Einstein Notation: Proofs, Examples, and Kronecker Delta

Рет қаралды 135,573

5 жыл бұрын

In this video, I continue my lessons on Einstein notation (or Einstein Summation Convention), by explaining how parentheses work in Einstein Notation. This is followed by an explanation of some Einstein Notation identities, non-identities, and the Kronecker Delta symbol.
This should wrap up the videos on Einstein notation, because in the next video on Tensor Calculus, I'm going to go more in-depth into actual Tensor Algebra!
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=1hRj...
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Twitter: / facultyofkhan
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Пікірлер: 77

@Jacob011 5 жыл бұрын

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 4 жыл бұрын

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

@lindale9717 3 жыл бұрын

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

@jacobmussi1684 5 жыл бұрын

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

@yudhisthir7247 5 жыл бұрын

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.

@jananeedhanasekaran4292 5 жыл бұрын

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

@independentchannel844 5 жыл бұрын

Half semester's tensor class in one sitting .

@anmol9096 5 жыл бұрын

Where does it help?

@sachinbs3961 5 жыл бұрын

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 5 жыл бұрын

I plan to!

@adamhawa1474 5 жыл бұрын

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

@krasserzocker3312 2 жыл бұрын

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

@sarahb5022 4 жыл бұрын

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

@FacultyofKhan 4 жыл бұрын

Awesome, thank you!

@PunmasterSTP 3 жыл бұрын

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

@wuyizhou 3 жыл бұрын

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

@feynstein1004 5 жыл бұрын

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

@pribyvate 4 жыл бұрын

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

@FacultyofKhan 3 жыл бұрын

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.

@f-heinze 5 жыл бұрын

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 5 жыл бұрын

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 5 жыл бұрын

These are also great lectures on Tensor Calculus imo: kzfaq.info/get/bejne/ns-JmryIr6vFfZ8.html

@ananashrabi3728 3 жыл бұрын

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

@nightstar1759 4 жыл бұрын

Very clear thank-you

@mikamikamusic7792 Жыл бұрын

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

@eirelavavedecc 3 ай бұрын

Thank you very much

@user-nd9tr7fm2s 2 жыл бұрын

Tnx sir, great explanation.

@aantoine5819 5 жыл бұрын

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

@himanshugaur4048 2 жыл бұрын

Thank you

@satyendravishwakarma1112 5 жыл бұрын

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

@skankhunt-zw6gg 5 жыл бұрын

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

@Aditya-pw7hm 5 жыл бұрын

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

@kenteo4857 3 жыл бұрын

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

@isreasontaboo 4 жыл бұрын

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

@achitame2372 3 жыл бұрын

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...

@josephhajj1570 4 жыл бұрын

is their a way to memorize them

@danielhoven570 4 жыл бұрын

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.

@iftikharullah3616 5 жыл бұрын

Kindly make a video on special and general relativity

@gabeparedes4944 5 жыл бұрын

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

@tanchienhao 5 жыл бұрын

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

@FacultyofKhan 5 жыл бұрын

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

@karthikragupathi3485 7 ай бұрын

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 ?

@markkennedy9767 2 жыл бұрын

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.

@oak6677 2 жыл бұрын

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

@aBigBadWolf 5 жыл бұрын

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

@jorgejimenez4325 4 жыл бұрын

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

@markkennedy9767 2 жыл бұрын

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.

@Antonio-xr8bj 3 жыл бұрын

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

@afroshirinlova3153 4 жыл бұрын

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

@AbhishekSachans 2 жыл бұрын

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

@ss-wl6zv 5 жыл бұрын

life saver

@drlangattx3dotnet 4 жыл бұрын

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.

@skankhunt-zw6gg 5 жыл бұрын

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

@skankhunt-zw6gg 5 жыл бұрын

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

@Octavarium666xyz 5 жыл бұрын

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 3 жыл бұрын

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

@cifria7051 5 жыл бұрын

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

@erickgudin 3 жыл бұрын

great

@prathameshnakhawa4975 4 жыл бұрын

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

@KhaosTy 4 жыл бұрын

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

@Kevin-qf2hs 2 жыл бұрын

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"?

@mahmoudsaleh9006 4 жыл бұрын

Thank you very much .. awesome material

@VinothKumar-qo7ry 5 жыл бұрын

Sir upload next videos

@blaze-pn6fk 4 жыл бұрын

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

@FacultyofKhan 4 жыл бұрын

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 4 жыл бұрын

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

@craig3.0 2 жыл бұрын

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?

@calyodelphi124 4 жыл бұрын

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 3 жыл бұрын

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. kzfaq.info/get/bejne/rNx1aNpkzrXYd6c.html). 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)!

@VoodooBoot 4 жыл бұрын

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

@tedsheridan8725 4 жыл бұрын

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 4 жыл бұрын

Good point! Thank you for the feedback!

@lesclassesmoyennes1442 2 жыл бұрын

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 (?)