The clever way curvature is described in math

Second channel video:    • What is "mean" about the mean curvature?  

How do mathematicians describe curvature of surfaces? There are two measures: Gaussian and mean curvatures, and both are useful in differential geometry, the study of surfaces and higher-dimensional manifolds (or lower-dimensional curves).

I know I have talked about Gaussian curvature before in this video:    • Theorema Egregium: why all maps are w...  , but I want to reintroduce it slightly differently with a fuller explanation of the shape operator. This will allow mean curvature in the picture, and is something that I want to focus on for the two future videos on minimal surfaces.

I deliberately didn't say principal curvatures, which are the eigenvalues of the shape operator. The eigenvalues are guaranteed to be real, and the eigenvectors must also be orthogonal, because the shape operator is real and symmetric. However, getting to the point where we can prove the shape operator is real and symmetric is a bit tricky (can be proved rather easily with computations, but I'm not sure how to do it "intuitively"); and getting from real symmetric matrices to real eigenvalues and orthogonal eigenvectors is another thing that I still don't know how to think about intuitively.

This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject.

Files for download:
Go to https://www.mathemaniac.co.uk/download and enter the following password: shapeoperator

More on my criteria on choosing videos in #SoMEpi: https://www.mathemaniac.co.uk/blog/my...

Sources:

Paternain’s differential geometry notes https://www.dpmms.cam.ac.uk/~gpp24/dg... (see pp. 28 - 33)

Visual Differential Geometry and Forms by Tristan Needham
For this whole series, I have not consulted this book, but it should be a nice resource anyway for the geometric intuitions.

Soap film images:
https://commons.wikimedia.org/wiki/Fi...
https://commons.wikimedia.org/wiki/Fi...

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If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.

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