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.
@@Tevso. I ended up taking and passing the FE for electrical and computer engineering, getting a job in RF engineering as an applications engineer, then went to R&D, and now I'm a design engineer. I had a clear vision for what I wanted long-run, then just set benchmarks I knew I'd have to attain. Once that was done, I had a map, and I just pushed as hard as I could toward the benchmarks. A lot of my friends after college got kind of, lame jobs, or took the first thing that came up and stayed there. There's nothing wrong with that. But they aren't exactly happy people now. I'd just say, set a big goal for what you want from your life, then set smaller, intermediate goals that will help you get there, and go for those. It's common to be indecisive about the future, but don't let that indecisiveness paralyze you.
We learned about it in math methods for physics. It's in Boas's undergrad math methods for physical sciences exactly the way Andrew explains in the video (Chapter 10 pages 508-511). Figured it would be pretty standard for most undergrad physics curriculums since its used so frequently. I guess It shows how much professors can vary. It also came up in QM2 when introduced to 3D quantum mechanics (Chapter 4 Griffith's if thats what you used). Also, whats QM3? Is it like a bridge to graduate quantum? Does it still stay in the realm of most undergraduate quantum textbooks?
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.
I share same emotions with you.I am a physics undergrad stud but I am going to crazy because physics professor know nothing about the real picture I think. They say always it is a definition and we simplify it by using notational tricks.DAMN! I don't believe them . Please show me a way to learn what things lie behind of these things.
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.
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 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!
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.
Which appears in that method because of the explicit formula for the determinant which incorporates summing over permutations and multiplying each element by the sign of the permutation
@@adelbertwalker1570 damn, this comment is from a long time ago lol. Anyway, I just wanted to point it out because in the video he said "a minus sign pops here for some reason" without bothering to understand what's happening.
@@vgmduarte Yes just started the study of Levi-Civita. I remember from linear algebra that computing the cofactors there is a minus 1 raised to the sum of the i-th and j-th indices. that is why you can have a minus sign.
Lol who would've expected that when searching RUclips 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!
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.
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.
The determinant is the volume of a parallelepiped with edges along a, b, c, right? a, b, c being vectors in R^3 that make up the columns of the matrix M = |a b c|. Another way to calculate the volume would be (a x b).c (x = cross product; . = dot product). this equation can easily be calculated by det(M) = epsilon_{i,j,k} a_i b_j c_k (you still have to sum over every index of course)
Whatever’s is most convenient, different notation could have been used, and in fact is used, but this is the agreed upon notation that is deemed easy and is universal currently, however in the future and new, better notation could come to play and replace all the old ones. Such as Roman numerals to that of the counting system we have today, 0! Having multiple definitions to which it suits best in whatever situation, and perhaps the triangle notation used to represent logarithms as shown in 3glue1brown’s vid. :D
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.
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?
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?
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.
@@AndrewDotsonvideos Andrew, yes, it’s the AXB-2. If I’m getting this right, the subscript number 2 refers to the second column of the 3x3. Right? If so, then that would mean we arre looking at columns 1 and 3 which are the “i”. And “k” components, not the “j” and “k” components. Incidentally, I learned working determinants the old fashioned way of blocking columns. It comes naturally, but only after my old German professors shamed me in front of the whole class for doing it wrong on an exam. Never did it wrong again.
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.
![Seungmin "그렇게, 천천히, 우리(As we are)" | [Stray Kids : SKZ-PLAYER]](http://i.ytimg.com/vi/kAzmhLHePqU/mqdefault.jpg)
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.
Where did you come from after all these years? I'm in the last years of undergraduate physics now, I'm very undecided about what to do. :(
@@Tevso. I ended up taking and passing the FE for electrical and computer engineering, getting a job in RF engineering as an applications engineer, then went to R&D, and now I'm a design engineer. I had a clear vision for what I wanted long-run, then just set benchmarks I knew I'd have to attain. Once that was done, I had a map, and I just pushed as hard as I could toward the benchmarks. A lot of my friends after college got kind of, lame jobs, or took the first thing that came up and stayed there. There's nothing wrong with that. But they aren't exactly happy people now. I'd just say, set a big goal for what you want from your life, then set smaller, intermediate goals that will help you get there, and go for those. It's common to be indecisive about the future, but don't let that indecisiveness paralyze you.
@@seacaptain72 I will consider your suggestions. Thank you very much for your reply.
@@Tevso. Good luck out there
We learned about it in math methods for physics. It's in Boas's undergrad math methods for physical sciences exactly the way Andrew explains in the video (Chapter 10 pages 508-511). Figured it would be pretty standard for most undergrad physics curriculums since its used so frequently. I guess It shows how much professors can vary. It also came up in QM2 when introduced to 3D quantum mechanics (Chapter 4 Griffith's if thats what you used).
Also, whats QM3? Is it like a bridge to graduate quantum? Does it still stay in the realm of most undergraduate quantum textbooks?
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.
Damn, I didn't see the cross product 'til college.
Yes! The cross product is definitely the coolest thing they teach in high school, just without all the cool stuff.
@@benjamincolson i found the application of ordinary differential equations the coolest from my high school years
I share same emotions with you.I am a physics undergrad stud but I am going to crazy because
physics professor know nothing about the real picture I think. They say always it is a definition and we simplify it by using notational tricks.DAMN! I don't believe them .
Please show me a way to learn what things lie behind of these things.
That is very interesting insight. Now I'm tempted to pick up on abstract algebra about all these structures.
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!
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.
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).
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
I never write comments on youtube videos, but I felt compelled to do so on this one. This video was amazing! Thank you!
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!
Fortunate to be learning physics in a time where videos like this exist. Cheers
A course on General Relativity! That's where I came across it. Thank you very much for this☺️
This just made your Tensor Calculus video on Covariant Curl so much easier! Bazinga!
in in my first Semester and just introduced this last week. Really needed this
Andrew really about to get me through graduate EM Theory with his math vids
Same bro
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.
Which appears in that method because of the explicit formula for the determinant which incorporates summing over permutations and multiplying each element by the sign of the permutation
Exactly! What he is doing is calculation the cofactors of the matrix. Minus 1 to the 1+2 power is negative 1
@@adelbertwalker1570 damn, this comment is from a long time ago lol. Anyway, I just wanted to point it out because in the video he said "a minus sign pops here for some reason" without bothering to understand what's happening.
@@vgmduarte Yes just started the study of Levi-Civita. I remember from linear algebra that computing the cofactors there is a minus 1 raised to the sum of the i-th and j-th indices. that is why you can have a minus sign.
I am taking a class in continuum mechanics and didn't get this until I went through you video. Nice presentation!
This is big kid talk, I'm no ready yet
Just had this 3 week into my Bachelors and I was so happy 😂
Lol who would've expected that when searching RUclips 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!
Your explenation was super easy, helped me a lot, Thanks !!
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.
My homework is done. My day is made!! Thanks man
Thank you!, thank you! I was struggling so much with my textbook but now I got it! I feel so relieved :)
I started watching Andrew's videos for the sketches. 2 years later I've come to this.
Extremely clear explanation thank you very much!
In high school, I calculated cross products by quaternion multiplication just to mess with our teacher
Josef Frühauf but y tho
This was an absolutely awesome video. I'd love to see more like it.
4 years later, hes a genius!
Nice methodic work; you are a solid instructor.
Thanks!
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
Very detailed explanation... Develop all the subindexes of the levi civita tensor in the cross product was really ilustrative
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.
Taught so eloquently...love it!
Still better than the WAP song on RUclips. Thanks Mate!!
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
You are my hero! I love you, so helpful!
Thank you so much, Andrew-sama!
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
Tutorial! Level of video is escalating.
and I love it
Extremely helpful video thank you!
Would you be able to show us how to calculate the determinant of a square matrix using the Levi-Civita symbol?
You just don’t use E,ijk. You use E, ij, where the same like ii=0. Look up kronecker delta which is for dot product but is similar
The determinant is the volume of a parallelepiped with edges along a, b, c, right? a, b, c being vectors in R^3 that make up the columns of the matrix M = |a b c|. Another way to calculate the volume would be (a x b).c (x = cross product; . = dot product). this equation can easily be calculated by det(M) = epsilon_{i,j,k} a_i b_j c_k (you still have to sum over every index of course)
How do mathematicians come up with stuff like this..... This is amazing❤️❤️
Whatever’s is most convenient, different notation could have been used, and in fact is used, but this is the agreed upon notation that is deemed easy and is universal currently, however in the future and new, better notation could come to play and replace all the old ones. Such as Roman numerals to that of the counting system we have today, 0! Having multiple definitions to which it suits best in whatever situation, and perhaps the triangle notation used to represent logarithms as shown in 3glue1brown’s vid. :D
brevity, elegance and usefulness
Man maths is all about adding
Every function in maths
No matter however complex
Can be imagined to have originated from adding things
Nicely explained keep it up 💪👍
Thank you for clear explanation, my head is spinning with tensor calculus
Thank you a lot! It is really helpful 😊
This video is in entertainment category, lol))
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.
My teacher taught us both the methods, 1st this one(Levi Cevita) to give us some intuition and then the Matrice method for faster calculations
What a teacher…be my teacher Sir
thank u so much this is very useful for us
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?
Could you do on pseudo vectors
Yes it is very helpful
Thanks Alot🤩
Thank you so 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?
good easy to understand
this video was great, thx a lot!
wow...cleared it up for me
Please, make a video about group theory..
Great, Now I understand it
Te amo, me ayudó mucho a entenderlo. :3
Thanks sir ..🙏
Holy cow this was awesome. Thanks!
Crystal clear......Thank you
thank you very much
The honestest of works
Thanks a lot for your help 😍😍💚💚
Massive thank you
thanks, nice video
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?
Thank You, Very Nice Vid
What kind of paper is that??
white
Muchas gracias
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.
Thanks.... Bro...
It helps me a lot
but how can you use this for triple scalar and triple vector product
Thank you sir.
we just did this in my modern physics class phy344 last week
What will be the j th and k th component....
Thank you so much!!
Just did this today!!
In the very last example, why isn’t the Sigma subscript i2k instead of 1jk?
For (AxB)_2? I think it says 2jk, it kind of looks like a 1 but I don’t loop my 2’s
@@AndrewDotsonvideos Andrew, yes, it’s the AXB-2. If I’m getting this right, the subscript number 2 refers to the second column of the 3x3. Right? If so, then that would mean we arre looking at columns 1 and 3 which are the “i”. And “k” components, not the “j” and “k” components. Incidentally, I learned working determinants the old fashioned way of blocking columns. It comes naturally, but only after my old German professors shamed me in front of the whole class for doing it wrong on an exam. Never did it wrong again.
Forgot to ask, what’s the next lecture after this one Levi-Civita? Couldn’t locate on RUclips.
I'm learning E&M and this concept confused me. Thanks for clearing it up!
Thanks bro!
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.
Good vid ty
Im glad using fingers is still applicable in a much higher level of math c:
what if there is one more character like Ax(BxC)
please help me i have exam and i cant get it
@glyn hodges it's never too late for knowledge
Ay yo bless me for my special theory exam tomorrow
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
isnt that basically what he said just the other way around?
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
Hope its going well. Goodluck on your finals.
Fresh fruit
about 4 mins in and why was i never taught that in statustics?
Nice
❤