Why you don't understand GREEN'S THEOREM -- Geometric Algebra, Calculus 3, Vector Calculus

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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.

[gijsb]

You sure went to a special kindergarten!

[danieljulian]

This is brilliantly concise. Edward Frenkel gave lectures for a semester of UC Berkeley's Math 53 (multivar calc) that were videotaped and put on YT, but only hints at this development without mentioning any defined concept (such as wedge product). Robert Ghrist even gives his online Calc 1 students a peek at the concept of boundaries, but doesn't note the next steps, not even in the Calc Blue series. It's not that I expect material offered openly to elaborate these next steps, but to come so close and then retreat seems a shame. And here you are with this wonderful presentation! I don't have enough math to go all the way there, but this is more than enough for me to see the outline of the advanced concepts. Beautifully done!

[rhumblinesnavalactionchann]

In my undergraduate years at UNSW, I took courses in Vector Algebra, but the theorems seemed ad hoc. I never became aware of any deep insight in these theorems. Geometric Algebra is what I should have been taught. Thank you. There are right ways and wrong ways to present mathematical concepts. I think Geometic Algebra will take us into the next millenium.

[TheJara]

What a clear cut presentation!! You made as clear as possible man...I am glad that I found your videos !!

[abnereliberganzahernandez]

this was so simple yet so accurate and elegant I cant believe it!

[peterhall]

Nice explanation. I'll check out your other work. Undergraduates are systematically short changed on these types of insights. It ’twas ever thus. Glimpses of some deeper structural reason for certain things are sometimes given but cannot be fully developed in the confines of a one semester course. And of course there is the tantalising expression: “The boundary of a boundary is zero”.

[Rockyzach]

Before watching the video, I just want to say that I don't understand Green's theorem because it's always taught at the end of the semester when you are trying to scramble to study for all your tests and get everything else done.

[Jon.B.geez.]

I might argue that integration and derivatives are indeed opposites functionally, but that their is a geometric perspectives of being opposites that is simultaneous/equivalent to their relation, and that has to do with the relationship between a volume and it's boundary, as expressed by density function (and understanding that density function is a dual game between the Radon-Nikodyn derivative vs Reisz Representation). Moreover, I might argue that their is even a 3rd different perspective on what is "opposite to what" in this deep theorem, and that touches on homology vs cohomology.

[Nylspider]

Brilliant video, but one small correction at 3:07 - the right hand side would not be written as f dx, but rather it would just be written as simply f, because f dx is a one-form, which isn’t integrable over a zero-chain (which is the boundary of [a, b] in this case). If we write f with no dx, then we have a zero-form, which can be integrated over the given zero-chain and is in fact the correct statement of FTC using differential forms

[jacksonstenger]

Great video, best explanation of Green’s theorem I’ve seen so far

[wonjunjang]

i think this might be my very first youtube comment. Thank you for making these videos. I just came across your channel and I'm looking forwards to watching all of your playlists.

[halneufmille]

"that we learned in kindergarten" I can't remember if we covered it after learning the square or the triangle.

[a.v]

Nice Video man! You deserve a Million subs. As a Computer science student, I found this very useful!

[Sidionian]

You're now an expert in Riemannian Geometry. Please post more of this stuff. In particular Ricci flows, curve shortening and surface minimization etc. I would like to learn more about Alexandrov Spaces. Please post more; you're great at explaining stuff.

[visualgebra]

Please make videos on Abstract algebra:Group theory.

[nektariosorfanoudakis]

So differentiating the form is adjoint to taking the boundary, the "bilinear form" being integration.

[KyleBroder]

👍 To support the channel, hit the like button and subscribe.

[julianbruns]

3:20
If you denote the boundary of [a, b] as {a, b}, don't you lose the information of the orientation of the boundary? (since this would be equal to {b, a}) The manifold you integrate over must be oriented and the boundary must preserve this orientation.

[qx]

At 11:55 why do we write w= Pdx + Pdy and not w = P ^ dx + P ^ dy. Or why don't we write dw = dP ^ dx + dQ ^dy ? Why suddenly there is this implicit wedge where we assumed a multiplication? Is the multiplication just a special case for a scalar functions ?

[Terieni-qc]

Great video! Why should I understand dxdy as a wedge though? Is there any intuition? (this is coming for non-math major!)