Transcript of gr-l06
==========


_0:11_: Books. Hello again and welcome

_0:15_: to lecture six of the GRE course,

_0:18_: and before we get going,

_0:20_: a couple of things about notes and things.

_0:25_: One is that thank you for those who filled

_0:28_: in things about the the emphasis things,

_0:31_: some very useful feedback there.

_0:32_: One of the things that was mentioned

_0:34_: was which I think it's very easy

_0:37_: to just think about audibility.

_0:39_: And I think the rumours against us there

_0:43_: because there's quite a low ceiling here

_0:46_: of soft material fabric at the back,

_0:49_: so there's not A and also there's

_0:51_: think because a shelf there a

_0:53_: bit of a poor impedance match.

_0:55_: So I think it is quite possible

_0:57_: that I'm yelling down there.

_0:58_: You can't necessarily be very well here,

_1:00_: so I should aim to project.

_1:03_: But if I'm failing.

_1:06_: At the back,

_1:07_: just shout out volume or something.

_1:08_: Just yell.

_1:10_: And I think we're going to handle that.

_1:14_: No, another thing that was mentioned was

_1:16_: the issue of just the volume of stuff.

_1:20_: And I have a sleep and and and how

_1:22_: how one navigates around those and so

_1:26_: I'll recap a bit and and remind you.

_1:29_: They believe each chapters aims and

_1:32_: objectives aims at the high level things.

_1:35_: The you know the point.

_1:37_: The objectives are the things which

_1:39_: are also useful but are less exciting,

_1:41_: but are accessible in the sense that

_1:44_: there are things which you can I

_1:46_: can ask you to do and in an exam,

_1:48_: and those structure the you should have

_1:52_: those in mind when reading through

_1:55_: reprocessing the notes after the lecture.

_1:59_: If you look at the exercises

_2:00_: at the back of the chapters,

_2:02_: a lot of those are keyed to

_2:05_: fairly specific objectives.

_2:06_: So you you,

_2:07_: you,

_2:07_: you can see all this exercise is in

_2:10_: the service of that objective and look

_2:12_: through things with that in mind.

_2:14_: Another issue is that it's

_2:17_: sometimes difficult to.

_2:22_: It is. It is a defect of the

_2:23_: notes that there are a lot of it.

_2:25_: It's something a little hard to

_2:26_: see what the the key thing is.

_2:28_: So what I've spent some time last

_2:31_: weekend doing is adding to the notes

_2:33_: some sort of key points in selective

_2:35_: sections and in the slides that

_2:38_: you'll see now at the end of these

_2:40_: sections that are point, point,

_2:41_: point of just the highlights of that.

_2:44_: Now having said that,

_2:45_: I'd encourage you not to look at

_2:48_: those because I encourage you.

_2:49_: You go and look at and and and and

_2:51_: do the same exercise for yourself

_2:53_: because picking a section and going

_2:54_: what am I supposed to be getting out

_2:56_: of this and writing that down is by

_2:59_: itself a very useful exercise because

_3:01_: seeing standing back a bit and seeing

_3:04_: what was what was the point of that,

_3:06_: I get it is a useful exercise

_3:08_: but you see my version there as

_3:10_: well in in the in the slides.

_3:13_: I've also decided to put the slides

_3:15_: up ahead of time rather behind.

_3:17_: There's barely,

_3:18_: there's there's essentially nothing.

_3:20_: The slides that are.

_3:23_: That isn't in the notes.

_3:24_: But it might be, depending how you

_3:26_: want to do what you would scribble on.

_3:28_: You may want to use that.

_3:29_: And I've also put up the as

_3:31_: I said in the model posting.

_3:34_: I've also put up the compendium

_3:36_: of all the exercises.

_3:38_: Usual thing is not useful to do these

_3:39_: to look at the answers too quickly.

_3:41_: But your Honor, students know.

_3:43_: You know that and you can do

_3:45_: what you like with those.

_3:46_: You can undermine yourself if you want,

_3:49_: and I'm sure you won't.

_3:51_: And so that I I put that up just to

_3:55_: remind me that those things are there.

_3:59_: Not entirely obvious.

_4:00_: You have to click on that

_4:02_: tiny little arrow to see the.

_4:05_: The contents are rearranged a bit

_4:07_: just to make it look less huge.

_4:09_: There, there, there are still there.

_4:10_: They're just basically three-part

_4:11_: three sets of of things in there,

_4:13_: but in a couple of different formats.

_4:17_: Any questions?

_4:21_: OK, there there were other useful

_4:24_: points in the the feedback.

_4:28_: Which I will. Moreover,

_4:31_: and I think of at various points,

_4:35_: so that's not the only feedback.

_4:37_: Other things you can think of to

_4:40_: see mail me or it's all good. OK,

_4:43_: where we got your last time was the end of.

_4:49_: Section of of three one and

_4:52_: we're just encroaching on.

_4:55_: The differentiation we basically

_4:57_: ran out of time, so if I can find.

_5:01_: Five slides few. You moved.

_5:08_: And we got.

_5:14_: But so far there so there's a you

_5:15_: know one of those key point things

_5:17_: that that that that I thought the

_5:19_: the key point from that section.

_5:21_: So what I'm going to do this time is talk

_5:24_: about how you do how do differentiation.

_5:28_: In our of bases, in a case where the

_5:34_: basis vectors that that were using

_5:37_: to create our coordinates where

_5:39_: they are changing over the space,

_5:41_: we're first going to do that in flat space.

_5:45_: And then we're going to do

_5:46_: it in curved space.

_5:47_: And the surprising thing is

_5:49_: that that second step turns out

_5:51_: to be the easier of the two.

_5:53_: You think it was the other way around.

_5:57_: So you have done this.

_5:60_: This is to some extent another

_6:02_: case where you've done this before,

_6:04_: but not in this notation.

_6:06_: So there's a slight notational issue.

_6:07_: And there's also slight stepping back

_6:09_: and seeing what it was you did before,

_6:10_: because what you did before was

_6:13_: things like you've seen the Laplacian

_6:15_: in spherical porous bit of a mess.

_6:18_: But it's a bit of a mess because R,

_6:21_: Theta and Phi basis vectors are

_6:23_: changing as you move over the

_6:25_: the space and so the when you

_6:28_: differentiate the components of a

_6:30_: vector in those coordinates you have

_6:33_: to do all sorts of stuff to get the.

_6:37_: The the coordinate independent

_6:38_: change in this vector as you move

_6:41_: as it moves around the space.

_6:42_: So, so, so this is about

_6:44_: differentiating a vector field.

_6:46_: You've got a vector field of

_6:47_: for example the electric field.

_6:49_: Are you asking how does that

_6:51_: change as I move around the space?

_6:54_: And that and and the Laplacian

_6:55_: is part of the.

_6:56_: It's part of the answer.

_6:59_: It was simplified set of things

_7:00_: a bit and talk instead just

_7:02_: of plain polar coordinates.

_7:03_: Let's start off simple and work up

_7:06_: so the plane polar coordinates.

_7:09_: Are um.

_7:14_: Near Project source camera.

_7:23_: Not that far. Not that far.

_7:27_: Playing public coordinates are.

_7:33_: ER. And E Theta. And we want

_7:39_: to ask and and those are.

_7:44_: Derived from the.

_7:48_: The basis vectors?

_7:49_: The Cartesian basis vectors.

_7:54_: ER. Is. Costita. Having this

_7:58_: done just to get the signs right.

_7:59_: Ex plus. Sine Theta EY&E,

_8:05_: Theta is equal to minus R.

_8:08_: Sign. Theta EX plus.

_8:12_: Are Cos Theta EY? No. Uh-huh.

_8:17_: But you may say I don't remember

_8:20_: that being there. And you don't.

_8:23_: That's because the basis vectors of

_8:25_: the the the plane polar basis vector

_8:28_: you're used to are orthonormal vectors,

_8:32_: where specifically the R is removed

_8:34_: in order that these both be squared.

_8:37_: However, this is in some sense the more

_8:40_: natural one without that correction.

_8:42_: And we can go into what?

_8:44_: Into why that, why there is,

_8:45_: but that are as they're

_8:48_: deliberately and inconsequentially.

_8:49_: It doesn't really matter how I define my,

_8:52_: my, my, my, my, my basis vectors,

_8:54_: just in this case,

_8:55_: these ones are orthogonal but

_8:57_: not orthonormal. For reasons.

_8:59_: Which we can talk about.

_9:01_: If need be.

_9:02_: Now we want to ask how do

_9:05_: these change as they go,

_9:07_: as we move around the space?

_9:09_: And that's not hard.

_9:10_: We do things we we can say D by DROFER is.

_9:18_: Well.

_9:21_: Yard doesn't vary.

_9:21_: It doesn't depend on our at all.

_9:25_: D Theta is this honest with me?

_9:29_: Either. You know, funny echo,

_9:32_: I'm not sure it's just.

_9:34_: Whose speech anyway D by D Theta.

_9:37_: Of, ER, it's going to be.

_9:48_: Yeah, it's going to be. A minus costs.

_9:54_: Theta EX plus sine Theta E. Why?

_10:03_: The other way. You have school.

_10:09_: Thank you my sine Theta EX plus Cos

_10:14_: Theta EY which is just 1 / r E.

_10:21_: Teacher. And so on.

_10:24_: So we we we can just just walk

_10:27_: through those four possibilities,

_10:30_: differentiating the the radial and

_10:33_: tangential basis vectors with respect to R

_10:36_: and Theta and get expressions which are.

_10:39_: The remaining two are D by Dre. Theta.

_10:44_: Equals 1 / r, E Theta and D by D Theta.

_10:50_: E Theta is equal to.

_10:53_: Made our ER and and and one of the

_10:55_: reasons why orthonormal coordinates

_10:57_: are good is because you your your void

_10:60_: these extra factors of factors of R.

_11:05_: So no surprises there.

_11:08_: Now what happens if we pick?

_11:10_: You pick a vector V and ask how does that?

_11:15_: Change as we move around,

_11:18_: as we move radially.

_11:19_: So as you move away from the origin,

_11:21_: how does how does the vector V change?

_11:26_: Um, that's uh. Ebitdar.

_11:31_: Of VR plus. Breathe easy.

_11:37_: Peter. And which is? The DVR by

_11:48_: Dre R + V RDERBYDR plus and and so on.

_11:52_: And notice, by the way,

_11:53_: that I'm I'm slipping in a

_11:58_: notation here that. I'll use.

_12:01_: Occasionally I'm indexing the basis vector.

_12:05_: And the component with the symbol

_12:07_: R rather than an index 123.

_12:10_: So that's sort of a slightly

_12:12_: slangy way of of indexing the arc,

_12:15_: the arc component and the Theta

_12:17_: component of of the basis vector of

_12:19_: put that on on that slide as well.

_12:26_: OK. And?

_12:31_: Or we can just write that

_12:33_: in in index notation as a

_12:39_: DDIBYDREI plus.

_12:43_: The IDE I by Dr. And. Ohh question.

_12:53_: With respect to our like in this case,

_12:56_: because it could be. Or or or or Theta.

_12:60_: So what we can do is maybe more

_13:03_: generic and exactly I think

_13:05_: you suggest and say DV by D XI.

_13:08_: Is going to be the VP of

_13:12_: called JD VI by DXJE I + V.

_13:19_: ITE.

_13:22_: Uh. You know.

_13:28_: Right. Aye, aye. By DXG.

_13:35_: Yeah. So is that what you meant?

_13:38_: Yeah. And we could have written

_13:41_: that down from the outset.

_13:43_: I'm just sort of easing us into

_13:45_: into that expression from me

_13:46_: because I just written that down.

_13:51_: OK, but notice. That's a victor.

_13:56_: But asking how does that vector

_13:58_: change as you move around and and if

_14:00_: the vector starts off here in this

_14:02_: position and here in this position,

_14:04_: then there's there's there's

_14:05_: a change in that vector.

_14:07_: So that change in the vector is a vector.

_14:11_: Which is a number. Times a vector.

_14:15_: A number times a vector.

_14:17_: In other words, this DEIBYDXJ is,

_14:20_: not entirely surprisingly,

_14:21_: also a vector.

_14:24_: Which vector is it? It's a vector.

_14:27_: Then we can express that vector

_14:29_: in terms of the basis vectors.

_14:33_: So if you're right, DE.

_14:36_: I by DX J. It's some some.

_14:42_: Components.

_14:46_: In multiplying the basis vectors and

_14:50_: we write the components. Gamma K. GI.

_14:57_: I think I've written IG. Sorry IG.

_15:02_: And gamma here is called the Christoffel

_15:04_: symbol or christophel symbols.

_15:06_: No one seems to be quite clear

_15:08_: whether a symbol or symbols,

_15:09_: the crystal symbols are nothing more than.

_15:13_: The one second, nothing more than the

_15:15_: components of that vector in that basis.

_15:21_: Basis the same as. Yes it is.

_15:25_: So. So we're just,

_15:27_: there's just we're picking a

_15:29_: different, a different index.

_15:32_: Yeah, so it's it's, it seems,

_15:33_: a bit of vectors. This, this,

_15:35_: this is a vector in that space.

_15:37_: And so we're seeing if it's

_15:39_: a vector in that space,

_15:40_: then it's expressible in terms of

_15:42_: the basis vectors in that space.

_15:46_: And we could also have just written

_15:47_: that down from scratch with the there.

_15:49_: There's nothing was stopping us doing that.

_15:52_: But this is a this is a motivating.

_15:59_: This is mogamma. 50%.

_16:04_: So good. It's. It's a, yes.

_16:08_: So this is a number. Exactly.

_16:10_: It's a set of N by N by N numbers. OK.

_16:18_: Is that number, and yeah,

_16:22_: yeah, so this isn't a tensor.

_16:25_: It's it's not the components

_16:27_: of a geometrical thing,

_16:28_: it's just a set of N by N

_16:31_: by N numbers. Thank you.

_16:36_: So we can write so, so if we.

_16:42_: If we go back a bit and and ask.

_16:47_: DERBYD. Theta I wrote. Will make that

_16:54_: EDEDEE 1 by DX. Two calling RX1 and Theta.

_17:02_: And Theta equals X2. That will be. Um.

_17:11_: Gamma one. 12 E 1 plus. Gamma 212 E 2.

_17:22_: I'm just illustrating what the

_17:24_: what what that some looks like.

_17:27_: And from above and and and we could

_17:30_: look back a page discovered that.

_17:33_: The the the the ER by the Theta is 0 * E.

_17:40_: 1ER plus 1 / R.

_17:44_: He 2IN other words, gamma 1/2.

_17:48_: Is equal to 0 gamma 212 is equal

_17:52_: to 1 / R and that's how you you

_17:55_: you we we calculate what the these

_17:58_: Christoffel symbols are for a

_18:00_: particular set of basis vectors.

_18:02_: It's a turning the handle thing.

_18:06_: A bit tedious, but it's the sort of

_18:09_: thing it's very easy to test, like,

_18:11_: very easy to do, and several of the the the,

_18:14_: the question of of the exercises are

_18:17_: encourage you to just turn that handle and.

_18:20_: There are no thrills in turning that handle.

_18:22_: It's just a matter of of practicing

_18:25_: doing so and not getting lost.

_18:27_: OK.

_18:31_: And and.

_18:34_: Uh. So that.

_18:41_: That means that in, for,

_18:42_: for, for plane pollers.

_18:46_: Gamma One woman two is 0 Gamma 2 and two.

_18:51_: Is it same as gamma 2 to one and gamma?

_18:55_: 122 If you could monitor and we'll

_18:59_: also write that sometimes as gamma.

_19:02_: Are. R Theta equals 0 gamma Theta

_19:07_: R Theta equals gamma Theta Theta

_19:10_: r = 1 / R and gamma R Theta Theta

_19:14_: equals minus R and again this is

_19:17_: a slightly slangy notation which I

_19:20_: hope is is clear but by our I mean.

_19:25_: These would match up.

_19:30_: So I'll, I'll just.

_19:35_: So does that mean so any questions on that?

_19:40_: That's mostly notational.

_19:41_: Another notational section

_19:43_: telling you something you.

_19:44_: Again, the idea is this is

_19:46_: telling something you do know,

_19:47_: but in different notation.

_19:51_: But it allows us to define

_19:53_: the covariant derivative.

_19:54_: Because what we have,

_19:56_: if we if we go back a bit we we have the.

_20:02_: V by DX J is equal to D

_20:07_: VI by DX JEI plus. The.

_20:16_: IDEI by DX J but we know

_20:19_: that that is equal to gamma

_20:25_: KIJK.

_20:27_: And so if then we decide to

_20:30_: renew, relabel that as. Uh.

_20:38_: DVI by DXJ. He. Aye. And.

_20:47_: And instead of eyes right keys instead

_20:50_: of keys right eyes so VK. Gamma. I.

_20:59_: KGEI. No, all I've done there.

_21:03_: Is that these?

_21:07_: Dummy these repeated indexes are

_21:09_: dummy indexes, the eye and the key,

_21:12_: so there can be anything.

_21:14_: So I've decided to rewrite them

_21:16_: simply with different letters.

_21:18_: There's no that that's exactly

_21:20_: equivalent expression,

_21:20_: but what it means is I can take

_21:23_: this EI out of there and get

_21:28_: DVIBYDXJ. Plus Ek gamma

_21:34_: IKJEI. And discover. That.

_21:42_: Right, right. So.

_21:43_: So what we have there is this is a.

_21:46_: A vector. With components. That.

_21:52_: And I'm going to write those

_21:54_: components in a particular way.

_21:55_: Would write those components as VI,

_21:58_: semi colon, J. EI.

_22:03_: And equal to VI comma J.

_22:08_: Plus VK Gamma I KKJJ.

_22:15_: EI where this notation VI semi

_22:17_: colon subscript semi colon G refers

_22:20_: specifically to that expression.

_22:23_: And the the the notation of

_22:25_: just introduced the VI comma.

_22:27_: J refers to the the plane.

_22:31_: Usual derivative of DVI by the exchange,

_22:34_: so that's DVI by DX J.

_22:37_: And like that.

_22:40_: That's the last notational bad surprise.

_22:43_: OK, punctuation in subscripts.

_22:46_: I'm sorry,

_22:47_: I didn't make it up.

_22:50_: And. G is a little tricky to write neatly.

_22:55_: I I I agree. My handwriting improved

_22:58_: massively when I started doing this.

_23:03_: No.

_23:07_: Umm.

_23:12_: Uh, you know, I have a quick question here.

_23:19_: This illustration of that as that.

_23:23_: And. So the the key point of the of

_23:26_: of previous section were that the

_23:27_: basis vectors vary over the space.

_23:29_: We knew that. And that variation can

_23:32_: be characterised using this notation

_23:34_: using the Christoffel symbols.

_23:38_: Um, so. A quick question.

_23:41_: What sort of thing is

_23:44_: determined brackets in the

_23:46_: expression and that expression?

_23:48_: Who's he with scaler?

_23:51_: Who said it was a vector?

_23:54_: Who said it was a one form?

_23:56_: Tensure. A matrix.

_24:01_: Have a brief chat about.

_24:46_: OK. With that reflection in mind.

_24:55_: Who would say that was a scalar?

_24:57_: OK, a vector. One form. Tensor. Amatrix.

_25:07_: Two of those answers sort of are correct.

_25:11_: In one sense, yes.

_25:12_: This this this is a scalar because there

_25:15_: is the for pick and I pick an IG and key.

_25:19_: Pick an iron key, and yes,

_25:20_: there's a number which corresponds to this,

_25:22_: so yes. But at the same time.

_25:26_: For a reason which I'm about to elaborate on,

_25:28_: this turns out to be a tensor as well,

_25:31_: because or the components of a tensor.

_25:35_: Because.

_25:39_: What we have here is our. Thing

_25:43_: which linearly depends on the. Um.

_25:49_: Right.

_25:52_: I'm going to go through the, the, the.

_25:54_: That's actually in the way I expressed on

_25:57_: in my notes because rather than busk are

_25:59_: possibly confusing answer, but it is. And.

_26:06_: So um.

_26:15_: The key thing is.

_26:18_: That this this vector, this vector here.

_26:22_: Is. Proportional to the.

_26:26_: Basis vector. Each egg.

_26:30_: If you made EG twice as big,

_26:33_: you make all the components half the size,

_26:36_: and that vector would would would change

_26:38_: inside inside the cornely accordingly.

_26:40_: So there's a a proportional

_26:41_: relationship between those things.

_26:43_: So this is a a thing which

_26:46_: depends on the variation of.

_26:48_: V around the plane and the size of the

_26:53_: depends on the size of the vector,

_26:56_: which corresponds to D by the XJ.

_26:60_: Which is the basis vector.

_27:03_: So what we can do?

_27:05_: Is we can define.

_27:08_: Are A11 tinger.

_27:15_: Nabla V.

_27:18_: And we'll define that by saying that

_27:21_: the action of that on the vector.

_27:24_: East. How do I call it EG so the

_27:30_: 11 tensor so it takes a vector

_27:34_: argument and A1 form argument?

_27:36_: And the action of it on on there. Yeah.

_27:43_: Right.

_27:47_: I'm, I'm, I'm, I'm, I'm, I'm gonna

_27:49_: write it in a way other way around,

_27:50_: in a way to just make sure

_27:51_: it's consistent with my notes.

_27:53_: Just just the the the

_27:55_: distinction is important but.

_27:56_: The action of that one one

_27:58_: tensor when we give it when we

_27:60_: give it 1 vector as argument.

_28:01_: One basis vector as argument is DV by DX J.

_28:09_: As a victor. So.

_28:12_: We're seeing let there be a tensor.

_28:16_: Which is related to V which we're going

_28:19_: to call the covariant derivative tensor.

_28:22_: And our definition of that tensor is through

_28:25_: is through the slightly indirect way.

_28:29_: Remember that the tensors A11 tensor

_28:31_: takes A1 form and a vector as argument.

_28:35_: And if we give the vector argument to

_28:37_: that covariant tensor, I'll get a moment.

_28:39_: If we give a as the vector

_28:42_: argument to that tensor,

_28:43_: one of the basis vectors,

_28:45_: then by Fiat we say the value of

_28:48_: that tensor is this vector here,

_28:50_: and being a vector is something

_28:52_: which takes A1 form of argument.

_28:53_: So there is a missing one form

_28:55_: argument in both in both places.

_29:04_: Yep.

_29:07_: The acoustics in this room are not good.

_29:13_: Right. And I it's we could

_29:17_: pick any basis vector here,

_29:20_: So what we've said. And and.

_29:26_: If you remember the basis vectors. Are.

_29:32_: EI equal to D by DX I.

_29:36_: And so we'll just pick a random

_29:39_: basis vector in this case,

_29:41_: EG and we're seeing if we apply

_29:43_: EG could be anyone picked, EG,

_29:45_: then we get the derivative of

_29:48_: that vector V with respect to

_29:51_: the corresponding coordinate.

_29:53_: Coordinate function. OK, right. And so.

_30:01_: What we have here then?

_30:05_: That's as I've said this this notation

_30:08_: with the dots and and and and missing

_30:11_: arguments is that your conventional,

_30:13_: the there isn't really a completely

_30:14_: conventional way of writing of writing these.

_30:16_: But one Commissioner,

_30:17_: we are writing these is to

_30:19_: write that as nabla, EG. V.

_30:24_: Where we're rate the.

_30:26_: This this vector argument

_30:28_: as a subscript there.

_30:33_: And. We that we write this visually often,

_30:38_: that of course we want to abbreviate

_30:40_: it so we end up writing we now blog.

_30:44_: V. And that is it. It tends to be a

_30:48_: short version of that, and that is.

_30:51_: Just means that it means that this tensor

_30:55_: nabla V with one argument filled in.

_30:58_: In a in a slightly funny place.

_31:01_: Written down is like money. Please. So.

_31:08_: So if we add so we have a tensor nabla V.

_31:12_: Want to ask what other?

_31:14_: So stepping back a bit, what other

_31:16_: components IG of that tensor?

_31:21_: And. What we do is we now

_31:25_: have a V and fill in Omega.

_31:28_: I egg. But we know that that is

_31:34_: also written as nabla, EG um.

_31:46_: V and the ith component of

_31:48_: that by filling in the. And.

_31:54_: So so by by filling in this Omega I we're

_31:58_: extracting the ith component of this.

_32:01_: Vector here of this vector here.

_32:04_: So again the ith component of the of this.

_32:07_: Napa V EEG, which we also

_32:11_: write as an Apple ID. P.

_32:17_: So G. Aye, I'm and that and in

_32:21_: particular by comparing it with

_32:24_: on the previous sheet that is.

_32:26_: VI. G. So it's important to look

_32:31_: at to to to be clear what we're

_32:33_: looking at the very stages here.

_32:35_: So Navisa is a tensor.

_32:38_: We're asking what are the IG

_32:40_: components of that tensor?

_32:42_: How do we do that?

_32:43_: We plug in Omega and and and and egg.

_32:47_: You know the two are the two one form

_32:50_: and vector arguments of that thing.

_32:52_: But we know what that is.

_32:54_: We can we just notationally change.

_32:57_: That and there. To this.

_32:60_: The Omega I is extracting the

_33:02_: ith component of the result,

_33:04_: so it's the ith component of this

_33:07_: vector because so this is by now.

_33:09_: This by now is a vector.

_33:11_: We rewrite that as.

_33:15_: For convenience, not nabla EG,

_33:18_: but just nabla G.

_33:20_: And we know from from this.

_33:24_: That the other way we write that

_33:27_: is with this VI semi colon J.

_33:30_: Which expands. To this.

_33:34_: So there's quite a lot

_33:35_: packed into that line.

_33:38_: It's we'll we'll take a

_33:40_: couple of goals through it,

_33:41_: but I think it's important that you

_33:43_: understand what different thing,

_33:44_: what different things are happening

_33:46_: in each of those equal signs.

_33:48_: So make sure that you're that

_33:50_: you have a story in your mind for

_33:52_: what that equals sign is doing.

_33:53_: That equal sign is doing,

_33:54_: that's equal sign is doing,

_33:56_: and that equal sign is.

_33:59_: Question.

_34:02_: It's it's a semi colon at the end, yes.

_34:05_: Yeah. What the meaning of

_34:07_: ohh that that semi colon is.

_34:10_: Have this. Yeah. So this.

_34:17_: the I semi colon G.

_34:19_: Is this expression here?

_34:21_: So is VI comma G where we've

_34:24_: introduced the rotation?

_34:26_: Iconology is just that, it's the derivative

_34:30_: of the ith component of the vector.

_34:34_: With respect to the XJ, nothing, nothing.

_34:37_: Covariant just goes straight forward.

_34:40_: That's real four thing plus the other term.

_34:42_: So that's and and that is the.

_34:46_: The the components of the

_34:47_: 11 tensor which is nabla V.

_34:53_: How is it the one one?

_34:54_: If you vote, you buy the vector space,

_34:56_: the vector position, so so.

_35:01_: With both of those empty.

_35:03_: But there's a on one format and and

_35:06_: a vector thing. If we fill one in,

_35:09_: then that thing there becomes a vector.

_35:13_: Because that thing had it has

_35:14_: a single one form argument,

_35:16_: so shouldn't it be a 0?

_35:18_: So, so so that is a 01 tenger.

_35:23_: That is a. But.

_35:28_: It's a it's A10 tensor.

_35:31_: It takes a single one form of argument,

_35:34_: so that takes up one form of

_35:36_: argument that takes A1 form

_35:38_: and a vector as argument.

_35:41_: So, so, so the nabla.

_35:44_: The the tension. Is it?

_35:47_: That thing is 11. It won't.

_35:49_: Whole thing is 10 once we

_35:51_: fill in one of its arguments.

_35:55_: Then what we what we're left

_35:58_: with is 1 unfilled argument.

_35:60_: And therefore it's a vector.

_36:04_: So remember that if you have a, a.

_36:11_: A11 tensor. It's a move.

_36:14_: Takes 1 vector one, one form

_36:16_: and one vector as argument.

_36:20_: So sitting there with two holes at the top,

_36:22_: ready to have things dropped in

_36:24_: handle turned a number come out.

_36:26_: If we put me into one of those holes.

_36:28_: There's only one hole left over.

_36:31_: In this case this this thing here. Nabavi.

_36:35_: Is the thing with two holes at the top.

_36:38_: Had one form and a vector hole one,

_36:41_: but we have filled one of those holes.

_36:44_: So what we're left overall?

_36:47_: Is a thing with one form whole on.

_36:52_: Open as it were.

_36:53_: In other words, a vector.

_36:57_: Which vector? We're seeing it's this, victor.

_37:00_: And I'm carefully writing here

_37:02_: the argument of this vector.

_37:05_: It being a vector,

_37:06_: it has a single one form hole. OK.

_37:10_: So. This is, it's like we've sort of

_37:14_: reversed into this definition of what?

_37:16_: Of what the tensor is.

_37:17_: We've said there shall be a tensor. And with.

_37:23_: Attention with attention with two slots,

_37:24_: yes, one form and and and and and a vector.

_37:30_: It is empty, yes.

_37:31_: In other words, this thing here is a vector.

_37:34_: This thing here is a vector,

_37:36_: which we've illustrated by showing

_37:38_: that there's a single one form shape

_37:40_: slot which which we filled in.

_37:43_: And we know what that is.

_37:45_: And so this whole business here is.

_37:48_: It is doing a bit of a notational dance to

_37:52_: to discover what the components of this.

_37:55_: Tensor.

_37:56_: There's one tensor.

_37:57_: It corresponds to this thing that

_37:59_: we've just that we worked out earlier.

_38:04_: OK. That might need going back

_38:07_: back through again, but a bit of

_38:09_: digestion you will have a little bit.

_38:11_: Well, it's another sort of thing that

_38:12_: you can sit back in the bathroom,

_38:14_: think, think, think your way through.

_38:15_: I think you have to stare at it for a

_38:18_: while and get some illumination that way.

_38:21_: Just to clarify implementation,

_38:23_: so if you ever give us a nobler, yeah,

_38:26_: then something after the Nova and then you

_38:29_: either superscript or subscript something,

_38:32_: that means that it's you have added

_38:35_: a basis something onto the and the

_38:37_: thing that comes after the number.

_38:41_: If that makes sense.

_38:42_: I know where you're going,

_38:44_: but I think that in this case.

_38:49_: You should think of that.

_38:51_: I think here Nabila isn't

_38:53_: really being an operator.

_38:54_: I mean, it's not with an operator and isn't,

_38:56_: but in this case there isn't a

_38:59_: nabla operator that acts on things.

_39:02_: We we ask what is the

_39:06_: covariant derivative of of V?

_39:08_: How can we talk about how V changes

_39:11_: as we move around the space?

_39:13_: Then. There's a tension there.

_39:16_: We've said there's a tensor which gives

_39:18_: that which which gives that, that,

_39:20_: that, that, that, that, that answer.

_39:21_: We could call it Q, we could call it Omega.

_39:24_: We call it anything we're like,

_39:26_: but the way we the the name we give it.

_39:29_: It's a 2 character name if you like.

_39:31_: There's a number of.

_39:33_: OK, so I can have like A1 form,

_39:35_: that's nabla A1 form,

_39:36_: and then I have an I substitute

_39:39_: because I have added the one form.

_39:43_: We're going to complete the command

_39:45_: derivatives of 1 forms and momently.

_39:48_: But I think right now the useful

_39:49_: thing to think about is is is is

_39:51_: not to think of this as an operator.

_39:53_: This is A2 symbol name for the tensor.

_39:58_: Which corresponds to V.

_40:00_: Which is the,

_40:02_: which describes how that tensor varies.

_40:04_: How how how that vector varies.

_40:05_: That vector field varies as

_40:07_: you move around the space.

_40:09_: So I think the next bit may

_40:11_: answer your question, I I think.

_40:17_: Keep this is moving less quickly

_40:19_: than I'd rather had hoped, but.

_40:24_: Another thing that you might see is no

_40:28_: might about it you will see is if you ask.

_40:38_: Specifically annotation like. And.

_40:43_: Nabla X number subscript X or V?

_40:47_: And what does that mean?

_40:51_: It. It means as we can see here.

_40:55_: This tensor nabla V.

_41:00_: With this X argument.

_41:03_: X is a vector is important X

_41:07_: i.e I so that is going to be.

_41:12_: Excuse me? XI, Tableau V.

_41:16_: Till the E. Aye. Which is. XI.

_41:26_: VI semi colon GE.

_41:31_: G. To.

_41:35_: And is X IVG semi colon IE.

_41:43_: Gee. I would have done there.

_41:46_: So here this this tensor what

_41:48_: character is is it's. It's.

_41:51_: It's vector argument is written

_41:53_: conventionally subscript.

_41:54_: In other words,

_41:55_: it's the it's the vector argument

_41:56_: we supplied to this one one tensor.

_41:58_: Then you need your argument so we can

_42:01_: take the XXI out and and have this,

_42:03_: but we know what to do with that.

_42:04_: You're turning the handle here

_42:06_: and we get X IVJ semi colon I

_42:09_: e.g so this is is the.

_42:11_: So this is the comment derivative of.

_42:16_: Of of of V with with argument X

_42:20_: rather than one of the basis of

_42:23_: vectors is asking. How does the?

_42:29_: The house how what this is doing

_42:31_: is is is essentially asking how

_42:34_: does the the vector field V vary

_42:38_: as you move in the EEG direction.

_42:40_: This is asking how does the vector feel

_42:42_: very as you move in a more general,

_42:44_: more general, different direction.

_42:46_: So we started off talking

_42:47_: about the support terms.

_42:48_: That's just a slightly more

_42:50_: general remark based on that.

_42:54_: I want to get into this section um.

_42:59_: So.

_43:02_: What we've done it remains to define

_43:05_: a tensor field associated with the V.

_43:08_: So V is a as a vector field,

_43:10_: it takes a different value at

_43:12_: at each point in the space.

_43:14_: We've managed to find a tensor field,

_43:16_: and there was a 11 tensor which takes a

_43:17_: different value at different points of space.

_43:19_: We've called that tends to field

_43:21_: the covariant derivative and would

_43:23_: define it in such a way that the

_43:26_: answer gives us when we put in its

_43:28_: two arguments is the amount that the

_43:30_: speed at which the vector field V

_43:32_: varies at that point in the space.

_43:34_: Remember vectors you put in

_43:35_: a vector about tensors,

_43:37_: you put in a vector in one

_43:38_: form and you get a number out.

_43:39_: In this case,

_43:40_: the answer to the question when you

_43:42_: put these two things in is how fast

_43:45_: is what's the ith component of the

_43:48_: variation of V as you change X.

_43:52_: G.

_43:56_: So so when I when I talk to the beginning

_43:59_: about this idea that Victor was our

_44:02_: attention was a a real valued function

_44:04_: of vectors in one and one forms.

_44:07_: This sort of thing is the thing I'm

_44:09_: saying attention is the put vector.

_44:11_: Wonderful. Then turn the hand it

_44:12_: comes in a number and answer.

_44:14_: In this case that's what the answer is.

_44:16_: Now we can also in the last five minutes. Uh.

_44:24_: Umm.

_44:27_: The point of a covariant um.

_44:31_: A covariant derivative is that

_44:33_: its coordinate independent.

_44:35_: We've described this.

_44:38_: Using arbitrary coordinates.

_44:40_: But the answer is not a

_44:42_: coordinate dependent thing.

_44:44_: That's the point of what's

_44:44_: the point of all this?

_44:45_: We're we're we're able to

_44:47_: answer question how do vary in

_44:50_: a coordinate independent way.

_44:51_: But if you remember.

_44:54_: If we had a function.

_44:58_: That's a scalar field. Then when we.

_45:03_: Obtaining the gradient of

_45:05_: that scalar field. We got.

_45:09_: A1 form answer which was was also

_45:12_: because a scaler is coordinate

_45:14_: independent which is also the the thing.

_45:17_: Also here is coordinate independent

_45:19_: so for functions. The coordinate?

_45:22_: The covariant derivative.

_45:23_: Of a function is just that,

_45:26_: it has it been just that. Great.

_45:29_: Operator, we we've seen before.

_45:33_: And um.

_45:40_: I'm not going to go through.

_45:42_: I invite you to look at. Equation.

_45:46_: The discussion just don't really

_45:47_: have time to go through in detail.

_45:49_: Equation 3242526 where we discover that

_45:53_: the covariant derivative of one form.

_46:01_: Is Pi semi? G is equal to Pi comma G minus.

_46:12_: Gamma. And. Key IGP.

_46:20_: OK, which looks just like the

_46:24_: expression for the derivative

_46:26_: of the derivative of a vector,

_46:28_: except there's a A minus sign.

_46:32_: Here where there's a plus sign for

_46:34_: the for the vector and there's a

_46:36_: quick derivation of that there which

_46:38_: I'm not going to explain effort

_46:39_: on and and similarly with with.

_46:41_: With the sensitive higher rank you you

_46:43_: end up with one plus and 1 minus sign

_46:46_: for each of the one form and vector.

_46:51_: Are they? Are they measuring

_46:53_: on the on on the next page?

_46:55_: And the last thing to to remark is

_46:58_: that the Leibnitz rule. Does apply?

_47:02_: Well, the key to make use of this PK, PK.

_47:06_: That's that's the commander

_47:08_: of a vector of a scalar,

_47:10_: which is a contraction of those things,

_47:12_: and it does end up being indeed P.

_47:17_: He semi colon GVK. Plus PKV.

_47:25_: The key thing called G.

_47:28_: Actually the library's rule in that

_47:30_: case and I I mentioned that just to

_47:32_: reassure that still is is the case,

_47:34_: it would be a strange and disturbing

_47:37_: derivative operator which

_47:39_: didn't where that wasn't true.

_47:41_: So the key thing here I think.

_47:45_: Blah blah blah.

_47:47_: Not over that.

_47:51_: Measuring why the work

_47:53_: very is used is useful.

_47:55_: Which will put over this and

_47:57_: ohh yeah that's the point.

_47:59_: When we take the derivative

_48:00_: of a higher rank tensor,

_48:02_: we get one plus and 1 minus for each of the.

_48:07_: Of the indexes,

_48:08_: I think it is useful to to just quickly

_48:11_: go through these key points because

_48:13_: we've covered a lot in this section.

_48:16_: It all, it is a bit of a hair dryer

_48:18_: lecture this one which is disturbing,

_48:21_: but there are.

_48:25_: All the time the question being

_48:27_: answered had been had been

_48:28_: afraid of straight forward one.

_48:30_: Given a vector field, how does it vary?

_48:33_: How can you talk about how it

_48:35_: varies as you move around the space

_48:37_: in a coordinate independent way?

_48:39_: And we'll talk about those defined

_48:41_: what the components of that,

_48:43_: we'll talk how that and there is

_48:46_: a tensor A11 tensor which gives

_48:48_: you the answer to that question.

_48:51_: The question then comes how do we

_48:53_: find the components of that tensor,

_48:54_: the IJ components of that tensor?

_48:57_: We discovered by slightly reversing

_48:59_: into it that we could do that

_49:01_: by simply considering the, the,

_49:03_: the, the, the,

_49:04_: the steps involved in differentiating

_49:06_: the components and then the basis.

_49:09_: Picture of our vector.

_49:12_: And we had this interesting rather

_49:14_: strange notation with semicolons,

_49:15_: discovered the covariant derivatives of

_49:17_: vectors of scalars and one forms to be

_49:20_: something we had sort of seen before.

_49:22_: And although I had rather hoped to

_49:23_: get on to the to the next section,

_49:26_: I have managed to get through this

_49:29_: section without rebellion on your part.