3. Summation Convention (Einstein Notation)

追加: 2026-05-07

[00:00:13]Good morning and welcome to this course on the foundations of continuum mechanics. Today we'll be talking about the topic on summation convention which is also known as Einstein notation. So let's get into it.

[00:00:31]So in this lecture today we'll be talking about the summation convention.

[00:00:35]I'll be defining that and taking you through lots of examples which will help you understand the convention in detail.

[00:00:42]It's very important to understand when and when we cannot apply this rule so that you get comfortable with the rule with the use of tensors.

[00:00:52]So in the last lecture towards the end we had briefly written that a vector V can be written as in its expanded form as the sum of the three components which uh in using the sum notation can be written as sum i going from 1 to 3 vi ei.

[00:01:20]So let me draw the three-dimensional cartesian coordinate axis O X1 X2 X3 where E1 E2 and E3 are the unit vectors.

[00:01:46]So now uh using Einstein's notation uh which is also called the summation convention in texts we will learn how to simplify this expression. Let us write the definition of the rule first.

[00:02:18]So the rule says that if an index in a mathematical term appears exactly twice.

[00:02:26]So let's underline the key aspects here.

[00:02:30]So first it's an index in a mathematical term. If it appears exactly twice and that's very important then a summation is implied over it. Okay. So what that means is that uh let's look at uh the term we just talked about. So the index in this case is I.

[00:02:52]So if I was to write this expression vi. So let's test the summation convention on this expression.

[00:03:05]So where's the index? The index is I.

[00:03:10]So I'll call this this is called the index.

[00:03:17]Does this appear exactly twice? The answer is yes. Uh does it appear in a mathematical term? So yes, this is uh a mathematical term. So vi ei is a mathematical term.

[00:03:35]So if the index I in this mathematical term is appearing exactly twice then from the summation convention summation is implied over it. So what that means is it implies that there is a sum on the index i where the index i goes from 1 to 3 because we live in a three-dimensional world vi.

[00:04:00]So this is what the Einstein notation or the summation convention says.

[00:04:08]So as I mentioned last time this is a notation we have chosen to use it. So we are applying that through the definition. So let's uh learn or understand the notation through a few more examples today.

[00:04:29]So this was the first example that uh we used to understand the notation. So let's take another example.

[00:04:42]So the term we have is ui vi. So let's write expand ui vi.

[00:04:53]So let's check through the definition.

[00:04:56]So the definition of the Einstein notation of summation convention says that if an index in a mathematical term appears exactly twice. So in this case the index we have is I which appears exactly twice in this mathematical term.

[00:05:12]So that means that a summation is implied over it. So I can write this as sum i going from 1 to 3 ui vi.

[00:05:28]All right. So this expands to u1 vub1 plus u2 v2 plus u3 v3.

[00:05:37]And as you may already have guessed that uh this is nothing but the dotproduct of the two vectors u and v. So u and v are vectors because we notice that u has one index and v has one index. So they are both vectors and this quantity ui vi actually denotes uh the initial notation representation of the dot products between u and v.

[00:06:05]Okay. We'll uh go into details about vector and tensor products later but uh this is just for noting down. Now remember that this is a notation. It's not a rule of mathematics um where uh we write ui vi as an expanded form. It is just making our life easy. It's making it convenient. So we choose to use it.

[00:06:29]So in this course as I mentioned last time we will be using the summation convention or the Einstein notation. So whatever expressions we write if we have an index appearing exactly twice in a mathematical term then a summation will be implied over it. It's not a choice whether we want to apply a summation or not summation is implied over it. So UI vi in our case or in the case where this convention is used will always expand to u1 v1 plus u2 v2 plus u3 v3.

[00:07:03]So the index that we have used here, the index uh I that we used here is uh actually called a dummy index.

[00:07:18]Now the meaning of the word dummy means not real.

[00:07:25]So the index I does not have a meaning by itself. So that's also out. there's no meaning.

[00:07:38]So what that basically says is that if you have the expression ui vi and you expand that which uh basically expands to u1 v1 plus u2 v2 plus u3 v3. So the index i did not have any uh real significance here. it was purely a dummy index which ultimately expanded to something uh in an expanded form.

[00:08:04]So from that logic I can also write ui vi as J VJ because from the Einstein's notation or summation convention a summation is implied over it and when I apply a summation from J going from 1 to 3 it expands to the same form this or I could use an index K. So it becomes UK VK.

[00:08:28]All right. So uh the index uh indices I J or K in these expressions I've written have no meaning by themselves. So they are called dummy indices.

[00:08:44]We also use the word repeated index.

[00:08:54]So the index I is uh repeated here. So it's called repeated index. We we'll be using the word dummy or repeated often in this course.

[00:09:07]So uh to understand this notation properly I suggest to you that you think of different cases. Okay just imagine what if this happens what if that happens. So uh try to think of all the cases possible with different kinds of expressions. Try to think think of exceptions and this is how through reasoning and arguments you will learn how to use this notation. And so I'll be giving you examples definitely a lot of examples today but uh I urge you to please uh think about different cases yourself which will help you a lot to understand this notation.

[00:09:44]So let's uh study a few more examples right so is UI vi the same as vi UI? So I request you to please pause this video and think about it.

[00:10:05]Right? So I'm sure you have thought about it by now.

[00:10:10]So let's uh check it out. So let's see the right hand side.

[00:10:14]So vi ui is the sum i going from 1 to 3 v i ui because from the summation convention we can write uh or this is summation implied over the index i because i is a repeated or a dummy index here. So this expands to v_sub_1 u1 plus v2 u2 plus v3 u3.

[00:10:40]So v_sub_1 and u1 and v_sub_2 u2 uh and v_sub_3 and u3 they're all scalers here remember scalers are zero tensors that only require magnitude to define uh describe them. So here v_sub_1 it's a component of a vector. So which is a scalar. So I can change the order. So I can write v_sub_1 * u1 as u1 * v_sub_1.

[00:11:05]Similarly the second mathematical term becomes u2 * v2 and the third becomes u3 * v3. So this is nothing but summation i going from 1 to 3 ui vi. and applying the summation convention again in reverse. So this becomes ui vi and remember i is a dummy index. I could have written uj vj doesn't matter. So the answer to this is yes ui vi is the same as vi ui.

[00:11:45]So let's look at more examples.

[00:11:50]So in this case uh the mathematical term that we have is ui vi wi.

[00:11:57]The question is what does this expand to using the Einstein notation? If it expands to anything. So let's uh again check the definition. So the definition of uh Einstein notation or summation convention says that if an index appears exactly twice in a mathematical term. So the index we have in this case is just an index I. Uh does it appear exactly twice? And the answer is no. It does not appear exactly. It's actually appearing thrice. So I appears so the rule cannot be applied.

[00:12:46]In fact, because we are using uh the Einstein notation, we are not allowed to use an index more than two times in our mathematical terms. So that's a very important point. So we'll write it down.

[00:13:15]Right? So it can only appear either once uh that will be when it's called a free index.

[00:13:31]I will talk about it uh later in the lecture today. And it can appear twice in which case it is called a dummy index where it expands using the summation convention rule.

[00:13:48]But it cannot be applied or it cannot appear more than twice because it creates confusion because if it appears three times uh like in this case um if I just look at this part I it might seem that it can expand to the sum but then there's an I left or if we apply this to viwi then there's a confusion. So for that reason we are not allowed to use an index more than two times in a mathematical term in uh if we plan if we choose to use this notation which we have chosen in this uh text as a next example.

[00:14:37]So in this example we have two terms in the expression and we have to expand this using the summation convention. So this is the first term and this is the second term. So I'll write it down that we have two terms in the expression.

[00:15:12]So let's go back to the definition of uh the summation convention. So if an index appears exactly twice in a mathematical term. So remember uh this whole thing is a mathematical expression or an algebraic expression but we looking for terms for the application of the summation convention.

[00:15:37]So in this case we can individually apply the Einstein notation or the summation convention to the two terms.

[00:15:46]So in the first term when we apply the summation convention this expands to a summation i going from 1 to 3 ui vi and the second term expands to summation i going from 1 to 3 vi wi. So now we can expand this So this is the expanded form of the expression in this example.

[00:16:29]So what you should not confuse here is that the index i appears four times because that is not the case. Remember we are always talking about mathematical terms for the application of the summation convention. So in this case the index I appears exactly twice in each mathematical term. So the summation convention can be applied. So just remember that I does not appear four times in the term. It only appears twice in the term. So as I said that these are the kind of examples we're looking for for uh improving our understanding of the summation convention.

[00:17:07]Let's now look at another example.

[00:17:18]So we have one mathematical term here and we'll see if we can apply the summation convention here. So if I look at the index I, it appears exactly twice in this mathematical term. And similarly the index J this also appears exactly twice.

[00:17:51]So I can apply the summation convention although I'll have to apply it individually over I and J in the same term.

[00:18:00]So first uh let's apply it over i and then because it's multiplied with wj so we'll apply the summation over j and then now we can expand the whole expression So this is the expanded expression. I won't multiply the terms because uh we'll have u1 v1 multiplied to all these three terms. So there'll be a total of 3 * 3 nine mathematical terms in this algebraic expression. But uh I'm sure you can do it yourself. Uh so we can see that using this uh summation convention or the Einstein notation, it makes things uh quite compact in terms of presentation and that was exactly the reason that Einstein chose to use this notation in his relativity paper.

[00:19:12]Let's ask us a question.

[00:19:23]So the question asks whether uj vj * wi ei is the same as ui vi wj ej. And by now I'm sure you have started to uh figure out that uh they're exactly the same because the indices i and j are dummy indices. So it doesn't matter whether I uh swap them in the expression because we always have to think of what the expression will look like once we expand it completely. And we can easily see that since J and I are dummy indices. I can do it in steps in fact to show you that uh in the LHS let's do it.

[00:20:13]So you have uj vj * w i ei. So what I'll do is I'll replace j with k because remember it's uh simply a dummy index.

[00:20:26]So it doesn't matter whether I write u j vj or I write uk vk so that they are exactly the same. And then what I'll do is I will replace i with a j here because uh I can do that j is u i is a dummy index. So say so is J.

[00:20:48]So WJ EJ and in the next step since uh um I has is gone I can replace K with an I because it'll still work and you can see that this is equal to the right hand side. So this uh clearly shows us that uh these uh two expressions or these two terms are exactly the same. Um so we can u swap the dummy indices because of the nature that they are dummy or repeated.

[00:21:31]So now let's uh try to apply or test this notation for tensors of higher order.

[00:21:40]So till now we've been working with vectors but uh why not apply to second order tensors.

[00:21:57]So the problem says that for a second order tensor sigma j expand sigma i i.

[00:22:04]So first uh let us check whether our definition of the summation convention will work here. The definition says that uh if a ma if an index appears exactly twice in a mathematical term so is sigma ii um is it a mathematical term? Well yes it's a mathematical term. So u we can uh check for the repeated index and also the index i appears exactly twice.

[00:22:41]So if an index appears exactly twice in a mathematical term a summation is implied over it. So what that means is that this is equal to some i going from 1 to 3 sigma i i which can further be expanded to sigma 1 1 + sigma 22 plus sigma 33. So if you look at sigma as a matrix uh the kind of representation we had shown yesterday or in the last lecture uh you can easily see that this is the sum of diagonals of the sigma matrix.

[00:23:31]In fact uh this has a name. It's a very important quantity as we'll see later in the course. um it's an important quantity mathematically as well as uh from a physical point of view in continuum mechanics this is called I'll write in bold it's called the trace of the tensor sigma i j so sigma i i is the trace of sigma ej which is basically the sum of the three diagonal terms of the tensor so if I want to um write the trace I can simply write sigma ii now is Is that equal to sigma JJ?

[00:24:09]Well, the answer is yes because it doesn't matter whether I use J or an I or a K. They're all dummy indices. They will expand to the same expression eventually.

[00:24:22]Right? So, we have learned now that the summation convention can be applied to the indices in a second order or higher order tensors as well.

[00:24:34]So let's look at a slightly uh more complicated term which involves a vector and a tensor.

[00:24:47]So vi sigma i j. So we have to expand this. So again we'll do it quickly now.

[00:24:54]So we have a mathematical term where the index i is an index which appears exactly twice. So this means that summation is implied over it. And when I expand uh the expression I get a v_sub_1 sigma 1 j plus v_sub_2 sigma 2 j plus v_ub3 sigma 3 j. So that's the expanded form of this expression which only has one term in this case. So you can see that uh the final expanded expression has still got a J in it and there's nothing we can do about J because J is appearing exactly once in each mathematical term and which is called a free index.

[00:25:46]So J is a free index here. So it can take any value 1 2 or three. Uh it does not expand here. So if you were writing this expression vi sigma iig j u it won't be complete until you specify the value of j otherwise it stays general uh in the sense that you will have to substitute a value of j for it to have a meaning uh from a physical point of view. So J can take a value 1 2 or three. There's no expansion happening over J because J is a pre-index not a dummy index.

[00:26:34]So I'll summarize.

[00:26:37]An index can either be free if it appears exactly once um in a mathematical term. Uh I'm not writing that again and again and it's repeated or dummy if it if it appears exactly twice.

[00:27:09]So let's uh take another example and this is going to be the last example in this lecture.

[00:27:17]So let's take a higher order tensor CI JK. So cig is a third order tensor and uh we have to expand the term ci ji.

[00:27:41]Now we can see that uh it's a mathematical term like you had in sigma i. Um this is a term. It has an index I that appears exactly twice.

[00:27:54]So we can write it as a sum of I going from 1 to 3. C I J I which becomes C 1 J1 plus C2 J2 plus C 3 J3 and J is uh a free index.

[00:28:12]So uh depending on the value of j that you have in the expression um that was given to be expanded uh the value of j will be replaced in this otherwise you can keep in general in the um inj.

[00:28:27]So what we have seen is that the indices uh can repeat uh in different um elements like uh we had previously uh in this case where uh v has an index i and sigma has a index index i. So the repeated index is appearing in different quantities or it can appear as subscripts in the same uh tensor which is happening here or if that happened in sigma ii.

[00:29:01]So that uh is uh going to be the end of this lecture. I hope uh you have been able to understand the summation convention and as I said please try examples think of situations where you could potentially break this notation and this is what will give you the uh strong understanding of this notation.

[00:29:23]It's a very important notation as we'll see when we apply further algebraic and u calculus based operations using this notation. uh so one has to get a good hold over it eventually uh for an understanding of continuum mechanics.

[00:29:39]So let's uh summarize today's lecture.

[00:29:47]So we learned about the summation convention.

[00:29:55]We did a lot of examples and we learned about the difference between two types of indices. One is called the free index and the other is called a repeated or a dummy index.

[00:30:18]So in the next lecture we will start applying uh various vector and tensorial operations using this uh initial notation with the summation convention in place and we'll also in the process define some important identities which will help us simplify mathematical expressions involving tensors um for uh writing more compact and uh more um more richer notations in continuum mechanics.

[00:30:50]So let's do a quick concept check for uh this lecture. So the first problem says is ui vi plus vj wj the same as ui vi plus vi wi. Right? So think about it whether they are the same. And the second problem it says that we have to expand this term which is vi * sigma iig j. So please uh always go back to the definition if you have any confusion. In fact I suggest in the beginning you should go to definition of summation convention to ensure that you have a strong understanding and there's no confusion as such. So thank you and I'll see you in the next lecture.