
Differential Forms and Integration (2008) [pdf] - luu
https://www.math.ucla.edu/~tao/preprints/forms.pdf
======
kachnuv_ocasek
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...](https://mitpress.mit.edu/books/functional-differential-geometry)

~~~
pmiller2
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.

------
FiberBundle
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.

~~~
tgb
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](https://en.wikipedia.org/wiki/Planimeter)
[2]
[https://en.wikipedia.org/wiki/De_Rham_cohomology](https://en.wikipedia.org/wiki/De_Rham_cohomology)

~~~
FiberBundle
< 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.

~~~
GaussBonnet
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...](https://math.berkeley.edu/~wodzicki/H185.S11/podrecznik/2forms.pdf)

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".

------
bmitc
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.

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

[https://www.amazon.com/Visual-Introduction-Differential-
Calc...](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)

~~~
sixbrx
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.

~~~
0xddd
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).

------
BlackFly
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.

~~~
jordigh
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.

~~~
BlackFly
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.

------
commandlinefan
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
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.

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

------
throwlaplace
this is his article in the princeton companion to mathematics

[https://www.amazon.com/Princeton-Companion-Mathematics-
Timot...](https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-
Gowers/dp/0691118809)

a great (even if expensive) math book

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

