Einstein Summation Convention: an Introduction
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- čas přidán 18. 05. 2024
- In this video, I introduce Einstein notation (or Einstein Summation Convention), one of the most important topics in Tensor Calculus. Einstein notation is a way of expressing sums in short-form; repeated indices are used to denote the index that is summed over.
I describe the 4 major rules of Einstein notation, as well as the definitions of free and dummy indices. I also discuss some important information related to these major rules.
Questions/requests? Let me know in the comments!
Prerequisites: The videos before this one on this playlist: • Tensor Calculus
Lecture Notes: drive.google.com/open?id=1qgQ...
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"Family show" oof can't wait to sit down and watch some tensor calculus with the fam later tonight.
Tensor Calculus and Chill
Admittedly, family entertainment is pretty dumb.
me too!
Or is it
It's like farting under the quilt. Secretly she loves it.
Mnemonic: a DUMMY INDEX really is a SUMMY INDEX !!!
Spiderman + Elsa vs Tensor Calculus [gone wrong!!]
'top 10 anime showdowns - watchmojo'
I love you man
Sincerely,
An engineering Student
Me : * tries to understand Einstein notation *
Also me : but why would you torture yourself like that?
I think they must have made sense to people who tried working with a whole bunch of awkward to write sums, got used to the constant patterns, and then just started writing the patterns. Also, I watched this video and half of the next one until I figured out that he had actually explained near the beginning of this video that these were ways of writing sums of monomials. I thought he had spent an inordinate amount of time explaining rules for things whose meaning he hadn't told us until he started talking about theorums and I realized that he had told us what these things meant.
its actually genius, information density per caracter is so much higher
Genie before Einstein notation: “There are three rules”
Genie after Einstein notation: “There are four rules”
😂😂😂
No words would be sufficient to express my gratitude for this amazing playlist.
Everything is to the point without missing any details.
Great introduction to tensor calculus, looking forward to the next lectures in this series :)
That handwriting is 🔥! Thanks for the video!
Excellent video. Highly instructive and very well presented. Looking forward to sequels. Thank you.
Thank you very much for this series of lectures. Very succinct and clear to understand. Looking forward to the next lecture.
Simply superb. I'm about to delve into the massive (pun intended) book "gravitation" and needed a quick recap of enstein's notation and tensor calculus. So far, a must-watch, your videos are perfect.
Nice handwriting, too
From France, big thanks
Thank you for explaining the notation convention! I got really confused with other videos on this topic because nobody bothered to take a second and explain this!
Your way of explaining is quite structured. I like that! Thanks!
This was exactly the explanation I was looking for!
Thank you so much for such beautiful lectures. I'm enjoying them.👏
Explained 3 lectures in 9 minutes. Thanks!
you are writing super fast, a big salute for your explanation
U nailed it man....clear cut and simple
This was so informative. Thank you!
Best video regarding this topic! Thanks
Extremely helpful. Thank you
Excellent work!
This is great supplement to my math methods class thank you!
You give a pretty clear explanation!
Thanks for doing it!
very well explained. The convention makes more sense to me now. What "summing over" means, for example.
Thank you very well explained
thank you so much for the logical, simple explanations! Mahn ive been so confused over this, now i understand. Giving you a like for every video from now on. On the side, i appreciate the writing being at the same speed which you speak at, thats cool af.
- 3rd year astrophysics student
@@ameerbux8908 in which university?
@@llayashree3974 Witwatersrand
Please more content. I love these
Thank you. This helped me understand Einstein notation.
always amazing
thank you very much kind sir. Your video has been a great help for me
Amazing video.. Very well explained..
Thank you for doing this!
Love these presentations..finally understanding Tensors......
Glad you like them!
Holy that was so good
Very concise and helpful explanation. Thanks.
Glad it was helpful!
Your handwriting is amazing!!!
Really thank you sir 💝
Sir, thank you so much for this Video. I dont understand why university lectures don't sum up the material like that and talk endlessly about irrelevant bs
I love this! thank you!
Sir, please upload more video on Einstein notation
Thank you so much!!
I think it would probably have been easier to follow if you started out explaining coordinate transformations or whatever else needs summation using old fashioned Σ summation notation, and then, only after that, introduced Einstein Summation Notation. That way, we would have an idea of what he was thinking. This video makes a little more sense to me after I watched someone explaing tensor transformations without really using Einstein Notation.
very helpful, thanks.
Plz make a video disscussing special and general relativity
Amazing! By the way, what blackboard software are you using? i love it! :)
loved it
Thank you! From Trinidad
At about 3:50 you speak the word term and write the word expression. I believe the written part, "the free index only occurs once in the expression," is correctI and the the spoken version, "the free index only occurs once in the term," is incorrect.
I am also assuming, for example, that abc + def is an expression with two terms abc and def.
Other then that this is a well done and very useful and clear lecture. Thankyou.
Computer Scientists: They're just SIMD variables!
Nice video
Thank u sir
Thanks
Unrelated but you write quite well using the mouse. What tool/program is that?
Smoothdraw + a bamboo tablet.
legend
Thanks a lot
When I first watched this video, it was like when they teach you matrix operations in high school without explaining what the point of the matrices and operations even is in the first place (so that there isn't any possible way for you to get any intuition about it and have to just memorize the rules). This situation continued until half-way into the next video, when I you were talking proving and not proving the identities and non-identities, and that we could prove it to ourselves by adding up terms. At first I was like, "Prove what!?! Components of what!?! Right and left side of what!?! You haven't even told us what any of this means yet!" Then I realized that you actually had; it just went by so quickly and you went into the weeds of arcane notational rules so fast that I forgot about it.
I encountered Einstein summation when I was working on local self-attention in machine learning, in which we need to deal with a 6D tensor.
Is 6D rank of the tensor here?
Great!
4 year olds watching calculus vids? heck yes!
the free index is component? (of for example a matrix?)
4:52 - in Rule 3 example, in the first term, ' i ' shouldn't come twice right as it is a free index?
May I ask what do use to create these videos? what do you write on a tablet?
I am pretty late onto the scene, but no doubt that this is an excellent presentation in both content and lucidity. However, I do have a doubt and if true, may look like nit-picking, but for accuracy, shouldn't the statement under Rule 1 - "i is a free index... occurs only once in the *expression* ..." be actually "... once in a _term_ .." and similarly, reg. replacing a dummy index in "a) Not already in the *expression* .."
This is for second order, right?
From the Machine Learning perspective this notation is actually pretty natural, which the exception of the reasin for 'super' and 'sub' indexes.
How did he manage to write so neatly while being so fast.
In the 8:07 example, can anyone please explain to me why is is it wrong? I mean, is it necessary for a dummy index to repeat in order to call it a dummy index.
So, rule 3 has the limitation of less than 3 indices by term considering a 3D space, or is it the same for any dimension?
Rule 3 says that you can't have the *same* index occur three or more times in a given term. If you have a 5-D tensor for instance, you're allowed to write it as a_ijkmn*b_i (with 5 different indices - i, j, k, m, n). However, you can't write something like a_ijkmn*b_ii, where the i occurs 3 times. Of course, with higher dimensions, you will require more indices to describe your tensor.
Hopefully that clarifies things!
sir you just said free index only occurs once in a single term but in rule 3 you have allowed for 2 free indexes in a single term ,isn't that a contradiction?
0:19 yes I too make my 4 year old watch videos on tensor notation and general relativity.
Guys could someone give me a textbook where I can learn this? And tensor calculus in further
U r genius
Next Video Plzzzzzzzzzz
What is the reson behind rule no. 3? Why does it exist?
I needed that Spider-Man Elsa joke so much 😂
Great :-)
Why is it that the free index has to be present on both the sides??
What is the difference between the superscript and subscript in the notation? Are they the same thing? i.e. a_sub(i)*a_sub(i)=a_sup(i)*a_sub(i)?
The indices in different positions denote different vector types. By most usual conventions, a subscript denotes a component of a row vector, while a superscript denotes a component of a column vector: en.wikipedia.org/wiki/Einstein_notation#Mnemonics
Hope that helps!
Does there exist a physical quantity which is not a tensor?
I can't think of any: pretty much every physical quantity falls into one of the tensor categories, either scalar, vector or rank-2+ tensor.
Possibly you didnt follow this from 'THE BEGINNIG' or ... >> I see it costs too much in usa -- leonardo
@Faculty of Khan Hmm but if we're considering invariance then isn't only the spacetime interval supposed to be invariant? Then the space and time separations are co-ordinate dependent and thus not tensors.
is rule three unbroken for higher order tensors too? just asking for fun.
Yes! For higher-order tensors, like rank-3 for instance, we'd be using another unique index like k (e.g. A_ijk).
if you're summing over indices in einstein notation its one index up and one index down. not two indices down. at least that's the case here in europe, but i don't think that the summation convention is different in the US.
What do you mean exactly by one index up and one down? Like instead of a_ij*b_j, you use a_ij*b^j?
we always wrote it like a^j_i*b_j if you're summing over j. like a_ij*b_j is one member of a sum, when multiplying a matrix A=(a_ij) and a vector b=(b_j) for example, while a^j_i*b_j is the whole sum
Interesting. I also agree that your convention probably doesn't have any regional ties, but I've seen it used in a bunch of places, so it's likely a book-dependent thing. Here's a couple of examples in Wolfram MathWorld where the index being summed over is in the subscript both times: mathworld.wolfram.com/EinsteinSummation.html
I also found a stackexchange question addressing your comment, and it turns out that's it's a book/author-dependent convention: physics.stackexchange.com/questions/262237/einstein-summation-convention-one-as-upper-one-as-lower
As another example, the books I'm using right now (Schaum's Outline + Boas Mathematical Physics) don't really use the one upper and one lower convention.
But yeah, I agree with you that it might be more conventional to use one index up and the other down, but this is just an introductory video. I feel like when we get deeper into Tensor Algebra and operations involving tensors and vectors, the need to use both upper+lower indices will arise with the introduction of contravariant and covariant tensors.
Hope that clarifies things!
Yeah, I don't think the placement of indices matters until you stop using Cartesian tensors.
Upper indices correspond to the components of a tensor that are in the vector space itself while the lower indices usually respond to the indices that arise from dual functionals. If the space has no differences between the two, then they don't matter. Otherwise they done. I have done no degrees at UC Berkeley. lol.
Index? More like hintex! Thank you for making and posting this video that explains something so confusing in such a straightforward way!
This is somewhat similar to Numpy array broadcasting rules...
I feel like Einstein notation can be a bit hard to read. When you see an expression in it, the "form" of the expression doesn't immediately jump off the page. Why not make the dummy index just a *? (I could be overlooking something).
So
a_*j b_j would mean
a_1j b_j + a_2j b_j + a_3j b_j
This would make the difference more apparent, I think, and reminds one of mathematical notation from homological algebra. It seems to me that having to look for duplicates is more prone to errors, but also just doesn't jump of the page in clarity as quickly.
It also seems to me that trying to make things coordinate free as often as possible seems way better.
about to do a module in heat transfer and i need to understand this thank you for this
Thanks for saving me from the Spider-Man Elsa videos 😊
Best sentence///
This is faculty of khan signing out
niceeeeeee vedo
To be honest, I wouldn't want to say "script" more than needed. It's a pretty intense syllable.
How do you write so fast you wizard
It is hard to see the things written in deep blue and green
This is not how Einstein summation should be used. It should always be used to contract a contravariant (upper) index and a covariant (lower) index. Even in Euclidean rectilinear coordinates where there's no real distinction between vectors and covectors, you still should raise or lower indices when you which to use the Einstein convention, to make the summation explicitly clear. Repeated lower (or upper) coordinates should be interpreted as free indices.
You are right this notation is Funky annoying
Honestly I'm laughing here on this rules, because I have no idea of what he's talking about. He's explaining a bunch of rules and I don't even know what is a_i super j...
Need some examples :/
That's what the next video is for!
Faculty of Khan Beautifully presented and explained otherwise. Thank you for your uploads!
Dear mister Fac.(khan),
If a four year old were to stumble on your video, he would definitely understand, But do add some example exercises so that the information could sink in better.
"a sub i" and "a sup i" -- even fewer syllables. ;-)
Yea, but people can mistake the 'sup' for 'sub' since they sound similar.
@@FacultyofKhan Pronounce it as 'soup' as it is in 'super'
@@gilrutter9481 It's still much easier to tell the difference if one word is one syllable and the other two syllables, since those two vowels are kind of close to each other (very close if you're from Northern England), and the two consonants are both very similar between the two words and are positioned so the two words have similar "shapes" (so to speak). I think one measly syllable per superscript is a small price to pay for having one less thing to be confused about.