Lecture 7: Integration (Discrete Differential Geometry)

[undrak]

I literally took 2 university courses with all proofs regarding the topic of manifolds and k-curves, etc... But I've never really seen a point in differential forms and integrating over them. I could prove general Stokes theorem for manifolds with boundaries on compact carrier but couldn't see any analogy at all and appreciate what it all means. You have really changed my point of view about whole subject of differential geometry. Thank you so much.

[yizhang]

Your analogy between an integral and a weighted area is so good.

[LimeHunter]

0:32 Integration and Differentiation

1:38 Integration of Differential k-Forms
1:40 Review—Integration of Area
2:59 Review—Integration of Scalar Functions
4:48 Integration of a 2-Form
6:57 Integration of Differential 2-forms—Example
8:55 Integration on Curves
13:08 Integration on Curves—Example

17:15 Stokes’ Theorem
17:31 Boundary
19:50 Boundary of a Boundary
20:55 Review: Fundamental Theorem of Calculus
22:16 Stokes’ Theorem (I)
23:34 Example: Divergence Theorem
27:02 Example: Green’s Theorem
29:49 Stokes’ Theorem (II)
30:17 Fundamental Theorem of Calculus & Stokes’
31:47 Why is d ◦ d = 0? (I)
34:28 Why is d ◦ d = 0? (II)
35:56 Integration & Stokes’ Theorem - Summary

38:25 Inner Product on Differential k-Forms
38:33 Inner Product—Review
41:06 Euclidean Inner Product—Review
42:18 L2 Inner Product of Functions / 0-forms
46:30 Inner Product on k-Forms
49:57 Inner Product of 1-Forms—Example

52:09 Summary
52:30 Exterior Calculus: Flat vs. Curved
54:28 Exterior Calculus—Summary

[rodrigoaldana]

Thanks, Prof. Crane. I have enjoyed all the lectures so far. However, I haven't understood some things in this one, and I hope that you (or anyone else) can help me. My questions are related to translating the notation of the covectors dx dy to actual differentials dx, dy, which can be integrated (when and how is this step perfromed)

-At 6:23, the motivation behind the integral of a 2-form is that it is the summation of the many 2-forms applied to a pair of vectors along with the whole region \Omega. However, I don't see what the vectors used in the two forms at 8:31 are. Mainly, I'm lost at the motivation behind the interpretation of the 2-form dx^dy as a scalar dxdy inside the integral, given that dx and dy were supposed to be just basis forms (covectors). Maybe I missed it, but when did we start to give the meaning of differentials (little displacements that can be integrated) to the "forms" dx and dy, which were just covectors. My thought is that we make a parallelepiped spanned by the vectors u=(dx)d/dx and v=(dy)d/dy where here dx and dy are small displacements in the directions d/dx and d/dy. This parallelepiped has area dxdy, and we can measure such area by the form dx^dy. Hence, we use dx^dy to measure this parallelepiped obtaining dx^dy(u, v)=dxdy since dx(d/dx)=1, dx(d/dy)=0, etc... However, this reasoning seems artificial to me (since we didnt't take advantage of the notation dx, dy for the basis covectors, instead we used other displacements dxdy different from the basis covectors). It would be better to have defined dx(d/dx)="dx" (where the second dx is the displacement and not the covector) instead of dx/(d/dx)=1 as we did before (move the displacements dx and dy to the definition of the covectors dx and dy instead of the parallelepiped u^v ).

-At 14:00 you show the integral of a 1-form \alpha. However, I don't understand what the systematic way of converting the "abstract" integral "\int_{S^1}\alpha" to the integral using the limits and the "differential" ds is. In particular, I don't understand where the ds came from. Note that in this case, we use "dy" not as a differential but as a covector, as opposed to what we did in my previous question (in my previous question I pointed out that in the video dx^dy was interpreted just as dxdy and integrated over).

I hope that you see this. These lectures have been very inspiring and have clarified many things I learned many years ago and haven't understood until now. Thank you so much.

[AkamiChannel]

Would be great to have a final lecture in the playlist about the history of how the subject was developed. Was it Elie Cartan who majorly did a lot of the legwork in putting this stuff together?

[bocckoka]

Question about the fundamental theorem: how do the dimensions fit there, eg in the one dimensional case, you would get a measure of area, while the difference of the endpoints is a measure of length? I think this generalizes to higher dimensions, the boundary is always one lower. How should we think about this?

[MatesMonchis]

At the slide at 52:00 I'm worried that if you follow the same steps with alpha = beta you get -1, while we would like to get a positive value. I wonder if it would be better to define the inner product as 《α、β》= ∬α ∧(＊β)

[brunomera]

Beautiful inspiring lecture! Thanks for sharing! Concerning notation. You introduced the L2 inner product of forms as the integral of \star form \wedge form. But if we take in the open rectangle in the plane the form dx, doesn't this give dy\wedge dx as the integrand? That would be in the opposite orientation of the usual one and the inner product would fail to be positive definite, right? Maybe I missed some of the conventions.

[rakhimovv]

Why at 11:34 the right hand side (discrete sum) is not multiplied by len(curve)/N where N=number of samples. Is it because tangent t_i is not unit but already has length?

[realzey]

So in 3D standard surface integrals of vector fields equal the integrals of 2-forms (Hodge-star of the corresponding 1-form) on surfaces. Am I right?

[sarvagyagupta]

Your example in 43:00, is that inspired by KL Divergence? Maybe that's why we can say that it doesn't obey inner product?

[felipekersting]

First of all, amazing lectures! At 11:49, the professor defines the integral as being the tangent vector applied to the differential form at each point p_i. However, I don't see how this converges when i->∞, since there is no 'differential'! By looking at the sum shown in the slide, it seems that when 'i' goes to infinity (i.e. when the length of each curve segment goes to 0) the sum does not converge since it does not depend on the curve length. Wouldn't it be [sum of alpha_pi(t_i) * s] where s is the length of the curve segment? In the next slide the 'ds' appears in the resolution.

[TiredGradStudent]

I think at 35:50 there is a typo in the product rule. There should be a + just before (-1)^k. Great lecture series nevertheless!

[unspeakablevorn]

So the "inner product" here given for functions obviously doesn't work as an inner product - integration isn't well behaved enough, there are nonnegative functions other than zero with zero as their integral. They're not continuous though. Question though is: what *is* a sensible inner product?

[columbusmyhw]

Fubini's theorem says, integrating over the unit square, say,
∫_0^1 ∫_0^1 f dx dy = ∫_0^1 ∫_0^1 f dy dx.
But with differential forms, we have dx∧dy = −dy∧dx, so surely
∫_0^1 ∫_0^1 f dx∧dy = −∫_0^1 ∫_0^1 f dy∧dx.
Why do we get different answers?

[AerysBat]

At 52:00 the negative value makes me uncomfortable, I feel like something went wrong :\