What is a manifold?
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- Опубліковано 10 жов 2015
- A visual explanation and definition of manifolds are given. This includes motivations for topology, Hausdorffness and second-countability.
If you want to learn more, check out one of these (or any other basic differential geometry or topology book):
- M. Spivak: "A Comprehensive Introduction to Differential Geometry"
- M. Nakahara: "Geometry, Topology and Physics"
- J. W. Milnor: "Topology from the differentiable viewpoint"
This video definitely is not "noob-friendly".
I think one should know a little bit of elementary differential geometry of curves and surfaces before starting with manifolds, I understood the video but for me it's more like a recap than an intro for anybody who totally don't know any thing about it
thanks a lot. I thought I was the only one ưho couldn't grasp it.
You may wanna check out WhyBmaths to get this
@@rokujadotorupata4408 Meanwhile my Multivar Calc class is covering manifolds and im shedding tears
Those who wish to learn manifolds will learn the contents in this video sooner or later, so this helps to some extent even if they don’t know words such as second-countable and Hausdorff.
I'm a physicist and teacher and I took a differential geometry course 6 years ago. Nowadays I'm repeating everything I've learned and I slowly grasp what's really important for me to understand manifolds. This video suffers from the same problem every mathematical textook has for this topic. You define a manifold by several terms, highlighting smoothness above everything else as if it's the most important thing but in the end I have no clue how to do operations on a concrete example of a manifold.
The most important terms for physicists are the Riemann metric, the Levi-Civita-connection and the Riemann curvature tensor and for that you don't really need to know what Hausdorff, homeomorphisms, smoothness and all the other abstract terms really mean. A good introduction of manifolds should use concrete examples like a sphere, a torus or a cylinder and explain those terms for these specific examples.
Another problem I face with mathematical introductions is lack of motivation of the definitions. Most mathematicians just define a concept, state a theorem and prove them but they don't explain why a concept is important or why a generalisation is necessary. Before defining a manifold you first have to learn that it is a generalisation of a submanifold of Euclidean space (curve, surface, hypersurface, etc.). There you have to learn what a tangent space is, what curvature means and the theorema egregium, which states that curvature can be measured without referring to the surrounding space. Only then you have the motivation to understand that a manifold is the generalisation of submanifolds by ignoring the surrounding space completely.
This video ignores all these important aspects and does just what every mathematics textbook do, which is bad teaching.
There is a really nice book Geometry of Physics by T Frankel (intro to differential geometry). It has everything that you mentioned.
The mathematical definition of a manifold: A subset of R^n locally diffeomorphic at all points to a neighbourhood of the origin in R^m, where m
I laughed
I have a math degree. This made sense to me. But honestly, I don't think this would help a non-PhD level phycisist
I'm an undergraduate engineer. It was probably the clearest explanation I've seen yet. (Granted, I just took a course on differential geometry.)
I think folks are getting stuck on a arbitrary terms like homomorphism and Hausdorff. Knowledge of these **aren't needed** to understand 1:34 - 2:24, which is the most critical but also intuitive part of the video. Just pause and try and figure it out!
Eric Regina You are right. ..it is not student friendly
A math students can easily understand this
>im a bio major, math is easy, bio is hard.
under grad.phys. here . it helps 😊😊
If the point of this was to take someone with a good math background but almost no differential geometry and give them a fast introduction to the key concepts and terminology so they can understand other material dense with differential topology terminology (e.g. the SageManifold software documentation), then this was perfect. That is what I was looking for and this superbly met that need. My background is electrical engineering undergrad and some theoretical physics. It took a few viewings, but so what - it is only 3:50 long. Amazingly great job taking a very complex topic and lots of terminology and reducing it down to a clear, short video. Thanks for doing it!
Man, this made it even more complicated!
I think one should know a little bit of elementary differential geometry of curves and surfaces before starting with manifolds, I understood the video but for me it's more like a recap than an intro for anybody who totally don't know any thing about it
What is a manifold?
It's a geometry with certain properties (as much as i understood today)
1 manifold= a curve is a 1manifold if every portion of it will look like a line after zooming in. like a circle, the shape "8" is not manifold since the intersection will look like x no matter how much you zoom in (and not a line)
similarly 2 manifold = a surface is 2-manifold if every portion of it will look like a square or other planer surface if you zoom enough. like a sphere
and same goes for higher dimensions. like 3 manifold (our universe) will look like 3d Euclidean space if we zoomed in (as we observe it because we are so small).
I kinda understood the video, but could not quite put it together. It was like reading wikipedia, but with a bit more pictures. But your explanation Neeraj made everything clear for me. Cheers!
@@matousak2 where is my cookie
@neeraj kumar : thnx man. Now this makes sense.
Kaecilius un bowl and foil
Mpd un
Coming from somebody with a degree in Mathematics, I never fully understood the notion of manifolds when studying Topology, but this video (in particular the introduction) elaborated on concepts I've been confused about for years; and within a matter of minutes and it made complete sense! Thank you very much!
So you're saying there is no spoon!
I have a short presentation on manifolds due tomorrow for my Differential Equations class. This video saved my life! Thanks.
Sounds cool, maybe I'll understand it someday :))
0:40 : f(xw +yv) = wf(x) + vf(y)
@0:42 with the map space table there is a typo|error in with the "linearvector space" line: its v*f(y) not y*f(y).
You weren't kidding with the name of the channel. This video is for physicists.
This video is very helpful! I'd love to see more.
+Daniel Estrada I agree, this video is very interesting and informative, and also comprehensive and succinct, but it was not 'easy' for me to digest. So it is perhaps not 'beginners' material.
:-) Speed dating was the concept that came to my mind after the first viewing.
It is something to return to for later repeated reference.
I r4eally like this video, the graphics and teaching style are exceptionally good! This visual approach is easy to follow and understand. I'm looking forward to more videos like this : )
undergrad here, man, I was so LOST, thank you so much for this explanation and the book recommendations, pretty cool
c student in 'high school' maths here, this was the clearest explanation
lol
This was great, can you do more videos on differential geometry and/or differential topology?
If someone knows that a linear map, vector space, group and group homomorphism is, then they will have an idea of what a manifold is.
You really don't need to understand those at all. Those are analogies. Anyone without a math background should just watch 1:34 - 2:24.
He has great handwriting
This is such a nice review of manifolds. Would be great to have another one on: Riemann Manifolds. :)
I don't have a PhD in physics, yet this helped. In fact, one of the best videos on the topic. I would like to see more of this material on geometry and manifolds.
If someone could help me. I am looking for the geometric forms, spirals that grow forward and then return at the same point (and grow (if it possible ) backward), and so on? A Kind of the loops in the growth in exponential expansion.
As someone with highschool math what I got from this is that a manifold is numbers that are related in some way. Might be cool to cover this in more that just 4 minutes.
Nice video. To the list of references at 3:35, you might want to add Chris J. Isham "Modern Differential Geometry for Physicists". He takes a modern coordinate-free approach from the start, and I found his chapter introducing topology helpful in developing intuitions. He also mentions some less studied ideas from topology, including the lattice structure which I found to be insightful and related to certain notions from computer science (locale theory, discussed in Steven Vickers "Topology via Logic")
I think of map projections as an application of this. The earth's surface is a spheroid, so a region of this surface is mapped via the coordinate function (the projection method: Mercator, Mollweide, Cylindrical, Azimuthal Equidistant, etc.) onto a region in R^2 (a flat map).
I wish there would be more videos. Please do much more videos to make this video's content to be grasped by a hobbiyst. Please consider it, because style (both narration and visualisation) in this video is perfect.
Though I’m just a computer science undergraduate with bad mathematical basis but great curiosity in geometry, I find it really helpful to make me understand manifold, a complex concept in topology and computer graphics. Also, could someone offer me some great books about manifold and its relationships to computer vision and computer graphics. Much thanks!
The first and second criteria are not needed, since the local homeomorphism to R^d implies them, right?
The way you convey these complex subjects is just beautiful. This is the way mathematics SHOULD be taught
I hold a graduate degree in mathematics and 7.5 years in machine learning, cloud and devops. While I can still understand the language model we use to explain math, I truly believe it’s overdue for a revolution. To turn AI research to practical applications, we must reinvent mathematics itself
*Sees M. Spivak*
*Flashbacks to countless hours spent on his incredible but really difficult Calculus book*
I know basic maths like additon and multiplication. So how can i fix the manifold on my car ? i didnt understand anything shown here.
more videos would be awesome!
Amazing explanation thank you for this
Excelente trabajo (good job)... por qué no seguiste haciendo vídeos?
you should have started with defining curves and surfaces in R^n by a parameterization function the simplest way to define a curve,surface .. manifold in general , say that the function should be nice or smooth and have no jumps and introduce it in a visualizing way , then introduce the idea of dividing the surface which is a set of points into subsets with a nice function for each one and say a manifold is all about mappings, you didn't explain accuratelly how to define nice functions or what kind of sets can have nice functions "the topological part" but you still explained the easy idea behind the manifold
Thank you, this was good.
Awesome video!!!!
More videos like that pls
please make a video when you can explain easily..everything that u said went above my head
I think one should know a little bit of elementary differential geometry of curves and surfaces before starting with manifolds, I understood the video but for me it's more like a recap than an intro for anybody who totally don't know any thing about it
at 1:09 that Hausdorff condition; is that a way of saying that the/a manifold cannot intersect with itself? There is another video [much less succinct than this one] in which it asserted that an M dimensional manifold can be embedded in in a space of 2M dimensions [but seemingly not of less than 2M]
+Mark Peaty For the best idea of what the Hausdorff condition gives you, I would suggest you looking up some typical examples of non-Hausdorff spaces, as well as objects which satisfy the other conditions of manifold that are not hausdorff: en.wikipedia.org/wiki/Non-Hausdorff_manifold. These are cases we would like to rule out. Also, the definition of a manifold is very important because it gives us partitions of unity, which we don't get without the paracompactness. Partitions of unity are fundamental in many geometric constructions and proofs.
Thank you!
Please you should prepare more videos because you're awesome 💚
Excellent quick introduction!
I Like your video. Well summarised.
Now I think I'm very high on the `R` scale - possible R³ cubed - the G reek letters arrested my development in the understanding
this video solve my problem,,, thank you sir
More videos please!
I would gladly contribute subtitles in portuguese and/or spanish for my students to enjoy it. However, this functionality is disabled due to the fact that the "community contribution" is not enabled in the video. Please consider changing this!
bro im high as fuck i don't know why im here but im learning something today i guess
At least 2 or 3 months of introductory Topology in 3 minutes!
YESSS exactly what I was looking for!
Yes please more!
Helpful review!
As a physicist trying to learn crazy math shit, I am convinced that true mathematicians don't do actual math (what they do is even harder). Never stop with the 'out theorem must work in some open set that is infinitely differentiable and comes with a family of terms that are elements of some other crazy subset'.
it's clear. Now I understand it. It's subject related to coordinate transformation so that you are study things which are not easy to be expressed in vector form
if my brainlet unederstand : this is to transfer data set of a 2 dimensional plane into a 3 dimentional plane while retaining the information of the location while keeping a smoothness to the graph?
Essentially yes but it also generalizes to higher and lower dimensions. Also topology typically starts in dimension 3(topologically “3D” is considered the first dimension that we care about)
Great intro.
Its good sir. Thanks
It seems a nice cogent intro to me.
That escalated quickly
just great
im dumb af but im learning that jack manifold grind
That shed light on the obscure proofs of my differential geometry class : ] Especially what we call a smooth map between manifolds.
So am i right in saying that a homeomorphism could be seen as morphism on the category of manifolds, and continuous maps a morphism on topological spaces?
Hence diffeomorphisms that we see everywhere in that pretty abstract course.
Perhaps someone could do a similar video on Lie derivatives and differential forms
Espetacular...
awsome video
what does hausdorff, second countable, and homeomorphism mean?
I think one should know a little bit of elementary differential geometry of curves and surfaces before starting with manifolds, I understood the video but for me it's more like a recap than an intro for anybody who totally don't know any thing about it
Well
......
Got it
i liked this video to feel cool.
I came here, bcoz I am confused while calculating normals for my mesh in opengl
This is great
cool
Bro, I just got here because of funny minecraft man
very cool
I heard you, now explain it like I'm an idiot
Pro tip: Set the speed to 0.5
It was wonderful! It helped me to get confused! Thank you!
Hmm don’t understand but seems cool
sorry but that doesnt help
Isnt this definition of smooth manifold? whats the difference of smooth manifold and manifold?
it's not very intuitive, I would say
!!! Mommy?
I'm a homeless fentanyl addict and watch this without understanding anything. I like knowing that I lived and died without ever grasping higher knowledge of the exosapiens
You really have a big heart for student through your very intuitive explanations! This is extremely helpful and I wish I had you as a teacher! I think we students wouldn't wanna c u going after the end of the lesson! ;-]
why did i even try to understand this
I wonder sometmes if mathematics hasn't taken a wrong turn somewhere....
it takes the turns mathematicians take in their research...if you think there is an area that needs work, get involved in it
I defy anyone to watch this video and then explain what a manifold it.
I cant believe I understood all of that... I'm only 1/3 of the way throught a Topology & Analysis II class and all of that made total sense....
I love the simplicity of the topic. It usually takes a one or two semesters of Differential Geometry, Tensor Calculus or Topology to describe what the video describe in 3.50 minutes.
s1dguy No it doesn't.
Housed Off.
Awesome, this vedio really helps! I now understand what is manifold
this could be a bit on adult swim
It doesn't explain thing well. I am confused.
Hello sir... thank you for fig. orianted explinatio...
Sir i want to do this types of videos, for that i need which software you used to create this video plzzz can you help me?..
And we mainly use these create videos for the students
I'm afraid of bears.
It'd make more sense on a flat earth map
听不懂
Nice video but Bro you love making simple things hard
I am convinced this is completely made up to fool me
I am currently too stupid and/or ignorant to understand this 🤦🏻♂️
Good video, but sorry that I was overwhelmed when the definition, which is of a whole page, come out.
I am really stoned. What am I doing here..
I'm more confused after watching this video