i don't know what people are talking ab when they complain about not bieng able to follow. this is amazing. they help create a solid intuition. These are good quality videos, don't doubt it
I absolutely love what you guys do and can't wait to see you grow more, been subscribed since the Riemann-Stieltjes integral video, will absolutely be here for every video!
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. 👍🏻
Note that you can always use (in Principle!) the Nash Extrinsic Theorem to substitute Intrinsically defined (reasonable) Manifolds by The extrinsic Rn for a sufficiently higher n since any "smooth" manifold is always embedded in a sufficiently higher dimensional euclidean space,
6:02 Mathematician: Thy Notations make me go bananas Physicist: Thy pure math Jargon makes me feel like a fish out of water! Mathematician: You extensively try to be i.
@@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 ...
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.
Am I right in thinking this way about metric tensor: " The metric tensor defined on a manifold at a given point takes a vector in the tangent space and gives the infinitesimal distance if one were to travel in that direction on the manifold"?
@@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
@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 😎
So, the "core" of differential geometry is, in the pursuit of trying to do calculus on manifolds, we instead do calculus on local, Euclidean approximations of the manifold.
@@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.
6:15 Just to clarify, a manifold is always the surface of the n dimensional manifold and never the volume? So “in” always refers to embedded in the manifold surface?
@@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)
It is nice when we can parametrize the curves on manifold M . What about solving a PDE on a domain D which is a 3-dimensinal manifold for engineering purposes (let's say the navier stokes equations) ? . When solving these equations, we engineers and in general the whole science community relies on discretization of the domain using points and interpolating between then or use splines but on the points themselves the boundary conditions , initial conditions and so forth need to be satisfied . After this they are solved numerically . What has been bothering me since i started uni and now in the master's is ' What if we find a way to parametrize any curve,surface,solid ? Could this bring as closer to analytical solutions of the Navier stokes equations such that we don't rely on the very expensive numerical methods used on supercomputers ?' . Of course parametrizing the domain is one thing and the nonlinear operator of the navier stokes is another thing ...
Good video. Intresting topic. Okk, thumbnails.. Like this channel should grow because its good content... You can make it a bit engaging.. Its like your attention slips out of vid for some reason.. its not that attention grabber..
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....
Hmm if you occupy a viewpoint on a coastline you can obtain sufficient evidence for global spherical curvature. Even more evident on the lunar surface. A pedantic quibble admittedly but worth a mention
No. This is certainly well-meant, but i fail to see the point of the video. You explain terms like “euclidean space” and require operations like “composed with” as known. I do not see what kind of user would require instruction on the first item while being versed in the second. You come about fresh and cool, but the didactic mistakes you make are just about the same as those of a standard uni instructor… Also, math never gets easier by not putting it on the blackboard.
@@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
i don't know what people are talking ab when they complain about not bieng able to follow. this is amazing. they help create a solid intuition. These are good quality videos, don't doubt it
@@willy8285 thanks for the encouragement Willy, it really helps us to keep going 💪🏻
I absolutely love what you guys do and can't wait to see you grow more, been subscribed since the Riemann-Stieltjes integral video, will absolutely be here for every video!
@@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)
Note that you can always use (in Principle!) the Nash Extrinsic Theorem to substitute Intrinsically defined (reasonable) Manifolds by The extrinsic Rn for a sufficiently higher n since any "smooth" manifold is always embedded in a sufficiently higher dimensional euclidean space,
6:02 Mathematician: Thy Notations make me go bananas
Physicist: Thy pure math Jargon makes me feel like a fish out of water!
Mathematician: You extensively try to be i.
What a cool video! I’m going to go straight to the manifold video now.
@@drybowser1519 thanks!!! Let us know what kind of content you are interested in 😎
Wow
You guys have grown a lot since I last clicked on your videos.
Congratulations.
Loved this video btw.
Thanks!!! Your comments are always nice, and they really encourage us to keep going 💪🏻😎
Great job
Very clear and coherent explanation of the subject
@@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 ...
Brings back 1st year memories. Now after a Masters this seems very normal. Good video for beginners.
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.
A joy to watch, thank you.
@@farrasabdelnour your welcome, Farras. Thanks for the encouragement 😎
Am I right in thinking this way about metric tensor: " The metric tensor defined on a manifold at a given point takes a vector in the tangent space and gives the infinitesimal distance if one were to travel in that direction on the manifold"?
@@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
Amazing explainer as always. But, I have a very small complaint, don't switch too often between each other sometimes it's difficult to follow.
@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 😎
So, the "core" of differential geometry is, in the pursuit of trying to do calculus on manifolds, we instead do calculus on local, Euclidean approximations of the manifold.
@@christressler3857 exactly! And then we do it with all the local charts (and their intersections), also called local coordinates.
Sehr gut
@@manfredbogner9799 Danke fürs Erkennen! 😎
Why paraboloid is not embedded in R3...Every point on it's surface can be given a 3 tuple for it's x , y and z coordinate...
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.
Great topic
@@user-wr4yl7tx3w thanks!! Differential Geometry is one our favorite areas in math. Let us know what else you’d like us to post about :)
6:15 Just to clarify, a manifold is always the surface of the n dimensional manifold and never the volume?
So “in” always refers to embedded in the manifold surface?
@@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)
Wow what a great video really
@@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!!!
It is nice when we can parametrize the curves on manifold M . What about solving a PDE on a domain D which is a 3-dimensinal manifold for engineering purposes (let's say the navier stokes equations) ? . When solving these equations, we engineers and in general the whole science community relies on discretization of the domain using points and interpolating between then or use splines but on the points themselves the boundary conditions , initial conditions and so forth need to be satisfied . After this they are solved numerically . What has been bothering me since i started uni and now in the master's is ' What if we find a way to parametrize any curve,surface,solid ? Could this bring as closer to analytical solutions of the Navier stokes equations such that we don't rely on the very expensive numerical methods used on supercomputers ?' . Of course parametrizing the domain is one thing and the nonlinear operator of the navier stokes is another thing ...
Good video. Intresting topic. Okk, thumbnails..
Like this channel should grow because its good content...
You can make it a bit engaging..
Its like your attention slips out of vid for some reason.. its not that attention grabber..
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 😎
Hmm if you occupy a viewpoint on a coastline you can obtain sufficient evidence for global spherical curvature. Even more evident on the lunar surface. A pedantic quibble admittedly but worth a mention
tldr: its just various bignesses next to each other
No. This is certainly well-meant, but i fail to see the point of the video. You explain terms like “euclidean space” and require operations like “composed with” as known. I do not see what kind of user would require instruction on the first item while being versed in the second. You come about fresh and cool, but the didactic mistakes you make are just about the same as those of a standard uni instructor…
Also, math never gets easier by not putting it on the blackboard.
Its a little hard to follow. I think I should read the document
@@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