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Comment by 0xddd | Hacker News Reader

Can anyone recommend a good introduction to differential geometry and forms? Does something analogous to "Visual Complex Analysis" exist for the topic? I have been curious to learn for a long time but, for whatever reason, always lose my way at some point with articles like this. I come away with some feeling that I understand what's going on and yet I can't say I have any concrete intuition for what a form or a manifold is despite knowing the formal definitions. I feel like applied examples would help, but at this level of math that seems to entail going on a side quest to learn a lot of difficult physics first. (Or alternatively, doing a lot of proofs, but that feels futile without having a tutor/mentor to check them.)

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I just took Keenan Crane’s course on Discrete Differential Geometry at CMU, and the slides and lecture notes are available online for free. Due to COVID, the second half of the semester’s lectures are available on YouTube, and they are really a goldmine (Keenan is a wonderful lecturer!). The coding exercises are in JS as well with a lot of base code to work with, so they are quite accessible and you get to focus on the geometry.

The figures on the slides are really great. Hope this helps:

http://brickisland.net/DDGSpring2020/

Wow, this is beyond what I could've hoped for. Especially the coding exercises, which make up for the other massive difficulty in self-studying--a lack of solutions to verify the work on written problems.

A Visual Introduction to Differential Forms and Calculus on Manifolds by Jon Fortney.

https://www.amazon.com/Visual-Introduction-Differential-Calc...

Thank you! This looks great and surprisingly affordable for an academic textbook. It seems like a lot of the good material for teaching these topics at the undergraduate level has come out rather recently.

I got this recently from Springer directly, and just as a psudeo-warning, this is a "print on demand" book, at least the one I got was (it said so when I ordered it, so I was properly warned). Now, the print is actually pretty high quality and so is the binding, and it's a large and beautiful book. My only complaint is the paper of the pages is a bit thin, like regular printer paper stock, as opposed to the thicker glossy paper I was hoping for and that would be usual for a book this size. When you're leafing through it and a page is lifted, you can often see the content on the opposite side showing through. That can be distracting and may bother some people.

BTW: If you buy from Springer, you get a free pdf of the book immediately while you wait for your physical copy, because of the delay for print on demand. They say you don't actually "own" the digital edition (can't remember the exact wording), but I can vouch that it's not time-limited. It's a very good deal.

Having gone through two chapters now, I also feel the need to caution others that the amount of typos in this book is simply jaw-dropping. The conceptual explanations in the text are generally excellent, but it is simply impossible to get through a page without hitting a substantial number of mistakes. I'm left wondering if there are errors I'm not catching on my own that are going to affect my understanding. I really hope a cleaned up second edition is on the horizon (hopefully with answers to some of the in-line exercises).

I listed references elsewhere.

https://news.ycombinator.com/item?id=23270163

Edwards' first three chapters give a wonderfullly intuitive exposition of forms and their application to integration.

Tu's book is a rigourous study of smooth manifolds and differential forms. His exercises are approachable, and his book is the most expedient to the full theory of differential forms.

As a quirky intuition pump, I recommend Geometrical Vectors by Gabriel Weinreich. The Fortney book mentioned in another comment is a nice, visual book, and there are other references in the replies to the comment I linked.

Edit: I'm an idiot.

Lol you linked the article that this hn post is a link to

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