For those who get lost around 14:30, the RHS of that equation is actually the Leibniz formula of a determinant. So that the inner product between m-forms can be more neatly rewritten as: = det(), where on the right-hand side is the p-q component of the determinant. And you may then understand the following examples bearing in mind that Michael is expanding this determinant term by term.
at 19:33 for anyone confused about symmetric groups, see Contemporary Abstract Algebra ~ Gallian. {(1),(1 2)} is cycle notation ((1): 1->1, 2->2; (1 2): 1 -> 2, 2->1; ). Even and Odd refers to the amount of 2 cycles each element ( each element is a permutation) can be broken into. (1) = (1 2)(1 2), thus 2, 2 cycles, thus even, (1 2) = (1 2), thus 1, 2 cycle, thus odd. Hope that helps, thanks so much for this series, it's helping me with this FEM course and Hodge Laplacian.
The matrix defined in 12:08 is actually just the Gram matrix for those elementary 1-forms and the formula in 15:20 is the determinant of the matrix whose entries are specifically only the entries in the Gram matrix corresponding to the multi-indices of the m-forms that you're taking the inner product of.
Have you reviewed hestenes geometric algebra? It allows for the volume component to act as an imaginary element and so we get a real algebra containing the complex algebras as sub algebras. Additionally the geometric product is ^ + . And is written xy = x^y +x.y on the vectors, so the algebra is graded on all dimensions - it enables spinous and relativistic sub algebras as well. Very interesting and very natural.
We've already seen that b^a can be written in terms of a^b (multiplying by -1 depending only on their ranks). This means that you could write that expression as (*a) ^ b = w, which is a bit more easier to remember I guess. Any particular reason why you presented it in a different way?
Is there any reason for you to introduce the extension of the inner product using that formula instead of using the determinant of the matrix A_ij = ? The result should be the same, but I guess most folks are pretty familiar on how to compute determinants.
In the Better definition of the Hodge star, do you obtain different Hodge stars for different k-form inner products? Or is it always the same Hodge star operator, regardless of *which* k-form inner product is used?
Does the inner product formula come from more general representation theory? This reminds me of Young's Tableaux/Tabloids and the rep theory of the symmetric group
Shouldn't be the product in 16:00 symmetric positive-definite? The matrix in there has < 0, so it can't be positive definite. Or is all this series with views to pseudo-riemannian geometry?
At 17:40 Why not just say you used the foil method? Great video and easier to read. The 3 form. Seems new. :) But Seems similar to: x1+y3+z6=n (But no memory of using a foil method for it so it's nice see the method being defined. )
For those who get lost around 14:30, the RHS of that equation is actually the Leibniz formula of a determinant. So that the inner product between m-forms can be more neatly rewritten as:
= det(),
where on the right-hand side is the p-q component of the determinant. And you may then understand the following examples bearing in mind that Michael is expanding this determinant term by term.
Thanks, even though I understood Michaels explanation, yours is significantly simpler
at 19:33 for anyone confused about symmetric groups, see Contemporary Abstract Algebra ~ Gallian. {(1),(1 2)} is cycle notation ((1): 1->1, 2->2; (1 2): 1 -> 2, 2->1; ). Even and Odd refers to the amount of 2 cycles each element ( each element is a permutation) can be broken into. (1) = (1 2)(1 2), thus 2, 2 cycles, thus even, (1 2) = (1 2), thus 1, 2 cycle, thus odd.
Hope that helps, thanks so much for this series, it's helping me with this FEM course and Hodge Laplacian.
Can't wait for the diff-forms Maxwell's Equations formulation!
At 10:49 the equal sign= should be corrected as + plus sign.
..inded, the most brillant exposición in this subject in the whole internet.. at least in wéstern civilization !!
Great video!
Please continue making these videos on more advanced topics, even if people, sadly, don't watch them very much.
The matrix defined in 12:08 is actually just the Gram matrix for those elementary 1-forms and the formula in 15:20 is the determinant of the matrix whose entries are specifically only the entries in the Gram matrix corresponding to the multi-indices of the m-forms that you're taking the inner product of.
Have you reviewed hestenes geometric algebra? It allows for the volume component to act as an imaginary element and so we get a real algebra containing the complex algebras as sub algebras. Additionally the geometric product is ^ + . And is written xy = x^y +x.y on the vectors, so the algebra is graded on all dimensions - it enables spinous and relativistic sub algebras as well. Very interesting and very natural.
We've already seen that b^a can be written in terms of a^b (multiplying by -1 depending only on their ranks). This means that you could write that expression as (*a) ^ b = w, which is a bit more easier to remember I guess. Any particular reason why you presented it in a different way?
Is there any reason for you to introduce the extension of the inner product using that formula instead of using the determinant of the matrix A_ij = ?
The result should be the same, but I guess most folks are pretty familiar on how to compute determinants.
i know i’m necroing but the reason is the same as why m-forms are evaluated the way they are, and what the exterior product represents
27:35 Isn't = 3?
27:54 also should be -1 I'm sure
Yes.
Yes.
Wouldn't that gnarly formula simplify significantly if we just wrote it as the determinant of a Matrix via Leibnitz's Formula?
It already is Leibnitz formula
Which playlist does this video belong to?
So I can see the videos for understanding this one...
Hello, click here: th-cam.com/play/PL22w63XsKjqzQZtDZO_9s2HEMRJnaOTX7.html
Differential forms of course th-cam.com/video/PaWj0WxUxGg/w-d-xo.html
Glenn wouda Thanks!
In the Better definition of the Hodge star, do you obtain different Hodge stars for different k-form inner products? Or is it always the same Hodge star operator, regardless of *which* k-form inner product is used?
Could you make a video on Hodge cycles and on Hodge conjecture?!!
But *why* is that the lift of a 1-form inner product to an m-form inner product?
28:41
Does the inner product formula come from more general representation theory? This reminds me of Young's Tableaux/Tabloids and the rep theory of the symmetric group
Shouldn't be the product in 16:00 symmetric positive-definite? The matrix in there has < 0, so it can't be positive definite. Or is all this series with views to pseudo-riemannian geometry?
No, there are exceptions where you do not have a positive-definite psuedometric, such as lorentz space
asnwer dude hasn't posted yet? Time to fill in. asnwer = HODGE.
Take a shot every time he says "wedge" 🤣
(do not. I will not cover your hospital fees💀).
if i don’t really understand any of this but i want to what video should i watch you think?
Try the Differential forms of course:
th-cam.com/video/PaWj0WxUxGg/w-d-xo.html
@@FranFerioli thank you!!
Great video.
I'm not ready for THIS approach, had to see the video twice and still wondering how the big scheme should look.
Great Videos! :D
Did anyone else start at the top by thinking they had to go back and watch the Reca I video first? Only me, I suspect.
What do you even mean by example implies the definition :O
At 17:40
Why not just say you used the foil method?
Great video and easier to read.
The 3 form. Seems new. :)
But Seems similar to:
x1+y3+z6=n
(But no memory of using a foil method for it so it's nice see the method being defined. )
This is about the point where I wish I was smarter.
🔥🔥🔥
That was indeed a good place to stop
Someone probably wrote a Python program for this.
Mathematica. It was very easy.