and/or Calculus 3 courses (from where the notion is inculcated usually) (probably even the guys who tried solving the Brachistochrone Prob. didn’t think that way 🤔) 1:36
@mathieulacombe3438 thanks Mathieu, it means a lot to us!! We learned that people like to see concrete examples, so thanks for letting us know that it also helps you. We are thinking about making a video only with examples
Hey thanks for the video. 6:53 in the video an expression for the instantaneous velocity at p is given as Vp=d(phi) *gamma(t)/dt || eval t=to; which seems very obvious, and the next step after taking d( )/dt of both sides of the expression shown at the bottom at 5:45. This all makes sense to me. However, again at 6:53, this Vp expression is equated to some kind of expanded form of the d(x)/dt portion of the expression [ d(x1(t))/dt,...,d(xn(t)/dt] || eval t=t0. This right side of the equivocated expression is what is confusing me. How are both of these expressions equivalent? The right side of the equation appears to be an expression of the entirety of x(t) (because it is in the same form as x(t) = [x1(t),....xn(t)], yet the left side is also equated to the Vp which is supposed to represent, not a family of solutions in a general sense, but simply the solution for this relationship at this point p. ultimately what it appears is that the Vp is equated to both the evaluation of the differential for a single value, and the general form of the derivative (that goes through x1,....xn). Please help me understand how these can be equated like this. Thanks again
@@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?
I think the exemple isnt really showing the previous points because you havent defined a map (the coordonates) from M ->R2 instead you have used the embedding of M in R3 to calculate v(t)
I was about to comment this, in differential geometry there is nothing done in the manifold itself, that's the whole point of the heavy machinery/theory developed, they just did plain old calculus on the surface and dish the whole idea of local homeomorphism. To be fair, only pure Mathematicians would dig into this properly, are they physicists?
@dibeos At 3:47, must the red curve be start and end within the open set U, i.e. from gamma(a) to gamma(b) ? Or does the open set U must be larger than or contain the red curve in the manifold ?
This is great, thank you! It would be also nice to see some practical example of this, or some math exercise which involves manifolds. And also, I don't quite understand, why do we need this conversion from M to Rn at the first place. I understand the reason in general, but in this this specific example, do we need the transformation in order to compute the derivative? Can we compute it in M?
@@bashbarash1148 the whole point is that we only know how to perform derivatives of functions the go from Euclidean space to Euclidean space. So, the function that is actually differentiated here is the one that goes from R (“time”) to the the coordinates of the manifold mapped in R^2 (in this case). After making sure that the function going from R to R^2 is well-defined we can perform the derivative. Also, when the manifold is more abstract we talk about R^n instead of R^2. About the exercises, we want to make a video only about exercises and examples related to the theory of this video. I think people will enjoy it 😎
Nice gentle intro for someone like me who just learned about "Riemann Geometry" the other day. 😃 So, by "tangent space" are you referring to the collection of all the tangent planes, one for each point on the manifold?" I know very little linear algebra, but I suppose the "basis" for each plane would work in the usual way: two basis vectors can describe the entire plane. And then you need one of these for each point on the manifold?
@@jammasound Exactly! In differential geometry, each tangent space at a point on the manifold has a basis, which can be represented by tangent vectors. Differential forms, however, are elements of the cotangent space (the dual of the tangent space). These differential forms act on the basis vectors of the tangent space to yield scalar values.
@@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 😎
In the example, the manifold is 2-dimensional, right? And its shown embedded in 3-dimensions. Shouldn't the velocity calculus be done in R^2? And not R^3?
0:02 No. Not every space that “looks like a patch of rectangles stitched together” is a Euclidean space. Such spaces are LOCALLY Euclidean manifold. Moreover, if you take such a space and zoom out, you are not guaranteed to see that the space is actually a sphere. Making statements like this anathema in mathematics; you know better so say it better.
Seems like the perfect time to learn Manim 😉 Don't get me wrong, I like the video and the content is wonderful, but those graphics are not. 😏 Quite distracting actually.
@polyhistorphilomath hahaha yeah, after rewatching the video I also noticed that. Weird, but… I guess you can find a beautiful metaphorical meaning for it
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I haven't seen such a PERFECT explanation of this topic as I saw in this video!
Differential Geometry and Riemannian Geometry are very difficult topics and require a huge amount of time and dedication to learn.
Thank you so much, and please continue making this excellent content!
@@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!
For an abstract subject, this really is as intuitive as it gets
Nah it was really crazy how i understood most of this like wtf
Love that you're making differential geometry more accessible! Hopefully you will clarify confusing points or mistakes along the way. For example, at 5:53 it seems you have conflated the local [composite] path with the local coordinate axes.
Excellent introduction. It is succinct, yet sufficiently thorough. I am quite impressed! Thank you for making this!
@logansimon1272 Thanks for the nice comment Logan! Let us know what parts of math you are interested in so that Sofia and I can post videos about them. 😎
i cant stop crying for joy.... thank you
@@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 😄
I think the discussion can't be simplified than this...
Very clear...and serving as a first step in the understanding of differential geometry...
It could be simplified with multivectors, directed integrals and other tools of Geometric Calculus as developed by David Hestenes and others.
@BlueGiant69202 that's great 👍
It is very far from our emotions
How good that you teach math to everyone🤓
@User_2005st thanks!!! It means a lot to me and Sofia😊
I love watching these types of videos
@@UA-camVideos-jk4nf thanks, really!! Could you please tell us what kind of videos you’d like us to post about? 😎
These are the only ones bringing the purity of mathematics and tools of physicists to the world out there through animation
*and “speed of a curve” is a forgotten notion in most texts and/or Calculus 3 courses (from where the notion is inculcated usually) (probably even the guys who tried solving the Brachistochrone Prob. didn’t think that way 🤔) 1:36
@@dean532 that’s awesome, Dean, yeah I think that in general textbooks lack concrete examples, and that’s a huge problem
I really liked the example with the parabola it makes the idea a bit more concrete mathematically. Cant wait for your next video💪
@mathieulacombe3438 thanks Mathieu, it means a lot to us!! We learned that people like to see concrete examples, so thanks for letting us know that it also helps you. We are thinking about making a video only with examples
Excellent presentation, thank you.
1:23 The verse was dope
@@badasswombae thanks 😎 we try hard to make dope verses
Please tell us more about tangent spaces!
Amazing video as always!
Can you make videos about Hopf Fibration, 1-forms and hyperfunctions?
Superb work! Keep doing it!
@@MathwithMing thanks for the nice comment, as usual, Ming! 😄
i wanted to learn that thanks
@rewixx69420 we will make one about the coordinates of the tangent space, which are actually differential forms… I think you will like it too 😉
Hey thanks for the video.
6:53 in the video an expression for the instantaneous velocity at p is given as Vp=d(phi) *gamma(t)/dt || eval t=to; which seems very obvious, and the next step after taking d( )/dt of both sides of the expression shown at the bottom at 5:45. This all makes sense to me. However, again at 6:53, this Vp expression is equated to some kind of expanded form of the d(x)/dt portion of the expression [ d(x1(t))/dt,...,d(xn(t)/dt] || eval t=t0. This right side of the equivocated expression is what is confusing me. How are both of these expressions equivalent? The right side of the equation appears to be an expression of the entirety of x(t) (because it is in the same form as x(t) = [x1(t),....xn(t)], yet the left side is also equated to the Vp which is supposed to represent, not a family of solutions in a general sense, but simply the solution for this relationship at this point p.
ultimately what it appears is that the Vp is equated to both the evaluation of the differential for a single value, and the general form of the derivative (that goes through x1,....xn).
Please help me understand how these can be equated like this.
Thanks again
@@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?
That's beautiful.. Thank you
@@MS-cj8uw thank YOU 😎
I think the exemple isnt really showing the previous points because you havent defined a map (the coordonates) from M ->R2 instead you have used the embedding of M in R3 to calculate v(t)
I was about to comment this, in differential geometry there is nothing done in the manifold itself, that's the whole point of the heavy machinery/theory developed, they just did plain old calculus on the surface and dish the whole idea of local homeomorphism. To be fair, only pure Mathematicians would dig into this properly, are they physicists?
@dibeos At 3:47, must the red curve be start and end within the open set U, i.e. from gamma(a) to gamma(b) ? Or does the open set U must be larger than or contain the red curve in the manifold ?
Great video ❤
@MathsSciencePhilosophy glad you enjoyed it!!! 😊 😎
please tell us more about the basis of the tangent space!! :)
@@jnaniify yessss we are preparing a video specifically about it 😎 thanks for letting us know that you’d like to watch it 😁
This is great, thank you!
It would be also nice to see some practical example of this, or some math exercise which involves manifolds.
And also, I don't quite understand, why do we need this conversion from M to Rn at the first place. I understand the reason in general, but in this this specific example, do we need the transformation in order to compute the derivative? Can we compute it in M?
@@bashbarash1148 the whole point is that we only know how to perform derivatives of functions the go from Euclidean space to Euclidean space. So, the function that is actually differentiated here is the one that goes from R (“time”) to the the coordinates of the manifold mapped in R^2 (in this case). After making sure that the function going from R to R^2 is well-defined we can perform the derivative. Also, when the manifold is more abstract we talk about R^n instead of R^2. About the exercises, we want to make a video only about exercises and examples related to the theory of this video. I think people will enjoy it 😎
Do a video showing that the tangets at a point form a vector space.
@@mathunt1130 yesss, we will do it! Thanks for letting us know that you are interested in it 😎
great video!!
@@tomasnuti9868 thanks Tomás! Please tell us what kind of videos you’d like us to post about 😎
Nice gentle intro for someone like me who just learned about "Riemann Geometry" the other day. 😃 So, by "tangent space" are you referring to the collection of all the tangent planes, one for each point on the manifold?" I know very little linear algebra, but I suppose the "basis" for each plane would work in the usual way: two basis vectors can describe the entire plane. And then you need one of these for each point on the manifold?
@@jammasound Exactly! In differential geometry, each tangent space at a point on the manifold has a basis, which can be represented by tangent vectors. Differential forms, however, are elements of the cotangent space (the dual of the tangent space). These differential forms act on the basis vectors of the tangent space to yield scalar values.
@@dibeos Gotcha. Its gonna take me some effort to understand cotangent space, but at least I know tangent space now. 😅
can you do manifolds over arbitrary fields?
@@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.
In the example, the manifold is 2-dimensional, right? And its shown embedded in 3-dimensions. Shouldn't the velocity calculus be done in R^2? And not R^3?
❤
0:02
No. Not every space that “looks like a patch of rectangles stitched together” is a Euclidean space. Such spaces are LOCALLY Euclidean manifold. Moreover, if you take such a space and zoom out, you are not guaranteed to see that the space is actually a sphere. Making statements like this anathema in mathematics; you know better so say it better.
muito bom, joão e maria
@@ramaronin Obrigado Ramon 😎
@@dibeos kkkkk
With regards
Ohh now I get it! -Not me
❤❤❤
@@thecritiquer9407 I love your critiques, The Critiquer 😎
2:42 “If we want to pick a specific point, how would we know its location?” Ummm… We’d know its location because we “picked” it.
“Well, we’d need to create a neighborhood of points” … How would that help if we apparently don’t know the location of any given point?
Some people don't know even the word manifold, so this is not for every one
BRO HOW WAS ALL OF THIS UNDERSTANDABLE with only knowledge of d/dx thats crazy
based video
Seems like the perfect time to learn Manim 😉 Don't get me wrong, I like the video and the content is wonderful, but those graphics are not. 😏 Quite distracting actually.
yay I'm the 243rd viewer!!
@Rio243tothenegativeone hopefully only one of the first ;)
"We can pick a point inside you."
お断りします。
@polyhistorphilomath hahaha yeah, after rewatching the video I also noticed that. Weird, but… I guess you can find a beautiful metaphorical meaning for it
The example was plain old calculus, you ignored the whole thing you explained and did not use the concepts of differential geometry lol.
Calculus on a manifold is a bit disingenuous of a title but I’ll let it slide
The field of study is called calculus on manifolds.... Please don't embarrass yourself
@@adityakhanna113 Like I said I’ll let it slide but it’s a bad title 😉🧐
fret not everybody, @ValidatingUsername is letting it slide this time
Talk about arrogant
@@JohnDoe-sl6mb Follow the curve of an epsilon thick surface and do rate of change calculations in/on the manifold boundary layer 😄