8 ms·

Differential Forms and Integration (2008) [pdf]

145 points by luu 6y ago

bmitc 6y ago
For 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.
- Y_Y 6y ago
  Spivak's Calculus on Manifolds (aka little Spivak) is another really good treatment.
  - Koshkin 6y ago
    Reads like Baby Rudin. (Good for some, bad for others.)
- nextos 6y ago
  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: http://matrixeditions.com/#vec http://matrixeditions.com/#vec
  - bmitc 6y ago
    That is indeed a very good book. Although I would say that some of the notation in it is non-standard, for better or for worse. Tu is the most consistent author I have ever seen with notation, and that matters a lot in smooth manifold theory and differential geometry. Another good book is Advanced Calculus: A Geometric View by James Callahan.
    - noch 6y ago
      > Another good book is Advanced Calculus: A Geometric View by James Callahan. Thank you for this! For those of us who had real difficulty with Advanced Calculus, Callahan's methodical, visual, generous approach is deeply felt and appreciated. I did not know of this book until now and immediately found myself absorbed. It's embarrassing to admit, but as one who loves mathematics yet seems to struggle and stagnate more often than everyone around me, I often want to ask for, and indeed need, a bit of hand-holding. Callahan is a wonderful guide in that sense. Thanks again.
      - bmitc 6y ago
        No problem. It is a great book. Definitely check out Edwards' book I mentioned above as well. It is a gem of a book. Although it doesn't use matrices and instead uses linear expansions, it is still brilliant. The first three chapters give an exposition of the theory, and then the next three go back and prove things. So if anything, take a look at the first three chapters and then the later ones on applications and extensions. It also has a geometrical viewpoint.
0xddd 6y ago
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.)
- poydras 6y ago
  A Visual Introduction to Differential Forms and Calculus on Manifolds by Jon Fortney. https://www.amazon.com/Visual-Introduction-Differential-Calculus-Manifolds-ebook/dp/B07QLN9B1C/ref=sr_1_3?dchild=1&keywords=visual+geometry&qid=1590139975&sr=8-3 https://www.amazon.com/Visual-Introduction-Differential-Calc...
  - 0xddd 6y ago
    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.
  - sixbrx 6y ago
    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.
FiberBundle 6y ago
I 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.
- deleted 6y ago
  [deleted]
- pfortuny 6y ago
  The thing with differential forms is that (to me) they look a bit magical (why those sign changes? why alternate?). Only when thinking of volume forms do you begin to understand it (the determinant being the paradigm of volume element, etc.). I think some grassmannian computations would be good in this context but, on the other hand, they would become very cumbersome very soon. As someone says below: Spivak's Differential Calculus on Manifolds is exceptionally good.
  - JadeNB 6y ago
    Incidentally, I'd argue that the determinant is a paradigm of a covolume element: you feed it a volume element and get out a number. (More technically, it lives in the top exterior power of the cotangent bundle, not of the tangent bundle.)
kachnuv_ocasek 6y ago
On a related note, and seeing that some people around here are interested to learn the topic from different angle: Gerald Jay Sussman (among others, co-author of Structure and Interpretation of Computer Programs) has co-written a book on differential geometry called Functional Differential Geometry. It builds up the theory of differential geometry using Scheme, just like SICP and SICM, and is a fantastic read for programmers wanting to grasp the topic in a more familiar language. It is available for download for free on the publisher's website: https://mitpress.mit.edu/books/functional-differential-geometry https://mitpress.mit.edu/books/functional-differential-geome...
- pmiller2 6y ago
  That is damn interesting, and probably worthy of its own submission. FYI for everyone looking for the download link, it's a bit hidden, so here it is: https://www.dropbox.com/s/t3si4b99ijqyhyk/9580.pdf?dl=1 https://www.dropbox.com/s/t3si4b99ijqyhyk/9580.pdf?dl=1 Licensed under Creative Commons BY-SA, too.
BlackFly 6y ago
I don't think the distinction between the signed and unsigned integral exists for the most general integral, the Henstock-Kurzweil integral. (I could be wrong, but an orientation seems to always be implied in being able to compute a Riemann sum over a tagged partition.) This distinction is probably related to the Lebesgue integral's inability to integrate functions unless they are absolutely integrable (since it needs to be able to compute the positive and negative components and take the difference, which being finite to make sense, must be absolutely finite) and is distinct from the Henstock-Kurzweil integral which works directly with a tagged partition of the underlying set (which implies an orientation) and is able to integrate (some) functions which are not absolutely integrable. Nevertheless, moving to differential manifolds introduces problems for the Henstock-Kurzweil integral becames the local orientation in charts does not always induce a sensible global orientation. However, integration on a manifold isn't so much integration of a function of several variables as it is integration of several functions of several variables. That one needs additional machinery to deal with the "several functions" part is unsurprising. I can't recommend studying the Henstock-Kurzweil integral enough! Strangely enough, despite being more general, it is far more approachable than the Lebesgue integral.
- mehrdadn 6y ago
  Wikipedia says there's an even more general integral (https://en.wikipedia.org/wiki/Khinchin_integral https://en.wikipedia.org/wiki/Khinchin_integral)? https://en.wikipedia.org/wiki/Henstock%E2%80%93Kurzweil_integral https://en.wikipedia.org/wiki/Henstock%E2%80%93Kurzweil_inte...
  - BlackFly 6y ago
    General in the sense that it integrates more functions. The Denjoy integral is equivalent to the Henstock-Kurzweil integral in that respect (Theories of Integration - The integrals of Riemann, Lebesgue, Henstock-Kurzweil, and McShane, Kurtz & Swartz).
- jordigh 6y ago
  I'm not sure gauge integrals are really "the most general integral". As you say, it doesn't work as well as the Lebesgue integral in multidimensional settings, does it? It also needs a bit of a tweak to give you the analogue of Stieltjes integration so you can unify sums and integrals, I believe.
throwlaplace 6y ago
this is his article in the princeton companion to mathematics https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809 https://www.amazon.com/Princeton-Companion-Mathematics-Timot... a great (even if expensive) math book
- apricot 6y ago
  Thank you! I was getting really strong "you've read this before" vibes from the article but couldn't place the source.
del_operator 6y ago
The Stein and Shakarchi Real Analysis text is a great read for learning measure theory
commandlinefan 6y ago
It makes me feel good to know that Terence Tao spends any time thinking about integration - I would have thought that would be like a normal person spending any time thinking about adding and subtracting.
- Syzygies 6y ago
  Funny you should say that. In elementary school I tested well for abstraction but exactly in the 50th percentile for arithmetic skills. Just like rejecting how people taught me to tie my shoelaces, and figuring out something for myself, I fixed my arithmetic deficiencies. I went on to a PhD in math and I'm now a professor. People still think I tie my shoes funny, and I add funny. I think for myself. It's a crippling misconception that talent is natural. Michael Jordan made himself the athlete he became; many people had his body but never got as far. Good mathematicians take conscious control of how they learn and think. Our tendency to go "meta" isn't restricted to math; it's applied to ourselves.

145 points by luu 6y ago

For 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.

Y_Y 6y ago
Spivak's Calculus on Manifolds (aka little Spivak) is another really good treatment.
- Koshkin 6y ago
  Reads like Baby Rudin. (Good for some, bad for others.)
nextos 6y ago
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: http://matrixeditions.com/#vec http://matrixeditions.com/#vec
- bmitc 6y ago
  That is indeed a very good book. Although I would say that some of the notation in it is non-standard, for better or for worse. Tu is the most consistent author I have ever seen with notation, and that matters a lot in smooth manifold theory and differential geometry. Another good book is Advanced Calculus: A Geometric View by James Callahan.
  - noch 6y ago
    > Another good book is Advanced Calculus: A Geometric View by James Callahan. Thank you for this! For those of us who had real difficulty with Advanced Calculus, Callahan's methodical, visual, generous approach is deeply felt and appreciated. I did not know of this book until now and immediately found myself absorbed. It's embarrassing to admit, but as one who loves mathematics yet seems to struggle and stagnate more often than everyone around me, I often want to ask for, and indeed need, a bit of hand-holding. Callahan is a wonderful guide in that sense. Thanks again.
    - bmitc 6y ago
      No problem. It is a great book. Definitely check out Edwards' book I mentioned above as well. It is a gem of a book. Although it doesn't use matrices and instead uses linear expansions, it is still brilliant. The first three chapters give an exposition of the theory, and then the next three go back and prove things. So if anything, take a look at the first three chapters and then the later ones on applications and extensions. It also has a geometrical viewpoint.

0xddd 6y ago

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.)

poydras 6y ago
A Visual Introduction to Differential Forms and Calculus on Manifolds by Jon Fortney. https://www.amazon.com/Visual-Introduction-Differential-Calculus-Manifolds-ebook/dp/B07QLN9B1C/ref=sr_1_3?dchild=1&keywords=visual+geometry&qid=1590139975&sr=8-3 https://www.amazon.com/Visual-Introduction-Differential-Calc...
- 0xddd 6y ago
  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.
- sixbrx 6y ago
  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.

FiberBundle 6y ago

I 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.

deleted 6y ago
[deleted]
pfortuny 6y ago
The thing with differential forms is that (to me) they look a bit magical (why those sign changes? why alternate?). Only when thinking of volume forms do you begin to understand it (the determinant being the paradigm of volume element, etc.). I think some grassmannian computations would be good in this context but, on the other hand, they would become very cumbersome very soon. As someone says below: Spivak's Differential Calculus on Manifolds is exceptionally good.
- JadeNB 6y ago
  Incidentally, I'd argue that the determinant is a paradigm of a covolume element: you feed it a volume element and get out a number. (More technically, it lives in the top exterior power of the cotangent bundle, not of the tangent bundle.)

kachnuv_ocasek 6y ago

On a related note, and seeing that some people around here are interested to learn the topic from different angle: Gerald Jay Sussman (among others, co-author of Structure and Interpretation of Computer Programs) has co-written a book on differential geometry called Functional Differential Geometry. It builds up the theory of differential geometry using Scheme, just like SICP and SICM, and is a fantastic read for programmers wanting to grasp the topic in a more familiar language. It is available for download for free on the publisher's website: https://mitpress.mit.edu/books/functional-differential-geometry https://mitpress.mit.edu/books/functional-differential-geome...

pmiller2 6y ago
That is damn interesting, and probably worthy of its own submission. FYI for everyone looking for the download link, it's a bit hidden, so here it is: https://www.dropbox.com/s/t3si4b99ijqyhyk/9580.pdf?dl=1 https://www.dropbox.com/s/t3si4b99ijqyhyk/9580.pdf?dl=1 Licensed under Creative Commons BY-SA, too.

BlackFly 6y ago

I don't think the distinction between the signed and unsigned integral exists for the most general integral, the Henstock-Kurzweil integral. (I could be wrong, but an orientation seems to always be implied in being able to compute a Riemann sum over a tagged partition.) This distinction is probably related to the Lebesgue integral's inability to integrate functions unless they are absolutely integrable (since it needs to be able to compute the positive and negative components and take the difference, which being finite to make sense, must be absolutely finite) and is distinct from the Henstock-Kurzweil integral which works directly with a tagged partition of the underlying set (which implies an orientation) and is able to integrate (some) functions which are not absolutely integrable. Nevertheless, moving to differential manifolds introduces problems for the Henstock-Kurzweil integral becames the local orientation in charts does not always induce a sensible global orientation. However, integration on a manifold isn't so much integration of a function of several variables as it is integration of several functions of several variables. That one needs additional machinery to deal with the "several functions" part is unsurprising. I can't recommend studying the Henstock-Kurzweil integral enough! Strangely enough, despite being more general, it is far more approachable than the Lebesgue integral.

mehrdadn 6y ago
Wikipedia says there's an even more general integral (https://en.wikipedia.org/wiki/Khinchin_integral https://en.wikipedia.org/wiki/Khinchin_integral)? https://en.wikipedia.org/wiki/Henstock%E2%80%93Kurzweil_integral https://en.wikipedia.org/wiki/Henstock%E2%80%93Kurzweil_inte...
- BlackFly 6y ago
  General in the sense that it integrates more functions. The Denjoy integral is equivalent to the Henstock-Kurzweil integral in that respect (Theories of Integration - The integrals of Riemann, Lebesgue, Henstock-Kurzweil, and McShane, Kurtz & Swartz).
jordigh 6y ago
I'm not sure gauge integrals are really "the most general integral". As you say, it doesn't work as well as the Lebesgue integral in multidimensional settings, does it? It also needs a bit of a tweak to give you the analogue of Stieltjes integration so you can unify sums and integrals, I believe.

throwlaplace 6y ago

this is his article in the princeton companion to mathematics https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809 https://www.amazon.com/Princeton-Companion-Mathematics-Timot... a great (even if expensive) math book

apricot 6y ago
Thank you! I was getting really strong "you've read this before" vibes from the article but couldn't place the source.

del_operator 6y ago

The Stein and Shakarchi Real Analysis text is a great read for learning measure theory

commandlinefan 6y ago

It makes me feel good to know that Terence Tao spends any time thinking about integration - I would have thought that would be like a normal person spending any time thinking about adding and subtracting.

Syzygies 6y ago
Funny you should say that. In elementary school I tested well for abstraction but exactly in the 50th percentile for arithmetic skills. Just like rejecting how people taught me to tie my shoelaces, and figuring out something for myself, I fixed my arithmetic deficiencies. I went on to a PhD in math and I'm now a professor. People still think I tie my shoes funny, and I add funny. I think for myself. It's a crippling misconception that talent is natural. Michael Jordan made himself the athlete he became; many people had his body but never got as far. Good mathematicians take conscious control of how they learn and think. Our tendency to go "meta" isn't restricted to math; it's applied to ourselves.