Differential geometry of curves and surfaces

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1. jupedApr 2020
   Never read it, but the preview pages look pretty neat. I might pick it up for a fun read. Might want to pair it with something like do Carmo's Differential Geometry of Curves and Surfaces (which is quite accessible) for the mathematical side.
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2. pmalyninNov 2018
   Differential geometry of curves and surfaces by Do Carmo
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3. fusiongyroMay 2018
   I'm in a similar situation to you. I have been really enjoying the 3blue1brown videos on Youtube. He has a sequence on calculus and linear algebra and both of them are worth watching and thinking about before going through a book. I also recently started watching some of Dr. Norman Wildberger's math lectures. He's a finitist crank, but most of his class lectures are great despite this. This inspired me to get some more textbooks and try to go through them. I'm pleased if I can make it through a chapter at all on any time scale, and I think having low expectations is probably healthy.

   One thing a friend of mine said, which I think has been very good advice, is to get several books on a single topic. Eventually every author will lose you and you'll get stuck; having alternate discussions will help you get through it. This is easy to do and inexpensive if you pick up some Dover math books, but I've been making heavy use of the local academic library. The math books you want to read are not in high demand at the library! You can get three or four and see if any of them are good enough to warrant a purchase later on, because you probably won't get through them in a month or whatever.

   I find that for many topics there is a really good text. Calculus by Spivak is a great example, it straddles the line between calculus-in-college and analysis. Every topic seems to have a few really good books like this one, and there are often books that will take a totally different approach, like H Jerome Keisler's nonstandard calculus book using infinitesimals.

   I used to see it as a real problem that I was learning math outside class, but more and more I see it as a benefit, because you can pick up the stuff you want at the resolution you want and benefit from the best books rather than whatever the publishers are bribing professors to use. Going at your own pace, you're not going to go through as much stuff as quickly, but you will actually _really_ learn it. I've spent the last three weeks or so thinking about the construction of the real numbers... in a classroom setting, you would be forced to get through this quickly to get on with the rest of the curriculum, even if you aren't interested in the rest of it.

   I think both our roads are eventually going to lead us to differential geometry, and the only thing I know about that is that there appears to be a very good book on Amazon (Tapp, Differential Geometry of Curves and Surfaces), and that you may want to avoid older books that use the older notation for it. I have heard great things about the book Gravitation, but I'm totally afraid of it, not ready to go there yet. Also check out Physics from Symmetry, that book looks amazing to me but I haven't read it yet, just flipped through the contents, but it might be exactly what you're after, since it discusses the math right before applying it to specific areas of physics.

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4. pmalyninDec 2016
   One book that I've read recently that really stands out is Differential Geometry of Curves and Surfaces by Manfredo do Carmo. It's a bit outdated, and at times hard to read, but it has a lot gems that aren't well know (there is a bit at the beginning on estimating the length of a curve by counting the number of intersecting lines).

   I'd recommend it if you're looking for some math reading.

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5. ajtullochJun 2014
   I think a functional (or CS) perspective is an interesting approach, but I think you can lose some nice results in introductory differential geometry by following just this course. For example, the Gauss-Bonet theorem doesn't appear to be covered, which is an incredibly beautiful result linking the geometry and topology of manifolds.

   For a more classical introduction to differential geometry requiring only multivariate calculus and some real analysis/point set topology, Do Carmo's "Differential Geometry of Curves and Surfaces" is a great textbook.

   I put together a summary (key definitions/theorems) from an undergraduate course following Do Carmo at [2].

   [1]: http://www.amazon.com/Differential-Geometry-Curves-Surfaces-...

   [2]: http://ajtulloch.github.io/PDFs/MATH3968LectureNotes.pdf

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6. defenMar 2018
   See https://en.wikipedia.org/wiki/Crofton_formula

   Also see page 41 of Differential Geometry of Curves and Surfaces by Manfredo Do Carmo

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