Differential Geometry: A First Course in Curves and Surfaces [pdf] | Hacker News Reader

Differential Geometry: A First Course in Curves and Surfaces [pdf]

math.franklin.uga.edu

147 pointsibobev8 comments

This is a nice textbook. We used it when I was an undergrad in Brazil and those were some of the most fun classes I've ever had. It gives you a ton of intuition and understanding for classical (extrinsic) differential geometry, and I feel that this experience was very helpful later in learning Riemannian geometry and General Relativity. It also helped a bit when I took an intro course to computer graphics. Solid stuff!

Those are (updated) notes for a course with description (taken from prof. Shifrin's homepage):

> This is an undergraduate introduction to curves and surfaces in R3, with prerequisites of either MATH 2270 (2500) and MATH 3000 or MATH 3510(H). The course is a study of curvature and its implications. The course begins with a study of curves, focusing on the local theory with the Frenet frame, and culminating in some global results on total curvature. We move on to the local theory of surfaces (including Gauss's amazing result that there's no way to map the earth faithfully on a piece of paper) and heading to the Gauss-Bonnet Theorem, which relates total curvature of a surface to its topology (Euler characteristic). As time permits, we'll discuss either hyperbolic geometry or calculus of variations at the end of the course.

Suddenly HN has decided that they're not visual learners anymore. Too bad, Keenan Crane from CMU has the best video course series on Discrete Differential Geometry:

https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1...

https://www.cs.cmu.edu/~kmcrane/Projects/DDG

Those are some awesome videos. You'll notice that by the end of the series, he's using advanced notation everywhere as well. And you'll notice the original PDF has lots of pictures too. The use of calculus notation is required to be precise about differential geometry. It is not some kind of elitism, it's just the most effective way to communicate abstract ideas without misunderstanding. I hit a wall in my math understanding until I was willing to embrace the symbols, rules and manipulations of good notation.

I'd go further and say that mathematical and physics notation is beautiful unto itself. The evolution of symbols used to transmit abstraction is incredible. They are way of exploring patterns which can then be refined and layered using proven notational magic. So cool.

This resonates much more with with than the above PDF that jumps right into the first definition.

I took differential geometry during undergrad but had to drop out of the course as i thought i would fail the final. Great professor, but too much for my 20 year old brain. Id like to try it again in retirement.

I was fortunate enough to take this course with Ted Shifrin at UGA, he is an incredible professor (now retired), and it was one of my favorite courses when I was in undergrad.

For those who don't know, differential geometry is used in the most elegant definitions of physics - quantum field theory.

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