Differential Forms | What is a differential m-form?

We define the notion of a differential m-form and look at some examples.

Books I like:

Abstract Algebra:

Differential Forms:

Number Theory:

Analysis:

Calculus:

[fattosan]

I like how your brain reboots everytime you make a mistake or say something wrong. Great videos!

[quantabot]

I made it to 8, will take another one tomorrow, This is the best math channel which actually teaches stuff...

[paulhan]

Dude your videos are literally top notch. Keep it up you're doing great work

[nikkatalnikov]

pure gold, concise and clear. thank you very much for the entire playlist. last videos of the series are about more profound topics, and yet quite helpful for beginners.

[byronwatkins]

At 2:30, if the f_I are functions, then it would seem more consistent with previous math to call omega a differentiable (rather than differential) m-form...

[davidgillies]

At 11:00 it is helpful from a pedagogic standpoint to denote omega as a multilinear _functional_ i.e. a function that yields as its output a function.

[TheXterminator]

These videos are incredible. Differential forms life saver, thank you very much.

[cncqqyw]

I'm Japanese and a first-year college student, so I don't understand mathmatics words. But I try to understand detail and improve English level.

[edwardhuff]

"Obviously two more terms"... Last time you said that you ran a correction, but that was for 2-forms on a 4D tangent space. 1 4 6 4 1 and all that...

Of course by "n k-forns in bD" I mean "space of k-forms on a b dimensional base space has n = (b choose k) dimensions" or "there are n linearly independent k-forms in any basis of the space of k-forms."

{1}: one zero-form in any dimension base space, since (b choose zero) is always one.

{dx, dy, dz, dw}: four = (4 choose 1) 1-forms in 4D base space.

{dx∧dy, dx∧dz, dx∧dw, dy∧dz, dy∧dw, dz∧dw}: six = (4 choose 2) 2-forms in 4D.

{dx∧dy∧dz, dx∧dy∧dw, dx∧dz∧dw, dy∧dz∧dw}: four = (4 choose 3) 3-forms in 4D.

{dx∧dy∧dz∧dw}: one 4-form in 4D, isomorphic to ℝ if the base space is orientable. Else isomorphic to
ℝ/(x ≈ |x|) on a 4D Klein bottle.

[manthing]

All i'm gonna say is great video and expectung for your next video, especially if it's about differential forms.☺

[byronwatkins]

These are great! Unfortunately, I cannot seem to find them in order...

[chaebeomsheen]

I appreciate your videos. ♡ Keep up the good work!

[s.l.]

5:42 Should have been 4(4)-8(-2) = 32. Oh no's.... -8 bro! 😅 Love your videos.

[youssefamen]

Please do 2006 IMO problem 5 and thx for your great work 👌👌

[helloitsme]

all this stuff reminds me a lot of the jacobian, does it have any connections?

[borisbryant.]

I am a virtual apprentice by you it on.

[shaunmodipane]

are Differiantial forms still projecting vectors

[oni]

3:00 shouldn't that base point be $p\in\mathbb{R}^3$ instead of the tangent space $T_p\mathbb{R}^3$?

[Reliquancy]

Why can’t I flip any of your videos to landscape on my phone? It seems to be just your videos...

[AlbertTheGamer-gksn]

Challenge: Given this vector field in 7:57 but with xydydzdw, find the flux across the surface of the unit tesseract centered at the origin of this vector field.



Answer:
Φ = ∰ F ∙ dS
= ∰ (F ∙ N)dS
=⨌(div F)dV
div F = ∇ ∙ F
F(x, y, z, w) = <P(x, y, z, w), Q(x, y, z, w), R(x, y, z, w), S(x, y, z, w)>
Q(x, y, z, w) = S(x, y, z, w) = 0
R(x, y, z, w) = 2
P(x, y, z, w) = xy
dS = <dydzdw, dzdwdx, dwdxdy, dxdydz>
F ∙ dS = xydydzdw + 2dwdxdy
div F = y
Φ = ⨌ydV = ⨌ydwdzdydx = 0.5