Transcript of gr-l05
==========


_0:10_: How do I get everyone this is Spectra?

_0:14_: Five, I'm fairly sure,

_0:16_: and we're gonna start on part

_0:19_: part three of the of the note.

_0:22_: Up to this point,

_0:23_: we've laid quite a lot of groundwork.

_0:26_: The we have the scaffolding,

_0:30_: or to switch metaphors,

_0:32_: we've acquired some basic tools

_0:35_: and learned how to use them.

_0:38_: The next section is Part 3.

_0:40_: Manifolds, vector vectors,

_0:42_: and differentiation is all about

_0:44_: acquiring some more specialized tools,

_0:47_: specifically a new way of defining

_0:49_: vectors and that one forms and tensors.

_0:52_: And the key thing.

_0:54_: Discovering how to differentiate.

_0:57_: Because the the end the goal here.

_1:00_: Is Einstein's equations which

_1:02_: will come to the next part which

_1:05_: are a differential equation.

_1:07_: On a current manifold.

_1:09_: So we have to learn how to do that

_1:13_: differentiation in order to set up a.

_1:17_: What language do differentiation?

_1:19_: For?

_1:19_: We can set up a differential

_1:21_: equation which we then solve very

_1:24_: briefly at the end of part four and.

_1:26_: In multiple cases in G2 next semester.

_1:32_: Um, anything else I could mention,

_1:35_: the audio notes are in the podcast

_1:38_: that's linked from the Moodle.

_1:40_: The notes are you have to

_1:42_: date with part three and the.

_1:45_: But overview videos of part three

_1:49_: are up and I have been putting

_1:52_: up some of last year's zoom based

_1:55_: lectures up on the the stream.

_2:00_: Site, whatever it's called page,

_2:03_: which you may or may not have found useful.

_2:06_: You're just out of curiosity,

_2:07_: has anyone looked at those and.

_2:09_: I don't look at those.

_2:11_: No pool. A treatment store.

_2:14_: They may or may not be useful to you.

_2:17_: I am interested in on all of these things.

_2:20_: The the version of the the the overviews,

_2:22_: the the audio, the video.

_2:25_: I am clear interested if you

_2:27_: back I think you have been sent

_2:30_: and emphasis questionnaire.

_2:32_: You have right and these are useful.

_2:34_: They are looked at and paid attention to,

_2:36_: so I encourage you to fill

_2:39_: that out in a useful fashion.

_2:41_: Any questions?

_2:45_: OK, let us get going.

_2:53_: So as usual, the aims, objectives,

_2:57_: the high level things and

_2:59_: the more detailed things.

_3:01_: I'm not going to go through them.

_3:02_: You can see them. I I put them

_3:03_: up there in order that I can.

_3:08_: The understands these aims

_3:09_: are all understands XYZ that,

_3:11_: that, that is the goal here,

_3:13_: and there is quite a long list of objectives.

_3:16_: Most of the the the bit of the course

_3:18_: that's most straightforward examinable.

_3:20_: Is this part?

_3:21_: And that's and there's quite a lot

_3:23_: of things you can do in in in this

_3:25_: part I mean it looks for feeling

_3:26_: for bidding thing but if you walk

_3:28_: through the if you walk through the.

_3:30_: The exercise is then you take you take

_3:33_: most of those off so that that look

_3:35_: that look at the most forbidding of the

_3:37_: sets of objectives of the various parts.

_3:39_: But it shouldn't.

_3:41_: I do aim for it to be as

_3:44_: horrible as it looks.

_3:46_: Whether I achieve that aim

_3:48_: is another thing, OK?

_3:52_: So in terms of pacing,

_3:55_: I do plan to get all the way through

_3:58_: section 3.1 in this lecture.

_3:60_: If you do 3.2 and 3.3 in the next lecture,

_4:02_: the 3.4 to the end in the lecture 7,

_4:05_: then I'm doing very well.

_4:07_: I think I won't probably won't manage that,

_4:10_: but we but we have we have we have

_4:14_: about lectures worth of slippage

_4:15_: because I want to be about 3 on Part 4.

_4:17_: So we have we shouldn't panic if

_4:19_: we're not that. That's the plan.

_4:23_: I'm gonna fall behind.

_4:24_: Keep chattering. OK, step one,

_4:25_: we're gonna define vectors. Step 2,

_4:27_: we're gonna learn how to differentiate.

_4:30_: So the first thing is the tangent vector.

_4:34_: Now you learned about vectors in school,

_4:37_: and you learned how to

_4:39_: differentiate them at university.

_4:41_: And.

_4:41_: A lot of the ways in which

_4:43_: you learn how to do,

_4:44_: how to do that differentiation using the

_4:47_: vector calculus that Gibbs developed at the

_4:49_: end of the century was using components.

_4:51_: And you can carry on doing

_4:54_: that in this context.

_4:55_: And that's how GM differential

_4:57_: geometry was introduced for a long

_4:60_: for a large chunk of the 20th century,

_5:03_: you still do that render,

_5:04_: for example, is one of the texts

_5:06_: which takes that approach.

_5:08_: I find it a bit confusing because

_5:10_: focusing on the, on, on, on the.

_5:13_: Um, and the components too early.

_5:16_: Gives them a status that we are at the same

_5:19_: time wanting to deny what we want to see.

_5:21_: The components aren't important and yet

_5:22_: we talked the component all the time.

_5:24_: So the other more modern way

_5:25_: of of doing this,

_5:26_: which was I think first pushed

_5:28_: my missing thought,

_5:29_: Misner,

_5:29_: Thorne and Wheeler in the 70s is to

_5:31_: focus on the geometry and that's

_5:33_: the approach we're taking here.

_5:34_: So I mentioned that just to

_5:36_: highlight that there are two

_5:38_: quite different approaches.

_5:39_: One I think I'd rather old fashioned,

_5:40_: one which is where incidentally the terms

_5:43_: covariant and contravariant come from,

_5:45_: blah blah blah,

_5:46_: we're not going to introduce those terms.

_5:47_: Yeah,

_5:48_: I think I I mentioned somewhere

_5:50_: the the relationship between them,

_5:51_: but they roughly correspond

_5:52_: to vectors in one form.

_5:56_: What we're going to talk about

_5:58_: is we're going to stick with the

_6:01_: that keep things as grim as.

_6:04_: Quota free as long as possible,

_6:06_: which isn't very long,

_6:07_: but we're gonna start there and

_6:10_: talk about the manifold. No.

_6:12_: Is there any pure mathematicians in the room?

_6:14_: You might want to sort of, you know, not,

_6:17_: not not close your eyes at this point,

_6:19_: but I'm going to be.

_6:23_: Playing slightly fast and loose from

_6:24_: the point of view of a mathematician,

_6:26_: and I'm going to be rather

_6:27_: rigorous from the point of view

_6:28_: of of your working physicist,

_6:30_: so I'm going to aim somewhere

_6:31_: in the middle there.

_6:33_: And there are differences,

_6:34_: some differences in terminology between

_6:35_: what's conventional in differential

_6:37_: reused for GR and differential.

_6:39_: Used my petitions.

_6:40_: They're not important differences,

_6:41_: they're just slight differences in.

_6:44_: Path that has curve.

_6:45_: I'll make you three complete.

_6:47_: The manifold is a set of points.

_6:51_: It said with with with very little extra

_6:54_: structure and that petition would would

_6:56_: would want to ask exactly what structure has.

_6:58_: At this point we're not gonna worry about

_6:60_: organization has valid construction.

_7:01_: In particular, there's no notion

_7:03_: of distance in the manifold.

_7:05_: The structure it has is that

_7:08_: locally it's like RN it,

_7:10_: locally it's like a a flat

_7:13_: Euclidean N dimensional space.

_7:15_: What does that mean?

_7:17_: It means that you can define a map.

_7:21_: From any local area of the manifold

_7:24_: to our end to end dimensional space.

_7:27_: And it's, it's, it's like in the

_7:30_: sense of how you move around,

_7:32_: how you'd lay out the different directions.

_7:35_: It's like including space.

_7:37_: And I'll say a little more

_7:39_: about what that means,

_7:41_: what that means.

_7:43_: So the manifold isn't just a set

_7:46_: of points, a ragbag of points.

_7:48_: It has just that much structure that

_7:50_: lets you think of it as locally our own.

_7:54_: And. Our goal is to add little

_7:57_: structure to it in a controlled way.

_8:02_: Now, so that's this is the manifold here.

_8:06_: This this of curvy thing here. And.

_8:12_: What we gonna call a chart is a set of

_8:15_: functions from a point of the manifold.

_8:18_: To the real line.

_8:20_: So I said to functions XI. Which?

_8:23_: Turn a point of the manifold into a number.

_8:28_: And a set of these would call a chart.

_8:32_: And there are any of them.

_8:33_: If this is like like RN, OK.

_8:41_: It might be that these are are

_8:42_: defined for the entire manifold.

_8:43_: It might be the defined for

_8:44_: a subset of the manifold.

_8:46_: The point is that round about

_8:47_: about a point P and our three-point

_8:49_: P there's a set of functions.

_8:55_: So each of these functions is

_8:57_: a map from M the manifold to R.

_9:05_: No, we're all gonna imagine

_9:06_: a path in the manifold.

_9:08_: We're just a set of points in the manifold.

_9:10_: It's there, there, there, there,

_9:11_: there, there, there, there, there,

_9:12_: there, there, there, there. OK.

_9:16_: And we're going to find a curve.

_9:18_: Which is like a path,

_9:20_: except that is it a path which

_9:23_: also has a function which maps the

_9:26_: real lane to a point on that path.

_9:29_: So a curve is a parameterized

_9:31_: path if you like.

_9:33_: So two curves which follow the same path

_9:36_: but with different parameterizations.

_9:38_: Did this curve maps 1 to that point

_9:41_: on the curve and this other curve,

_9:44_: another curve maps 1 to a

_9:45_: different point in the curve.

_9:46_: There are different curves

_9:47_: even though the same path.

_9:51_: OK, you don't have to remember these terms,

_9:54_: these these verse terms long-term.

_9:56_: I need them. I need to define these terms

_9:58_: in order to get through the next section.

_10:06_: So what I meant is if we have a curve.

_10:10_: Which takes the real line or

_10:12_: sectional real line to the manifold?

_10:14_: And our coordinate function,

_10:15_: we're going to call it,

_10:17_: which takes the manifold to the real line.

_10:21_: Then X of Lambda of T.

_10:25_: Is a function which takes

_10:26_: through line through line.

_10:29_: OK.

_10:32_: It was a set of mappings from the curve

_10:35_: parameter to the set of coordinates XI.

_10:43_: Or let's. Out on this one.

_10:50_: What that means is. XI.

_10:54_: Is a function of Lambda.

_10:56_: Which is a function of T.

_10:59_: Or in other words,

_11:00_: XI is our function which maps through line.

_11:04_: Through line. And what that means

_11:08_: is we can differentiate it.

_11:09_: Already we can do.

_11:11_: Already we're talking differentiation.

_11:13_: And we're going to talk about the the

_11:16_: the set of of quadratic functions XI as

_11:18_: a reference framework coordinate frame.

_11:20_: All those various words mean the

_11:22_: same the same thing in the in.

_11:24_: In this context,

_11:24_: we're going to distinguish between.

_11:29_: So the next step is to define a function.

_11:35_: Which maps the manifold to the real line.

_11:37_: So an example of these Xis are an

_11:40_: example of a function which which

_11:42_: maps the the manifold to the line.

_11:44_: But we're gonna pick an arbitrary function.

_11:48_: As the manifold goes to. Through line.

_11:53_: And we're going to talk about

_11:55_: the functions which takes.

_11:59_: The point P through line.

_12:03_: The function which takes.

_12:04_: How do I write this

_12:06_: Lambda of T through line?

_12:09_: The function which takes X.

_12:13_: I of. Lambda of T.

_12:18_: And these are all, if you've been

_12:20_: precise about it, different functions.

_12:24_: But we're going to talk about as if they

_12:25_: were the same function, but we're going

_12:27_: to label them all F as sort of pun.

_12:30_: OK, it's at this point it's, but you

_12:34_: realize that there are different things.

_12:35_: That's a function from from M to R

_12:39_: that's a function from also from MTOR,

_12:41_: because Lambda is the point of the manifold.

_12:43_: This is a function from,

_12:45_: you know, N numbers XI.

_12:48_: You are the different functions,

_12:49_: but they're basically the same function,

_12:51_: just in a way that.

_12:54_: We're going to elide for the moment.

_12:57_: So this function F of Lambda of T.

_13:01_: What happens if we differentiate it with

_13:04_: respect to T or this function F of XI?

_13:09_: Of Lambda of T what happens we

_13:11_: differentiate that with respect to T.

_13:14_: Well, the. Checking the derivative

_13:16_: of F with respect to T.

_13:20_: Is some. Of a DF by DX I.

_13:26_: Txi by. DT. Ohh no.

_13:32_: Yeah, OK.

_13:36_: But that's true of any function F.

_13:39_: So we can write down instead just D by DT.

_13:44_: Either some. Of Dxi by DTD by DX.

_13:55_: And there's two things we can do

_13:57_: at this point. One of the things

_13:59_: we can do is change variables.

_14:01_: So let's instead talk about key A,

_14:04_: which is right way up T. Divided by E.

_14:12_: So. D XI by DT is equal to a DX I by DT.

_14:23_: So that D by DA is going

_14:26_: to end up being a. DX I by.

_14:33_: It's a T.

_14:36_: D. By DX I. OK, that's one thing.

_14:42_: I'm gonna come back to that in a moment.

_14:45_: And. No. Imagine.

_14:49_: There's not just dysfunction

_14:52_: that this curve Lambda.

_14:55_: There's a curve going through the point P.

_14:56_: There's also a curve.

_14:59_: You this so, so, so so here's our point P.

_15:03_: There's our Lambda. Of tea and.

_15:08_: There's a mu of another another curve

_15:11_: which goes through the same point P.

_15:14_: They don't meet anywhere else,

_15:15_: but they both do cross

_15:17_: each other at the point P.

_15:21_: Let's ask what is E? D by D.

_15:27_: DF by DS plus B, DF by D.

_15:32_: T let's not worry about

_15:33_: why we want to do that.

_15:36_: Let, let let's think about that.

_15:41_: Look just. Here look at this.

_15:46_: We'll see that's. Some. Of. XD.

_15:56_: DXIBYD. Yes. Plus BD XI by DT.

_16:06_: ADF by D. To.

_16:13_: Which screen is shown? What?

_16:15_: Because they can't read.

_16:16_: What's going on behind here?

_16:18_: Ohh, sorry, right?

_16:20_: You can't write dranoff. Good idea.

_16:28_: You know, I'll just show both.

_16:29_: Thank you. Yeah. Yeah. If I'm going

_16:32_: off the page or not legible then,

_16:34_: then you do exactly that. Do show.

_16:38_: So we have, we can end up with that.

_16:44_: But. That's a number. I said relative,

_16:47_: so there will be. Another curve.

_16:52_: Called the Tau of what are.

_16:58_: The derivative of which with

_16:59_: respect to R is just that.

_17:03_: Aye of D XXI by Dr.

_17:11_: Would be. And that's

_17:14_: equal to DF by. Yeah. But.

_17:22_: That's a lot of maths, which is the roots.

_17:24_: Don't worry too much about writing it down,

_17:27_: but the point what we've done

_17:29_: here is interesting and important.

_17:31_: Because what we've we've done is

_17:34_: discovered that whatever. DBT is.

_17:41_: Umm. 8 times it. No.

_17:48_: Um. Equals a.

_17:54_: DT8 times it.

_17:56_: It is another thing of the same type.

_17:59_: It's also a derivative.

_18:01_: In other words, DBT is. Think of.

_18:03_: This can be multiplied by the by a scalar

_18:06_: to get another thing of the same type.

_18:08_: We've also discovered.

_18:10_: That you can add.

_18:12_: Some multiple of DDS which is one

_18:15_: of these things and the multiple of

_18:17_: DBT is one of these things and get

_18:19_: DDR with another of these things.

_18:21_: In other words.

_18:23_: These derivatives defined in that way

_18:27_: satisfy the axioms of a vector space.

_18:30_: So This is why we're very genetic

_18:32_: about the the the definition of

_18:34_: a vector space in the last part.

_18:36_: In other words, these objects DBDT,

_18:39_: which you're familiar with and there

_18:42_: isn't anything really exotic there

_18:43_: you're you're you're you're you're

_18:45_: familiar with that sort of thing.

_18:46_: I haven't done,

_18:47_: I haven't pulled any fast one

_18:49_: that's that is nothing more

_18:50_: than what you think it is.

_18:52_: But it obeys the accent of vector

_18:54_: space and so we can see it's a vector.

_18:60_: And the rest of this course,

_19:02_: that's exactly what we're going to do.

_19:03_: We're going to say this is a vector.

_19:07_: And when I talk about a vector from now on,

_19:09_: that's what I mean. OK.

_19:13_: And So what that means is that we

_19:16_: can write do things like write again.

_19:21_: V.

_19:24_: Some I'll provector.

_19:32_: Is. DB DB duty at P Now notice

_19:38_: that vector depends on two things.

_19:41_: It depends on the curve.

_19:44_: To which T is the parameter,

_19:45_: so that vector V.

_19:47_: Is related to the Lambda curve.

_19:51_: Who's prompter is T? OK.

_19:54_: So it's sort of it's it's sort of like

_19:60_: you know what that what that means.

_20:02_: Is that? V is called the tangent vector.

_20:06_: It describes how much the the the function.

_20:10_: So so yeah, therefore V.

_20:16_: Applied to F and that's a fight,

_20:18_: and regularly.

_20:19_: That is a funny thing.

_20:21_: It looks like anything to write is.

_20:24_: DBDT. At P applied to F.

_20:30_: Which is equal to.

_20:32_: So what we would find to

_20:34_: be equal to DF by? DT at.

_20:41_: TOP. This looks like a

_20:44_: strange thing to write.

_20:45_: This V applied to F.

_20:48_: All it is is an A weird way of

_20:51_: writing a differential operator.

_20:53_: OK, it's a differential operator

_20:56_: which apply to F so that debt

_20:60_: PF is defined to have the value.

_21:02_: The value of this operator

_21:03_: applied to F is the derivative.

_21:06_: How fast F is changing when you

_21:09_: change T at the value of T which

_21:13_: corresponds to the point B.

_21:15_: And we call it the tangent

_21:16_: vector because it you can think

_21:19_: of it as lying along the curve.

_21:22_: Lambda of T.

_21:23_: And providing the answer to the question,

_21:26_: how much does this function F

_21:29_: which is applied which is defined

_21:31_: across the whole manifold?

_21:32_: How does it change as you

_21:34_: move along this curve?

_21:36_: Land Lambda of T How does it

_21:39_: change when you change T?

_21:41_: So I'm here.

_21:42_: F is being a scalar field in

_21:45_: the sense that I defined it.

_21:48_: Last part is a field in the sense that it

_21:51_: has a value for each point in the manifold.

_21:54_: So distinguish maps the manifold

_21:56_: to the through lane.

_21:58_: In this case I was wondering.

_22:01_: What? What is?

_22:03_: You have to FB right?

_22:05_: So yeah, sorry I interrupted myself.

_22:08_: So this, this, this,

_22:09_: this V here depends on two things.

_22:11_: It depends on the.

_22:14_: On the curve it's standard vector to a curve.

_22:18_: But but it's defined only.

_22:20_: At the point P on that curve.

_22:24_: So this operator here is is

_22:26_: defined only at one point.

_22:30_: OK, at the point P,

_22:32_: so if we go back to. Umm. Uh.

_22:40_: Or do you? Oh sorry there.

_22:42_: Alright, so that's the the value

_22:44_: of T which the value of T on the

_22:47_: curve which corresponds to point P.

_22:50_: So it's it's sort of the the

_22:53_: backwards map there. So at the so.

_22:57_: This vector here is defined at the

_22:60_: point P at the point on the curve.

_23:04_: The the value of T which

_23:06_: corresponds to that point P.

_23:07_: So as you as you move T along

_23:10_: here you move along this curve.

_23:13_: And you have a function which you go to the

_23:16_: map the manifold to the to to the real line.

_23:20_: As you move along that curve,

_23:21_: the value of that function at the

_23:24_: at the corresponding point changes.

_23:26_: You're asking how much does it change

_23:27_: as you go on that particular curve.

_23:29_: So this V corresponds to that curve.

_23:32_: At that point.

_23:35_: You have to point you to the

_23:36_: local nature of the mind.

_23:40_: We could only have it at a point

_23:43_: because all of the. And the only.

_23:48_: No.

_23:51_: I don't want to resist saying we can

_23:53_: all have it at that point because

_23:54_: that's only defined at that point,

_23:55_: because that's not very illuminating.

_23:58_: We applied only to point at that point.

_24:02_: I think because the yes.

_24:07_: If we're talking about this family of curves.

_24:10_: The, the, The, the A Lambda of T,

_24:12_: the MU of S, the top of our and so on.

_24:14_: The only thing that they have in common

_24:16_: is they all go through the point P.

_24:19_: So we are concerned at this point

_24:22_: with the whole set of curves.

_24:25_: Which go through the point P.

_24:28_: And for each of those curves.

_24:30_: There is one of these

_24:31_: tangent vectors defined.

_24:32_: Which is seeing how much of

_24:36_: the picking up the derivative.

_24:41_: Along that curve at that point.

_24:43_: So for each of these,

_24:45_: did that sort of seem like

_24:46_: an answer to the question?

_24:50_: For each of these curves,

_24:52_: there's a tangent vector.

_24:54_: And so that there's a a

_24:56_: set of tangent vectors,

_24:58_: which we call the tangent plane.

_25:01_: At that. At that point.

_25:04_: So the set of these derivative operators.

_25:08_: V would you define only at point P?

_25:12_: Is a set which is one.

_25:15_: That picture is A2 dimensional flat space,

_25:18_: which is called the tangent plane

_25:20_: at the point P of the manifold M.

_25:22_: And that is called the tangent plane

_25:26_: is seeing that although this looks

_25:28_: like it's laid out on the manifold.

_25:30_: It really touched the manifold at one point,

_25:32_: if you like.

_25:33_: So all the the points that are

_25:37_: all the points in that tangent

_25:40_: plane correspond to vectors V.

_25:42_: But they're not perfect.

_25:43_: But they don't correspond to.

_25:45_: They don't join points in the manifold,

_25:47_: they are purely at on the tangent plane.

_25:49_: They're purely defined at the point P.

_25:54_: And a key thing.

_25:55_: And this might also come back to

_25:57_: what you're thinking is that if

_25:59_: we do all this at a point. Cute.

_26:02_: And what else in the manifold?

_26:04_: Then we get a different tangent plane.

_26:07_: Tangent plane TQM.

_26:10_: Which is a completely different plane.

_26:11_: It's pretty different space the

_26:13_: the the vector is defined in that

_26:15_: different tangent plane at the point

_26:17_: Q somewhere else have nothing to do.

_26:19_: With the vectors in the

_26:21_: tangent plane of the point P.

_26:25_: We don't want that.

_26:27_: Because what we do want to end up with is

_26:31_: be able to ask how do the vectors? How?

_26:34_: How do the rates of change at this point?

_26:37_: How do they relate to rates of change

_26:38_: at that other point in the manifold?

_26:40_: So we do want to talk about how we go

_26:43_: from one tangent plane to the next.

_26:46_: We can't do that yet. We will.

_26:48_: We need to find a way of

_26:50_: connecting this tangent plane,

_26:51_: that tangent plane we'll discover.

_26:52_: The way we do that is through

_26:54_: a thing called the connection.

_26:55_: Conclusion the. But we can't do that yet.

_27:01_: And they're not all going in One Direction,

_27:03_: right? Yeah, yeah, that's right.

_27:06_: So, so, so this, this cover is going

_27:08_: there another cover here whatever.

_27:10_: And a constant each is, is, is the.

_27:15_: The that this vector V gives

_27:16_: the answer to the question.

_27:18_: How much does do things change as you head

_27:21_: along that that that particular curve?

_27:24_: So for all the curves that go through

_27:26_: the point P, each one of them is a.

_27:30_: A vector.

_27:35_: The pending. Yes, look at the curve.

_27:43_: Do they change like with the plane?

_27:48_: And I think the so yes,

_27:51_: you can imagine a number of a number

_27:53_: of of curves here which go all the

_27:56_: way through through the the manifold.

_27:58_: So I think the ones we're interested in

_28:01_: are the ones which go through the point P.

_28:04_: So of all the curves that

_28:05_: go through the point P,

_28:06_: each of those has a corresponding

_28:08_: vector in the tangent plane.

_28:12_: So we're not talking about

_28:13_: all the possible curves,

_28:14_: only the curves which go through.

_28:17_: So I think that is against

_28:19_: circling around your, your,

_28:20_: your question why are we talking about

_28:21_: why we're talking about because we're,

_28:23_: we're picking out what we're

_28:24_: looking at, what happens at P.

_28:31_: Ohh.

_28:33_: They should expand 3.

_28:37_: 3. Should also be or.

_28:41_: Yeah, the tangent plane is has the same

_28:44_: dimensionality as the at the manifold.

_28:46_: Yeah, I mean in this picture.

_28:49_: The manifold is 2 dimensional.

_28:53_: And and to the tangent plane,

_28:54_: but and and and so I think that the

_28:57_: the the name I think comes from that.

_29:00_: So, so yes, it's a bad name,

_29:03_: but it has the pick.

_29:05_: The good the picture is of of

_29:07_: a a flat plane just touching

_29:08_: the manager at one point.

_29:12_: Key. Um.

_29:20_: OK. And So what?

_29:23_: But that's so moving on.

_29:26_: And with this in mind. We can.

_29:30_: Look back at this expression here.

_29:33_: And reread this.

_29:38_: With.

_29:41_: And five.

_29:47_: As being. Me too, I thought.

_29:53_: Define defining a set of basis

_29:56_: vectors which are D by DXI. ATP.

_30:03_: And write down DDT. A TP is a sum.

_30:11_: Of.

_30:15_: DXIBYD.

_30:18_: TID by DXI. Which is um,

_30:23_: how do I write this?

_30:26_: Yeah, it's V i.e. Aye.

_30:31_: Jumping back to the tation of our

_30:33_: previous of the previous previous part.

_30:36_: So all I'm doing here is rewriting

_30:39_: something which was what we saw before,

_30:41_: calling it a tangent vector with this

_30:44_: notation here and identifying that as a

_30:47_: basis vector and that as the component VI.

_30:50_: And as you can see the the

_30:52_: index is sort of matched in the

_30:54_: sense that a raised index or the

_30:56_: denominator corresponds to a a a

_30:58_: load index in this so the summation.

_31:01_: Mention still hangs together in that sense.

_31:07_: At which point we can import all

_31:09_: of all of the understanding we

_31:12_: got from the last part of how.

_31:14_: Vectors and components and stuff work.

_31:19_: Which is good.

_31:21_: Next thing is we'll talk with vectors.

_31:24_: So you are asking yourselves,

_31:26_: So what about one forms?

_31:27_: What one forms come in?

_31:30_: One forms come in when we

_31:32_: ask how functions vary.

_31:37_: So again, we're going to consider

_31:39_: a a function. On the manifold

_31:42_: function F arbitrary funct.

_31:46_: And. We're going to define

_31:49_: A1 form field called. DF.

_31:57_: And. That we were going to find that

_32:02_: one form field is by its action on

_32:05_: a vector one formed remember turn.

_32:07_: Vectors into numbers.

_32:10_: So we're going to define this one form.

_32:12_: By asking what its action is when applied to.

_32:21_: DBDT at P.

_32:25_: And we're going to see,

_32:26_: we're going to define.

_32:28_: So remember a function you're

_32:30_: used to seeing a F of X equals

_32:33_: mathematical expression.

_32:34_: All function is is a rule for going from.

_32:37_: Domain to range.

_32:39_: And our rule here is given this picture.

_32:46_: The value of that is DF

_32:49_: by DT at the point P.

_32:55_: And this DF. Is the gradient.

_32:59_: So-called of the function F

_33:01_: the one form which the gradient

_33:03_: one form of the function F.

_33:06_: Which is a wonderful field.

_33:07_: In other words, it is A1 form which is

_33:10_: defined at each point in the manifold.

_33:12_: The value of which it obtained by definition.

_33:15_: By definition by applying the applying

_33:18_: it to a vector in a particular.

_33:22_: Tangent space to give that

_33:24_: to give that result. Um.

_33:31_: Do uh, what does that look like?

_33:34_: I'm also do things like write that as the F.

_33:40_: Thank you. Umm.

_33:49_: What blows right that is that we

_33:52_: picking up this notation that I

_33:54_: briefly mentioned for the contraction

_33:55_: between one form and a vector.

_33:57_: Um, if we write this?

_34:04_: In component form, then we see the DPDT.

_34:11_: Is uh.

_34:15_: This. PDF is applied to.

_34:18_: We could DDT is that that is operator V.

_34:23_: We know from last part that that is it's VI.

_34:27_: DF. EI. Which is VI. Yep.

_34:39_: Yeah, full derivative F by D. XI.

_34:43_: So we we can see a component version

_34:47_: of that same thing. Been that way.

_34:51_: And and similarly if we ask so

_34:56_: each of the coordinate functions.

_34:58_: Remember I said I I I had the picture there

_35:01_: of the coordinate functions which Map XI

_35:03_: which map the manifold to the real line.

_35:06_: Each of those is is is also a function.

_35:08_: So each of those has a gradient

_35:11_: one form associated with it.

_35:13_: Written something like.

_35:16_: The. X. Alright, yes, XI.

_35:22_: And we apply that to one of

_35:25_: the basis vectors DVDX I.

_35:27_: We get. The

_35:34_: XXGXIBYDXJ. Part of relative and if

_35:37_: the the the various quarter function

_35:39_: are independent of each other.

_35:42_: Then that will be one if

_35:44_: I is equal to J and 0.

_35:46_: Otherwise on or the chronicle delta.

_35:51_: So which is reassuring,

_35:52_: that's telling us that our basis.

_35:55_: One forms. The the these gradients,

_35:60_: the gradients of the coordinate functions

_36:03_: are indeed dual to the basis vectors.

_36:06_: As we demanded our last time.

_36:10_: So all the stuff we learned

_36:13_: in the last part about.

_36:15_: Basis vectors basis one forms

_36:17_: component blah blah blah blah

_36:19_: blah can now apply to these.

_36:21_: Species, vectors and these pieces one forms.

_36:24_: So we can import all that

_36:25_: technology and move on.

_36:29_: Any questions about the question?

_36:36_: Yes, I probably am, but so

_36:38_: the point should be up there.

_36:42_: So so yes, that's that's that's a

_36:46_: good point, because all of this,

_36:48_: all of the the the vectors in one

_36:51_: form we're talking about here are

_36:53_: in the tangent plane. Thank you.

_36:55_: So that's all I at a point P.

_36:58_: So so we've sort of we've we've

_37:00_: sort of left manifold behind and

_37:02_: and and most of the stuff we're

_37:04_: going to do immediately is in the

_37:06_: tangent plane at a point P so I

_37:08_: so I I will sometimes miss out the

_37:09_: the the P there we have to remember

_37:12_: that that that that's the space

_37:14_: that we're talking about here the

_37:16_: vector space that we're talking

_37:17_: about here is the tangent plane.

_37:19_: Thank you.

_37:25_: Yeah.

_37:29_: OK, next. We're good progress here.

_37:36_: So one of the things we did last

_37:38_: in last part was change basis,

_37:39_: change of mind, what the basis

_37:42_: functions were the. The.

_37:46_: This function here. Is arbitrary.

_37:50_: We can change our mind about what

_37:53_: that coordinate function should be.

_37:55_: So if we do that.

_37:59_: Then we will write.

_38:02_: E. I bar you go to.

_38:06_: Deep by DXI bar. At peak.

_38:13_: And. This new basis.

_38:15_: Will be related to the old one.

_38:19_: By. Lambda I I bar E. Aye.

_38:28_: And and. I think that there may be an

_38:34_: exercise which allows you to to to to

_38:36_: reassure yourself that Lambda I I bar.

_38:38_: Is be sure to get things right, we up.

_38:43_: Me.

_38:47_: No. Um is.

_38:53_: So we would have to help himself

_38:55_: is the XI by DX. I bought.

_39:01_: Sorry, I got off the boat.

_39:04_: So I I encourage you to go through

_39:08_: that section of the that that last

_39:10_: section is slightly slower to reissue,

_39:12_: which I'm not pulling a fast one here but.

_39:15_: There's nothing surprising

_39:16_: particularly writing here.

_39:18_: The the Lambda that we saw last time

_39:21_: is just the relationship between two of

_39:24_: the of the arbitrary arbitrarily chosen

_39:26_: basis functions coordinate functions.

_39:32_: We are making excellent progress.

_39:35_: We'll move on to section 3.2,

_39:36_: slightly ahead of schedule.

_39:38_: And I think last, I think last

_39:41_: questions about that. So we've.

_39:47_: Yeah. What, what, what,

_39:47_: what, what with them?

_39:50_: Define a useful vector space.

_39:52_: Because the vector space we're able

_39:53_: to pull in all the, all the the.

_39:55_: Vectored at one form tensor manipulation

_39:58_: technology from the last part,

_40:00_: so we now move forward with it. OK.

_40:07_: So we've defined vectors.

_40:09_: We've defined 1 forms.

_40:11_: And now we want to start defining

_40:14_: differentiation of vectors.

_40:15_: In other words,

_40:16_: how given a vector in a?

_40:19_: The electric field vector C.

_40:22_: How does that vector vary?

_40:24_: How does that vector change as

_40:25_: you move around the space? OK.

_40:27_: Because we want to ask things

_40:30_: like how does the argumentum?

_40:32_: Teacher change as you move around the space,

_40:35_: because that's going to tell us.

_40:38_: It turns out how the coverage of

_40:41_: the space it turns out changes

_40:43_: remove around so we've got we've

_40:45_: got we're going to ask yourself the

_40:47_: question how do you take intentions.

_40:48_: How do we manage the to calculate

_40:50_: with the the the the way that tends

_40:53_: to change as they move in space?

_40:55_: We're 2 steps.

_40:57_: We're going to first of all find

_40:59_: out how to do that in flat spaces.

_41:01_: And then discovered that the

_41:03_: step going from there to doing

_41:05_: differentiation in curved spaces.

_41:07_: It's quite a big deal,

_41:09_: but it's a smaller step than you think.

_41:11_: Which is nice.

_41:20_: OK.

_41:27_: So yeah, blah blah.

_41:34_: Well, I ask that question here.

_41:35_: What type of thing is?

_41:38_: Is it is this is this.

_41:41_: Just quickly, who's able to scaler?

_41:43_: A vector. A1 form. Potential.

_41:47_: The matrix had we hang up yet?

_41:51_: Quite a little there. OK.

_41:52_: Talk to each other and I

_41:55_: think that question is?

_42:22_: OK, with a bit of reflection,

_42:24_: then ask the question again.

_42:26_: Who's it with the scaler?

_42:29_: The vector one form.

_42:33_: Tensor. Matrix. And.

_42:41_: It's just scale.

_42:43_: It's just number. Because.

_42:44_: Because. If you look at that,

_42:47_: I think at the end here,

_42:48_: but for me it's just a derivative.

_42:54_: This whole thing is a complicated

_42:55_: way of writing that derivative.

_42:57_: And it's answering.

_42:58_: It's answering the question, how much

_43:00_: does the function F vary as you change T?

_43:03_: Just number F the number.

_43:05_: So as you change T the number changes.

_43:08_: You move from different the manifold.

_43:09_: How much will it change?

_43:11_: OK, and V is a vector.

_43:15_: Is a function actually function so,

_43:18_: but the action of this operator V.

_43:22_: On if.

_43:24_: Is this there is this thing applied to

_43:28_: F which is the derivative of DF by DT?

_43:32_: So if you wanna be picky then used attention.

_43:35_: We could all scale attention too,

_43:37_: but that's slightly.

_43:39_: It's not the simplest answer you can give.

_43:43_: So I see this.

_43:44_: It seems it feels it feels

_43:46_: a bit like a trick question.

_43:48_: But I'm saying that there's slightly

_43:49_: less to this than meets the eye.

_43:51_: That looks a really exotic thing.

_43:52_: You've never seen a vector written

_43:54_: next to a function before you think?

_43:58_: So what this what?

_43:59_: So what this means is that vectors

_44:01_: in this context are slightly dual.

_44:05_: Meaning. They're both pointy things.

_44:08_: And what that's a good thing to

_44:10_: have in your head and they're

_44:12_: also differential operators.

_44:13_: Which should be applied to functions

_44:15_: to give to give numbers. OK.

_44:17_: They're both things you I need

_44:19_: both things you had at one time,

_44:21_: and which what?

_44:22_: Which sense of the of that I

_44:24_: I'm I'm I'm I'm referring to

_44:25_: in a particular context will

_44:27_: depend on what I'm doing with it.

_44:28_: So sometimes I'll be manipulating

_44:30_: these vectors like vectors,

_44:31_: but components, all that stuff.

_44:33_: Sometimes like this there are an

_44:35_: operator which applies to a funk.

_44:41_: And and you've seen that that's

_44:43_: just a summary of other things.

_44:47_: What can you see?

_44:49_: The same question and what sort of thing is?

_44:52_: D XI by DT. In that expression.

_44:56_: Wish it was a scalar. A vector.

_45:00_: A1 form. Tensor. Matrix.

_45:06_: Hadn't read hand but hand up yet,

_45:09_: but still haven't found it yet.

_45:11_: Don't you have a brief chat?

_45:13_: What is DX IDT?

_45:36_: And with that reflection,

_45:39_: who she was the scaler.

_45:41_: Who is he? Was a victor. Who?

_45:44_: She was in one form. Tensor OK.

_45:49_: There's sort of two Oz here.

_45:55_: Strictly it's just it's just a scalar.

_45:57_: It's number. It's the derivative of

_45:59_: this function X of X of I with respect

_46:02_: to T is how much did that that that

_46:05_: coordinate function change as you change T?

_46:07_: So it's just a number again

_46:10_: because XI is just a function.

_46:13_: But of course these are also the

_46:15_: components of function and and.

_46:17_: And as I said last time,

_46:19_: sometimes I'll start the equivocate

_46:20_: between the components of a vector

_46:22_: and the and the vector itself.

_46:23_: So if you said vector, OK, not long.

_46:26_: But strictly, that's just a number.

_46:31_: XI is a function.

_46:32_: It's there's north of them.

_46:34_: That ain't functions,

_46:35_: but in each case the how much that

_46:38_: function changes as you change

_46:40_: T as you go along the curve.

_46:42_: It's just a number.

_46:48_: Um. Yeah, goodbye you abuse of notation.

_46:53_: I was somebody. You think of VI as

_46:54_: being the vector, but but it's not.

_46:58_: Um. There are dual. There's that.

_47:09_: Not sure I'm getting out there.

_47:11_: Um. I think I mean really

_47:15_: tricky tricky there but.

_47:18_: OK. I'll just to start you off. I'm going to.

_47:27_: Mention. That you have seen.

_47:33_: Yeah, don't do that.

_47:37_: No, let's just stop there.

_47:38_: Let's stop a little early.

_47:40_: I think it means we can go right

_47:42_: into a secondary .2 next time,

_47:44_: which I think is next.