Manifolds #3 - Transition Functions

In this video I discuss transition functions in more detail. If a portion of the manifold is covered by more than one chart the points in that region will have images in both charts. The transition function gives us a way to express the coordinates of a point in some chart in any other given chart that covers the same point.

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[shubhamgothwal]

@WHYBmaths Why can't we just map both the open sets to the same Rn? Why did we need two separate Rn's?

[insising]

My appreciation for this video's existence is immeasurable. This stuff is completely incomprehensible from the perspective of someone self-studying from a textbook. Having these illustrations makes these functions as simple as breathing, while the textbook is just an alien language of random functions and subscripts strung together.

[jeremyaguilar]

Ive been trying to teach myself this stuff from a book. So having someone writing down what im seeing in the book and explaining the notation was awesome

[petermarshall]

Wonderful set of lectures … just superb

[PravNJ]

Brilliant series mate. Helping me a lot.

[paichethan]

He is either exceptionaly good at teaching or exceptionally good at maths.

[sofiyavyshnya]

Great visualizations! Really help explain the content in a more intuitive way

[jeremyaguilar]

Thanks for making these videos man. Their good and i found this info very useful.

[sofiyavyshnya]

I just noticed that the bottom of your board has a drawing of a dog. How cute!

[finaltheorygames]

Hey this is really good. I really appreciate this explanation. Keep up the videos.

[flooreijkelboom]

I really love your vids man! Thanks so much for making these

[h.h.c]

Most introductory script I found on manifolds found their content so basic that they were lacking in visual material. I wished your approach was standard, it really pushed the point for me (still in manifold of course ;-)

[houhoutrad]

Great work!.however I do have a question: why do we need to go to Euclidean space R^n in order to do calculus on manifolds?

[jackgaul]

@WHYBmaths how would we represent p (the point on the manifold)? I get how we can give it values and describe it on the two different charts using either (x1, x2) or (y1, y2) but what about while its "on" the manifold. Like how do you talk about the elements on the abstract manifold before you map them into a chart?

[JohnSmith-vqho]

Quick question: wouldn’t the transition function have its domain as the image of Ux, not the entire R^2 space? I’m referring to what you have written at the bottom of the board around 14:53, just above the dog.

[LemoUtan]

Makes it clear how those arrow and text diagrams can actually be trusted!
But isn't it more conventional to write the 'composed' form of φᵥ(φᵤ⁻¹(p)) as φᵤ⁻¹⚬φᵥ, i.e. the two φₖ (in this case k being x & y) swap places?

[datsmydab-minecraft-and-mo]

This series seems a bit similiar to the book Geometry and Topology by Mikio Nakahara, are you basing this series on it?

[chun-hohung]

I might have missed in the video: Is it implicitly required that, for a (smooth) transition map to exist, each chart map must have an (smooth also?) inverse?

[hugogallardo]

I feel like I joined a cult. I love your vids :)

[darrenpeck]

Does the 8nverse transition function have to be continuous?