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Differential Forms and Integration (2008) [pdf]

math.ucla.edu

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Re: Differential Forms and Integration (2008) [pdf]

#21

Post by enriquto »

FiberBundle wrote: post #9

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 m…

> 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.

Re: Differential Forms and Integration (2008) [pdf]

#22

Post by monktastic1 »

0xddd wrote: post #3

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 mani…

Edit: I'm an idiot.

Re: Differential Forms and Integration (2008) [pdf]

#23

Post by throwlaplace »

monktastic1 wrote: post #22
0xddd wrote: post #3

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 mani…

Edit: I'm an idiot.

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

Re: Differential Forms and Integration (2008) [pdf]

#24

Post by GaussBonnet »

FiberBundle wrote: post #9

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 m…

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.

Re: Differential Forms and Integration (2008) [pdf]

#26

Post by Syzygies »

commandlinefan wrote: post #25

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.

Re: Differential Forms and Integration (2008) [pdf]

#27

Post by JadeNB »

tgb wrote: post #20
FiberBundle wrote: post #9

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 m…

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…

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.

Re: Differential Forms and Integration (2008) [pdf]

#28

Post by JadeNB »

enriquto wrote: post #21
FiberBundle wrote: post #9

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 m…

> 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 a…

> 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).

Re: Differential Forms and Integration (2008) [pdf]

#29

Post by enriquto »

JadeNB wrote: post #28
enriquto wrote: post #21

Earlier quoted context omitted.

> 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 a…

> 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 vecto…

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.

Re: Differential Forms and Integration (2008) [pdf]

#30

Post by [deleted] »

tgb wrote: post #20
FiberBundle wrote: post #9

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 m…

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…

[deleted]
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