I have been searched for intrinsic geometry definitions through out the whole web, but they were didnt helped to get a clear idea. Thanks to Danial Chan, the way he is explained is super cool. Loads of stars for your way of explanation .

I did enjoy this adventure in pure mathematics! Thank you!!

Your approach is so clear and so direct, it's inspiring!

Thank you sir ,you are a great professor !

Thank you! This video exactly responds to the questions i had!

Manifolds shall be taught in high schools already. It is too important nowadays.

Thank you so much, this is the only satisfying explanation that makes sense to me, awesome content

🎈 example=God level

thank you !!

I like the test for curved surface at 7:42

Hi, Daniel. Congratulations on your lectures. You are a very good teacher. I have a question. In your definition of smooth manifold the homeomorphisms where all defined on the d dimensional unit ball and I have always seen this definition using arbitrary open sets. I suppose this definition is fine since any open set on a d-dimensional space is locally homeomorphic to the d dimensional unit ball. Am right?

@StatelessLiberty

5 жыл бұрын

You can show the two definitions are equivalent like this. Let's show that your definition implies his definition (since the other direction is trivial). Given your point p in your manifold, suppose you have a homeomorphism from an open neighbourhood U of p to an open subset V of R^n. Then consider x to be the image of p under this mapping. You can put an open ball around x (contained in V) and the consider the preimage of the open ball, which will also be open by continuity and will contain p. If you restrict your homeomorphism to this preimage of the open ball, you have the definition of a manifold used in the above video.