I honestly don't understand manifolds and only studied Physics and math until Differential Calculus in uni undergrad, but you guys made it super simple to understand this level of math at MY level. Now I feel like studying more of this after watching. Thanks to both of you!
@10011011110 thanks for the nice words! Our goal is to slowly build up to more complex concepts (while starting from the most intuitive things) 😎 Glad it helped you!
Hey, so the math might seem really odd and difficult but it’s really just the surface of shape in what ever dimension and that surface is literally all of the space any movement can take place in the “manifold calculus”
Both of you are very brilliant people who can explain various issues very simply and this shows the depth of your insight. I am sure that your future works will bring various fields under the microscope and light!
I'd be interested in a video zooming in on that leap to the chain rool. Somebody's gotta type it. So I do. Great job, guys! Nice visuals, great explanations!
Very cool video! I was surprised when I saw how few subscribers you have, this is very well done! Keep up the good work, I hope you get big because you deserve it
Yup Tangent spaces literally put the “differential” (irrespective of d^n or position dependability) into geometry! Any of you studied (NCG) C* by any chance?
I'm sorry. I want to support this channel but it needs a lot more work I learned this stuff a long time ago, and it seems to me that your explanations are a bit misleading and/or maybe even wrong. In classic textbooks (like Lovelock and Rund, say) tangent spaces are defined differently, They are spaces of contravariant vectors associated with a point on a manifold. So what are these contravariant vectors? It turns out they are what is called 'derivations', which are functions from the space of continuous functions in the manifold to the reals. I understand that is hard and abstract, but that is what the tangent vectors really are.... A tangent vector actually measures how much any continuous function on the manifold would change for the given increment and direction of that tangent vector,(Analogously to how it is in a 2d surface embedded in 3d space.) The set of those tangent vectors is the tangent space. Thanks for the opportunity to give feedback. No offense..
I honestly don't understand manifolds and only studied Physics and math until Differential Calculus in uni undergrad, but you guys made it super simple to understand this level of math at MY level. Now I feel like studying more of this after watching. Thanks to both of you!
@10011011110 thanks for the nice words! Our goal is to slowly build up to more complex concepts (while starting from the most intuitive things) 😎 Glad it helped you!
Hey, so the math might seem really odd and difficult but it’s really just the surface of shape in what ever dimension and that surface is literally all of the space any movement can take place in the “manifold calculus”
Both of you are very brilliant people who can explain various issues very simply and this shows the depth of your insight. I am sure that your future works will bring various fields under the microscope and light!
@@plranisch9509 thanks for the nice words!!! 😎👌🏻let us know what kind of content you’d like us to post about. Thanks for the encouragement again
I'd be interested in a video zooming in on that leap to the chain rool. Somebody's gotta type it. So I do.
Great job, guys! Nice visuals, great explanations!
@@benjamingoldstein1111 thanks for letting us know, Benjamin!!! We will do it 😎👌🏻
@@dibeos Cool!
I really like Differential Geometry and think this is a really good explanation. Thank You!
@@letstree1764 thank you so much for the encouragement!!!! 😎
Very cool video! I was surprised when I saw how few subscribers you have, this is very well done! Keep up the good work, I hope you get big because you deserve it
@@rathalas_enjoyer thank you so much!!! It really means a lot to us…
The set of tangent vectors to à point of a line/curve/surface, collectively it is a vector space.
Please make a video on fiber bundle 🙏
Yup Tangent spaces literally put the “differential” (irrespective of d^n or position dependability) into geometry!
Any of you studied (NCG) C* by any chance?
Once again... Excellent work!
@joelmarques6793 once again, excellent comment! thanks for encouraging us 😎👍🏻
i'm exited to study this at college then saying that i know everything because of you, thank for both of you
@@mouha003 thanks for the encouragement, and we really hope to be very useful!! Let us know how we can help 😎
amazing video!! which software u use to make those animation please?
I'm currently working on a math project for uni
@@RayaneAoussar thanks!!! We just use keynotes
Awesome.
I have a question. How did you do the images for the pdf file?
@davidake8604 thanks! They’re just images from the video but in black and white
Top!
Cool
Dude you are absolutely mogging in the thumbnail
@@connorcriss thanks, I do my best to seduce people into math 😏
I'm sorry. I want to support this channel but it needs a lot more work I learned this stuff a long time ago, and it seems to me that your explanations are a bit misleading and/or maybe even wrong.
In classic textbooks (like Lovelock and Rund, say) tangent spaces are defined differently, They are spaces of contravariant vectors associated with a point on a manifold. So what are these contravariant vectors? It turns out they are what is called 'derivations', which are functions from the space of continuous functions in the manifold to the reals. I understand that is hard and abstract, but that is what the tangent vectors really are....
A tangent vector actually measures how much any continuous function on the manifold would change for the given increment and direction of that tangent vector,(Analogously to how it is in a 2d surface embedded in 3d space.)
The set of those tangent vectors is the tangent space.
Thanks for the opportunity to give feedback. No offense..