Einstein Notation: Proofs, Examples, and Kronecker Delta

[@Jacob011]

I once used the Einstein notation to try to simplify notation of my calculations. Sometimes it worked for me, but I kept running into problems. Now I know why. I wasn't aware there are all these identities and non-identities.

[@gaussiano3388]

Right? I was reading an introduction to Einstein Summation, but nothing of this is in the books!

[@lindale9717]

Yeah! Therefore, this video is the best! XD So informative!

[@yudhisthir7247]

The best video in the youtube for einstein notation. I have studied it in my grad but not clear on this. Thankyou for providing such unique and concrete video.

[@sachinbs3961]

Please continue this. Also your playlist has all the maths needed to understand general relativity. It would be great if you did that too in some point of time. Great effort. TY

[@FacultyofKhan]

I plan to!

[@jacobmussi1684]

Just found this. PLEASE DO MORE. I need more of this in my life. I really hope you come back to this topic at some point

[@jananeedhanasekaran4292]

Waiting for the next video. Thanks a lot. The tensor calculus video series has helped a lot so far.

[@independentchannel844]

Half semester's tensor class in one sitting .

[@anmol9096]

Where does it help?

[@adamhawa1474]

Of all the tensor videos and wikipedia pages i've read your series is the only one that I understand

[@krasserzocker3312]

never thought about the importance of the difference between dummy indeces and free ones... that changes a lot. Thank you for this very "simple" and good explanation! helps my for my bachelor thesis

[@feynstein1004]

All we had to do was follow the damn notation, cj 😂

[@pribyvate]

best comment i've seen on a maths video, well done sir

[@FacultyofKhan]

I'll have two free indices, a free index large, a dummy index with extra dip, a notation identity, two Kronecker deltas, one with cheese, and a large soda.

[@sarahb5022]

You explain things very well. This is the first time I am learning this and it makes sense :)

[@FacultyofKhan]

Awesome, thank you!

[@PunmasterSTP]

Another phenomenal video! I tried to learn about Einstein notation before, but nothing came close to laying things out this clearly.

[@wuyizhou]

Who in the world are you? How can a university student explain such a range of topics so well? Keep up the amazing work!

[@f-heinze]

Thank you for this video! I really like your teaching style! Are you going to make videos about covariant and contravariant tensors with these funky upper and lower indices in this video series here any time soon? Or do you have any recommendations where to learn this stuff easily? I need this for my particle physics stuff that starts in october. Greetings!

[@FacultyofKhan]

Glad you like it! I'm going to be making the videos at some point, but maybe not in time for October since I have other series going on. In the meantime, I'd recommend going over Schaum's outline of tensors (yes, not that rigorous, I know) to give you an introductory overview.

[@legendary_egg]

These are also great lectures on Tensor Calculus imo: kzbin.info/www/bejne/npq8mIyNf6qUg80

[@mikamikamusic7792]

I'm first year highschool and I'm able to understand because you explain things so well, thank uu

[@jorgejimenez4325]

Holy cow this is complicated. I love it! Probably gonna start learning this after getting into more linear algebra stuff. Thank you for this señor

[@iftikharullah3616]

Kindly make a video on special and general relativity

[@gabeparedes4944]

I would really like this too! That's the reason I've been following along with the Einstein Notation videos

[@tanchienhao]

me too, please consider doing a series on general relativity. i believe it will benefit many

[@FacultyofKhan]

Rest assured that they're coming in the future! That's what my tensor calculus and differential geometry series are for!

[@ananashrabi3728]

Thank you so much for the explanation sir! It helped a lot!

[@aBigBadWolf]

Is it good practice to use einstein notation? It seems a non intuitive notation that might lead to read and write errors.

[@drlangattx3dotnet]

every other explanation of Einstein notation convention uses upstairs and downstairs as related to contravariant and covariant objects, components or bases. I find this demonstration a little confusing because of that.

[@eirelavavedecc]

Thank you very much

[@karthikragupathi3485]

I didn't quite understand when you said during solving the fourth identity " by combining the dummy indices and applying the Einstein's summation convention" how do we derive the last statement ?

[@satyendravishwakarma1112]

Please sir make more videos on tensor i am waiting for that

[@nightstar1759]

Very clear thank-you

[@himanshugaur4048]

Thank you

[@markkennedy9767]

In the first non-identity, are the left or right hand sides alone legitimate expressions in Einstein notation. I suspect not since there are different numbers of i and j in the two terms but can you confirm if they are indeed meaningless.

[@kenteo4857]

Can I ask what is the application or software you use to write down all these notes? thanks alot

[@Kevin-qf2hs]

I think the reasoning you provided for 3rd identity is not true. You said since i and j are dummy indices, then i and j can be switched, so aij = aji. However, can't i use the same reasoning for the 2nd non-identity by saying "since i and j are dummy indices, i and j can be switched between x and y"?

[@markkennedy9767]

Is there any shorter/more rigorous way of proving the fourth identity without having to write out the terms with i,j from 1 to 3 in full.

[@danielhoven570]

I watch these videos at 1.75 speed so I'm forced to draw out what he's saying in my head instead of trying to read it. It is actually a great memory tool, as well as a fun thrill if you're into that sort of thing. I knew tensor calculus from a classical non-Einstein notation perspective, and needed a succinct definition of the notation.

[@aantoine5819]

At the beginning you say combine terms outside the parentheses but do you mean the factors outside parentheses?

[@user-nd9tr7fm2s]

Tnx sir, great explanation.

[@calyodelphi124]

Unless there is another video in which you have explained these two things, and it's not obvious which one it is, there are two things that this video and the previous one it continues completely failed to explain: 1) What is the difference between a subscript index and a superscript index??? Some of the formulae that you have written hint that there's some sort of fundamental difference between the two, and that the only thing that seems to ignore that is the Kronecker Delta. Is (x_i x^i) the same as (x_i x_i) and (x^i x^i)? 2) What happens if a term has three or more vectors/tensors that all share a common dummy index? Something like (x_i a_ij b_ik)? Does that still violate Rule 2 of Einstein notation? If so, how is this incongruence resolved?

[@FacultyofKhan]

1) The difference between the subscript index and superscript index comes into play when discussing contravariant and covariant vectors/vector components. Contravariant components are specified with the superscript, while covariant components are specified with the subscript. My future videos explain this (e.g. kzbin.info/www/bejne/rKeoZqppnrSnfdU). 2) From my previous video, the same index cannot occur three times or more in the same term, so in your example (x_i a_ij b_ik), you would need to change one of the i's to another index depending on what you're trying to specify. Hope that helps (and sorry for the late reply)!

[@tedsheridan8725]

These are really good videos! Though I'd encourage you to adopt a more natural / conversational speaking style. The equal emphasis on every syllable and lack of breath / pauses makes it sound a bit robotic and off-putting.

[@FacultyofKhan]

Good point! Thank you for the feedback!

[@achitame2372]

In Identities 2, 3 and 4.. How will one determine the dummy and free index...?? As each term contains 2 numbers of j and i...

[@oak6677]

5:49 is it not necessary for an equation to have a free index?

[@craig3.0]

Wait, non-identity 3 is exactly the same as identity 4, all that's different is the equals/not equal sign. I'm going to assume that non-identity 3 is incorrect. Am I missing something here?

[@prathameshnakhawa4975]

At 5:47 the third identity aren't a12 and a21 different so how is the identity true

[@KhaosTy]

You can switch the indices i and j because both are dummy indices.

[@ss-wl6zv]

life saver

[@isreasontaboo]

Wait I thought you could only do contractions on covariant and contravariant indices.

[@cifria7051]

I don't understand step 1, why is i - 2, j -1, and etc.?

[@Antonio-xr8bj]

Can someone help me with the proof of the first non-identity please?

[@skankhunt-zw6gg]

@04:16, couldn't we say that the last two non-identities are invalid because there was not any free index present?

[@skankhunt-zw6gg]

Is there a rule like for a valid expression in Einstein notations, there should be free index present?

[@Octavarium666xyz]

You don't need free index everytime. For example trace of 2nd order tensor E is written like this "tr(E) = Eii" which is tr(E) = E11 + E22 + E33.

[@FacultyofKhan]

You don't have to have a free index present for every term (e.g. a_j b_j is valid), as Ales mentioned!

[@josephhajj1570]

is their a way to memorize them

[@erickgudin]

great

[@afroshirinlova3153]

Where cronecker delta is used ? plz explain it in details.

[@AbhishekSachans]

Suppose you are asked to build a matrix out of diagonal elements of matrix [Aij], how do you do it? You write M = Aij * Delta_ij

[@skankhunt-zw6gg]

@01:50, what if k also was =3? Would that be an invalid expression?

[@Aditya-pw7hm]

The equation is valid, though it may or may not have a usage in physical phenomenon (none that I know of) as the equation was probably chosen randomly to describe the Einstein's convention. It would just be that the equation would contain a summation of ALL the indices along it's range

[@anilphilip546]

but why do you write i-2, i-1? it is misleading.

[@mahmoudsaleh9006]

Thank you very much .. awesome material

[@lesclassesmoyennes1442]

There seems to be a contradiction between Rule 3 (no index may appear more than twice insise a given term) and the above example: delta(ij) xi xj = delta(ij) xi xi ... ...Of course you did not write that; I did. In fact you wrote directly: delta(ij) xi xj = xi xi , which is fine by me. Strictly speaking, one may also write as I just did: delta(ij) xi xj = delta(ij) xi xi , by nature of delta. In other words, there is an ambuguity, unless you impose some additional rule, for the sake of consistency (?)

[@blaze-pn6fk]

@9:23 shouldn't the RHS be 0, coz i != j.

[@FacultyofKhan]

The statement before says 'when i =/= j', so when i =/= j, the RHS is zero. However, in the one situation where i = j, we get what I put in the RHS. j (and i too) is a running index that's being summed over, so there must eventually come a point where j and i are equal.

[@blaze-pn6fk]

@@FacultyofKhan thank you, I assumed it was just i not equal to j. But it's the sum.

[@ahsanulhaq8056]

3rd identity doesn't make sense because aij is not same as aji

[@VoodooBoot]

9:20 For the example. You need i and j to be in the same range, right?

[@VinothKumar-qo7ry]

Sir upload next videos

[@thomasfisherson]

ARGHHHHHHH why is Einstein notation so COMPLICATED?

[@annesanila1897]

Too many ROLES, the nature finds the simplies way - leonardo