Manifolds 11 | Projective Space is a Manifold

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This is my video series about Manifolds where we start with topology, talk about differential forms and integration on manifolds, and end with the famous Stoke's theorem. I hope that it will help everyone who wants to learn about it.

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#Manifolds
#Mathematics
#Differential
#LearnMath
#Stokes
#calculus

I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[iliTheFallen]

I do not know whether you follow a particular book or not; but I have never come across such a nice and intuitive explanation about projective spaces for the last six years of my advance math studies! I learned it from the people who teach algebraic geometry in general and I never saw anyone who teaches manifolds, touching projective spaces. Besides, algebraic geometry people's way of conveying it to the audience is not intuitive at all :( Thank you so much for this new perspective (the way you do it using the equivalence relation) into it; even though your sole purpose is just to give an example :)

[mahmoudalbahar]

Thank you for your great videos and thank for your cool way of saying hello.

[maxwangwang]

thank you much for making this video, your videos make me more intuitively understand these abstract definitions in the books. Both the Imagesyou drew and the examples you gave complete my understanding of this Knowledge.❤❤❤

[tensorfeld]

Manifolds!

Important in Physics, Dynamical Systems, Differential Geometry, Topology, ...

:D

[StratosFair]

Great video, as always ! It's really nice how you carefully build up from things introduced in the previous videos so we can go back to them and get a deeper understanding of the concepts.

Quick question : can't the atlas you use for S^n be chosen smaller ? For S^2 for instance it suffices to take U_1 plus and U_1 minus to cover the whole sphere, I suspect it is the same for the higher dimensional spheres although it's quite harder to visualize

[PunmasterSTP]

Projective space? More like "It's time to face"...the fact that I'm almost at the end of this playlist. I'm sorry to be reaching the end of something so amazing!

[wishcow]

This is a technicality but you should intersect your open sets with S^n, or take only x's which are in S^n, when defining the atlas on S^n.

[nirshahar]

Thanks for the great content!

Btw, cant we choose the chart for V_i in the same way we chose it for U_{i, -} (since they are homeomorphic)? Why do we have to go through the slope?

[larrygiebel]

Hey, great video!
I would like to use this approach of showing that the projective space is a manifold in one of my assignments. I didn't find any sources with this approach that i can quote, do you follow a particular book or do you know any sources with this approach?
Thank you very much!

[ILManent]

Thanks for the great series!
I have a question (maybe I'm missing something):
You introduced V_i and h_i in order to demonstrate that (P^n(R), {V_i}) is a topology and in particular that it is locally eucledian with h_i, right?
If so, why is (P^n(R), {V_i}) a topology? In the 2D case V_1 intersect V_2 is not an element of {V_i}. Am I missing something?

[camilon]

Thanks for the manifolds series, it is a very buitifull topic! I have some question about the projective space: Is there a way to visualize it or to "embedd it" on the 3d space for example? (When we take n=2 I guess) Doea this is space has a application in physics or something lile that or is more like a abtract mathematical example? Thanks again for the series!!!

[xmeansnop]

Why is it second countable if x is element of R? arent there uncountably many equivalence classes?

Thanks by the way for the great courses, I am fairly new to this concept

[japedr]

8:06 This is of course correct, but my OCD tendency tells me to write it in terms of trigonometric functions :)
That would be something like (cos(atan m)), sin(atan m)), where m is the slope (x1 prime), and with the proper signs, etc.
Great video as always.

[StratosFair]

I'm struggling to prove that V_i' is open. Any tips ?

[yeast]

If RP^1 is S^1 with antipodes identified, aren't U_i, + and U_i, - the same sets?