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!
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.
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
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.
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.
Hmm took me sometime to understand it, my question was how those twice written are the summation while single written are free. The logic is pretty simple, if you are writing a summation equation you will likely have multiple different variable with same index being summed. If you have a variable which point to specific location, it should be a unique index.
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.
So if a 2-tensor can be represented by a matrix along with a coordinate-system specification, then a matrix on its own could be seen as a 2-tensor, assuming unit-coordinates are the implied coordinate-system.
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 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* .."
Are there any instances when using Einstein notation that a double index does NOT mean summation? Such as maybe when one instance appears in the Levi-Civita symbol times another coefficient (i.e. a)? I have one case in a text book where I have epsilon_LMN times X_i,L but it doesn't seem to make mathematical sense for L to have more than one value.
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.
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!
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.
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.
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?
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...
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!
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.
@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.
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.
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.
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!
"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.
I love you man
Sincerely,
An engineering Student
Mnemonic: a DUMMY INDEX really is a SUMMY INDEX !!!
That handwriting is 🔥! Thanks for the video!
No words would be sufficient to express my gratitude for this amazing playlist.
Everything is to the point without missing any details.
Spiderman + Elsa vs Tensor Calculus [gone wrong!!]
'top 10 anime showdowns - watchmojo'
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, i will also be showing my little brother and sister this amazing video
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!
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
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
Excellent video. Highly instructive and very well presented. Looking forward to sequels. Thank you.
you are writing super fast, a big salute for your explanation
Genie before Einstein notation: “There are three rules”
Genie after Einstein notation: “There are four rules”
😂😂😂
Great introduction to tensor calculus, looking forward to the next lectures in this series :)
Thank you very much for this series of lectures. Very succinct and clear to understand. Looking forward to the next lecture.
Explained 3 lectures in 9 minutes. Thanks!
Your way of explaining is quite structured. I like that! Thanks!
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
@@ameerbux78666 in which university?
@@llayashree3974 Witwatersrand
This was exactly the explanation I was looking for!
Your handwriting is amazing!!!
Phenomenal video. Cleared up my questions before I even had them lol
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 well explained. The convention makes more sense to me now. What "summing over" means, for example.
U nailed it man....clear cut and simple
Hmm took me sometime to understand it, my question was how those twice written are the summation while single written are free. The logic is pretty simple, if you are writing a summation equation you will likely have multiple different variable with same index being summed. If you have a variable which point to specific location, it should be a unique index.
Love these presentations..finally understanding Tensors......
Glad you like them!
This is great supplement to my math methods class thank you!
Thanks for doing it!
Computer Scientists: They're just SIMD variables!
Very concise and helpful explanation. Thanks.
Glad it was helpful!
You give a pretty clear explanation!
Best video regarding this topic! Thanks
Thank you so much for such beautiful lectures. I'm enjoying them.👏
thank you very much kind sir. Your video has been a great help for me
Thank you. This helped me understand Einstein notation.
4 year olds watching calculus vids? heck yes!
This was so informative. Thank you!
Wow! Love it man!
@@MuhammadFahadAli-yz8wg Glad you like it!
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.
I needed that Spider-Man Elsa joke so much 😂
Holy that was so good
Excellent work!
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?
Please more content. I love these
How did he manage to write so neatly while being so fast.
Extremely helpful. Thank you
Thank you for doing this!
Thank you! From Trinidad
always amazing
Amazing video.. Very well explained..
Thank you very well explained
So if a 2-tensor can be represented by a matrix along with a coordinate-system specification, then a matrix on its own could be seen as a 2-tensor, assuming unit-coordinates are the implied coordinate-system.
Thank you so much!!
Sir, please upload more video on Einstein notation
Amazing! By the way, what blackboard software are you using? i love it! :)
I love this! thank you!
Damn this guy is on 🔥🔥
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.
From the Machine Learning perspective this notation is actually pretty natural, which the exception of the reasin for 'super' and 'sub' indexes.
very helpful, thanks.
Plz make a video disscussing special and general relativity
Unrelated but you write quite well using the mouse. What tool/program is that?
Smoothdraw + a bamboo tablet.
legend
Nice video
loved it
0:19 yes I too make my 4 year old watch videos on tensor notation and general relativity.
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* .."
Thanks for saving me from the Spider-Man Elsa videos 😊
Thank u sir
Are there any instances when using Einstein notation that a double index does NOT mean summation? Such as maybe when one instance appears in the Levi-Civita symbol times another coefficient (i.e. a)? I have one case in a text book where I have epsilon_LMN times X_i,L but it doesn't seem to make mathematical sense for L to have more than one value.
4:52 - in Rule 3 example, in the first term, ' i ' shouldn't come twice right as it is a free index?
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.
To be honest, I wouldn't want to say "script" more than needed. It's a pretty intense syllable.
Thanks
about to do a module in heat transfer and i need to understand this thank you for this
Thanks a lot
Best sentence///
This is faculty of khan signing out
the free index is component? (of for example a matrix?)
Great!
U r genius
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.
Next Video Plzzzzzzzzzz
You are right this notation is Funky annoying
Great :-)
Guys could someone give me a textbook where I can learn this? And tensor calculus in further
you could try a student's guide to vectors and tensors by daniel fleisch
Thanks, I'll try it
Index? More like hintex! Thank you for making and posting this video that explains something so confusing in such a straightforward way!
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?
I'd rather my 4 year old just be a regular kid and watch spiderman than watch this.
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...
What is the reson behind rule no. 3? Why does it exist?
This is somewhat similar to Numpy array broadcasting rules...
This is for second order, right?
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!
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.
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.
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.
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.
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!
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).