Transcript of gr-l02
==========


_0:09_: Hello and welcome to Lecture 2 of the Greg.

_0:14_: This is lecture 2 out of 11.

_0:17_: I trust you will all have made

_0:19_: your apprise yourself of a copy

_0:21_: of the part of the Part 2 notes.

_0:23_: The plan is to get through these this

_0:27_: part on vector sensors functions in.

_0:31_: A bit under 3 lectures may

_0:33_: or may not manage that,

_0:34_: but that that that that's the plan.

_0:37_: There's not a huge amount

_0:39_: of news of huge new stuff.

_0:42_: Well, there is.

_0:45_: Most of us have in this you probably

_0:47_: have seen before in some form.

_0:49_: Probably not in this form,

_0:51_: and it might be an alien,

_0:53_: a bit alien to you in the

_0:55_: way we're approaching it.

_0:56_: But the aim of this chapter is to

_0:59_: get everyone on the same page really.

_1:02_: So if you've seen him this stuff before,

_1:04_: good for you. If not,

_1:07_: make sure that that you you want to keep up.

_1:12_: As I mentioned last time,

_1:14_: those of you with a fairly who's

_1:17_: whose background includes a fair

_1:19_: amount of of pure ish maths might

_1:21_: well find this deplorably informal.

_1:24_: Those of you who's background hasn't

_1:26_: included a parameter punish maths

_1:28_: might find this rather bracing.

_1:30_: OK, so I don't think it's

_1:32_: over in between your, your,

_1:34_: your your your expectations.

_1:36_: But as I also mentioned last time,

_1:38_: this is the first of the two middle parts,

_1:41_: which is mostly mathematical,

_1:42_: allowing us to get back onto

_1:45_: the physics in the last chapter.

_1:47_: Because the payoff of this maths is merely

_1:51_: maths, the payoff is to allow us to.

_1:54_: Talk about gravity in the physics

_1:58_: of gravity in a very powerful way.

_2:02_: OK. Any questions about that?

_2:06_: Or speculations,

_2:07_: or thoughts or feelings you wish to share.

_2:11_: OK then let's proceed.

_2:16_: So the as before, there are aims,

_2:21_: objectives, high level,

_2:22_: fairly high level things.

_2:24_: The aims are not very there

_2:26_: aren't very many aims to this,

_2:27_: to this section,

_2:28_: because this is a fairly mechanical

_2:29_: section in the sense it's about technique.

_2:31_: There are stuff you're going to

_2:33_: be learning here about how to do

_2:35_: certain particular maths so that so

_2:36_: you will use all this stuff later,

_2:38_: but there there aren't great physical

_2:40_: insights awaiting you in the next 3 lectures.

_2:44_: But quite a lot of objectives,

_2:45_: so there's quite a lot of things you'll be

_2:48_: able to do after this this part I and as I.

_2:51_: As I said last time,

_2:52_: the distinctively aims

_2:53_: objectives is aimed at the point

_2:55_: objectives of the party tricks.

_2:56_: The things that are fairly

_2:58_: straightforwardly accessible is the point.

_3:02_: Thank you.

_3:06_: This first section on linear algebra will

_3:09_: be some sort of revision to most of you.

_3:12_: OK, you will have seen a lot of these things

_3:15_: before at earlier stages in your education.

_3:18_: If not, go back to your notes from

_3:21_: previous years and make sure you

_3:23_: are fully up to speed with them.

_3:25_: One thing that I'm going to,

_3:28_: I think otherwise isn't on this slide,

_3:29_: but which is going to come

_3:31_: up again and again.

_3:32_: Is the notion of linearity and linearity in

_3:35_: this context means a quite specific thing?

_3:38_: Linearity in this context means.

_3:45_: A function. F of XC it can be any

_3:49_: function you think of, it's linear. If.

_3:58_: If and only if F of 2X.

_4:01_: If you could two up F of

_4:04_: a * X equal to AF of X.

_4:09_: That's what linearity means in this context.

_4:11_: It doesn't mean a straight line graph,

_4:12_: although that's also referred

_4:13_: to as a linear graph.

_4:14_: It means if you if you multiply

_4:17_: the argument by a by a scaler,

_4:19_: the result is multiple sclerosis.

_4:21_: But that's not true of every function,

_4:23_: for example.

_4:26_: If X = X, ^2 is not linear.

_4:31_: Because F of of 2X is equal to two

_4:33_: X squared which is equal to four X

_4:37_: squared which is not equal to two X ^2.

_4:40_: But you'll be OK.

_4:43_: So I'm belaboring that point because that.

_4:48_: This this term linear has possibly slightly

_4:51_: different conditions in different areas,

_4:53_: but returning to these points

_4:55_: here about vector spaces.

_4:57_: I will come back again again to

_4:58_: this notion of a vector space.

_4:60_: And a vector space. Is.

_5:02_: Anything.

_5:04_: Which satisfies these five properties.

_5:07_: I'm going to go see a little

_5:08_: bit more about those,

_5:09_: but not too much more,

_5:10_: OK?

_5:14_: The vectors that you're

_5:15_: familiar with pointy things,

_5:16_: the things you learned about in school.

_5:18_: Those are an example of the

_5:20_: elements of a vector space,

_5:22_: and they're they're the, hence the name.

_5:24_: They're the approach type

_5:26_: example of a vector space.

_5:28_: You can add vector together.

_5:30_: This vector plus that vector

_5:31_: gives another vector.

_5:32_: OK, you know that, right?

_5:34_: So that's a property.

_5:36_: There exists an element, a zero at 0,

_5:38_: and you add that zero to anything,

_5:41_: you get the the thing you

_5:42_: started with back backwards.

_5:43_: So so a + 0 is a.

_5:45_: There exists an identity element.

_5:49_: Inverse for every vector,

_5:50_: there's a vector that points

_5:51_: in the opposite direction.

_5:53_: We add up to the identity.

_5:58_: Multiplication and in vector you can

_6:01_: double it and you get another vector

_6:04_: which is the same direction and

_6:07_: and twice as long. And there is a.

_6:13_: If you multiply that by one.

_6:16_: You get this thing better.

_6:17_: Nothing surprising here.

_6:18_: Is the tribute of that that that

_6:21_: multiplication by rules is retributive.

_6:22_: So there's nothing there that's

_6:23_: surprising to you, I trust.

_6:27_: Good. But I'm, I'm, I'm,

_6:30_: I'm listing them because these

_6:32_: are very general properties.

_6:33_: And anything that refers that

_6:35_: satisfy those properties is a

_6:37_: vector space in the sense we're

_6:38_: going to be talking about. OK.

_6:43_: So. Hold on to that thought.

_6:48_: I'm going to also assume.

_6:51_: That you know about these

_6:54_: various things here and.

_6:56_: London Stadium dimension

_6:58_: the dimensionality of space.

_6:60_: Idea basis set spans of space.

_7:04_: Existence, component, blah blah blah.

_7:05_: The Chronicle just symbol and

_7:07_: you may have seen before the

_7:09_: Chronicle death symbol is just a.

_7:12_: I'll be good to stop

_7:13_: switching the back and forth,

_7:13_: but the coronary death symbol is.

_7:16_: Dot IG is defined as being one.

_7:21_: If I = g and 0 if I is not equal to G.

_7:26_: What's the definition of the chronicle delta?

_7:29_: I'm positive, definitely.

_7:32_: This just means that the.

_7:35_: In a product.

_7:37_: Of of of two of of two things

_7:40_: is is not negative OK?

_7:45_: And some stuff about matrix algebra.

_7:48_: That the matrices have an inverse,

_7:50_: a trace, a determinant and so on.

_7:55_: Have you you you you have come

_7:58_: across these words before.

_7:59_: Yeah, OK. And anyone who's feeling

_8:01_: uncertain at this point, you know,

_8:03_: quick trip to your to your last

_8:05_: year's notes would be a good idea.

_8:07_: And I'm not gonna depend on an intimate

_8:08_: knowledge of these things or having

_8:10_: them at the front of your head,

_8:11_: but I am going to assume that you can

_8:13_: look those up and remind yourself

_8:15_: of the details when necessary.

_8:16_: Are there any questions with that?

_8:19_: OK.

_8:22_: So.

_8:26_: I'm so matrix algebra.

_8:29_: You know what means algebra good.

_8:30_: I'm sure you know all the Metra. Umm.

_8:40_: OK, quick question.

_8:43_: Consider yourself square and by emissaries.

_8:46_: Is that a vector space?

_8:48_: Kind of Lucy, yes.

_8:50_: And those who say no.

_8:53_: Excellent, right?

_8:53_: It is a vector space.

_8:55_: You don't think of square

_8:57_: matrices as vectors.

_8:59_: But the squeamish sees

_9:01_: satisfy all those axioms.

_9:04_: You can multiply it by you

_9:06_: a scalar times a matrix.

_9:07_: A square matrix is a square matrix.

_9:09_: You can add 2 square matrices

_9:11_: together and get a square

_9:12_: matrix of the same rank.

_9:14_: They all zeros matrix.

_9:15_: That they all zeros matrix can

_9:16_: be added to any matrix to get

_9:18_: the same matrix started off

_9:19_: with their existing identity.

_9:23_: There's a negative of every matrix.

_9:25_: And matrix and.

_9:29_: The addition, the mosque addition

_9:32_: are distributive like that,

_9:33_: therefore therefore and that

_9:34_: therefore it's important.

_9:36_: Therefore the set of square and

_9:37_: by matrices is a vector space.

_9:42_: Each for each end, so two by

_9:46_: two matrices and three by three

_9:48_: matrices are not in the same space.

_9:50_: You can add 2/2.

_9:54_: 2 minutes to get three matrix.

_9:55_: They're in different spaces,

_9:57_: but they're each separately vector spaces.

_9:58_: OK. Make sure you're

_10:01_: comfortable with that notion.

_10:07_: OK.

_10:11_: Now I'm going to talk about tensors,

_10:14_: vectors and one forms.

_10:16_: You probably have heard about

_10:17_: tensors at some point before,

_10:19_: but you've never really had to you,

_10:21_: and you may have had to to

_10:22_: wrangle with them a bit,

_10:23_: but certainly I remember to.

_10:26_: Tensors being a slightly exotic

_10:28_: thing that was happened in bits of of

_10:33_: continuum mechanics and I think some

_10:35_: slightly exotic classical mechanics.

_10:37_: It was it was a thing that was clearly

_10:39_: quite powerful, slightly magic.

_10:41_: Let's not think about it too much, OK?

_10:45_: That's fine.

_10:46_: Now, if we're tensors, really matter.

_10:49_: So here is a an important use of tensors,

_10:53_: and so we're gonna we're gonna be

_10:54_: talking about them a lot from now on.

_10:58_: Tensors. Are um?

_11:06_: Before putting up the next slide,

_11:07_: I'm going to see the that tensors are. And.

_11:13_: See? So we're going to label.

_11:17_: I don't want to put next slide

_11:19_: up quite yet because I want

_11:21_: to have something up first.

_11:23_: Ohh yeah, and I'm going to introduce

_11:25_: tension in a fairly axiomatic way,

_11:26_: in a fairly mathematical way.

_11:27_: I'm not going to give you

_11:28_: examples and then say, oh,

_11:29_: and now we call these tensors

_11:31_: because see these are the the

_11:32_: definitions of tensors because I I

_11:35_: think that although that can be,

_11:37_: it can feel a bit sort of bit

_11:39_: mathematical as an approach,

_11:41_: it does emphasize that tensions

_11:43_: are fundamentally rather simple

_11:44_: things and that simply saying.

_11:47_: Attention, each rank of tensor rank in

_11:50_: the moment is an element of a vector space.

_11:53_: By just saying that, I've told you

_11:55_: quite a lot about what tensors are.

_11:57_: OK, so a tensor.

_12:00_: And what's your tensor T?

_12:03_: Will have a rank.

_12:05_: Well,

_12:05_: I'll see what this means in a minute.

_12:08_: MN so that we'll have a a rank

_12:13_: MNMNR moment, and each M instead

_12:15_: of MSN tensors is a vector space.

_12:17_: Well, that means you have two

_12:19_: tensors of that rank together.

_12:21_: You get another one of that rank you can,

_12:22_: there's a theory element, and so on.

_12:27_: And I'm going to give

_12:29_: certain of these tensors.

_12:30_: Well, two you 2 two sets of

_12:33_: these tensors a special name.

_12:35_: So now always get these nervous

_12:38_: in these three wrong around

_12:41_: the set of 10 tensors.

_12:43_: We're going to call.

_12:46_: Vector. And the set of 01 tensors.

_12:55_: We're going to call 1 forms now.

_12:58_: This is a slight,

_12:59_: slightly unfortunate naming collision.

_13:03_: And these are all.

_13:04_: These are both elements of vector space.

_13:07_: But it is usual to call this this

_13:11_: set of tensors, to call them vectors.

_13:15_: So in future, gonna talk about vector space.

_13:19_: I mean or or an element of a vector space.

_13:21_: I mean the very general thing that

_13:24_: obeys the axioms of of vector space.

_13:27_: We're talking about vectors.

_13:28_: I mean one of these.

_13:30_: OK, so the point is that there are.

_13:34_: For each M&N which are greater or equal to 0.

_13:39_: There is a set of objects called tensors.

_13:43_: Which satisfy that each of those sets

_13:46_: satisfy the the acting of a vector space.

_13:49_: These are two two examples of that set.

_13:52_: With special names.

_13:54_: With those definitions.

_13:58_: An MN tensor is a function.

_14:02_: We did linear in each argument.

_14:05_: In the sense of which I mentioned before,

_14:08_: which takes M1 forms and N vectors as

_14:12_: arguments and map them to a real number.

_14:16_: OK, question. What does one form mean?

_14:20_: Right and right, examples.

_14:23_: So I'll come to examples shortly because

_14:26_: I'm going to stick with the abstract

_14:29_: in your terminology to begin with,

_14:31_: and then I'm going to bring an example,

_14:32_: some examples question.

_14:35_: N vectors and then M vectors and then one

_14:39_: for the right 10 vector, because ohh,

_14:43_: that's a good point, yes, yes.

_14:47_: That does seem to be the one that

_14:48_: does seem to be the wrong way around.

_14:49_: It's not the wrong way round,

_14:50_: but it does seem to be the wrong way round.

_14:52_: Enough that it will cause confusion, right?

_14:54_: And This is why I checked which we around.

_14:56_: I I I wrote this because

_14:58_: I always get it wrong.

_14:59_: OK, it doesn't actually matter hugely much,

_15:01_: but well spotted the duty

_15:03_: to be the wrong way around.

_15:04_: OK, but I'll come back to that mode.

_15:08_: OK, so the point is, it's a function.

_15:12_: Now a function is a machine which

_15:15_: turns one thing into another.

_15:18_: This function.

_15:22_: Here. Is a machine which takes a number.

_15:28_: And give you back it's, it's square,

_15:30_: OK, so it's a number which maps.

_15:34_: Real line. To the real line.

_15:38_: OK, so that's a good.

_15:39_: That's a good way of it.

_15:41_: Sounds like a slightly baby way

_15:42_: of thinking about functions.

_15:43_: There's really good way

_15:44_: of thinking functions.

_15:44_: Functions are machines which take

_15:46_: one or more things of 1 type and

_15:48_: turn them into another and tensors.

_15:50_: Take. One forms and vectors as input.

_15:54_: You think about machine has

_15:55_: holes in the top with which are

_15:57_: one form shape or vector shape.

_15:59_: Turn the handle and outcomes a number,

_16:01_: not anything else.

_16:02_: A number,

_16:02_: something something in R and the real line.

_16:05_: OK, and it's linear.

_16:07_: So you you put two of the of

_16:10_: these things and the number is

_16:12_: twice what you started with.

_16:14_: OK. Any questions?

_16:17_: So those are really good questions.

_16:19_: Any other questions about that so

_16:20_: but don't really good questions,

_16:22_: the answer to which is coming soon.

_16:24_: Any other questions? OK. Um.

_16:31_: So I think we want some pictures here.

_16:34_: So here are some some.

_16:38_: Some tensors, no. There's a slight

_16:40_: informal notation I'm going to

_16:42_: be using here. For attention.

_16:45_: And what what example we're using?

_16:50_: And I'm going to. See?

_16:53_: This to be consistent with this.

_16:57_: Thank you. Alright, ohh by the way,

_16:59_: I'm going to write vectors.

_17:02_: Typically with an overbar.

_17:05_: And one forms. With the children. OK.

_17:08_: So I'll be feeling consistent with that.

_17:13_: No, absolutely consistent but

_17:15_: fairly consistent with that.

_17:16_: OK, So what this this is going to

_17:19_: be rather suggest from station.

_17:21_: So this tends to T is a.

_17:26_: 21 tensor or one? Yes, at A21 tensor.

_17:32_: Which take it which means it takes.

_17:36_: It has three arguments 2 one

_17:38_: form shaped arguments and

_17:40_: one vector shaped argument.

_17:42_: And it turns that into.

_17:45_: A A real number.

_17:46_: So if we give that tensor.

_17:54_: 3 arguments.

_17:57_: The answer is a real number.

_17:59_: OK, that's that's what that

_18:01_: that's what I mean when I when

_18:03_: you talk about definition.

_18:05_: Yeah. So we're keeping this

_18:06_: abstract at the moment.

_18:07_: Examples are main mode.

_18:14_: But you can also partially apply things.

_18:17_: So there this this here

_18:20_: attention or questions. Confused?

_18:23_: Like when you put the? Alright.

_18:27_: Yes, yes. It's just this intends

_18:31_: to to suggest a sort of empty slot.

_18:34_: So as a whole a machine with with

_18:36_: those three holes in the top.

_18:38_: And that's not a,

_18:39_: that's not a formal notation,

_18:40_: that's just to guide the eyes that were.

_18:45_: So that's that. That's a A21 tensor.

_18:49_: 21 form shape tends to one form shaped holes

_18:51_: and one that shaped hole. I put in. 114.

_18:58_: What I have left is also a tensor.

_19:01_: It it I think which had one one form

_19:03_: shaped hole and one vector shaped

_19:06_: hole A1 form argument and a vector

_19:08_: argument that is S is A-11 tensor.

_19:11_: So by partially applying.

_19:13_: By partially filling in.

_19:15_: The arguments of the tension.

_19:17_: We can turn one type of vector,

_19:18_: a true 1 tensor,

_19:19_: into 111 tensor in this example.

_19:23_: We can do that more than once.

_19:27_: That he he will fill in the one form

_19:29_: shaped one, the one form argument,

_19:31_: one of the one form arguments,

_19:32_: the vector argument.

_19:33_: And we have something which has our, a, our.

_19:40_: A single one form argument.

_19:43_: You know a single just single

_19:45_: one form argument is A10 tensor.

_19:49_: A vector. So that's so we

_19:53_: could write this as a vector.

_19:56_: OK. Um, so we could give

_20:00_: names to these other things,

_20:03_: you know, but we don't.

_20:04_: There's there's no need to give

_20:07_: them the other things because

_20:08_: the the vectors in the one forms

_20:11_: are the things that we sort of

_20:13_: have to give names to in order

_20:14_: to create the definition you

_20:15_: saw in the last slide.

_20:17_: Thank you.

_20:20_: Um.

_20:23_: And similarly if in this example we.

_20:28_: Said talk about tea.

_20:32_: Omega. Sigma.

_20:37_: And and and and and and feel to

_20:38_: fill in the vector ship argument.

_20:40_: That would be a thing

_20:42_: which has 01 form argument.

_20:43_: Zero will zero open one form

_20:45_: argument and one vector argument.

_20:47_: In other words that is A1 form.

_20:54_: Because all the one for means is.

_20:56_: It is something which has a.

_21:00_: AS01 form argument and one vector

_21:02_: shaped argument such as that.

_21:04_: OK. So, and I say we're keeping

_21:08_: this abstract at the moment.

_21:10_: Examples come to the moment.

_21:13_: The last point relevant here is.

_21:18_: In one form. As you recall, is our.

_21:24_: 01 tensor. It takes a single vector

_21:26_: as argument and gives a number.

_21:32_: A vector. Takes a single one form

_21:34_: of argument and gives a number.

_21:36_: Now there is nothing to say

_21:38_: that those are the same number.

_21:40_: These are just functions.

_21:41_: They're functions which take this

_21:43_: thing and turn it, which is a number.

_21:46_: They don't have to be equal.

_21:47_: We are always going to

_21:48_: assume that they are equal.

_21:50_: OK, you can talk about this sort of this

_21:53_: sort of stuff without that assumption.

_21:56_: It makes things harder.

_21:57_: We don't need extra hardness, so in this

_21:59_: context this will always be the case.

_22:01_: We'll always constrain.

_22:04_: And so this this acts as a constraint

_22:06_: on the functions that we allow here.

_22:09_: So, so these functions,

_22:10_: these these these one form,

_22:12_: these functions are not arbitrary

_22:13_: in that sense.

_22:14_: We'll all they'll always have that

_22:16_: reciprocal property and well,

_22:18_: sometimes write that with

_22:19_: these angle brackets,

_22:20_: it just means either of those things

_22:22_: and they're really use angle brackets.

_22:24_: Sometimes it's just emphasise

_22:26_: the symmetry of this.

_22:27_: And that's true for all one

_22:29_: form and vector arguments in GR.

_22:32_: OK, that's a mathematician

_22:34_: would not have that bit.

_22:38_: If I'm working that,

_22:38_: that,

_22:39_: that,

_22:39_: that's that's always gonna be the case.

_22:42_: How we're doing so far?

_22:47_: Makes a lot more sense checking down

_22:49_: round when you go through your notes

_22:51_: afterwards it it will be illuminating.

_22:56_: This this is.

_22:57_: Quick mathematical defense that that

_22:60_: we are laying on the definitions here.

_23:05_: I think it's worth stressing that.

_23:08_: This definition is doing a lot of work.

_23:11_: All the things that I've said.

_23:13_: Your last 10 minutes so are just

_23:16_: immediate consequences of this.

_23:19_: Attention to function is linear. Which

_23:22_: takes these arguments about a real number.

_23:25_: The only thing I've said there isn't.

_23:28_: An immediate consequence of that is this

_23:31_: last point here, which is an extra. OK.

_23:34_: And all that depends on the statement that

_23:37_: the tensors are illness of vector space.

_23:40_: So you get all those other properties

_23:42_: you're being scooped up being provided

_23:44_: for free in a sense by that definition.

_23:46_: OK. So at this point,

_23:48_: we don't mean you do not have a picture.

_23:51_: I don't expect to have a picture

_23:53_: of what tensors are yet.

_23:54_: But you really know a lot about them.

_23:56_: OK. And that's that,

_23:57_: that's the the I think the the the sort of.

_24:00_: When you come back to think through it again,

_24:03_: that's with the clarity of a fully

_24:04_: activated approach comes from.

_24:05_: You already know a lot about them,

_24:06_: even though you have a picture.

_24:08_: OK.

_24:11_: The so these definitions

_24:12_: are doing a lot of work.

_24:15_: Key. Um and?

_24:23_: Umm. We have, and I'll mention just

_24:27_: a bit of terminology. This is all.

_24:29_: I will at this point freely,

_24:31_: freely refer to contractions

_24:32_: between vectors in one forms.

_24:34_: And I say they were contraction.

_24:35_: I mean that.

_24:36_: I mean applying of extra one form

_24:38_: that's contracting the vector P

_24:39_: the the vector A and the one form

_24:42_: one form P so you contraction.

_24:44_: That's what I mean.

_24:47_: There's more we can say about that,

_24:48_: but we're not going to. Ohh question.

_24:52_: In the example you're given for the.

_24:56_: Yeah, yeah, this is.

_24:58_: So the Q, is that the one form or is

_25:01_: that it's a one form with child over?

_25:04_: But that's also considered

_25:05_: as an argument for M.

_25:07_: So we considered as MN, so that's M&N.

_25:12_: And if and if I'm if.

_25:16_: I'm not sure that's a A21 tensor.

_25:22_: It's telling us that there are two

_25:24_: wonderful arguments, and one vector.

_25:25_: Is that what you meant?

_25:26_: OK, yeah, so that's that's an

_25:28_: example of a true 1 tensor, and.

_25:30_: And take your tea in each of both

_25:33_: cases is A21 tensor by partial

_25:35_: application which we turn into

_25:37_: A-11 tensor and A10 tensor. OK?

_25:42_: Sorry, what are the thoughts?

_25:47_: Which bracket? And in

_25:52_: practice. Throw some dots.

_25:56_: Ohh the brackets here.

_25:58_: And that alright, that I an important

_26:02_: question and that's just picking

_26:04_: up that that we've said T is a is a

_26:08_: is a is a a function. So just like.

_26:14_: Like here we're seeing F

_26:16_: of X is a function symbol,

_26:19_: brackets argument is a function,

_26:22_: and that's that's why we're

_26:24_: seeing that the we're big,

_26:26_: that's the same rotation.

_26:27_: So we're seeing the tensors are functions.

_26:30_: They are machines which turn

_26:32_: one thing into another.

_26:34_: In this, in this case,

_26:35_: it's a real number into

_26:37_: another real number.

_26:38_: In this case,

_26:39_: it's a function which turns

_26:40_: a number of 1 forms of

_26:42_: vectors into a real number.

_26:47_: The dot or the dots or?

_26:49_: I think that's similar question here.

_26:51_: That's just a fairly informal notation.

_26:53_: Just to sort of guide the eyes

_26:54_: what to to to to see that this is

_26:57_: a thing which has three arguments.

_27:01_: And these are, remember I said that the

_27:04_: idea of a function as being a machine

_27:07_: which has multiple holes in the top.

_27:09_: You put things in the top, turn the handle,

_27:11_: and now comes something else.

_27:12_: So in this case the that that that

_27:16_: tension is a machine which has

_27:19_: two one form shaped arguments.

_27:21_: One vector shaped argument.

_27:24_: When you put things in,

_27:25_: turn handle oakum sausage.

_27:31_: Ohh yeah, because one one is A1 form

_27:33_: shaped argument and one is a a vector

_27:36_: shape argument and the notation there

_27:37_: and informal notation intend to show

_27:40_: that what we have with they're so empty.

_27:43_: So so we'll we'll pre fills in one

_27:44_: of the one of the arguments and

_27:46_: we'll have two empty arguments.

_27:50_: Is it about to see?

_27:52_: Speaking. The two behind?

_27:55_: Yeah, they're empty.

_27:57_: They're 22 empty slots.

_27:58_: Two empty slots.

_28:03_: Yes you can, because remember

_28:05_: that these attentions as well.

_28:07_: That's just A10 tensor.

_28:09_: That's a 01 tensor.

_28:12_: So I think that with this

_28:14_: update donors comes from so.

_28:18_: A10 tensor. Takes 0 vectors as and one

_28:23_: one form as input and a 01 tensor takes.

_28:29_: 1. One form and 0101 forms and

_28:33_: one-on-one vector as argument,

_28:34_: so, but that's that's.

_28:35_: Hence that that's all the answer to the

_28:38_: upside downness there appeared to be

_28:40_: present in the definition of of tensors.

_28:44_: OK. And the question there, I'll call

_28:47_: you next and then we better move on.

_28:53_: 21 forms and the vector, yeah.

_28:56_: I'm assuming, yeah, that.

_28:58_: That could be anything you can have.

_29:03_: Exactly, yeah, yeah,

_29:03_: that, that, that, that.

_29:05_: That's just an attempt to keep

_29:07_: consistency between in each case.

_29:09_: It's sort of sort of notion of the same.

_29:11_: The same tensor.

_29:12_: Yeah, here's a question.

_29:17_: When we say so.

_29:20_: Why is it not?

_29:26_: So T is A21 tensor because it has two.

_29:33_: Slots two holes which are which are one form

_29:36_: shaped and one which is vector shaped. Um.

_29:42_: Ohh, but S yes, true.

_29:45_: So S with that,

_29:46_: you know you've got machine.

_29:48_: You've got one whole filled in.

_29:51_: You then have two.

_29:53_: A whole open one one form and one vector,

_29:59_: and so S is A-11.

_30:02_: Attention.

_30:03_: Because it S with that whole

_30:06_: field in has one one form

_30:09_: and one vector as argument.

_30:13_: But I should pressure. And.

_30:17_: So. A quick question.

_30:20_: Given an arbitrary tensor,

_30:21_: an arbitrary 11 tensor T.

_30:25_: What's the value of

_30:28_: T2P3A?

_30:31_: How many folks it was just just

_30:33_: quickly would say it was one.

_30:36_: 2. 3. Possible to see?

_30:42_: Haven't given up yet.

_30:45_: Have a chat to your neighbor.

_31:03_: And and why and why you yeah before

_31:06_: going and what are the rank of the

_31:09_: object to T what the rank of that

_31:11_: object and have a think and if S is

_31:15_: A02 sensor an SAB is 5, what is?

_31:19_: Have a chance to each of those questions.

_31:40_: OK, so give an operator one tensor.

_31:46_: Books.

_31:49_: So given an average 11 tensor,

_31:52_: what's the value of T2P3A?

_31:54_: Was it with one? 2. 3. 4.

_32:00_: Can't see. I think we didn't get

_32:03_: everybody putting their hands up,

_32:05_: but I'll, I'll, I'll make a,

_32:06_: I'll make, I'll system that later.

_32:08_: It's it's that. Because.

_32:13_: Since T is linear in each argument.

_32:17_: TF2P. Is 2 to FP.

_32:21_: TO of O P3 is 3 times. TV.

_32:26_: So two and three is 6. That's just

_32:31_: because I was entertained, Sir.

_32:33_: Therefore it's linear in each argument.

_32:37_: What's the rank of the object 2TP

_32:41_: empty space? And everyone shout out.

_32:49_: Yes, it it it takes it.

_32:50_: It takes 1 vector at argument.

_32:52_: Yeah, so it's, so it's it's just A1 form.

_32:56_: And if this is a.

_32:59_: And as you 2 tensor,

_33:01_: an SAB is 5 for a given A&B.

_33:05_: What is a SBA, BBA?

_33:09_: Do arguments live around?

_33:10_: Who said it was five?

_33:13_: Which it was minus 5.

_33:15_: Who's impossible to see?

_33:17_: Who had brand up yet?

_33:20_: Who was it was five.

_33:22_: Who was it minus 5? Who seemed policy?

_33:27_: Is it possible to see?

_33:28_: Because you don't know what the function is.

_33:32_: How many? What the function is?

_33:35_: It's a function.

_33:36_: Which will turn 2 vectors into a number.

_33:40_: But I think, I think the tension

_33:41_: I've told you that much.

_33:42_: It will be the answer when both

_33:44_: things are filled in a number.

_33:46_: But I've said nothing about what.

_33:50_: What this what this machine

_33:51_: does to the two vectors?

_33:53_: Could they ignores the second argument?

_33:56_: OK. So if it is the case that.

_34:01_: SAB and SBA are equal for all A&B.

_34:04_: Then the tension is known as symmetric.

_34:08_: It is the case that SAB is equal

_34:10_: to minus SBA and the tensor

_34:13_: is not anti symmetric.

_34:14_: But most sensors,

_34:15_: it's whatever it is.

_34:19_: So we won't see many tensors which.

_34:22_: Are so important to see we're mostly

_34:25_: dealing with symmetrical metric tensors.

_34:27_: But the distinction is important and

_34:29_: and the fact that this machine does

_34:31_: stuff to effect to it to its arguments

_34:33_: doesn't tell you what what it does.

_34:35_: OK, it. Uncertain about that.

_34:42_: You will have noticed I put these slides up.

_34:47_: Generally, after a little while after the

_34:49_: fact in the lecture notes thing I think

_34:51_: I don't put the slides up beforehand,

_34:52_: but after the the the part is finished,

_34:54_: I put the the size up in the

_34:56_: lecture notes folder on the Moodle.

_34:58_: Ohh, and just because it's absolutely say.

_35:02_: And I. Aspired to get the recordings

_35:06_: audio recordings up before now.

_35:09_: Hasn't happened. I indicate this

_35:11_: week so it should be up and I know.

_35:15_: OK, so I said that. Right.

_35:21_: I last thing about, um,

_35:25_: vectors is the notion of the and there's

_35:28_: also just another another definition.

_35:31_: Examples for me in the moment one last

_35:33_: definition, the idea of the outer product.

_35:36_: This circled cross. So given 2.

_35:41_: Two tensors where of whatever rank you want,

_35:44_: but let's in this example use 2 vectors.

_35:48_: The outer product.

_35:50_: Of these two two vectors this V cross W.

_35:55_: Is it tender?

_35:57_: Which tells you a lot about it.

_35:59_: Which has two arguments.

_36:02_: And and and the action of this outer

_36:04_: product on those two arguments.

_36:06_: Is they actually the first

_36:08_: one on the 1st argument?

_36:10_: Action second one of the second argument.

_36:12_: Those are just real numbers

_36:14_: multiplied together.

_36:16_: OK.

_36:17_: So that that that times there just an

_36:19_: ordinary times that that's the times

_36:20_: that you learned about in primary school,

_36:22_: OK, that's not an exotic times,

_36:24_: that is a company that's an exotic times.

_36:26_: I think other product that's just

_36:27_: the the the real product that's

_36:29_: that's that's the apple themes.

_36:30_: Oranges, right.

_36:32_: I'm not going to mention outer

_36:35_: products again for some weeks,

_36:37_: but this is the place to introduce that term.

_36:44_: So finally some examples.

_36:48_: Umm.

_36:53_: This is an example of a. Victor.

_36:58_: Nice simple vector A2 in in the plane it's

_37:03_: a vector which has a direction and length.

_37:07_: And it has it composed.

_37:09_: The space is spanned by two

_37:13_: basis vectors E1 and E2.

_37:16_: That's E1 and E2. And as you know,

_37:20_: I'm I'm stressing this just to

_37:22_: reassure you that stuff you know

_37:25_: is still true that vector A.

_37:26_: Can be broken down into components.

_37:28_: It's some number,

_37:29_: a real number times one piece of vector plus.

_37:34_: You know, cause they're both vectors.

_37:36_: That's a vector addition plus some

_37:38_: number times the other basis vector.

_37:41_: Nothing exotic there.

_37:42_: Those things are slightly exotic.

_37:44_: Is the placement of these

_37:46_: indexes A1 and A2 OK,

_37:50_: and they are there because these

_37:52_: are the components of a vector.

_37:54_: We'll see why that annotation

_37:57_: matters in a moment.

_37:60_: So that's just a vector.

_38:04_: OK. And we'll also. So there.

_38:16_: And we can write that as a equal to A1.

_38:24_: And that's how you remember I say in

_38:26_: school writing a column vector. OK,

_38:29_: we can also imagine in this context A1 form.

_38:39_: Being a rule vector.

_38:43_: And you you feel like familiar with the idea

_38:47_: of one form of vector turning of all vectors

_38:50_: called vectors and and all that stuff.

_38:52_: I'm in this context distinguishing

_38:54_: vectors and one forms. They can.

_38:57_: These two things are exist

_38:58_: in different spaces.

_38:58_: You have to, you have to.

_38:59_: You have to do something to turn

_39:01_: one of those into one of those.

_39:03_: We can contract these, we can track the

_39:07_: straight forwardly and the contraction.

_39:09_: Of P. And E. You would be fine.

_39:15_: That being P1U2. One trick I do apologize.

_39:21_: Ah, you NP21. I'm going to be,

_39:27_: yes, so apologies for that.

_39:31_: The components of 1 forms are are labeled

_39:35_: with the A1A2 and that as you know

_39:40_: will is going to be P1A1 plus P2. 2.

_39:46_: And since these components are numbers.

_39:52_: And that should ordinary,

_39:53_: you know, simple numerical.

_39:56_: That is a number. In other words,

_39:58_: the contraction of this one form

_39:60_: and this vector is a number.

_40:03_: And we've are our choice here.

_40:07_: Is our choice to define the

_40:10_: contraction in this way?

_40:12_: Now as you can see here.

_40:16_: We ask, OK, that's P applied to a.

_40:21_: What's a applied to P?

_40:23_: Well, we can't really use our experience

_40:26_: of of of row vectors and column

_40:28_: vectors to answer that question,

_40:30_: so we're just going to say ohh.

_40:31_: But of course we don't have to

_40:33_: because we know it's. HP print A. OK.

_40:37_: Through the contract, so we define. That.

_40:46_: To be the thing that we've just defined here,

_40:48_: the contraction, OK, so that's an

_40:51_: example of all the things that were

_40:54_: mentioned up to this point. And. This is.

_40:60_: These are both elements of vector space.

_41:02_: You can add two column vectors together,

_41:04_: a column vector, you can add two

_41:05_: row vectors together or row vector,

_41:07_: etcetera, etcetera.

_41:07_: All the other I can supply and

_41:11_: we have defined a contraction.

_41:14_: And so on.

_41:16_: And we can also define hierarchy question.

_41:24_: In this case, well, the.

_41:30_: That is because it when you

_41:32_: learn about about vectors,

_41:33_: I convectors vector stuff, it's.

_41:35_: It's cool you are told are all vector

_41:38_: just a convection outside. Yeah, yeah,

_41:41_: your tools are very straightforward.

_41:42_: We are going from one to the other.

_41:44_: And in this case we could do that,

_41:47_: but we're keeping them as separate things.

_41:49_: So we have, so I haven't said how you,

_41:51_: how you get, how you get.

_41:56_: So so so this.

_42:03_: I have said nothing about how you get that.

_42:05_: From that, OK,

_42:06_: that's just completely different thing.

_42:08_: OK, because I have, I've chosen not to.

_42:11_: OK, so the difference is that they are just.

_42:16_: The difference is we haven't is is that

_42:18_: we have abstained from the thing that

_42:20_: we learn about in school, which is,

_42:22_: which is how do you go from one to the other?

_42:23_: So I I wish we implicitly said

_42:25_: that the two things are,

_42:27_: are are the same sort of thing.

_42:28_: So we are we are holding you

_42:30_: back from that and saying these

_42:32_: separate things and and.

_42:33_: But but I I've linked them.

_42:35_: By this process of defining the

_42:38_: contraction between them and.

_42:39_: I can also define.

_42:44_: Our attention. Um.

_42:54_: Which in this example is square matrix.

_42:58_: And now I can apply a tensor.

_43:02_: To a vector. And get.

_43:12_: The.

_43:15_: Et cetera. I mean,

_43:16_: in usual things, super.

_43:17_: So the point here is that this

_43:20_: tensor is definable in this way.

_43:23_: It's a, a thing which takes a.

_43:26_: A single one form.

_43:28_: Of course could be we could left

_43:31_: apply a road matrix to this and

_43:33_: we can right apply column matrix to this.

_43:36_: And if we? To.

_43:43_: P1P2.

_43:47_: T1A2, we could let's apply

_43:49_: the row vector, right,

_43:50_: apply the column vector and

_43:52_: get a number. So this works.

_43:54_: We would that that is equal to.

_43:55_: Is A is a thing.

_43:59_: And on your line, if we partially apply it,

_44:01_: if if we give this tensor,

_44:03_: there's 11 tensor, just one thing,

_44:06_: what we get is another vector.

_44:08_: Which is a is a I think, which takes

_44:10_: one form of argument and gives a number.

_44:13_: And similarly,

_44:13_: if we're left applied at one form,

_44:15_: we'd get also get another

_44:18_: one form which is a I think,

_44:19_: which is a vector vector as argument, so.

_44:22_: So. The thing this this stuff real vectors,

_44:26_: column vectors,

_44:27_: matrices you're very familiar with.

_44:29_: It has been an example of of of

_44:32_: three different types of tensors,

_44:34_: which you didn't realize when

_44:35_: you learn about the school.

_44:37_: OK, so these ideas are not

_44:40_: the the terminology is exotic.

_44:43_: And we're going to use the

_44:44_: power of that terminology,

_44:46_: and the power of those debased definitions,

_44:48_: fairly freely hereafter.

_44:50_: But the things are examples of them.

_44:53_: Are not exotic.

_44:54_: There are things you you you you,

_44:56_: you you are familiar with.

_44:58_: And um.

_45:01_: Another point here is that I'm going to.

_45:04_: I want you to end up with.

_45:08_: And this will matter a lot.

_45:10_: In the next part,

_45:11_: we're going to hold on to the idea

_45:14_: of a vector as being two things.

_45:16_: I would hold on to the idea.

_45:17_: There's a pointy thing, OK?

_45:19_: Cause that's a good thing to hold on to you.

_45:21_: That's a clear idea in your head,

_45:23_: and you could hold on to that as a as a

_45:26_: as a notion to hold on to that thought.

_45:28_: I also want to stress that you can.

_45:34_: That you must be also be able to think of um.

_45:39_: I've lost an example of it.

_45:44_: To think of vectors as functions.

_45:49_: Think of a vector as a function

_45:51_: which takes one form as argument.

_45:53_: And turned a number. So yeah,

_45:55_: you have these two pictures in your head,

_45:56_: swapping in and out as we go on,

_45:59_: because they're both right. OK.

_46:03_: OK.

_46:07_: And I and I I I I jumped ahead of myself.

_46:10_: So so this will have to apply A1

_46:12_: form to to to to to our tensor.

_46:15_: We get a I think,

_46:17_: which is another victory.

_46:18_: So it's another another one form.

_46:21_: In this example. OK.

_46:28_: And.

_46:32_: In general, we'll also talk about fields.

_46:37_: In this context, a field is a.

_46:43_: This is under any mathematicians in the

_46:45_: audience you might want to close your ears.

_46:47_: Now I feel is a thing which takes

_46:50_: different values over the space.

_46:52_: So a scalar field is something

_46:54_: which takes different values in say,

_46:57_: the 3D space of this room.

_46:58_: So the pressure in the room

_46:60_: is a 3D is a scalar field.

_47:02_: At different points of the room,

_47:03_: the pressure, air pressure is different.

_47:04_: It's just a number.

_47:06_: On Vector field is something

_47:08_: which different vector values at

_47:10_: different points in the space.

_47:12_: So the magnetic field of the

_47:14_: earth is a vector field,

_47:15_: the different points in the space.

_47:17_: In the Space 3 space it takes different

_47:19_: values and each at each of those

_47:20_: points it it has a a vector value.

_47:22_: It has a middle of the earth has

_47:24_: a a direction and a strength.

_47:27_: OK, that's what field means in this context.

_47:31_: Um.

_47:34_: And we can then go on to

_47:37_: visualize these things. Were we?

_47:42_: Victors.

_47:46_: There's nothing complicated

_47:46_: about visualizing a vector.

_47:47_: It's a pointy thing.

_47:49_: OK, but we're going to also,

_47:50_: but we're going to visualize.

_47:53_: One forms.

_47:57_: I'm saying this because you

_47:58_: know to help you, to help you.

_48:01_: A need to thought rather it's a

_48:04_: movement actually calculated with

_48:06_: good would visualize 1 forms as.

_48:08_: Well, in 2D lines they have. A direction.

_48:14_: That this has a direction and length.

_48:17_: This set of planes has a direction.

_48:21_: And it has a sort of pitch. OK.

_48:25_: So there there's a magnitude to this, so.

_48:31_: That's in this in this picture.

_48:34_: That's a one form which is the same

_48:36_: direction but has a higher magnitude.

_48:38_: They're closer together.

_48:40_: Why closer together?

_48:41_: Because are we in this picture?

_48:44_: So this is not another example of of danger.

_48:46_: We we can imagine the contraction

_48:48_: of a vector in one form in this

_48:51_: picture is by laying the. Vector.

_48:54_: Across this one forms and saying.

_48:58_: This vector so goes crosses 2 lines.

_49:00_: It goes from one cross cross.

_49:03_: And it gives an answer of two

_49:04_: so that the contraction of that

_49:05_: one form and that vector is 2.

_49:07_: The direction of this one form 11234 is 4,

_49:10_: so that the contraction of that the same

_49:12_: vector with with with with the that the

_49:15_: larger one form gives a different number.

_49:18_: Tell the direction of one

_49:19_: form without using the right,

_49:20_: because it could be, it could be, yes.

_49:24_: That's not a very mathematically

_49:25_: very precise picture,

_49:27_: so I think that I think that.

_49:30_: We think of vectors is useful to have

_49:32_: this notion of a point of of an arrow.

_49:34_: You're just trying to get

_49:36_: your things in your head.

_49:38_: When I took one form useful to

_49:39_: have a a picture like that,

_49:41_: maybe you could color one side

_49:42_: of them once, one side the other.

_49:49_: Somewhere in between 21, yes, so.

_49:52_: Yeah, so the field, obviously,

_49:54_: I mean, I think we we don't want to

_49:56_: overstress the precision of this,

_49:58_: that, that, that.

_49:59_: That's just a picture. That's useful.

_50:01_: And also it means that things like.

_50:04_: Like this big sense.

_50:06_: So there's a A1 form field time stop.

_50:10_: There's one form field.

_50:12_: And like like the gradient lines on the map.

_50:16_: So so the gradient it turns

_50:19_: out to be modeled as A1 form.

_50:22_: In this case, the, you know, up uphill.

_50:24_: Has a direction,

_50:26_: and the closer these lines are together,

_50:27_: the steeper the slope is and if you.

_50:33_: Move, move like that.

_50:35_: Or like that, or like that.

_50:38_: In each case, you've crossed the same number

_50:40_: of grid lines or of of contour lines, sorry.

_50:42_: And so you've you've risen the same

_50:44_: amount of, you've claimed the same distance.

_50:47_: So in that sense the contraction

_50:50_: of those three different vectors.

_50:52_: With the one form field.

_50:54_: There's the same number.

_50:55_: So when I talk about of of vectors,

_50:58_: think of pointer pointy things.

_50:59_: We're talking of A1 form fields.

_51:01_: Think of the gradient lines of the gradient

_51:03_: lines on the country lines on a map.

_51:05_: And with that picture,

_51:06_: I'll let you go.

_51:07_: We should.

_51:08_: I think we've we've done not too

_51:10_: badly in getting just about right

_51:12_: just through this this part.

_51:13_: I'll see you again on Friday, I think.

_51:15_: Not in this room.

_51:17_: OK, all the other elected in

_51:19_: this room and other words,