Differential Geometry in Under 15 Minutes

[agspda]

My left ear says this was an amazing video! it's so excited to explain it to my right one tomorrow.

[dj]

Awesome video! I am not sure how much of it I understood, but it makes me think of how far geometry has progressed since Euclid's times in terms of its abstraction and technical sophistication.

[defaultlamplamp]

Turning on mono audio fixes the audio. Good content!

[WithinEpsilon]

Flashback to my Tensor Analysis class, taught by a physics professor. This is much better!

[alepica]

Windows Settings > Accessibility Options > Hearing > Turn on Mono Audio

[AdrianYang]

The idea around 1:49 is really smart: instead of compressing the two semi-spheres into 2-D circles, compressing the southern one into a 2-D circle, and then cutting and stretching the northern one onto the same 2-D plane so that the central circle is left as a hole (which is already occupied by the southern). Then since the northern pole is mapped to infinite numbers of points at an infinite distance, only it is not mapped onto the 2-D plane. Thank you for your video.

[manimusicka]

Such a great video with beautiful animations! Thank you Qilin!

[bugiairl]

Great video, helped a lot to understand this concept, I’d love to see you cover other subjects!

[mathe]

Man, you teach a Semester of DG in 15min. You are a genius

[eden]

A point on the animations--k-forms should be thought of as paralellopipeds, not simplices. Consider ||v wedge w||---it is the vol of the paralellopiped, which is twice the vol of the simplex.

[AlejandroMFilz]

My left ear enjoyed this, really cool!!

[mosti]

Brilliant opening experiment. Got me hooked right away.

[pyropulseIXXI]

Your due East line shouldn't be curved, because travelling due east or due west are not paths that fall on a Great Circle; they are generally called Rhumb lines or Loxodromes

[Naverb]

This is differential topology, not differential geometry. Stokes theorem is definitely cool and used from time to time in diff geom, but defining the exterior derivative does not require the existence of a metric

[altus]

You voice only comes out of the left channel!
Also, consider getting a stereo lapel mic and using beam-forming, this will result in much better audio-quality.

[hannibal]

I am a structural engineer. I only needed to take vector calculus at uni. The general form of stokes theorem wasn’t a part of our course. It is very satisfying to see this introduction with animations. Only one issue. Why the audio is panned to the left?

[winniedobrokot]

I was completely lost at 6:30 with line X = ∂/∂x + x ∂/dy. It looks like differential operator created as combination of multiplication, addition and differentiation named as X. But I don't understand how it related to visualized vector field or any vector field. The operator after application to some function of two variables gives gives just function, not two functions of vector components. Also voice description become ambiguous because "X" and "x" sound the same. And I don't understand everything after, because it based on this. For example, the next slide shows equality
∂/∂x(x ∂/dy) = ∂/dy
But ∂/∂x(x ∂/dy) equals to (∂/dy) + ∂/∂x(∂/dy) by differentiation of multiplication... And how it related to the vector field is still non-clear.

Next slide, some "forms" things are used without explanation what the forms are... And I lost again. The "dx" for me is "hieroglyph in the integral notation to tell what variable is used for integration" or "hieroglyph in the differentiation operator to tell what variable is used for differentiation" with some vague relation to infinitesimally small piece in the definition of integral and differentiation by limits. Or related to intuitive understanding of integral as "sum of small pieces dx" or differentiation as "division by small number dx", but it is intuitive, not formal, and I am not sure this "small piece" is "form".

So, maybe this video is useful to those who already know the subject to recall the whole subject, but I couldn't extract any knowledge after 6:30 because lot of unknown or implicit assumptions. For example, it hard to tell is empty space between letters means application of operator or is it multiplication when you are not in the context, because you want to learn the context.

Still, it was very interesting and useful part before 6:30 to see how arbitrary manifolds are tied to functions and researched by local "maps" of these functions. Thanks for great work anyway, I think if you consider that some implicit things are not evident for newcomers, it will make great educational video for newcomers too.

[lowerbound]

I sincerely appreciate your work. Thank your for the great insight and inspiration!! 😻😻

[trafyknits]

I'm so glad that there are brilliant people out there who make life easier for the rest of us. If progress was dependent on me, we'd still be wearing loin cloths and using spears to hunt our food.

[dionisiocarmoneto]

Why it was saved in just left ear? It becomes very tiresome!