Differential Forms | The geometry of multiplying 1-forms.

We discus the geometry of multiplying 1-forms with examples.

Books I like:

Abstract Algebra:

Differential Forms:

Number Theory:

Calculus:

[novagate]

Not hearing "That's a good place to stop" at the end of the video feels like a proof without QED or a black square

[someguy]

I studied mathematics and had various courses on differential geometry and aspects of lie-theory.
Feel like i am now getting a real understanding of some of this stuff for the first time - great content, keep going!

[ykkim]

You are natural to explain that the mathematical definition is so natural. :) Thanks for the great video!!

[maluithil]

17:00 mind blown. It's so satisfying seeing it all come together.

[tomatrix]

This is lovely to study after finishing vector calc and linear algebra. It brings everything together

[TheNiTeMaR]

You’re putting out some of the best mathematical content on YouTube right now!

I really struggled with my first manifolds course since we never received any intuitive explanations.. so this series is amazing.

[madhavshekharsharma]

hands down the best mind clearing explanation, thanks you!

[nric]

Please continue making this series! This really are great videos

[yeonhyungjun]

Oh my! After so much time spent on form in order to study general relativity and electromagnetism, I finally get the intuitive mental imagery of form contrast to the algebraic expression which I didn't understand, thank you, signed area explanation was brilliant

[jackhanson]

I'm sure I've commented before, consider this additional engagement! Truly a wonderful gift these lectures you've shared with us, thank you!

[manthing]

I loved to study this back at uni. I like this forms much more because they helped me understand what a gradient, rotor amd divergence are in higher dimensions, Also Green's theorem and Stokes' become much simpler to remember. Awsome video, keep it up.☺

[Gyroglle]

this is brilliant, and it pains my that this was not included at the start of my GR course

[apteropith]

I'm watching these for a better understanding of "n-forms", because they keep coming up, but I can't help but think that a formulation might be conceptually simpler (at least to me). I guess I've just never had much appreciation for this distinction between vector spaces and their duals.

[JernejBarbic]

Your videos are amazing. Very clear presentation. Thank you very much!

[calvinlee]

somewhat like the grassmann algebra approach i read in rudin better but I really appreciate seeing this perspective on things.
Thanks so much for doing this series!

[hershyfishman]

10:34 It's obvious, but just in case anyone is confused, he's not correcting himself to say it's not 2 but 3 vectors (like it sounds). He's just calling those vectors "3-vectors."

[gabrielh.martins]

I wish you were my college teacher... you know, officially.

[Koisheep]

I have studied differential forms both in undergrad and more recently in my master's. I know where your determinant formula comes from. But I never thought of like, going backwards and translating the "usual formula" for the exterior product given in terms of the tensor product and just, turning it back onto a determinant to see it as a signed "hypervolume" (quotes because I'm not sure if a complex value could be consider a hypervolume, heh).

[chaitanya.awasthi]

Hi Michael, thank you for a great series on differential forms.
I have a question at about 17:11 in the video. I understand the symbolic manipulation you performed to get the final expression for w1 (wedge) w2. However, in doing that, the term, say dx, went from being an element of a vector in TpR^2 space to a differential form that itself now operates on an element from the TpR^2 space. In other words, it looks like this symbolic manipulation somehow completely transformed the nature of the object 'dx'. Can you please elaborate on this a little?

[ObsessiveClarity]

For others confused near 16:30, as to the nature of dx and dy (are they no longer components of vectors? He's taking the wedge product of them?):
Here's what I think is going on.

EDIT: 18:42 describes exactly what I was saying here... oops.

I think we're supposed to, in this context, think of dx as the 1-form that maps a vector (v1, v2) to the first component: v1.
And think of dy as a 1-form that maps a vector(v1, v2) to the 2nd component: v2.
In other words, in the language he has used, dx takes (dx, dy) to just dx (abusive notation here: using dx as the name of a function and as one of the input vectors to that function).
And (again with abusive notation) dy takes (dx, dy) to just dy. Think of dx and dy (the 1-forms) as the "elementary 1-forms" just like you think of e1 and e2 as the elementary vectors in R2.

I'm guessing this leap is more natural if we understand more intimately what the dual of a vector space is.