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.

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.

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

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

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

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).

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.

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

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!

This was an incredible explanation, thank you so much!

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

This was an absolutely awesome video. I'd love to see more like it.

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

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

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

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

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

Thanks. I remember doing this in electrostatics - basically any permutations of indexes that go clockwise are 1, counter are -1 and no change is zero. My prof also combined it with kroniker delta somehow.

Very detailed explanation... Develop all the subindexes of the levi civita tensor in the cross product was really ilustrative

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

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

Thank you!, thank you! I was struggling so much with my textbook but now I got it! I feel so relieved :)

Taught so eloquently...love it!

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

Extremely clear explanation thank you very much!

My homework is done. My day is made!! Thanks man

You are my hero! I love you, so helpful!

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.

This is great! it makes me wonder if this method breaks down when the vector space is of dimension 4 or above and if so whether that gives a clue for why cross products only work in 3 dimensions or less.

Fortunate to be learning physics in a time where videos like this exist. Cheers

Nice methodic work; you are a solid instructor.

Thank you so much, Andrew-sama!

Extremely helpful video thank you!

Thanks so much for the video We studied this topic on my first day at university and it was not so clear but this video explained the idea well

Lol who would've expected that when searching KZfaq for a video to explain the Levi Civita symbol for my relativity assignment that I would find my way back to Andrew Dotson and have this video perfectly explain it!

Class mate, really helpful !

Thank you a lot! It is really helpful 😊

Holy cow this was awesome. Thanks!

Nicely explained keep it up 💪👍

this video was great, thx a lot!

Crystal clear......Thank you

Still better than the WAP song on KZfaq. Thanks Mate!!

Tutorial! Level of video is escalating. and I love it

In high school, I calculated cross products by quaternion multiplication just to mess with our teacher

4 years later, hes a genius!

There are only two ways to calculate the cross product in my knowledge storage before I watch this video. One is to solve determinant of 3 by 3 matrix, which is where the cross product comes from indeed, the other is what taught in high school, kind of dead algorithm also shown in beginning of this video. It's really helpful dude, and can we use this levi civita method to solve even higher dimensions?

Massive thank you

I started watching Andrew's videos for the sketches. 2 years later I've come to this.

Thanks.... Bro... It helps me a lot

Thank you so much 🙏

thank u so much this is very useful for us

Great, Now I understand it

Thanks a lot for your help 😍😍💚💚

so, one of the "failures" my vector analysis teacher had was not being able to show how to use the Tensor notation in a real problem? How do we use this when we have actual values for A and B?

The honestest of works

Yes it is very helpful Thanks Alot🤩

Thank you

I think this method is very efficient if you want to create a program to calculate the cross product, it definitely beats the usual tabular method

Thank You, Very Nice Vid

thank you very much

Can this be used to get an orthogonal 4vec from two 4vecs the same way that an orthogonal 3vec can be made from two 3vecs?

My teacher of physics didn't teach well this topic and i hate him for that because now i see the beautiful of levi civita symbol. Thanks x3 million

wow...cleared it up for me

Te amo, me ayudó mucho a entenderlo. :3

good easy to understand

Muchas gracias

but how can you use this for triple scalar and triple vector product

Would you be able to show us how to calculate the determinant of a square matrix using the Levi-Civita symbol?

Thank you so much!!

What a teacher…be my teacher Sir

thanks, nice video

This video is in entertainment category, lol))

Just did this today!!

Thank you sir.

Thanks sir ..🙏

Thanks bro!

Thank you for clear explanation, my head is spinning with tensor calculus

Just double checking my math, no pun intended. Is the order of i, j, k as presented on the levi civita symbol, and the commutative, associative, and distributive properties what determines the calculation of cross products? I'm comparing notes from what I've learned from Vector Calculus for Engineers. From my previous studies, summing over the i, j, and k by multiplying the one attributed to the vector by the following attributed to the scalars, thus cancelling out the one attributed to the vector, and lastly the order of which the aforementioned i, j, or k presented on the levi civita symbol is the order that they will be presented on the following vectors and scalars cyclically. Did I get that right? Is that basically what being learned here? Constructive criticism, com padres.

How do mathematicians come up with stuff like this..... This is amazing❤️❤️

Could you do on pseudo vectors

Nice!

What kind of paper is that??

What will be the j th and k th component....

I'm learning E&M and this concept confused me. Thanks for clearing it up!

There is an easier way of performing the Levi Cevita method: Write i, j and k (unit vectors) clockwise, whenever you want to multiply any 2 of them, if they are clockwise, put a +sign else(anti clockwise) , put a -ve sign and multiply the unit vectors.

we just did this in my modern physics class phy344 last week

Good vid ty

Please, make a video about group theory..

❤

what if there is one more character like Ax(BxC) please help me i have exam and i cant get it

thanks!

In the very last example, why isn’t the Sigma subscript i2k instead of 1jk?

In Italian Civita is pronounced like Chivita :D

This is some Gucci Stuff right here

There is actually another equation AiBj-AjBi=sum k £ijk(A×B)k

Nice

Bruh its a cheatcode at this stage🎉

learning this in methods and it's the first time i've ever seen a tensor. it's not the hardest thing but what angers me is i have absolutely no idea where levi chevita came from, it seems so random.

This is great, my physics teacher taught us this literally on the second week of first semester and i love it, btw im still a freshman :v

you just saved my ass