Manifolds #5 - Tangent Space (Introduction)
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- เผยแพร่เมื่อ 9 พ.ย. 2024
- In this video I give an overview of the concepts involved in constructing the tangent space. I briefly introduce the notion of a vector as a derivative, acting on smooth functions at a point to produce the tangent vector at that point.
For more detail on this construction see the next video!
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Hello, I am from India. I follow your all videos. Can you suggest me a basic book on smooth manifold. I am first learner.
I would suggest the first few chapters in 'Spacetime and Geometry' by Sean Carroll. Whilst being a relativity text it has a clear introduction to the concepts, starting with flat spacetime to build your intuition about tensors. The first chapters are also available on his website for free!
www.preposterousuniverse.com/grnotes/
A more rigorous mathematical treatment can be found in 'Geometry, Topology and Physics' by Miko Nakahara, but this is definitely an advanced textbook so I would read it with caution, although it still has plenty of useful stuff!
@@WHYBmaths Thank you so much sir
@@WHYBmaths Thank you so much for your helpful suggestions!
@@WHYBmaths Thank you sir , I'm also a viewers of your channel & it's really well.
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I like how you fast-forward though the stuff that would make me bored to make it easier to focus on the concepts you explain in a way that more-naturally flows instead of getting distracted by the time it takes to draw something!
Yes I find it helps the flow and also saves alot of time, plus gives me more time to think that I can sneakily edit out
Excellent video, your dog Lula is the real tangent hero ! Happy you have such a bond!
found your video to get and idea about tangent space for numerical optimization course for my masters , wanted to thank you for your clear easy to follow teaching and yeah :3 , will save it to my playlist so i can tell other students about it :) , thanks again and wish you all the best
greetings from Egyptian studying in Germany
I'm doing my masters in mathematical physics and your videos help soo much!
Thankyou I'm so glad they are helpful!:)
Thank you! This has been very useful in my Geometric Data Analysis class.
Commenting for support and to come back to this once I've covered my fundamentals :D
Hello, I've been binge watching this series and I must say ur method of teaching has helped me a lot. I'm writing my high school essay on spherical and cylindrical coordinate systems and watched this topology series for more insight. If possible, could you recommend books or any other source I could use to write my essay?
I really like the material! And the dog!
I really like your videos. Please keep it up and cuddle your puppy for me
Doggo understood it all! :-)
Thank you!
>implying you can teach mathematics
joke, this was a brilliant video dude x
Thumbs up , you deserve it. And doggy too
Good going. Plz solved some research problem
Yess i needed this intuition! thanks a lot!! ps: what is the name of those star-like thingies hanging from your blackboard?
Great stuff, mate. I just can't figure where you've introduced the vector space in this playlist?
Thank you! And I have a separate playlist that covers all of the vector spaces/tensors stuff!
@ 5:10 The phi(lambda) in the x,y-plane is NOT the same function as the phi(lambda) on the manifold M!
@ 7:51 In 2D we had a tangent-LINE to a curve, in 3D we had a tangent-PLANE to a surface and now in 4D we have some tangent-SPACE to some "volume" called a manifold. Never forget The Bigger Picture ;-)
Again my mistake, on that figure I should have called the line I drew f(phi(lambda))
And absolutely, this bigger picture is always good to keep in mind but also very hard to visualise beyond two manifold dimensions, and I probably should have emphasised that more, maybe in a future video!
Thanks for great explanations. Is this a hobby of yours? Are you a student or a teacher? I am a hobbyist
love the dog
Isn't C-infinity(M) a collection of functions? I don't understand how it is the "tangent" space of the manifold
How do you know that the tangents of all curves at the point lie in one plane?
Because the derivatives that produce the tangent vector form a basis for this space, we know that any vector must be a linear combination of such basis vectors, and so subsequently they all lie on the plane spanned by the \partial_x operators, intuitively, at the infinitesimal level the function can only be changing in some small amount in each coordinate direction, so the tangent plane at each point is just the linear space spanned by such small infinitesimal coordinate directions at each point on the function, hope that helps!
The question is incorrect. It’s not a plane, but a vector space. First, one needs to prove that the constructed objects are linear, such that for the sum of two objects and scalar multiplication, there exists a corresponding curve. Thus, the set of all curves forms a (tangent) vector space. Second, one needs to prove that the dimensionality of the tangent vector space is the same as that of the base manifold. Tangentiality itself arises from the way of constructing objects as derivatives, but this doesn’t play a crucial role.
Instant Sub
i coudnt understand the word durative u used for tangent vector.. explain plzzz
It's simply the derivative from regular calculus, in vector calculus the derivative is usually represented as some kind of gradient derivative vector, which in this case is equivalent to what I'm doing here, essentially using derivatives to produce the gradients to a function (which can be represented as the tangent vector to that function). Hope that helps!
book name ?
What's the dog's name??
Lula! She has an instagram, instagram.com/luladogislush
That dog has a Ph.D. in differential geometry
This is just like math class. It's like if I don't already know it, then I have no chance of following it. First question: what's a manifold? Second question: what's the relevance of defining the number of dimensions as "d"?
Hi sorry that you are struggling! Hopefully I can answer your questions. If you haven't seen my previous videos defining a manifold I'd encourage you to watch those, but I can summarise: Essentially, at the most basic level a manifold is a set, more specifically a set that has the structure of being a topological space (which essentially allows us to define the notion of continuity of maps on the set, and lets us talk about open subsets of the space. This set, which is an abstract quantity, just a set of elements that could be anything, but usually correspond to points in space or of some geometrical object, say the points on the surface of a sphere. These points are then given meaning by expressing them in terms of a chart, which is usually viewed as giving 'coordinates' to the space. This just gives us a concrete way to talk about the elements of our abstract topological space in terms of coordinates (lists of numbers).
Now about the dimension, essentially the dimension just quantifies how many independent coordinates you need to 'cover' the entire manifold, so for example every point on the sphere can be characterised by two numbers (longitude, latitude). Stated another way it can be viewed as the number of independent coordinate 'directions', specified by the coordinate basis vectors (and hence the dimension of the tangent vector space is equal to the manifold dimension). The dimension is useful in classifying various spaces, as dimension is a topological invariant. Also when we usually talk about manifolds we often like to think of them as being 'embedded' in some higher dimensional space. For example, the sphere is usually visualised as living inside a three dimensional space, however it exists entirely independently of the higher dimensional embedding space, this notion is made explicit by the dimension which allows us to identify manifolds that are submanifolds of some higher dimensional space. Hopefully some of that is useful, I also encourage you to go back and start from the beginning of the series for more!
This will henceforth be the canonical counterexample to anyone claiming that you won't learn anything from reading comments on youtube.
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Hey I've seen you in Mit quantum physics and Frederich Schuller quantum theory comments, nice to see you here again, haha.
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Hm, don´t you want c ∈ C^{infty}(I) = C^{\infty}(I,M) s.t. c(0) = p, where 0 ∈I, I ⊆ R at 2:33?