Transcript of gr-sup1
==========


_0:07_: Well, folks. This is the the, the,

_0:12_: the first of I think 2 supervisions,

_0:14_: but there's also tutorial.

_0:16_: Now, I've never been,

_0:17_: never been clear to me what the

_0:18_: difference is between these things

_0:19_: because it's not as if we do anything

_0:21_: particularly different in each of these.

_0:23_: I have various times trying to think

_0:25_: is there an exciting thing we could

_0:28_: do in in in this context, and I don't.

_0:31_: I've never really thought

_0:33_: of one that convinced me.

_0:36_: Because the point of all all those exciting

_0:38_: things to do in attribution computational,

_0:40_: is to find some way of

_0:41_: encouraging a conversation.

_0:44_: Because the the basic idea is you have

_0:47_: there are things you don't understand.

_0:50_: Good to ask and I can try and sort out those

_0:52_: problems which might be from last lecture,

_0:55_: I might be from lecture 1 because

_0:56_: there might be some of the there's just

_0:58_: being a block at the very beginning,

_1:00_: the whole thing, or maybe there's a

_1:02_: little bit of tuning at the end that

_1:04_: you that you just smoothed over.

_1:06_: So this is the latest version of

_1:08_: of doing the useful thing here.

_1:11_: Which is I want to put up a questions

_1:13_: in a bit of the padlet and so thank

_1:15_: you to those who did that.

_1:20_: So there aren't any of those.

_1:22_: No one liked any of them,

_1:23_: which is a pity, but that's not

_1:25_: actually quite in this context.

_1:27_: That's quite a useful button to like this.

_1:30_: You know, this is an idea.

_1:31_: I have this question too.

_1:32_: So maybe when we do this for the

_1:34_: next supervision of the tutorial,

_1:35_: I'll encourage you again to to do that,

_1:38_: but I think they are

_1:39_: useful way of proceeding.

_1:40_: Because always easier to ask a

_1:43_: supplementary question than it is to ask.

_1:45_: So the initial question but your

_1:47_: hand up I think you're proceeding

_1:49_: is to talk about these questions

_1:52_: here which I I think are.

_1:58_: I think that the the illustrate

_2:01_: quite small but important,

_2:03_: you know, Rd bumps here and I think

_2:06_: though we'll use those I hope,

_2:08_: as a way of asking for the for the questions,

_2:10_: digging into other things

_2:12_: that are a bit obscure.

_2:14_: And if you have any,

_2:16_: you feel free to add to add things

_2:17_: to that even while we're talking or

_2:19_: just put your hand up and see ask

_2:21_: question you the old fashioned way.

_2:22_: Just I have this question.

_2:25_: OK, let's go through those

_2:28_: vector in one form thing.

_2:30_: Yes, and thinking about this it is the

_2:32_: way that I introduce vectors 1 forms,

_2:35_: vector one form and tensors does

_2:38_: seem a bit self referential.

_2:39_: It seems as if it's it's

_2:41_: building on building on itself

_2:43_: before it has an opportunity.

_2:45_: And I think that's partly because the. Um.

_2:53_: Because the definition.

_2:54_: Of of a tensor is something

_2:57_: which takes N1 forms and vectors

_2:59_: as arguments, blah blah blah.

_3:03_: Is presented to you before

_3:04_: I've said what vectors in one

_3:06_: forms are. So it feels a bit.

_3:10_: You know, which is an inconsistency here.

_3:13_: It's something missing this

_3:14_: in this definition and I

_3:15_: think it's just a matter of.

_3:17_: I think it's just a matter

_3:18_: of terminology in that.

_3:21_: I think it's just a matter of terminology,

_3:23_: so let's let's go through that.

_3:25_: I think in, in, in in an order

_3:27_: which doesn't confuse things.

_3:31_: Vision. Can we see that? Nope.

_3:41_: Oh God.

_3:44_: So let's see a. I think the first thing.

_3:51_: Is to say that there are there

_3:54_: are there exist sets of.

_3:58_: Things which we'll call centers.

_4:01_: And there are multiple sets of these tensors.

_4:04_: Or rather, there are we which have

_4:06_: the structure first of all that

_4:08_: they exist in the vector spaces,

_4:10_: so each of those tensors of those spaces,

_4:14_: so we'll call it tensor.

_4:17_: And M. Uh. Is a.

_4:29_: Is a vector space.

_4:31_: And if you remember from the beginning,

_4:34_: Part 2, by simply saying that

_4:37_: I've said quite a lot about what

_4:40_: these things called tensors are.

_4:42_: OK, I said you could add them

_4:45_: together to get another tensor.

_4:47_: You can multiply by scalar,

_4:49_: there exists a a unit.

_4:53_: No, there's an inverse,

_4:54_: and there exists a 0.

_4:55_: So I've said a lot about these

_4:57_: things just by writing that down, OK?

_5:01_: And I'm saying that there are multiple

_5:03_: sets of the so for each of this NM,

_5:06_: and haven't said LMR yet,

_5:08_: this thing called tensor is a vector space.

_5:13_: And then I'm going to,

_5:15_: for reasons which will become,

_5:16_: you know, which will emerge

_5:17_: as important later on.

_5:18_: The definition, I'm going to say,

_5:20_: going to have special names.

_5:22_: To two of these spaces. So um.

_5:28_: 10. Tenters.

_5:32_: Are called.

_5:36_: Victors and 01. Called one.

_5:43_: Go 1 forms and that's just I'm doing nothing

_5:46_: other than giving a conventional name.

_5:48_: To those two vector vector spaces ohh and.

_5:56_: 00.

_6:01_: 00 the 00 vector space are,

_6:04_: we'll call those the the the members

_6:07_: of that vector space fillers.

_6:10_: We've done no math there.

_6:12_: We've just called.

_6:13_: We've labeled things.

_6:16_: And then and that and it's at

_6:18_: this point that we can in a sense

_6:20_: legitimately give the the the

_6:21_: definition that we sort of started off

_6:23_: with in in the notes that a tensor.

_6:30_: Is a function.

_6:35_: OK. So. I'm at that point seeing already

_6:40_: giving quite a lot of information is a

_6:43_: thing which takes things and turns them

_6:45_: into other things, and that sounds like

_6:48_: another baby definition of function.

_6:50_: It's usually that's trivial and silly,

_6:52_: but it's a very general thing.

_6:53_: So so you we are used to

_6:56_: thinking of functions.

_6:57_: We learn functions in school.

_6:60_: There are some of you.

_7:01_: There's something which things like X ^2,

_7:03_: you take a number and it comes

_7:06_: out with another number.

_7:08_: And so it's easy to fall into the

_7:09_: rut of thinking of a function as

_7:12_: something which manipulates numbers.

_7:14_: But a function in mathematical terms,

_7:15_: I think, which manipulates

_7:16_: things have done to other things.

_7:18_: So it's a very general notion.

_7:20_: So we're seeing a tensor.

_7:23_: It's a function, obviously.

_7:27_: Component.

_7:31_: So how to write the components of

_7:32_: potential in terms of the metric

_7:34_: that's weird on the line here.

_7:36_: So that's that we've because

_7:39_: you've several several a couple of

_7:42_: chapters at like half a chapter to

_7:43_: go forward before we get to there.

_7:45_: So we haven't got nearly as far

_7:47_: as talking about components yet.

_7:48_: So we know that there will be components.

_7:54_: Yeah, yes. So in in Part 2 we we go on to

_7:57_: talk about our components quite quickly.

_7:59_: So the fact that we are saying this

_8:01_: is a vector space means that we're

_8:03_: importing all that technology.

_8:04_: So we know that we will be at some

_8:06_: point talking about components

_8:08_: and basis and and bases.

_8:10_: So there will be a basis,

_8:12_: a basis which spans each of

_8:15_: these of these spaces.

_8:17_: So yes, so I think that yes,

_8:19_: we will be talking about

_8:21_: components eventually because

_8:22_: we've talked about vector spaces,

_8:24_: but that's still a little while yet

_8:27_: because we haven't yet even really

_8:28_: got a notion of what a tensor is.

_8:30_: And.

_8:34_: But it's important to be

_8:35_: able to do that, but not yet.

_8:40_: So it's a function.

_8:41_: Which is something to do other things.

_8:43_: So the question is.

_8:44_: What does it turn into what?

_8:46_: What does it turn into what?

_8:50_: And the domain.

_8:53_: Sorry, the range. Of each of all tensors.

_9:00_: Is. The real line.

_9:04_: So from all from all the things

_9:06_: that the attention could be,

_9:07_: we're now down to things which are.

_9:11_: We mapped to the real line.

_9:12_: We mapped R, not RN, just R.

_9:15_: So tension thing which takes things

_9:18_: and gives and turns them into numbers.

_9:20_: What's what's the domain of

_9:23_: of of of tensors? Um. The.

_9:30_: To mean of. Over. In. And.

_9:37_: Is uh.

_9:40_: Well, and and and I'll say put it.

_9:43_: I'll write this this way so that I can.

_9:48_: Link to to that question is.

_9:55_: And.

_9:58_: Papa bomb.

_10:02_: View one.

_10:04_: Cross.

_10:20_: Now I live in perpetual terror of

_10:21_: getting you the wrong way around, OK?

_10:23_: And and the wrong way around.

_10:25_: I I may I may correct myself in a moment.

_10:29_: So we're seeing that that's a that's a,

_10:33_: a, a formal way of saying that the.

_10:39_: Attention or not just has the

_10:42_: the range of through line,

_10:44_: but the things that go into it.

_10:45_: The things that turns into a real number are.

_10:51_: These things would say to

_10:52_: call 1 forms and vectors.

_10:57_: You said that vectors and one forms are just

_10:60_: functions that take something into a real.

_11:02_: Yes? But how can you have a denser

_11:05_: which is something that there is a

_11:08_: real into a real but? It doesn't.

_11:12_: It's very weird that you say that

_11:14_: the domain is waveforms in one form,

_11:16_: which are theoretically real numbers.

_11:17_: But you can't put the scale all right,

_11:19_: because yeah, make something into it, yes.

_11:23_: To another if if if I'm following you this,

_11:27_: this isn't the the result of of the

_11:31_: application of the of the of the of the

_11:34_: tensor, it's the the function itself.

_11:38_: So This is why I'm being quite out.

_11:40_: So this is a function which takes

_11:42_: functions as arguments. Yeah.

_11:45_: That's why. That's why I was being.

_11:48_: Very general about things and things.

_11:52_: Not scale when you apply them to.

_11:56_: So what this means?

_11:60_: A10 tensor. It's called a vector.

_12:02_: By this definition,

_12:03_: that means that 10. Tensor.

_12:10_: It's something which maps.

_12:13_: The set rather informal notation.

_12:18_: 01.

_12:21_: Suppose the set of.

_12:26_: Yeah, we call that a vector that maps

_12:30_: the set of 1 forms to the real line.

_12:34_: So attention is something which maps.

_12:35_: It's not not the not the result

_12:37_: of applying that effect function,

_12:39_: but the function itself.

_12:42_: So that sounds very strange.

_12:46_: A function which takes function as argument.

_12:48_: But that's in a sense is why we introduced.

_12:52_: It occurs to me, not just,

_12:53_: not just now talking about it, that

_12:54_: perhaps is why we we introduced this this,

_12:56_: this terminology vectors in one forms

_12:59_: quite promptly because what we want to

_13:02_: immediately start thinking of those of

_13:04_: those vectors 1 forms as as pointy things.

_13:07_: So in this sense,

_13:09_: what the attention is doing is it's taking.

_13:14_: And you know that.

_13:15_: Yeah, that's why we jump to.

_13:18_: Not this rather formal

_13:19_: way of thinking about it,

_13:20_: but we jump to the terminology where

_13:22_: we were talking about vectors in

_13:24_: one forms because at this point we

_13:26_: want to immediately start thinking

_13:27_: of those as geometrical objects.

_13:29_: So in that case I wanted you

_13:32_: to answer some of which takes.

_13:35_: A vector, no one form, sorry.

_13:38_: Enter into number.

_13:42_: Um.

_13:46_: So.

_13:52_: We did does that.

_13:54_: You're still not looking.

_13:55_: Totally convinced because you

_13:56_: said that it takes a function.

_13:58_: Yeah, but the scalar is a function.

_13:60_: But they can't put a scalar

_14:02_: into an argument as a no.

_14:04_: 0. Potential is a function.

_14:08_: Yes, but they can put it as

_14:09_: an argument to attention.

_14:10_: You can't know.

_14:12_: So, so sorry. Yes.

_14:14_: So, so this definition of tensor

_14:16_: says that we're going to restrict

_14:19_: ourselves not to potentially don't

_14:22_: take arbitrary tensors as arguments.

_14:24_: We're going to, we've said not only the,

_14:27_: the range of the changes in the real numbers,

_14:31_: but the.

_14:32_: Argument of tensors are going

_14:34_: to be not just uproot tensors,

_14:36_: but specifically.

_14:39_: One forms and vectors,

_14:41_: so of all the things,

_14:43_: so you don't get 10 get A20 tensor

_14:46_: as an argument to a tensor.

_14:49_: We say, we declare.

_14:51_: So it could be.

_14:52_: Otherwise you could imagine a structure,

_14:54_: and quite probably there are in the head

_14:56_: of mathematicians structures where where

_14:58_: you where you have what looked like tensors,

_15:00_: which can take 10.

_15:01_: You know other tensor tensors as arguments.

_15:06_: We don't do that,

_15:07_: so we we so so this is quite

_15:10_: a narrow definition of.

_15:11_: We've narrowed this definition general a lot,

_15:14_: first by restricting the range through line.

_15:17_: Secondly by restricting the arguments

_15:20_: to only vectors in one forms.

_15:23_: And we have divided up the

_15:27_: sensors into things which take.

_15:31_: And one of the other so,

_15:33_: so these so and so,

_15:34_: so the different so the 20.

_15:40_: 20 and two, one and 02 tensors

_15:43_: are all separate vector spaces.

_15:45_: In principle,

_15:46_: they have nothing to do with each other.

_15:50_: At this point I'm sorry to I'm I

_15:52_: think I'm wondering worrying I'm sort

_15:53_: of starting to you talk around in

_15:55_: circles enough there's making it so

_15:57_: more confusing than it than it is.

_15:58_: I think the thing I want to

_15:60_: go over is that yes,

_16:01_: the definition as presented in

_16:03_: in the notes does appear to be

_16:06_: uncomfortably self referential,

_16:08_: and it seems to start talking about

_16:10_: vectors before we say what vectors are.

_16:13_: And and. I think that what this might

_16:18_: illustrate is that if you try and avoid that.

_16:22_: By avoiding the word vector until

_16:24_: a little bit later,

_16:26_: you end up with an explanation,

_16:27_: which is possibly a little you know,

_16:30_: possibly more formally correct.

_16:31_: But a bit more confusing.

_16:36_: Does that. Feel better in a sense.

_16:44_: I'm not seeing any nods or

_16:47_: shakes of heads or tears, so.

_16:52_: I think because I mentioned

_16:54_: this this situation here with

_16:56_: with with the other products,

_16:58_: I think it's fairly natural to go on and

_17:01_: talk about the the tensor product here.

_17:04_: Because again, that's another

_17:06_: thing that I think is looks more

_17:09_: more exotic than it really is.

_17:13_: And it's a way of.

_17:17_: Building up a tensor from other tensors.

_17:21_: And you know, the other product

_17:23_: or the tensor product appears in

_17:25_: multiple contexts in mathematics.

_17:26_: It it basically means just

_17:28_: putting two things together.

_17:29_: Most of the cases where you you,

_17:31_: you you are an outer product,

_17:33_: you're you're just in jamming things,

_17:35_: you're not into leaving them,

_17:36_: you're not doing anything complicated,

_17:37_: you just going 1/2 and put them

_17:40_: network each other and and dealing

_17:42_: with the parents together.

_17:43_: That's the intuition that often

_17:45_: is behind the notion of the tensor

_17:48_: product or the product in mathematics.

_17:50_: So in this case.

_17:54_: The.

_17:57_: So if we had.

_18:01_: See function F of X is X ^2.

_18:06_: The function G of. X is equal to.

_18:11_: Um to. X + 1 for example.

_18:16_: That each of those is a function which maps.

_18:19_: I'm making this up as I go

_18:20_: along so many time myself.

_18:21_: Notes here and which maps

_18:23_: the real line through line.

_18:24_: Nice simple function,

_18:25_: nothing exotic about them.

_18:29_: But what about the function F?

_18:32_: Outer product G. As a function.

_18:36_: What does that mean?

_18:40_: We're going to see that the definition of

_18:42_: that outer product there in this context

_18:45_: for functions some, you know, similar.

_18:48_: It's similar to using other things is that?

_18:51_: Is a function. Which has two arguments.

_18:57_: And the. The way in which the definition

_19:02_: of the what this function does to

_19:04_: these two arguments is straightforward.

_19:06_: It is defined. As the first linear

_19:10_: product applied to the first argument.

_19:14_: Multiplied by the 2nd.

_19:16_: Applied to the other argument.

_19:19_: Which in this case is X ^2.

_19:22_: Times two y + 1.

_19:26_: And there's nothing more to it than that.

_19:30_: So the outer product in this context,

_19:32_: when you apply it to functions,

_19:33_: is just a way of composing,

_19:35_: because there's multiple ways you can compose

_19:38_: functions you know that you could talk about.

_19:41_: If after G. And that's.

_19:45_: Of X, which is F applied to G,

_19:49_: of X, and so on.

_19:51_: So there's multiple ways you can you can

_19:53_: you combine function into other functions.

_19:54_: You have you seen that notation after?

_19:57_: Yeah, so this is just one of the ways you

_20:00_: can make what you can combine functions,

_20:03_: and it means nothing more than that.

_20:05_: So. You make a two argument function

_20:09_: from 2/1 argument functions.

_20:10_: Specifically,

_20:11_: by multiplying the first argument,

_20:14_: the first function to the first argument,

_20:15_: the second argument,

_20:16_: the 2nd to set the first function to,

_20:19_: the to the first argument,

_20:20_: and the second function to the second

_20:22_: argument, and the numbers together.

_20:24_: And similarly,

_20:24_: that means that if you have a.

_20:29_: Vector V.

_20:30_: Which is is I'm which is A1 form of argument.

_20:35_: Outer product. Um, P for example.

_20:39_: Which is A1 form takes the vectors

_20:42_: argument and that is. A function.

_20:48_: Is a function which can take.

_20:52_: A1 form. And a vector as argument.

_20:56_: Let's call that that them, um,

_21:00_: Q&A. Which is such that by definition.

_21:05_: We apply the first function.

_21:07_: To the. First argument.

_21:09_: The second function to the

_21:12_: second argument and multiply the.

_21:14_: These are both numbers multiplied together.

_21:18_: So we're not doing anything there,

_21:19_: but we're not doing there except

_21:20_: we're just doing this case.

_21:21_: We're doing it with specific functions

_21:24_: which are vectors of one form,

_21:27_: and you can do this we don't.

_21:29_: But you can do this with

_21:31_: higher order tensors,

_21:32_: higher rank tensors as well.

_21:34_: So you could have a V cross T and that

_21:37_: if T is a is A2 argument tends to,

_21:39_: then V cross T would be a

_21:41_: three argument that tensor.

_21:43_: So this is the way of combining

_21:44_: lower first rank one object

_21:46_: into a rank two objects.

_21:50_: And So what? Yeah.

_21:55_: And and what this is what's happening

_21:57_: here with outer product here is

_21:59_: that also the idea of you know,

_22:01_: just in jamming multiple spaces together?

_22:04_: And and becoming the domain

_22:06_: of the of the Nam tensor.

_22:10_: So it's the same notion.

_22:12_: The sort of straightforward German.

_22:16_: And.

_22:19_: So if we look at that a

_22:22_: particular. Question there.

_22:25_: And. So what we're doing here

_22:29_: is just what we're doing here.

_22:34_: But for the special case where the

_22:36_: vectors we're talking about are.

_22:37_: Why is that blue, I wonder?

_22:41_: Oh, the special case where the vectors

_22:45_: in question are the basis vectors.

_22:48_: So there this this T.

_22:51_: Which now the you know,

_22:54_: there's a couple of things

_22:55_: going on here, one of which is

_22:56_: the question of of components.

_22:60_: This.

_23:04_: And. The the The this object El Cross E.

_23:13_: M. Cross EN. That's our function.

_23:19_: Which takes what at arguments? Someone.

_23:27_: 31 forms, yes. So let's call them and

_23:33_: P1P2. P. Three and, and and and that the.

_23:40_: Value of that. Function. It is therefore

_23:48_: A&I A31A30 tensor. Just purely because

_23:52_: it takes 31 forms of argument.

_23:54_: That's the definition and the and

_23:57_: and the value of that function

_23:59_: applied to those arguments is El

_24:01_: applied to P 1 * E M applied P.

_24:07_: 2 * E and apply to P. 3.

_24:15_: But we're doing a little bit

_24:16_: more here by saying the 10th were

_24:19_: actually want that's that's so.

_24:20_: So we're defined at a tensor

_24:23_: there AA30 tensor for each of

_24:27_: every possible value of LM&N.

_24:30_: The change that we actually

_24:31_: want in this case.

_24:34_: Is a linear combination of those.

_24:39_: So it's. E 1 cross one, cross one plus,

_24:43_: E some multiple of E 1 cross

_24:45_: one cross E2 plus plus plus E 1

_24:48_: cross one cross E 3 and so on.

_24:51_: So it's a large number.

_24:55_: Uh. Ohh sorry. The way is is is different.

_25:00_: That's OK. OK, switching switch,

_25:05_: switching to pressure in the 1st Place Elm.

_25:12_: NOFEL cross EM cross.

_25:16_: Omega. And. That's a linear

_25:20_: combination of n * n * N terms.

_25:24_: Where?

_25:28_: There are. Yeah, so? So there's n

_25:32_: * n * N terms in that sum. What?

_25:39_: Thanks. It was intended on that thumb.

_25:44_: And that means that when you

_25:46_: apply T to things. We apply.

_25:54_: We just drop these arguments.

_25:57_: You want one form vector argument into the.

_26:01_: Argument positions of each of

_26:04_: these things and straightforward.

_26:07_: Real number of application of

_26:08_: the results between them added

_26:10_: the result and in this case.

_26:12_: So there's there's two things happening here.

_26:14_: First of all, in top think we're

_26:17_: defining this tensor as the sum

_26:20_: of N by N by N outer product.

_26:24_: With a with a different coefficient,

_26:26_: a different a different multiple of each

_26:29_: of those which we can choose there. And.

_26:38_: Just because. I mean,

_26:41_: I mean the same thing in both,

_26:42_: in both cases. It just is.

_26:43_: It's it's the same thing, yes.

_26:45_: So I I I'm rather than writing.

_26:48_: And we'll have to avoid to to

_26:51_: to slightly avoid confusion.

_26:54_: I'm, I'm. I'm changing my mind

_26:55_: between right bringing across.

_26:57_: It looks a bit like an X. So yeah.

_26:60_: So there's two things happening here.

_27:03_: First, that's the.

_27:05_: At most places, and it's true.

_27:07_: It's just simple multiplication

_27:09_: of real numbers.

_27:10_: So that there's nothing, nothing,

_27:12_: nothing exciting there, that's just 2 * 3.

_27:15_: OK. Over there?

_27:18_: If so,

_27:19_: that's our sum of multiple terms with

_27:23_: coefficients given by this matrix here.

_27:25_: Essentially,

_27:25_: we'll call it a matrix in that way.

_27:29_: Key elements of

_27:32_: TLM&N with isolation convention.

_27:36_: It's a man. Well, I think right there

_27:41_: there it's we're thinking as a matrix what

_27:45_: we're going to discuss since when we.

_27:48_: OK, so temporarily we're going

_27:50_: to think of it as just matrix.

_27:53_: OK all all is there is a matrix

_27:56_: of numbers of north by N numbers.

_27:59_: And then when we ask what is the value

_28:03_: of that tensor, what the new make?

_28:05_: What is the number to which this

_28:08_: evaluates? We.

_28:11_: You know, we just drop in.

_28:15_: Section 1 forms if we specifically drop in

_28:19_: basis one forms and basis vectors. Then.

_28:22_: By the definition of product, we get this.

_28:27_: Well, these are simple.

_28:30_: Numerical applications.

_28:33_: The next step here is to recall.

_28:37_: That our choice.

_28:38_: Of 1 forms by choice or one form basis,

_28:42_: one forms are always going to be dual.

_28:47_: To our basis vectors, meaning that we

_28:50_: have we choose that the basis one forms.

_28:55_: Will always have.

_28:56_: In this situation,

_28:57_: will always evaluate to the Chronicle delta,

_29:02_: so that would be 0 unless L is equal to I.

_29:05_: When it will be 1.

_29:08_: So we can replace each of these terms

_29:10_: by Delta IIL, Delta LM, Delta NK.

_29:14_: At which point we can do the the three sums.

_29:17_: So if we now do the sum over over,

_29:21_: well, say L.

_29:24_: So we add so that is.

_29:25_: It is the summation sign in

_29:28_: front of that which sums for L.

_29:33_: M&N. From one to N. When we do that

_29:37_: for L that term there will be 0.

_29:41_: Except when L is equal to I.

_29:44_: So the only 10 that will survive from that?

_29:46_: Sum is when L is equal to I.

_29:50_: Who do the sum of M exactly the same.

_29:53_: The only term that survives is when M is

_29:55_: equal to J and we'll do the sum over N.

_29:58_: Go in terms of survive is the

_29:60_: one where N is equal to K.

_30:02_: Should we discover?

_30:04_: By virtue of the definition

_30:06_: of the outer product.

_30:08_: That when we do specifically this and

_30:10_: give not just arbitrary what one forms

_30:13_: of vectors as arguments for the tensor,

_30:16_: but specifically basis vectors in one forms,

_30:19_: what we get out turn the handle.

_30:21_: What we get out is this.

_30:23_: The Matrix was started off with.

_30:27_: Up there. And it's at this point we see ohh,

_30:31_: that's not just a a matrix.

_30:33_: We're going to see if that's

_30:35_: the components of the tensor.

_30:36_: And that's and and and

_30:38_: when we in that context,

_30:39_: we write this matrix carefully with

_30:42_: indexes staggered and and and it's so

_30:45_: although we started off saying well.

_30:47_: We discovered that this,

_30:49_: this set of numbers actually had a, a, A.

_30:52_: We could we think of it in a

_30:54_: slightly enriched way by thinking

_30:57_: of it as the the component,

_30:59_: because that's what they are of,

_31:02_: of the center.

_31:03_: So all tensors will have components.

_31:07_: So for any tensor.

_31:09_: That you do this to,

_31:11_: you could do the same thing

_31:13_: and get that answer there.

_31:15_: So that this doesn't depend on that.

_31:20_: For, that's true for any danger.

_31:23_: What's special about this?

_31:24_: Is that a particular way of

_31:27_: calculating a higher rank tensor?

_31:29_: Which we can see to be consistent,

_31:30_: because when we do this to that tensor,

_31:33_: what we get out is what we started with.

_31:36_: So there's more than so. Uh.

_31:40_: Now let's see could this right around?

_31:45_: Not all tensors are the outer

_31:48_: product of lower rank tensors.

_31:51_: But all tensors can be decomposed into

_31:55_: a sum of outer products like this.

_32:00_: Below the 2.9 equation,

_32:01_: but this is some of them by by.

_32:06_: M by M. Or because these are just.

_32:11_: Indexes was run from 1:00 to

_32:14_: 1:00 to end, so if you recall.

_32:24_: So when we see T KLM.

_32:32_: T = t LNEL. Because he. And cross.

_32:38_: When we go and what we mean is.

_32:43_: L = 1 to N. And.

_32:52_: OK, that's confusing, right? Tea.

_32:60_: LMNEL cross E. And cross E Omega

_33:05_: N so remember that the the the.

_33:13_: And the summation convention is

_33:15_: taking the summation for granted,

_33:17_: so there's north. And so I I see yes.

_33:22_: So the problem is I've I've I've

_33:23_: I've used in two different ways.

_33:28_: They have brought.

_33:33_: So I've used in two different ways, right?

_33:37_: Didn't even think of that.

_33:38_: So so yes, this this end here is a

_33:42_: dummy index as opposed to this end here,

_33:44_: which is obviously the the dimensionality

_33:46_: of the of the vector species.

_33:49_: Sorry, I I yeah,

_33:50_: I make it myself to rewrite that.

_33:57_: Questions.

_34:04_: Answer.

_34:06_: Space. Right. But which has more

_34:11_: dimensions than all, right? So if the.

_34:18_: That yes that's another restriction

_34:20_: here to the to to to what we.

_34:24_: Our initial definition definition

_34:26_: of tensors as functions to

_34:28_: take things and other things.

_34:30_: We restricted it in various

_34:31_: ways on the previous page,

_34:33_: but I another restriction is that

_34:36_: the vectors that you add. That you.

_34:42_: Put his arguments in here.

_34:44_: They have to have the same dimensionality.

_34:47_: That should be in our.

_34:51_: Vectors in an N dimensional vector space

_34:54_: the same dimensionality of the tensor.

_34:57_: So all of these one forms and vectors are

_34:60_: all in a four dimensional vector space.

_35:02_: 4 dimensional vector spaces,

_35:04_: same formation vector space. Umm.

_35:10_: No, they are all vector spaces

_35:13_: and each so each of the N&M.

_35:16_: Usually sets of M tensors.

_35:19_: Corresponds to a vector space.

_35:21_: And they all must have the

_35:24_: same number of dimensions. And.

_35:27_: Alright, this doesn't work.

_35:28_: But that, that, that's,

_35:29_: that's a good point,

_35:29_: which hadn't really occurred to me and now.

_35:36_: In our case. We are all we are.

_35:39_: We are. We're doing GR here, not math.

_35:41_: So we are primarily interested in and.

_35:44_: The case where the vector space

_35:46_: in question is Minkowski Minkowski

_35:48_: space or or the four dimensional

_35:50_: space-time were interested in.

_35:51_: So I think so I think I've sort of

_35:55_: in some parts of this explanation

_35:57_: I may have implicitly assumed that

_35:59_: energy equal to four all the way

_36:01_: through so but that's that's important

_36:03_: point is yet another way in which

_36:04_: we have the from the void range

_36:06_: of possibilities of tensors we've

_36:08_: narrowed that we've we've you chopped

_36:10_: off possibilities here here and

_36:11_: there and that's further constraint.

_36:16_: So if you.

_36:21_: Cancer, yes. In three dimensions.

_36:26_: How would you feel?

_36:30_: As you've chosen faces, right? Yeah.

_36:35_: If you have like a choice

_36:36_: of three basis vectors.

_36:41_: Ohh, is he right? Yeah, right.

_36:44_: Well, I think here if, if, if,

_36:48_: if I have heard you properly then the.

_36:55_: The way that you would.

_36:60_: So what you're doing here is this is for,

_37:03_: you know, arbitrary I GK.

_37:06_: So pick an IG K when you're when you

_37:09_: apply those three, the Omega one,

_37:12_: Omega 2E4, whatever you apply those

_37:15_: then what you get is is T 11124 here.

_37:21_: So if you were doing this by hand,

_37:23_: as it were, you'd never do this by hand.

_37:25_: You would work your way through

_37:28_: all of the the end dimensions here

_37:31_: and get the NMN numbers here,

_37:34_: and that would allow you to reconstruct

_37:36_: that that, that, that tensor.

_37:38_: But if you change your mind about what

_37:41_: the basis vectors and what one forms were.

_37:44_: You have to do the calculation

_37:45_: all over again.

_37:46_: And you get different numbers here,

_37:49_: which would apply to.

_37:52_: That would be right for the

_37:53_: different set of basis vectors.

_37:55_: 1 forms in there so that and.

_37:60_: I think this might be touching on the the the

_38:03_: question on your question that This is why.

_38:09_: The illustrating in a way or

_38:11_: touching on the way in which the.

_38:14_: The components are basically dependent.

_38:17_: You change your mind about the basis.

_38:20_: So you change your mind but need arguments.

_38:22_: So you end up getting different numbers here.

_38:24_: But.

_38:27_: Potential is the same in both cases.

_38:29_: And that's what it you know, in a way, what.

_38:34_: Is into an answer to the other question.

_38:38_: Geometrical objects.

_38:40_: So you can see something has an underlying

_38:42_: geometrical object associated with it.

_38:46_: I think there's a couple of things.

_38:47_: I, I there's a couple things I mean there.

_38:50_: One is. When I see that,

_38:54_: I mean this is not a a basis dependent thing.

_38:59_: There's a thing which isn't basis dependent.

_39:00_: Clearly that discussion there

_39:02_: would highly be dependent.

_39:03_: The point being that if you change your mind

_39:04_: about the basis of 1 forms, you change the,

_39:07_: the components you got out, but the.

_39:10_: What this ends up supporting is a notion

_39:15_: of the of the tense underlying tensor the

_39:17_: the the key on the left hand side there,

_39:19_: which isn't based dependent.

_39:23_: And so one sense of geometrical.

_39:27_: Is just it is some negative sense that by

_39:31_: geometrical I mean not basis dependent.

_39:34_: So it it, it, it's it's the same.

_39:37_: It has the IT has a sort of continuity

_39:40_: irrespective of the obesity you choose.

_39:42_: But the other sense in which I'm talking

_39:46_: about things be geometrical is that for

_39:49_: all these things there is the the way

_39:53_: that this man knowledge is built up.

_39:56_: There is a fairly natural interpretation

_39:58_: in terms of pointy things and and and and

_40:01_: and planes that we can use to think with.

_40:03_: And so the idea of a vector as a direction

_40:07_: and another size is a geometrical notion.

_40:11_: The idea of 1 forms as.

_40:15_: Are foliation of of the space which

_40:18_: has a direction basically and A

_40:20_: and a size in a geometrical notion.

_40:25_: And that matters because what

_40:27_: we where we would end up with,

_40:30_: we want to end up is talking

_40:32_: about generativity in a in an

_40:35_: approach where we're focusing

_40:36_: on the shape of space-time.

_40:39_: And the way we can do that

_40:41_: is it is in a way which.

_40:45_: Um. Are routes to that are route to

_40:49_: that which which doesn't involve

_40:51_: getting bogged does involve.

_40:53_: So the same old fashioned way

_40:55_: of introducing generativity and

_40:57_: different geometry is, you know,

_40:59_: start with components that beginning.

_41:01_: It says a vector,

_41:02_: something which transforms like

_41:04_: a vector and I never liked that

_41:06_: definition because it seemed that

_41:08_: that really seems self referential

_41:09_: in the in the sense that what the

_41:13_: the the idea is there is that.

_41:21_: Let's let's let's let's not let's

_41:22_: not go there because I don't want

_41:24_: to start you know teaching you.

_41:25_: I think we I think even talking

_41:26_: about I think is a slightly

_41:27_: mad way of approaching this.

_41:28_: But that was the same old

_41:30_: fashion way of doing it.

_41:30_: And then in the 70s it was with Mr Stone

_41:33_: Wheeler in the vanguard of this this

_41:35_: change to the way different different

_41:37_: geometry generativity was taught.

_41:39_: They are the ones who emphasized

_41:41_: the the geometry first approach.

_41:43_: And talking about and finally of the

_41:46_: things like tangent vectors and and

_41:48_: tensors defined in this rather abstract way,

_41:51_: I'm talking about privileging

_41:53_: them and interest them first,

_41:55_: and then from that just discovering

_41:57_: components and so the behavior of

_41:59_: components near the transformation vector,

_42:00_: like we it comes out,

_42:02_: comes out in the wash.

_42:02_: If you're like, it's a consequence.

_42:05_: And there's another way you

_42:07_: can talk about these things

_42:09_: called geometric algebra,

_42:10_: which is.

_42:14_: I don't want to see fashion,

_42:16_: but it's a bit of a hobby in some circles

_42:18_: just now and that's the way it was.

_42:20_: Really. Does see geometry at all

_42:22_: and find a way of talking and and

_42:25_: and talks about the the How you can.

_42:28_: Define an algebra for directions.

_42:30_: How do you combine two directions

_42:31_: to give another direction?

_42:32_: And so on. And it's very lovely

_42:34_: and possibly very powerful.

_42:36_: And there's some folk could bang on the

_42:38_: table and say this is how George be taught,

_42:40_: but you know, there's quite a lot of

_42:42_: mathematical knowledge which has to

_42:43_: be rewritten and rethought and so on,

_42:45_: you know, for that to work.

_42:46_: And this and this approach is OK.

_42:48_: So if you are feeling

_42:50_: under under underemployed,

_42:52_: if you're feeling bored,

_42:54_: then Google Geometric algebra and

_42:56_: you'll find some sort of lecture

_42:58_: notes and things about that.

_43:01_: It's very nice.

_43:01_: Do not distract yourself with that, OK?

_43:04_: But we always mention it because that.

_43:06_: Very, very much.

_43:07_: That's really going all the way

_43:09_: along a line which says Geometry

_43:11_: 1st and and how do you?

_43:13_: How do you?

_43:17_: Calculate with geometry

_43:19_: with geometrical objects.

_43:21_: Components are one way.

_43:21_: The other way we do, we do,

_43:22_: we do in this, in this case,

_43:24_: we the way folk do in this context.

_43:31_: So, so yeah, I suppose in that

_43:33_: sense I do mean two things.

_43:35_: Geometrical, I mean, I mean there's

_43:37_: this general approach which is.

_43:41_: Shapes first, directions 1st.

_43:42_: And when I see this is geometrical

_43:45_: object, I mean this is a

_43:47_: component independent object.

_43:48_: This is a basis independent object.

_43:52_: And yes. That's your question.

_43:58_: I think ohh. I've seen.

_44:02_: Can we go over like? Like basis.

_44:08_: Because I understand like.

_44:12_: OK.

_44:15_: More than.

_44:18_: Alright, OK. You know is that there is

_44:21_: a particular part of the of the of the

_44:23_: notes that you're thinking of there.

_44:31_: Ah,

_44:34_: OK. It's relating to.

_44:44_: So.

_44:50_: OK.

_44:52_: OK, so.

_44:57_: Is it? And this is it the point

_44:59_: before or after that statement

_45:01_: that we're talking about here?

_45:06_: I didn't read. Ohh yeah I think so.

_45:12_: Like it?

_45:15_: Considering.

_45:20_: Right, so Section 2 to 7, let's see.

_45:24_: Let's go back and see what I see there.

_45:29_: Changing business. OK, so.

_45:36_: I don't know the famous words.

_45:39_: It is easy to show. Um. Right, I I'm.

_45:49_: Thank you.

_45:51_: It's easy to show.

_45:55_: I'm going to try and come up

_45:58_: with that easy show live.

_46:01_: I'm not convinced I am.

_46:03_: Well, how do we do this? We. Um.

_46:13_: Well, let's let's have a go

_46:15_: and see what happens. So um.

_46:21_: Grade A. Ibar Lambda I, bar I. AI.

_46:30_: BJ bar equals Lambda J bar J.

_46:35_: PG. So if then we say T. Um.

_46:42_: Etcetera or I. Let's say we write.

_46:49_: T is our.

_46:53_: If you 20 tensor, so it

_46:55_: takes 2 vectors. So, um T.

_47:01_: IG will be equal to. Um.

_47:08_: TE i.e. G. Right.

_47:13_: OK, but this might work now if

_47:17_: that's we if we rewrite that as E.

_47:22_: I bar Lambda I. Bar i.e.

_47:28_: G bar Lambda G Bar G.

_47:34_: What I have done there is simply done this.

_47:38_: For the basis vectors EI and e.g,

_47:42_: SOEI will be EI expressed in the other basis.

_47:46_: We have an expression like this but.

_47:49_: We said that ohh yeah.

_47:51_: Another restriction on the on on the

_47:52_: functions of the vectors are the tensions

_47:54_: are is that they're linear in each argument.

_47:56_: We said that that's crucial.

_47:59_: And what that means is we

_48:02_: can take these numbers out.

_48:04_: So we can write that as.

_48:06_: Lambda I bar I. Lambda G Bar G.

_48:13_: Thank you, TEI bar.

_48:17_: EG bar.

_48:18_: Which is the Lambda I bar I Lambda G bar.

_48:25_: GTI Bar G. Both. Which I

_48:30_: think is what was required.

_48:35_: So all we've done there is use this

_48:39_: general property of expanding 1 vector.

_48:42_: In terms of I know the basis using

_48:47_: the transformation matrix there.

_48:50_: And. Use the linearity.

_48:54_: To take these numbers out.

_48:57_: Leaving expression there which we

_48:59_: recognize as a way of getting the

_49:02_: T in the components that the IGIBJ

_49:05_: bar components of T in the bar.

_49:09_: Precious. And I haven't.

_49:13_: I didn't draw my memory for

_49:15_: any of these things.

_49:16_: In each of these cases,

_49:18_: I Simply put in the eyes and

_49:20_: eye bars in the only place that

_49:22_: could go that was consistent

_49:25_: with the Association Convention.

_49:30_: The answer to. Getting very direction.

_49:36_: Invite previous. OK. I finally I

_49:41_: finally better be quick. I'm sorry.

_49:46_: Right, because the part of the

_49:48_: definition of the of the tensors

_49:51_: was they were functions which

_49:52_: are linear in each argument.

_49:54_: And what linear means what linear,

_49:56_: So what? Just to recap there,

_49:57_: what linear means F? Is linear.

_50:02_: Means that F of a X is equal to a. F of X.

_50:11_: So Lambda here are so each of the.

_50:15_: Each of the components of that matrix.

_50:18_: So that's that's an end by matrix.

_50:21_: For each of the components of it is a number.

_50:23_: So in this thing here each of the.

_50:29_: Lambda is a number and

_50:31_: therefore can come out.

_50:34_: OK, I'll see you. I think.

_50:36_: I see you next Wednesday and I

_50:38_: think we're in a different room.