Geometric Algebra -- What is area? | Wedge product, Exterior Algebra, Differential Forms

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I have not had the opportunity to teach mathematics as much lately, given the amount of focus I have given to my research. I enjoy the process of teaching and interacting with students and so have decided to do some online tutoring.
In this video, we discuss the wedge product -- an operation on vectors which gives us an understanding of area. This will be particularly fruitful when understanding the fundamental theorem of calculus in greater detail. This is part of an earlier video that I made concerning Green's theorem: • Why you don't understa... The audio quality was not the best, so I fixed the audio (I hope), and focused only on the part of the video which treats the wedge product. For more on so-called Geometric Algebra, this video has a swift introduction to geometric algebra: • A Swift Introduction t...
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These videos are separate from my research and teaching roles at the Australian National University, University of Sydney, and Beijing University.
Hi, my name is Kyle and I'm currently doing my doctoral mathematics degree in complex differential geometry under the supervision of Professor Gang Tian and Professor Ben Andrews.

[@KyleBroder]

👍 I have not had the opportunity to teach mathematics as much lately, given the amount of focus I have given to my research. I enjoy the process of teaching and interacting with students and so have decided to return to some online tutoring. If you're interested in personal help, I've posted my tutoring services on Fiverr: www.fiverr.com/s/dDYkBlz

[@PeeterJoot]

Your use of the wedge of i + j with itself to demonstrate that the wedge product is antisymmetric was beautiful! Very nicely done video!

[@lisannradmacher3139]

I was so close to a mental breakdown because of this wedge product. Thanks a loooooot for this video!

[@PeeterJoot]

@0:47 "From Kindergarten linear algebra". I had a 1st year engineering electromagnetism Prof who insisted that we knew many things (like Taylor series, which wasn't actually in the Ontario high school Calculus curriculum back in '92) from Kindergarten. I always hated how they made us watch Sesame Street in Kindergarten, so I tuned it out or played with other stuff. It's becoming increasingly clear that I missed an awful lot by doing that.

[@sk7w4tch3r]

underrated comment :)))

[@antoniusnies-komponistpian2172]

Yeah, I actually loved learning Taylor series with the Sesame Street. But the best was integration on manifolds. Finally I knew how to calculate the surface area of my Klein Bottle and how to distribute my donut to my infinitely many friends whose hunger values made up a converging series, in a way that the dough and the icing would converge proportionally.

[@Spirrie2002]

This was ace! I've struggled with this concept for years and never seen it like this and related to area before! 😮 When the formula for the determinant just falls out like that... glorious.

[@christelleaugustin1695]

I love how cross product, outer product, normals, determinants, are all "connected" (in a sense), and probably much much more i dont know about... what a fascinating topic! I wasnt smart enough to understand it from the web page i read, so your video helped a lot!

[@nemooverdrive760]

I was able to make sense of the whole determinant calculation procedure when thinking in terms of wedge products. Hope to see more geometric algebra stuff 👍

[@babai08_]

The part that I love the most about this is that you didn't assume i and j were orthonormal. This is exactly what I was looking for, I've been trying to understand the index notation for the divergence, and you've given me the first step. Thank you!

[@carterwoodson8818]

I have been wanting to see someone expanding on the wedge product 'bivector' for so long! This is great thank you!

[@l_a_h797]

This was very helpful. Thank you. I would like to push back against the use of morphing to go from one step to the next in a proof/demonstration. While it may look cool, it's actually much less helpful than showing the steps on successive lines. The latter lets you visually compare the two equations or expressions, to see how the second comes from the first. Morphing actually produces a gap during which you can't see *either* of the steps; then while looking at the last step you have to try to compare it with the previous step from memory. That being said, I appreciate the work and artistry that went into this video explanation.

[@kylenetherwood8734]

Everything else I looked at on this assumed the reader already knows what a wedge product is so thank you for explaining everything so clearly

[@KyleBroder]

Glad it was helpful!

[@johncrwarner]

Nice straightforward video linking the wedge product to the determinant / area with little hand-waving and building up the algebra

[@Anmol_Sinha]

This video is GREAT!!! Perfectly explained. The proof for i^j=-j^i was GENIUS

[@antoniusnies-komponistpian2172]

Why did we learn the wedge product in linear algebra 2 and analysis 3 without any geometric intuition? I finally understand the point of this, thank you!

[@TheTimeDilater]

Thanks a lot this was super simple yet intuitive!!!

[@Jon.B.geez.]

amazing little video. what a treasure/pleasure

[@ShredEngineerPhD]

Holy shit! I finally understood something related to differential forms! :) Thanks man!

[@larzcaetano]

Hello, Kyle! Your explanation was very, very easy to understand!!! I have this particular intuition that maybe studying math is about first exploring the idea and the motivation, instead of starting straight from the definition. You did exactly that, nailing it! Could you make videos about more geometric algebra with the same approach? Would love to see how you would explore it! Again, many thanks!

[@SphereofTime]

0:21

[@paulmarsh1930]

Thank you for the clear explanation!

[@ekasijohn]

😊 thanks I understand wedge of vectors is so similar to cross vector..

[@l_a_h797]

One thing I'm not clear on ... is the result of a wedge product a scalar? a vector? a bivector? something else? If it's just area, then I would expect a scalar, and this is supported by the statement u ^ u = 0. But the general formula for u ^ v ends up with (ad - bc)i^j, that is, a scalar times i^j. We could expect the area of i^j to be 1, but in that case, why doesn't i^j just drop out of the answer, giving us u ^ v = (ab - bc)? Or maybe i^j is a scalar but isn't necessarily 1, it just depends on what basis vectors we happen to be using? Or maybe we're using "Area" in a bigger sense, to include orientation, and it really is a bivector?

[@mrervinnemeth]

This is exactly the problem with this video. Just think about i⋀j as the unit bivector. Magnitude of 1 and orientation i → j. Of course, orientation makes more sense in higher dimensions.. Only the direction is meaningful in 2D. So in reality u⋀v is the unit bivector times the area of the parallelogram formed by u and v. Basically we got back to the definition of the bivector, which has a magnitude and an orientation.

[@robertwilsoniii2048]

Why not just use the product of the magnitudes of the vectors? ​@@mrervinnemeth

[@TheOzpad]

Brilliant ! Now it makes sense

[@MrWorshipMe]

Why is the area given by i^j and not j^i? Or would that depend on the sign of the product, and we would just switch order to get the possitive product?

[@JakubS]

Thanks, I was wondering what these were

[@mrpengywinz123]

I get that, if we assume \wedge has the properties of a product operation (ie, if we assume it will distribute onto vector addition) and that if we assume v \wedge v = 0 for any vector v, that it implies the anti-symmetry and determines the operation uniquely up to a scalar multiple of i \wedge j (in the case of 2-vectors). What I don't get is why we would *expect* the area operation to have the distribution property in the first place. Why are we motivated to think that this operation f(u,v) = Area(ru + sv : r,s \in [0,1]) would have the distribution property/could be seen as a vector product?

[@user-sl6gn1ss8p]

I was wondering the same. Just commenting to get notifications

[@KyleBroder]

I'll make an additional video on why we should expect the wedge product to be distributive :)

[@mrpengywinz123]

@@KyleBroder You're awesome!

[@wraithlordkoto]

Take a parallelogram with sidevectors A and B, now divide A into two colinear vectors A1 and A2; visually it is clear that the area of A^B = A1^B + A2^B, and by definition A = A1 + A2, therefore (A1+A2)^B = A1^B + A2^B. Using anti-symmetry easily proves the distributive case for left multiplication as well. So we logically require that our wedge product be a homomorphism (looks like distributivity, in this case preserving the operation of vector addition)

[@_BhagavadGita]

Very nice. Thanks.

[@wraithlordkoto]

Why is your video titled geometric Algebra with no hint of the geometric product or multivectors? Just wondering

[@KyleBroder]

The exterior algebra is not as well-suited to the KZbin algorithm.

[@wraithlordkoto]

@@KyleBroder I'll watch it either way, but you're right about the algorithm

[@BlueGiant69202]

Use of the 'Geometric Algebra' name is not restricted to the formalism developed by David Hestenes. It's been used as an alternate term for Clifford Algebras before Dr. Hestenes, PhD came along. E.g. The MIT Geometric Algebra course. It does cause a lot of confusion. I prefer the Hestenes Geometric Algebra primer to the formalism used in this video.

[@wraithlordkoto]

@@BlueGiant69202 Geometric Algebra is how clifford referred to his algebras before everyone else called them clifford algebras IIRC, I consider the names synonymous with the choice being to communicate intent of the algebra

[@lancequek5203]

What's the difference between this and cross product?

[@Roxor128]

Just a multiplication by the unit trivector.

[@rishikaushik8307]

Does it have an equivalent operator in higher dimensional vectors?

[@KyleBroder]

This (i.e., the wedge product) works (i.e., is defined) for any dimension.

[@chrstfer2452]

[@jongraham7362]

Why is (2,1) + (1,2) = (1,1)? Am I missing something?

[@jongraham7362]

This is great, by the way! I'd like to understand the wedge product better.

[@wraithlordkoto]

Must be a typo

[@yizhang7027]

Thank you!!!

[@johnhippisley9106]

Why can you assume the wedge product to be distributive?

[@KyleBroder]

I have a proof in my lecture notes on vector calculus, which I will make available shortly.

[@AbouTaim-Lille]

I need a link to download the book "Symplectic topology" By D. Hamann and D. Mcduff.

[@KyleBroder]

You can find it in any university library.

[@dominicellis1867]

So is the wedge product a generalization of the cross product. Is there a wedge operator on the del operator to make an exterior derivative?

[@lina31415]

Great video, though a little bit of intuition as to why the wedge product should be distributive would be nice.

[@KyleBroder]

Yes I agree. It's contained in my Vector Calculus textbook.

[@wraithlordkoto]

To get an intuition as to why, take the parallelogram visualization and break either side up into two or more vectors and try to find the area using the "cut up" versions

[@keyblade134679]

hi kyle, what do you think of differential geometry as an area of research right now? And also, are there plenty of industry (non academia) jobs for those that did their Phd in differential geometry? I know some areas of pure math like algebraic topology really dont have much non academia jobs, but since differential geometry has applications in robotics, and visuals, I would assume there are more industry jobs.

[@KyleBroder]

@AznWill789 The field of differential geometry is extremely active as an area of research. I'm not sure of the use of differential geometry outside of academia, but I'm sure there are a very large amount of uses. One that comes to mind is that general relativity is an application of Riemannian geometry. Hence, anything to do with aerospace engineering, G.P.S. systems.

[@mujtabaalam5907]

“Kindergarten Linear algebra” lol

[@ogunstega7348]

Thanks

[@KyleBroder]

No problem

[@redbaron07]

At 0:45 "Now of course from kindergarten linear algebra we know that the area of the parallelogram is just going to be given by the determinant of the matrix whose columns are given by u and v..." Try saying this casually in front of a "tiger mom" pushing her kid on a swing at the park. She'll sign the poor kid up for every math tutoring program in town!

[@MyMathYourMath]

this now makes sense why forms are represented by matrices, where you wrote u=ai+bj, v=ci+dj. Could you make a video on how to calculate d(a /\ b) for k,l forms a,b. Also, follow back plz :)

[@mndtr0]

Looks like skew (or also named pseudoscalar) product...

[@mathalysisworld]

wowwwww

[@hillcore6615]

💥💥💥💥💥💥💥💥💥💥

[@sahhaf1234]

Of course you are assuming that we have an orthonormal basis...

[@danieljulian4676]

The wedge product has many of the same features as the cross product, but spits out something other than a vector.

[@Roxor128]

They're related by the fact that if you multiply a k-vector by an n-vector in n dimensions, you get an (n-k)-vector as your output. For k=2 and n=3, you get a regular vector as your output. Multiply the output of the cross product by the unit trivector in 3D and you'll get back the wedge product. Which could be useful if you're writing code using geometric algebra and you have to call on older libraries written to work in terms of the cross product.

[@jameyatesmauriat6116]

What’s the meaning of wedge here?

[@matin.f139]

wait, "Kindergarten"? seriously?