Differential Forms and Integration (2008) [pdf]
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Differential Forms and Integration (2008) [pdf]
1–10 of 48 posts
Re: Differential Forms and Integration (2008) [pdf]
#2For anyone wanting to learn more about this, I highly recommend Advanced Calculus: A Differential Forms Approach by Harold Edwards and An Introduction to Manifolds by Loring Tu. The former reads almost like a novel and is a real treat of mathematical exposition. It's also a little quirky which is always nice. Tu's book is simply the gold standard of an introduction to the mathematics of manifolds and differential forms. It is the most concise and straightforward introduction to the full theory. It is also a wonderful book.
Re: Differential Forms and Integration (2008) [pdf]
#3Can 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.)
Re: Differential Forms and Integration (2008) [pdf]
#4Can 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 mani…
A Visual Introduction to Differential Forms and Calculus on Manifolds by Jon Fortney.
https://www.amazon.com/Visual-Introduction-Differential-Calc...
Re: Differential Forms and Integration (2008) [pdf]
#5Can 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 mani…
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:
Re: Differential Forms and Integration (2008) [pdf]
#6For anyone wanting to learn more about this, I highly recommend Advanced Calculus: A Differential Forms Approach by Harold Edwards and An Introduction to Manifolds by Loring Tu. The former reads almost like a novel and is a real treat of mathematical exposition. It's also a little quirky which is always nice. Tu's book is simply the gold standard of an introduction to the mathematics of manifolds and differential for…
Spivak's Calculus on Manifolds (aka little Spivak) is another really good treatment.
Re: Differential Forms and Integration (2008) [pdf]
#7Can 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 mani…
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 t…
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.
Re: Differential Forms and Integration (2008) [pdf]
#8Can 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 mani…
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.
Re: Differential Forms and Integration (2008) [pdf]
#9I took a Differential Geometry course in university. The course covered basic theory on smooth manifolds and Riemannian Geometry. While I did appreciate the beauty of the subject from a theoretical perspective, I have to admit that I somewhat lacked intuition for the material, especially with regards to differential forms. I understood that their properties make sense to define the notion of integration on a smooth manifold, but never really saw whether they had any purpose besides using them for integration. Even in their use in integration they were kind of a mystery to me, if I recall correctly we just defined integration on subset of R^n, where differential forms played the same role as d_{x_1}d_{x_2}...d_{x_n}, which is known from the riemann integral, but then to define integration on manifolds, we just used the pullback of the charts, which didn't alleviate any of the mystery and didn't add anything about why differential forms themselves are supposed to be important. I wish I had taken some physics courses, where vector calculus is used heavily and that might have helped some, but in the end I was somewhat disappointed, because even though taking the course certainly did make me a better mathematician, I still didn't have the feeling of really completely grasping the concepts, even though I could prove statements.
Re: Differential Forms and Integration (2008) [pdf]
#10For anyone wanting to learn more about this, I highly recommend Advanced Calculus: A Differential Forms Approach by Harold Edwards and An Introduction to Manifolds by Loring Tu. The former reads almost like a novel and is a real treat of mathematical exposition. It's also a little quirky which is always nice. Tu's book is simply the gold standard of an introduction to the mathematics of manifolds and differential for…
I love Hubbard & Hubbard, which is also great as it's an introductory text. It's been used often at Harvard Math 55 and some much simpler courses: