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. 😎
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
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 😎
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?
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.
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.
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For an abstract subject, this really is as intuitive as it gets
I think the discussion can't be simplified than this...
Very clear...and serving as a first step in the understanding of differential geometry...
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. 😎
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
1:23 The verse was dope
@@badasswombae thanks 😎 we try hard to make dope verses
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
How good that you teach math to everyone🤓
@User_2005st thanks!!! It means a lot to me and Sofia😊
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 😎
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?
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 😎
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?
Superb work! Keep doing it!
@@MathwithMing thanks for the nice comment, as usual, Ming! 😄
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. 😅
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 😉
Great video ❤
@@MathsSciencePhilosophy thanks!!! 😊 😎
great video!!
@@tomasnuti9868 thanks Tomás! Please tell us what kind of videos you’d like us to post about 😎
With regards
muito bom, joão e maria
@@ramaronin Obrigado Ramon 😎
@@dibeos kkkkk
❤
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.
❤❤❤
@@thecritiquer9407 I love your critiques, The Critiquer 😎
The example was plain old calculus, you ignored the whole thing you explained and did not use the concepts of differential geometry lol.
yay I'm the 243rd viewer!!
@Rio243tothenegativeone hopefully only one of the first ;)
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 😄