Vector Dot Tensor | (lesson 2) Index/tensor Notation: The Scalar Or Dot Product


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(lesson 2) Index/tensor Notation: The Scalar Or Dot Product


Okay, so this is the introduction lesson on the scaler or dot product for index notation. What we’re going to do is going to break down thing into all its components and show what it ends up equaling. And why it equals that, okay. So what we want to do here is first. We want to just rewrite the each vector is what they really are and always make sure that you have any information available to you so. I’m going to need to know this here and I’ll go back over this in a second. Hopefully you watch the video previous to this on the Kronecker Delta and we’ll come back to this. But first what I need to do is break down the scalar product because it will first of all, it’s easy to see that if we’re doing vector or tensor notation for vector calculus that we’re working in three dimensions, so we’re only going to we’re going to stay in r3 so everything’s going to be three, so this is going to be equal to vector a 1 comma, a two comma, a 3 dot ed with b1 comma b2 comma b3 okay, so we’re not going anything greater than that, So we know that that’s why for our Kronecker here. I and J go to one two and three because we’re only working in r3 so let’s do this, so let’s break down each component, so I’ve got vector a well. This is equal to a 1 comma, A 2 comma B a 3 Sorry and well. What is this equal to well? This is equal to a. I plus AJ. Plus, that’s a 1 I a 2 J plus a 3 K. If you remember that from, you know, unit vectors. But in vector notation, we wouldn’t want to use IJ and K. We would want to use E 1 e 2 and E 3 so this is e. This is also a 1 and let me do this. So this is a 1 comma 0 comma 0 plus 0 comma 8 2 comma 0 plus 0 comma 0 comma A 3 Okay, so all. I did was separate this out because the sum of these three things equals this. Then what I can do is I can factor out the a one, and there we go. One zero zero plus a 2 0 1 0 plus a 3 0 0 1 right, so I just broke this down, so it looks like this. And now this is equal to a 1 e 1 plus a 2 e 2 plus a 3 E 3 OK, so you don’t what you don’t want to do. Is you don’t want to write IJ and K anymore? Even though we know we know that IJ and K are unit vectors from previous courses. You don’t want to do that anymore. So because the IJ and K is going to be reserved for the index, So you don’t want to have you don’t have like a sub? I times I and so forth. We want to use a 1 e 2 and E 3 and now you could write a little half like this to indicate that that is a unit vector. But it’s generally considered that the knowledge is known that E is reserved for a unit vector. So you don’t need to write the hat. If your teacher wants you to then do it, you know? Make sure that you always do what the teacher wants to do, but for for the most part. You don’t really need to do that, all right, so that being said now we know that a is equal to that, and then we can say that will be then is equal to B 1 B 1 E 1 plus B 2 E 2 plus B 3 E 3 OK, and now we’ve just broken that down, so we’ve got these here and you can see that. This is 1 1 2 2 3 3 1 1 2 2 3 3 So let me let me write. That is then the vector a is equal to is equal to the sum as I equals 1 to N of a I times the Vector E I right, because this is 1 1 This is just the summation, so if I made N Equal 3 then this is what I would get and since we’re working in Vector Calculus, N is always going to be 3 so I could rewrite this. Then as the sum as I equals one to three of a I II, I is equal to a 1 a 2 a 3 alright, and it’s going to be the same for B. Okay, so but B we’re going to be, We’re going to use the index J. And now you see why we have the index. I and J U N 1 use ijk up there, so I just I just found a so. Let me write that over here. So vector a is equal to the summation as I equals 1 to 3 of a I e I okay, And then B is equal to the sum as J equals 1 to 3 of B J. E G okay. Now we’re doing the dot product between these things. So let me grab another color and a so a vector, a dotted with vector. B is going to be equal to the sum. Is I equals 1 to 3 of Ai E. I dotted with the sum as a J equals 1 to 3 of BJ E. J And now since these components Here are all scalars. Ai e I. These are all well a is. Ai is a vector, but since it’s a column vector and this here is a scalar, then I can. I can commute these things and make them associative. And so what that means then is. I can extract the scible first of all. The summation signs are commutative, so I can pull these summation signs out here, and this is where the Einstein Summation convention comes into play. So let me pull those out, so I get three and I equals three, and then J equals three, two three, so I can pull these out and this is. This is a this is being multiplied by each other. Now, right to the dot. Now, it’s going to be multiplied because the summation because each individual term is summed. Right, so we get the the summation symbols are commutative, so I can pull those out, and then I can then rewrite this as a I e I and then times BJ, EJ. All right, so I just. I just pulled the summation signs out. This is that where the Einstein Summation convention comes into play. If you see, there’s two indexes that are equal to each other like this, then you know, it’s a summation, and you can omit the summation signs which we’ll get to that later. This is more of an introductory lesson on what you need to be doing. With index notation, the whole point of index notation is so we don’t have to write all of this because you see. This is a huge mess of junk. We don’t want to keep writing on that, that’s. The whole point of index notation is that. I can show you this once. I prove this then later on in problems. I can show you that this equals what I’m about to show that it equals and skip writing all this. So that’s the whole point, but you need to understand where it comes from before you can use it, right, okay, So now a and B are scalar’s, right, They’re scalar’s look. Scaler scalar scalars so scalar’s commute. I can put them anywhere in the equation. I want they commute, but hopefully, do you remember what commute means if you have a times? B that’s equal to B times. A that means I can reverse the order That’s come commute commutation so a and B commute. Ian, I I and II J They. In this case they may or may not commute. You know, depending so, we won’t worry about that right now, but let’s just take into account. That summation sign is commute and a. I and BJ are scalars so they commute so I can rewrite This again. Is the summation and the summation here and that’s three three. I equals one. J equals one. And then I can rewrite this as a I the J and then this is e, I and II J and those are dotted with each other there, right, so dot product with those because this is a this is a vector of unit vector with the one in the I component and the one in the Jade component, right and well. What is that equal to e IE. J is equal to the Kronecker. Delta, okay, so this is I didn’t put this in my list yet. But II ie. J is the I J component. So let me let me do this here, so we can add to our list as we go along and write that if we have vector E. I dotted with vector E J that this is equal to the IJ component of the Kronecker Delta and the reason why is because well, if you if you do the dot product of two vectors that have unit one, you’re just going to be left with. You’re going to be left with. You know, I one one position and well, let’s look at the end, lets. Look here, okay, so if you have. Kronecker one one. That’s the I J component of the unit vector of the sorry identity matrix, which is the first component here so II I dotted with. EJ is just going to be one position and it since it’s it’s I J. That tells me it’s one component. It’s not a hole back there. It’s going to be one component because it’s a scalar, right. Let me let me just let me just break that down. This is going to. This is kind of a lengthy lesson because we’re just introducing to it. So let’s show. Let’s show again. Why this is why this is the Kronecker? Delta, so let’s say I’ve got. Let’s say I’ve got e1 so that’s one zero zero dotted with e2 So that’s I J. So this is One-zero. Well, this is going to be 1 times 0 plus 0 times 1 plus 0 times 0 is equal to 0 so you can see here. I is not equal J. So we get 0 and it’s a scaler, and now we get so if we did a 1 with E 1 we would get 1 0 0 dotted with 1 0 0 and you can easily see that that is 1 times 1 plus 0 so that is equal to 1 right there, and so that’s why this term here is the IJ component of your Kronecker, Delta or identity matrix, so I can rewrite that, and now this whole thing since this whole thing is a scalar, this now compete commutes. Each individual thing does not commute, but now that this whole thing is a scalar because the dot product puts out a scalar. You could put this anywhere you want it to, but we don’t really need to do that. I’m just saying that can alright so now I rewrite this. Then as the summation is, I equals 1 to 3 summation as I equals 1 J equals 1 to 3 times a. I be J Time’s Kronecker Delta IJ. So that’s I did component of the Kronecker. Delta now the whole idea. Now that we get, we’ve gotten to this point. So let me let me take this. I’m going to take this part here and rewrite on another piece of paper so that we can see what we’re working with. Because what I did is I just I just derived this thing here. Based on rules and definitions. And it’s probably it’s okay. If you’re still confused, it’s going to take about 10 times of working these problems to really get it unless you’re just naturally really good at things like this. So you know, just bear with us as we break through this. OK, so now. I’m going to rewrite what I just had. And that’s the summation is I equals 1 to 3 I equals 1 J equals 1 to 3 of Ai BJ. Kronecker, Delta. I J ok, so this is what we’re trying to get to right here. This is the whole point of index notation is that? I I can do everything that I just did to show that well. This here is let me rewrite a dot V right, This is equal to a dot B in in our three three-dimensional space, right, so the whole point of this is is that I’m simplifying this. And since we see that we have, I and I and J and J. We have two indexes that are the same, and we have some nations here from I. 1 2 3 we can omit this. We don’t have to write this because we it’s assumed that we know that that is going to be there because it’s just a waste of time to keep writing this, so we can just write this as Ai be J. Kronecker Delta IJ. OK, so this is what we’re breaking down to this right here is equal to the dot product, But you know this. Is, you know, depending on if you’re working in different dimensions and so forth, but we’re breaking it down and making it simpler and simpler. So what we want to do then is we want to? We can go ahead and we can show that since this is the this is the dot product and we can. We can go even further and break this down or we can just we could leave it at this or we could move. You know, there’s different directions, but lets so let’s take this to be what it is and let’s make. I equal to 1 this equal to 1 this equal to 1 and this equal to 1 and then then we have 1 2 1 2 1 3 1 3 and then 2 1 and so forth because it will this. This Kronecker Delta. You remember over here? The Kronecker Delta has nine different things happening with it, OK? So 1 2 3 4 5 6 7 8 9 because the Kronecker Delta is nothing more than the identity Matrix 3 by 3 and so because that’s true, we have to break this apart in every single component, so we have IJ as 1 1 Ij is 1 2 Ij is 1 3 and so on, so let me go ahead and do that, and therefore I’m going to get A DOT. B is going to be equal to well for the first Kronecker. We’re going to get a1 b1 Kronecker 1-1 and then this is a summation, right, It’s a summation, so we have to do the sum of all of these, so this is going to be plus a well. The second one is 1 & 2 So when you have a 1 B 2 Kronecker 1/2 and then we’re going to have 1 & 3 so this is going to be plus a 1 plus Times B 3 Kronecker 1 3 So we just did the first row. Now we’re still adding, because this is the summation from. I goes to 3 right 2 3 so I need to do it for all of these all 9 of these and let me just let me just write it all out. Let me get let me grab another color here so that we can kind of keep it organized. So this is the first one, and then it’s going to be plus. Well, if I go down the second row. It’s going to be 2 1 2 2 & 2 3 so plus a2 b1 Kronecker 2 1 plus a2 b2 Kronecker 2 2 plus a2 b3 Kronecker, 2 3 and then let me do the last one again. This is very tedious. This is why we use index notation. So we don’t have to write all this out all the time, so we get a3 b1 Kronecker 3 1 plus a3 b2 Kronecker 3 2 plus a3 b3 Kronecker 3 3 Okay, so you see, I just took each individual component from the position of the identity matrix that it’s in. Hopefully you watch the video prior to this on the Kronecker Delta. So you kind understand that and well, now we take a look and so well when Kronecker Delta is 1 2 that’s 0 and as you can see, so when they’re not equal, they’re 0 so I can go ahead and kick out a whole bunch of terms here, so we look at 1 ones that is equal to 1 1 2 is equal to 0 so this goes away. This goes away! This goes away! This goes away, this goes away and this goes away, so I’m left with a 1v1 Kronecker Delta 1 1 + a 2 B 2 Kronecker Delta 2 2 plus a 3 B 3 Kronecker Delta 3 3 And well, this is 1 right, so when they’re equal to each other, it’s 1 so I’m just left with this being a 1 B 1 plus a 2 B 2 plus a 3 B 3 Which if you recall is the dot product right from the beginning here you get the dot product of this is equal to a 1 times. B 1 plus a 2 times B 2 plus a 3 times B 3 so I think that’s a really really cool that that actually works and then so we can just say that this is equal to a I bi or you could say it’s equal to AJ BJ. Or you could say it’s equal to a K BK as long as the symbols are the same, then they’re equal to each other, So the reason why we just did All of that nonsense was the show that a dot. B can be rewritten as a base JBJ base J. And this comes up later on. If you have, you’re doing index notation with things like the cross product and things like that. And if you you end up with, let’s say you’ve got a j b k c j. D f. Something like that, and these are all scalars. Then you know they commute so you could rewrite this. As a j c. J and then b k-df. And you’ll see here that you have. J and J Here. So you know that this term right here is a dot C. And so that’s where it comes about youll. This is why we do this. Because when we’re doing index notation, we’ll notice that if we have two indexes that are equivalent, then that’s nothing more than the dot product, and that will simplify our life later on down the road. Okay, so that’s scaler or DOT product for index notation. I’m going to move on to cross product and we’ll do a bunch of examples.

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