Cross Products Using Levi Civita Symbol

Everyone has their favorite method of calculating cross products. Today I go over the way I was taught, and then a more formal way of doing cross products by using the levi civita tensor.

[seacaptain]

I'm coming to the end of undergrad physics. We got thrown to the fire here in QM3 and he just pushed this on us like we're supposed to know what's up. I have watched literally hundreds of physics videos over my education here, and this is the best one. There are no clear resources on this explained like you did it. I can't stress enough how much I appreciate this video. Thank you so much.

[theflaggeddragon]

The cross product is one of those things you learn in high school and don't really question it, but the concept is quite deep, related to representation theory and Clifford algebras. I remember seeing the determinant way of calculating and thinking "WTF how does that even work?". My teacher told us it was just a "notational trick" (I hate that phrase), but I knew that the reason was deeper.

3Blue1Brown had a video proving the determinant method and giving some intuition, but I wasn't satisfied since it didn't easily generalize or give me the deeper "why" picture.

Now that I'm studying Lie groups and Lie algebras, seeing the quaternions, cross product, and the determinant are all related by the normed division algebras, Clifford algebras, and even Hopf fibration and deeper math concepts, my mind is blown, and I feel rewarded for all these years I was deprived of that knowledge.

There are other "notational tricks" with the curl and divergence being represented as "dot" and "cross" products with the del operator, which is clearly absurd. The reasoning is also very interesting, and related to the exterior derivative operator studied in differential geometry.

[jososa]

After all these years, still the best video on the rel. between Levi-Civita symbol and cross product

[gregcoyle]

I was struggling over a textbook presentation of this - crystal clear, here. Thanks!

[chrisallen]

I saw how hard the math looked and thought there was no way I would follow, but that was surprisingly straightforward

[arjuntandon]

Thanks fellow NM bruh. You basically showed me the key thing I was missing in incompressible fluids and now I understand an entire semester (like actually understand, not just understand for the grade).

[sandrocaballero]

I am currently taking Analytical Mechanics, and do to my car not starting I missed the lecture that cover this method. I was confused trying to understand it, but watching this helped me get caught up. It makes much more sense.

[dreamelatte]

This was an incredible explanation, thank you so much!

[NinjaVsBear]

Am just learning about the Kronecker Delta and Levi Civita today, taking Math Methods for Physics I. So I’m still significantly confused, but your video helped a bit. Thanks man!

[ElliLovett]

i like how you always call your viewers smart before starting your videos ... it is very motivating ^^
..
I also like how you test it for one coordinate ' ok see it works!' ...and my brain goes back to the time we had to proof things like that, because my physicist self always was like 'it works for x so why shouldn't it work for y too? why do i have to proof things if i know they work? I'm not a math student.' :P

[johnq]

my god!!! i spend a week trying to understand this, your video is such a life saver

[alwaysbored]

A course on General Relativity! That's where I came across it. Thank you very much for this☺️

[gracetuttle]

I never write comments on youtube videos, but I felt compelled to do so on this one. This video was amazing! Thank you!

[dangernuke]

This just made your Tensor Calculus video on Covariant Curl so much easier! Bazinga!

[victorgabrielmoreleduarte]

The minus sign in the ey term appears because of the Laplace's method of taking determinants. The method works exactly as you said, except for that you can take any row or column as the pivots and the correspondent determinants will be multiplied by (-1)^(I+j), where i and j are the row and column indices of each pivot.

[paranira]

in in my first Semester and just introduced this last week. Really needed this

[Sambungus]

Andrew really about to get me through graduate EM Theory with his math vids

[sagedutorials]

I am taking a class in continuum mechanics and didn't get this until I went through you video. Nice presentation!

[Medumex]

This is big kid talk, I'm no ready yet

[ranmen]

Your explenation was super easy, helped me a lot, Thanks !!