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Differential Forms and Integration (2008) [pdf] (ucla.edu)
144 points by luu 12 days ago | hide | past | web | favorite | 45 comments





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-geome...


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

Licensed under Creative Commons BY-SA, too.


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.

> why differential forms themselves are supposed to be important

This concern means that you are looking at it backwards. Often, it happens in the opposite sense. You will find some problems that are hard to model or understand. Then, you realize that with quite a lot of effort you might be able to tackle them. And then, you learn about differential forms and see how they allow to express your problem very clearly and its solution becomes sort of immediate.

There is nothing mysterious about differential forms from the point of view of physics, but pure math texts often take this intuition for granted. If you are used to working with scalar and vector fields in space , you may realize that there are different kinds of each:

Examples of scalar fields:

(1) a potential (2) a density

Examples of vector fields:

(3) a velocity field (4) a flow (5) the gradient of a substance (6) the field of normal vectors on a surface (7) the field of tangent vectors to a curve (8) a field of "surface elements" filling the whole space

This list includes all particular cases of vector fields and differential forms of all orders on R^3 and on its sub-manifolds.

If you know the least amount of physics, you will realize that you can do some kind of integrals on these objects, but not all of them. For example, in the case of scalar fields, you can integrate a density over a domain, or you can evaluate a potential at one point (or more often, the difference of potential between two points). Thus, potentials are 0-forms and densities are 3-forms. And a similar reasoning for the vector fields, and 1-forms and 2-forms.


> This list includes all particular cases of vector fields and differential forms of all orders on R^3 and on its sub-manifolds.

I think that first 'all' shouldn't be there, right? That is, these are all possible orders of vector fields and differential forms (among which the various Hodge dualities permit lots of identifications) on submanifolds of ℝ^n, but they're not all the possible particular cases of such vector fields and differential forms, in the sense that there are plenty of other, different physical situations that lead to the same mathematics (which, as you argue, is why the concept is so useful).


I meant all in terms of mathematical models. I think I did not forget any case. In R^3 you have p-forms for p=0,1,2,3 and vector fields; those are respectively cases (1), (5), (8), (2), (4) above. Of course each "case" may have several different physical interpretations that are modeled by the same mathematical object.

If you feel that pullbacks by charts conceal the intuition, you can work with manifolds that are embedded in R^n. Differential forms are still the "right way" to do integration on these manifolds but now can be defined in terms of the coordinate system of R^n. Then you can do cute tricks like seeing that if you want to know the area of a region on the plane, you can compute it by integrating dx ^ dy over the area. OR you can compute it as x dy over the boundary since d(x dy) = dx ^ dy. (Where ^ is the wedge operator.) This means you can And x dy can be integrated mechanically by a planimeter [1]. And this is also how you would compute the area of a region in software given its boundary!

There's some other uses. They form the basis of De Rham Cohomology [2] which is a useful and computational way of describing topological properties of a manifold (recall how Stokes's theorem and friends show how the topology of a space constrains the integrals of differential forms).

Another thing is that they're specific kinds of tensors on the manifold. Tensors represent basically all the information we might be interested in about a manifold (for example, its curvature). Differential 1-forms are "dual" to vectors and are therefore important to building more complex tensors (higher tensors take in some number of vectors and 1-forms and output a value).

And just as regular integration and differentiation relates to solving of differential equations, differential forms are needed for differential equations that are on a manifold.

[1] https://en.wikipedia.org/wiki/Planimeter [2] https://en.wikipedia.org/wiki/De_Rham_cohomology


< Another thing is that they're specific kinds of tensors on the manifold. Tensors represent basically all the information we might be interested in about a manifold (for example, its curvature). Differential 1-forms are "dual" to vectors and are therefore important to building more complex tensors (higher tensors take in some number of vectors and 1-forms and output a value).

This was how differential forms were introduced in the course. I understood all of this from an algebraic standpoint, but I was lacking any geometric intuition for differential forms whatsoever. Say you have a k-form on some manifold and you evaluate it at some point which gives you an alternating covariant k-tensor. Then when you evaluate that at k tangent vectors at the point you get a scalar, does this scalar have any geometric meaning? Does it measure anything? Later when we did Riemannian manifolds and introduced the volume form that was at least a little more intuitive, as far as I remember, but general differential forms were intuitively a complete mystery to me. Also I kind of got their usefulness in an algebraic sense when we did some typical vector calculus calculations using the concepts of divergence and curl, but I didn't have much intuition for these concepts since I don't have a physics background and only worked with vector fields in this abstract setting. Unfortunately we did not cover De Rham cohomology. Thanks for your answer, I will take a look at planimeters.


The first 10 pages of the following link may be helpful; it shows probably the simplest concrete nontrivial 2-form:

https://math.berkeley.edu/~wodzicki/H185.S11/podrecznik/2for...

The first example there is: given a base point X and two vectors V,W based at X, the 2-form gives the "signed" area of the parallelogram spanned by V and W. Determinants (which measure n-dimensional parallelograms), when viewed as functions of their column vectors, have all the properties of differential forms.

Differential forms are a bit like generalized determinants and in a sense specify a way to measure something like an abstract volume in the neighborhood of a point of a manifold, in such a way that the Jacobian needed for changing coordinates is "built in".


It's an awesome example, but I think that the mechanics of a planimetre mean that, 'internally', it's integrating x dy - y dx (a vector at every point orthogonal to the position vector from a fixed origin), not just x dy. Of course the end result is the same (up to normalisation), as it must be.

The point of differential forms is that they give a way to express geometric theorems in a coordinate free way. Coordinates are seen as obscuring the pure geometric content of theorems. They are sometimes necessary artifacts of doing concrete calculations, but the idea is that geometry shouldn't depend on a choice of coordinates.

The important ideas can be found in pages 9-10 in this link:

https://www.math.ucla.edu/~tao/preprints/forms.pdf

Note halfway down page 9 we get a really clean equation for how to change variables (change coordinates) in an abstract way.

Also note the simple form that the general n-dimensional stokes theorem takes in terms of differential forms at the top of this page:

https://en.wikipedia.org/wiki/Stokes%27_theorem

That they allow the expression of substantial theorems in concise form is a clue that they are the "right" way to do differential geometry.


> [...] but the idea is that geometry shouldn't depend on a choice of coordinates

Indeed, in "Tensor Geometry" (Dodson & Poston), they note:

> Most modern "differential geometry" texts use a coordinate-free notation almost throughout. This is excellent for a coherent understanding, but leaves the physics student quite unequipped for the physical literature, or for the specific physical computations in which coordinates are unavoidable. Even when the relation to classical notation is explained, as in the magnificent [Spivak], pseudo-Riemannian geometry is barely touched on. This is crippling to the physicist, for whom spacetime is the most important example, and perverse even for the geometer. Indefinite metrics arise as easily within pure mathematics (for instance in Lie group theory) as in applications, and the mathematician should know the differences between such geometries and the positive definite type. In this book therefore we treat both cases equally, and describe both relativity theory and (in Ch. IX, §6) an important "abstract" pseudo Riemannian space, SL(2;R).


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.


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.)

Well yes: it is a volume form certainly. Totally right.

Vector Calculus by Marsden is a very good book for getting a grasp of the meaning and application of basic concepts such as Poisson and Langrange equations, and Maxwell equations. You will see that in R^3 those concepts can be represented graphically.

Regarding your last sentence, I'm thinking back to my differential geometry course, and I'm not even sure if we ever calculated an integral that wasn't equal to 0. Totally agree with the rest of the sentence though: I don't think I quite got it right away, but I felt better off for the experience.

There is a philosophical reason to distinguish forms, manifolds, and integration. When you talk about integrating a form on a manifold you concentrate on the result: the number you obtain. You almost do not notice that the form and the manifold exist somewhat independently. If you think of expressing forms via coordinates, you already need to have a manifold (so also a coordinate system) in place to even "define" the form. However this is not necessarily the case.

Let me be more specific:

Suppose your ambient space is $\R^3$ and you are looking at a vector field (let us say your space is full of water and the vector field models the velocity of the movement of water at every point). The vector field $V$ is a $1$-form, it exists.

Now suppose you insert a membrane (2d surface) into the water and want to compute how much water flows through it at any given moment in time. This is the "flow" of $V$ through your surface $S$.

If you go and look how to do this there are intuitive pictures and the computation reduces to 1) parameterize the 2d surface using 2 variables $(u,v)$ 2) compute some partial derivatives of the parameterization 3) wedge product them 4) take the dot product with $V$ 5) integrate in $u$ and $v$. At first this seems like magic but whoever is explaining the procedure draws a bunch of pictures to explain why this is reasonable and tries to convince you. Usually they eventually manage.

However this is only part of the story. You see, you have a map that inputs $V$(the vector field) and $S$ the surface and spits out a number. Furthermore this map is intuitively "continuous" in the sense that if you change $V$ a bit or $S$ a bit you do not expect the result to change too much. However if you try to prove this or explain this at any mathematical level, you run into trouble!!!

The reason is that the way you defined integrating the vector field DEPENDS on the parameterization, and worse, it depends on it at the first step of your procedure. If you have to membranes that are "close" how can you even think that their parameterizations be "close". You can't! Even the SAME surface can have drastically different parameterizations.

So clearly you need to abstract away the coordinates so you can talk about continuity, stability, perturbation.

Let us get back to abstract definitions. You know that you can integrate 2-forms on 2-manifolds (2d surfaces). You are used to having a 2-form DEFINED on a 2-manifold (so you don't really see the difference between integration and 2-forms). However we do know that we have this rather standard procedure of computing the flow of a vector field (1-form) through a 2-manifold (2d surface). How so? It seems that for whatever reason a vector field is ALSO a 2-form. And it is a 2-form just floating around R^3 in the same way a vector field (the velocity of water) exists independently of whether you are computing how much of it is flowing through a given surface.

So how is this the case? This is exactly an instance of Hodge duality. Since the ambient space $\R^3$ has a volume form (3-form) there is an intrinsic association from $k$ forms to $3-k$ forms (specifically, given a $k$ form the associated $3-k$ form is that unique $3-k$ form such that wedged with the original gives you the volume form).

So there you go! Given a vector field you have an associated 2-form in $\R^3$ that is there, by itself, without needing any 2-manifold to justify its existence. In practice if $V=(V_x,V_y,V_z)$ then the two form is $V_x dy dz + V_y dz dx + V_z dx dy$.

And if by chance it encounters a 2d surface it can naturally be integrated through it. The Hodge duality above actually expresses in a very concise form the multiple points on HOW to compute the flow (the procedure we started with).


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.

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


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.


> 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.


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.


Spivak's Calculus on Manifolds (aka little Spivak) is another really good treatment.

Reads like Baby Rudin. (Good for some, bad for others.)

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.)

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...


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).

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 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

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.


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.

They are the most general in the sense that there are Henstock-Kurzweil integrable functions that are not Lebesgue integrable and that other integrals that are also more general than the Lebesgue integrals are equivalent to the Henstock-Kurzweil integral.

It still works better than the Lebesgue integral in the multidimensional settings, since it is trivial to create a product f(x)g(y) of two functions which will not be Lebesgue integrable but is Henstock-Kurzweil integrable.

As for generalizations to generalized functions, my preference lies with Colombeau algebras over Schwartz distributions, in any case. Where at least there is an arithmetic of the generalized functions.



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).

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.

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.


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

this is his article in the princeton companion to mathematics

https://www.amazon.com/Princeton-Companion-Mathematics-Timot...

a great (even if expensive) math book


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



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