Lecture 4: k-Forms (Discrete Differential Geometry)

[williamsamosir]

Hi Keenan I just want to say thank you so much for uploading this lecture series, it's such a gift. I came from a 3D art and self-taught programming background, and I can't believe that today I am able to learn of such complex concept just through a youtube video, on the public domain. Even at this point it already transforms the way I look at the possibility of meshes, beyond just sculpting and modelling. Really excited to see where DDG can revolutionize the interactive and creative domain.

[brainxd]

I think I need to praise your way of explaining in EVERY single video. THANK YOU for this gem of a series!!

[adamhendry]

The best videos, bar none, for learning discrete differential geometry. You MUST watch these in order to understand it! Thank you Prof. Crane!

[shiv]

0:36 k-Vectors and k-Forms - Overview
1:42 Measurement and Duality
2:39 Motivation: Measurement in Curved Spaces
3:49 Vectors and Co-Vectors (Duality)
5:30 Analogy: Row & Column Vectors
6:55 Vectors and Covectors
7:37 Dual Space & Covectors
9:13 Covectors - Example (R3)
11:28 Covectors - Example (Functions)
14:41 Sharp and Flat
16:12 Sharp and Flat w/ Inner Product
18:27 k-Forms
18:41 Covectors, Meet Exterior Algebra
20:06 Measurement of Vectors
20:32 Computing the Projected Length
21:02 1-form
21:47 Review: Determinants & Signed Volume
24:21 Measurement of 2-Vectors
24:59 Computing the Projected Area
26:18 2-form
27:21 Antisymmetry of 2-Forms
30:22 Measurement of 3-Vectors
31:21 Computing the Projected Volume
32:45 3-form
33:48 k-Form
35:47 k-Forms and Determinants
37:47 A Note on Notation
39:31 0-Forms
40:31 k-Forms (Measurement in Coordinates)
41:47 dual Basis
43:59 1-form - Example in Coordinates
46:50 2-form - Example in Coordinates
49:47 Einstein Summation Notation
51:59 Sharp and Flat in Coordinates
54:01 Coming Up: Differential Forms

[brunomera]

Beautiful intuition and remarkable presentation of these abstract concepts! Thanks for sharing! These same kind of ideas also appear in the second quantisation for fermions in many body physics. The exterior algebra is, at least for me, the most natural and beautiful way to think about fermions. However, the subject is not usually presented like this and it makes some of the details obscure.

[ObsessiveClarity]

I can't believe how enlightening this is. Your explanation of the dual between row and column vectors was so damn good. I watched a video by 3b1b dedicated to the commutativity of the dot product describing this dual geometrically, but this was even more informative. Thank you.

[sakarapu]

You are just brilliant!! It is humanly impossible to thank you enough!

[familywu]

You are such an excellent teacher, Prof. Crane. Thank you!

[matveyshishov]

First of all, your videos are amazing in terms of deeply detailed and contextualized understanding. Covectors being measuring device really clicked for me when I first watched it (I'm rewatching a year later now).
I'm deeply grateful to you for creating these videos and sharing your insight and understanding!

Second, paradoxically, going in the usual direction, operating in arrows, areas, volumes, has been getting in my way, and I had major difficulty understanding many concepts. Until I went in the opposite direction, "it's an inner product and it corresponds to XXX in physical applications". So having that tangible context interpretation is key, but starting with that context turned out to be more of a nuisance. As a side effect, I can't think in the usual terms about the tangible concepts anymore in real life, but it's fun, not complaining.

[haluk]

Damn, you are the best teacher in the internet.

[ObsessiveClarity]

I wish I saw this 1 year ago. I have been struggling with k-forms and the wedge product and all of that for so long. Why? Because they never talked about k-vectors and this whole duality! Only mentioned the geometric meaning in passing. Thank you :)

[aaronkurtz]

These videos are incredibly helpful, thanks so much!

[jdsahr]

An amusing musical comment:

at 52:16 you give a musical diagram suggesting a motivation for the sharp and flat notation, and certainly your brief singing clearly illustrates your meaning.

However, in the first example of a sharp "raising the pitch", note that B# and C are the same note (pitch), so the same pitch should be sung for both.

The second example has the same issue, because Cf and B are, again, the same pitch.

Perhaps a better musical notation for this slide would have been
B B# (== C)
and then
C Cf (== B)

which, in fact, would have been somewhat closer to the point being illustrated.



Gratuitous comment:

Some diabolical composers feel compelled to write in 5 or 6 sharps (or flats) and will sprinkle some double sharps (or double flats) in the score to annoy the performer. In the case of k-vectors and k-forms, it doesn't make sense to write A## because once you have raised the index, you can't raise it further. In that sense the k-vectors and k-forms are like a piano with exactly two keys, a B and a C

[robertwilsoniii]

Wow. This is powerful. The pay off of learning all that hard multilinear algebra is that these dual vector things simplify down to the kronecker delta making calculations a breeze now. This must be the insight Einstein had about this notation.

[guangfuwang]

best ever for discrete differential geometry tutorial!

[theodorostsilikis]

It's a good thing that the notation (a^b)*(u, v) is used and not (a^b)*(u^v). turns out that those two are opposite in geometric algebra. (a^b)*(u^v) gets an extra minus sign from the (e1^e2) squared for being in the same space.

[maxwang]

46:26 thank you prof for being hopeful of this. 😂

[DoobooDomo]

Loving this series! I'm probably overthinking, but on the musical isomorphism slide the notes you used, B and C, only differ by half step. So, Bsharp is an enharmonic spelling to C (they are the same sound). Likewise for Cflat and B. Was this intentional?

[rudypieplenbosch]

This a a great and concise explanation 👏

[guyejumz]

Is there a generalization of k-forms where linearity is relaxed, but compatibility of scalar multiplication is kept? Something that would capture e.g. f(v) = ||v|| except with a sign correction somehow. It seems like it would be useful to define integration of the differential modulus of a curve which would make operations like line integral over a scalar field and surface area over a surface precise.