Hello and welcome to Lecture 2 of the Greg.
This is lecture 2 out of 11.
I trust you will all have made
your apprise yourself of a copy
of the part of the Part 2 notes.
The plan is to get through these this
part on vector sensors functions in.
A bit under 3 lectures may
or may not manage that,
but that that that that's the plan.
There's not a huge amount
of news of huge new stuff.
Well, there is.
Most of us have in this you probably
have seen before in some form.
Probably not in this form,
and it might be an alien,
a bit alien to you in the
way we're approaching it.
But the aim of this chapter is to
get everyone on the same page really.
So if you've seen him this stuff before,
good for you. If not,
make sure that that you you want to keep up.
As I mentioned last time,
those of you with a fairly who's
whose background includes a fair
amount of of pure ish maths might
well find this deplorably informal.
Those of you who's background hasn't
included a parameter punish maths
might find this rather bracing.
OK, so I don't think it's
over in between your, your,
your your your expectations.
But as I also mentioned last time,
this is the first of the two middle parts,
which is mostly mathematical,
allowing us to get back onto
the physics in the last chapter.
Because the payoff of this maths is merely
maths, the payoff is to allow us to.
Talk about gravity in the physics
of gravity in a very powerful way.
OK. Any questions about that?
Or speculations,
or thoughts or feelings you wish to share.
OK then let's proceed.
So the as before, there are aims,
objectives, high level,
fairly high level things.
The aims are not very there
aren't very many aims to this,
to this section,
because this is a fairly mechanical
section in the sense it's about technique.
There are stuff you're going to
be learning here about how to do
certain particular maths so that so
you will use all this stuff later,
but there there aren't great physical
insights awaiting you in the next 3 lectures.
But quite a lot of objectives,
so there's quite a lot of things you'll be
able to do after this this part I and as I.
As I said last time,
the distinctively aims
objectives is aimed at the point
objectives of the party tricks.
The things that are fairly
straightforwardly accessible is the point.
Thank you.
This first section on linear algebra will
be some sort of revision to most of you.
OK, you will have seen a lot of these things
before at earlier stages in your education.
If not, go back to your notes from
previous years and make sure you
are fully up to speed with them.
One thing that I'm going to,
I think otherwise isn't on this slide,
but which is going to come
up again and again.
Is the notion of linearity and linearity in
this context means a quite specific thing?
Linearity in this context means.
A function. F of XC it can be any
function you think of, it's linear. If.
If and only if F of 2X.
If you could two up F of
a * X equal to AF of X.
That's what linearity means in this context.
It doesn't mean a straight line graph,
although that's also referred
to as a linear graph.
It means if you if you multiply
the argument by a by a scaler,
the result is multiple sclerosis.
But that's not true of every function,
for example.
If X = X, ^2 is not linear.
Because F of of 2X is equal to two
X squared which is equal to four X
squared which is not equal to two X ^2.
But you'll be OK.
So I'm belaboring that point because that.
This this term linear has possibly slightly
different conditions in different areas,
but returning to these points
here about vector spaces.
I will come back again again to
this notion of a vector space.
And a vector space. Is.
Anything.
Which satisfies these five properties.
I'm going to go see a little
bit more about those,
but not too much more,
OK?
The vectors that you're
familiar with pointy things,
the things you learned about in school.
Those are an example of the
elements of a vector space,
and they're they're the, hence the name.
They're the approach type
example of a vector space.
You can add vector together.
This vector plus that vector
gives another vector.
OK, you know that, right?
So that's a property.
There exists an element, a zero at 0,
and you add that zero to anything,
you get the the thing you
started with back backwards.
So so a + 0 is a.
There exists an identity element.
Inverse for every vector,
there's a vector that points
in the opposite direction.
We add up to the identity.
Multiplication and in vector you can
double it and you get another vector
which is the same direction and
and twice as long. And there is a.
If you multiply that by one.
You get this thing better.
Nothing surprising here.
Is the tribute of that that that
multiplication by rules is retributive.
So there's nothing there that's
surprising to you, I trust.
Good. But I'm, I'm, I'm,
I'm listing them because these
are very general properties.
And anything that refers that
satisfy those properties is a
vector space in the sense we're
going to be talking about. OK.
So. Hold on to that thought.
I'm going to also assume.
That you know about these
various things here and.
London Stadium dimension
the dimensionality of space.
Idea basis set spans of space.
Existence, component, blah blah blah.
The Chronicle just symbol and
you may have seen before the
Chronicle death symbol is just a.
I'll be good to stop
switching the back and forth,
but the coronary death symbol is.
Dot IG is defined as being one.
If I = g and 0 if I is not equal to G.
What's the definition of the chronicle delta?
I'm positive, definitely.
This just means that the.
In a product.
Of of of two of of two things
is is not negative OK?
And some stuff about matrix algebra.
That the matrices have an inverse,
a trace, a determinant and so on.
Have you you you you have come
across these words before.
Yeah, OK. And anyone who's feeling
uncertain at this point, you know,
quick trip to your to your last
year's notes would be a good idea.
And I'm not gonna depend on an intimate
knowledge of these things or having
them at the front of your head,
but I am going to assume that you can
look those up and remind yourself
of the details when necessary.
Are there any questions with that?
OK.
So.
I'm so matrix algebra.
You know what means algebra good.
I'm sure you know all the Metra. Umm.
OK, quick question.
Consider yourself square and by emissaries.
Is that a vector space?
Kind of Lucy, yes.
And those who say no.
Excellent, right?
It is a vector space.
You don't think of square
matrices as vectors.
But the squeamish sees
satisfy all those axioms.
You can multiply it by you
a scalar times a matrix.
A square matrix is a square matrix.
You can add 2 square matrices
together and get a square
matrix of the same rank.
They all zeros matrix.
That they all zeros matrix can
be added to any matrix to get
the same matrix started off
with their existing identity.
There's a negative of every matrix.
And matrix and.
The addition, the mosque addition
are distributive like that,
therefore therefore and that
therefore it's important.
Therefore the set of square and
by matrices is a vector space.
Each for each end, so two by
two matrices and three by three
matrices are not in the same space.
You can add 2/2.
2 minutes to get three matrix.
They're in different spaces,
but they're each separately vector spaces.
OK. Make sure you're
comfortable with that notion.
OK.
Now I'm going to talk about tensors,
vectors and one forms.
You probably have heard about
tensors at some point before,
but you've never really had to you,
and you may have had to to
wrangle with them a bit,
but certainly I remember to.
Tensors being a slightly exotic
thing that was happened in bits of of
continuum mechanics and I think some
slightly exotic classical mechanics.
It was it was a thing that was clearly
quite powerful, slightly magic.
Let's not think about it too much, OK?
That's fine.
Now, if we're tensors, really matter.
So here is a an important use of tensors,
and so we're gonna we're gonna be
talking about them a lot from now on.
Tensors. Are um?
Before putting up the next slide,
I'm going to see the that tensors are. And.
See? So we're going to label.
I don't want to put next slide
up quite yet because I want
to have something up first.
Ohh yeah, and I'm going to introduce
tension in a fairly axiomatic way,
in a fairly mathematical way.
I'm not going to give you
examples and then say, oh,
and now we call these tensors
because see these are the the
definitions of tensors because I I
think that although that can be,
it can feel a bit sort of bit
mathematical as an approach,
it does emphasize that tensions
are fundamentally rather simple
things and that simply saying.
Attention, each rank of tensor rank in
the moment is an element of a vector space.
By just saying that, I've told you
quite a lot about what tensors are.
OK, so a tensor.
And what's your tensor T?
Will have a rank.
Well,
I'll see what this means in a minute.
MN so that we'll have a a rank
MNMNR moment, and each M instead
of MSN tensors is a vector space.
Well, that means you have two
tensors of that rank together.
You get another one of that rank you can,
there's a theory element, and so on.
And I'm going to give
certain of these tensors.
Well, two you 2 two sets of
these tensors a special name.
So now always get these nervous
in these three wrong around
the set of 10 tensors.
We're going to call.
Vector. And the set of 01 tensors.
We're going to call 1 forms now.
This is a slight,
slightly unfortunate naming collision.
And these are all.
These are both elements of vector space.
But it is usual to call this this
set of tensors, to call them vectors.
So in future, gonna talk about vector space.
I mean or or an element of a vector space.
I mean the very general thing that
obeys the axioms of of vector space.
We're talking about vectors.
I mean one of these.
OK, so the point is that there are.
For each M&amp;N which are greater or equal to 0.
There is a set of objects called tensors.
Which satisfy that each of those sets
satisfy the the acting of a vector space.
These are two two examples of that set.
With special names.
With those definitions.
An MN tensor is a function.
We did linear in each argument.
In the sense of which I mentioned before,
which takes M1 forms and N vectors as
arguments and map them to a real number.
OK, question. What does one form mean?
Right and right, examples.
So I'll come to examples shortly because
I'm going to stick with the abstract
in your terminology to begin with,
and then I'm going to bring an example,
some examples question.
N vectors and then M vectors and then one
for the right 10 vector, because ohh,
that's a good point, yes, yes.
That does seem to be the one that
does seem to be the wrong way around.
It's not the wrong way round,
but it does seem to be the wrong way round.
Enough that it will cause confusion, right?
And This is why I checked which we around.
I I I wrote this because
I always get it wrong.
OK, it doesn't actually matter hugely much,
but well spotted the duty
to be the wrong way around.
OK, but I'll come back to that mode.
OK, so the point is, it's a function.
Now a function is a machine which
turns one thing into another.
This function.
Here. Is a machine which takes a number.
And give you back it's, it's square,
OK, so it's a number which maps.
Real line. To the real line.
OK, so that's a good.
That's a good way of it.
Sounds like a slightly baby way
of thinking about functions.
There's really good way
of thinking functions.
Functions are machines which take
one or more things of 1 type and
turn them into another and tensors.
Take. One forms and vectors as input.
You think about machine has
holes in the top with which are
one form shape or vector shape.
Turn the handle and outcomes a number,
not anything else.
A number,
something something in R and the real line.
OK, and it's linear.
So you you put two of the of
these things and the number is
twice what you started with.
OK. Any questions?
So those are really good questions.
Any other questions about that so
but don't really good questions,
the answer to which is coming soon.
Any other questions? OK. Um.
So I think we want some pictures here.
So here are some some.
Some tensors, no. There's a slight
informal notation I'm going to
be using here. For attention.
And what what example we're using?
And I'm going to. See?
This to be consistent with this.
Thank you. Alright, ohh by the way,
I'm going to write vectors.
Typically with an overbar.
And one forms. With the children. OK.
So I'll be feeling consistent with that.
No, absolutely consistent but
fairly consistent with that.
OK, So what this this is going to
be rather suggest from station.
So this tends to T is a.
21 tensor or one? Yes, at A21 tensor.
Which take it which means it takes.
It has three arguments 2 one
form shaped arguments and
one vector shaped argument.
And it turns that into.
A A real number.
So if we give that tensor.
3 arguments.
The answer is a real number.
OK, that's that's what that
that's what I mean when I when
you talk about definition.
Yeah. So we're keeping this
abstract at the moment.
Examples are main mode.
But you can also partially apply things.
So there this this here
attention or questions. Confused?
Like when you put the? Alright.
Yes, yes. It's just this intends
to to suggest a sort of empty slot.
So as a whole a machine with with
those three holes in the top.
And that's not a,
that's not a formal notation,
that's just to guide the eyes that were.
So that's that. That's a A21 tensor.
21 form shape tends to one form shaped holes
and one that shaped hole. I put in. 114.
What I have left is also a tensor.
It it I think which had one one form
shaped hole and one vector shaped
hole A1 form argument and a vector
argument that is S is A-11 tensor.
So by partially applying.
By partially filling in.
The arguments of the tension.
We can turn one type of vector,
a true 1 tensor,
into 111 tensor in this example.
We can do that more than once.
That he he will fill in the one form
shaped one, the one form argument,
one of the one form arguments,
the vector argument.
And we have something which has our, a, our.
A single one form argument.
You know a single just single
one form argument is A10 tensor.
A vector. So that's so we
could write this as a vector.
OK. Um, so we could give
names to these other things,
you know, but we don't.
There's there's no need to give
them the other things because
the the vectors in the one forms
are the things that we sort of
have to give names to in order
to create the definition you
saw in the last slide.
Thank you.
Um.
And similarly if in this example we.
Said talk about tea.
Omega. Sigma.
And and and and and and feel to
fill in the vector ship argument.
That would be a thing
which has 01 form argument.
Zero will zero open one form
argument and one vector argument.
In other words that is A1 form.
Because all the one for means is.
It is something which has a.
AS01 form argument and one vector
shaped argument such as that.
OK. So, and I say we're keeping
this abstract at the moment.
Examples come to the moment.
The last point relevant here is.
In one form. As you recall, is our.
01 tensor. It takes a single vector
as argument and gives a number.
A vector. Takes a single one form
of argument and gives a number.
Now there is nothing to say
that those are the same number.
These are just functions.
They're functions which take this
thing and turn it, which is a number.
They don't have to be equal.
We are always going to
assume that they are equal.
OK, you can talk about this sort of this
sort of stuff without that assumption.
It makes things harder.
We don't need extra hardness, so in this
context this will always be the case.
We'll always constrain.
And so this this acts as a constraint
on the functions that we allow here.
So, so these functions,
these these these one form,
these functions are not arbitrary
in that sense.
We'll all they'll always have that
reciprocal property and well,
sometimes write that with
these angle brackets,
it just means either of those things
and they're really use angle brackets.
Sometimes it's just emphasise
the symmetry of this.
And that's true for all one
form and vector arguments in GR.
OK, that's a mathematician
would not have that bit.
If I'm working that,
that,
that's that's always gonna be the case.
How we're doing so far?
Makes a lot more sense checking down
round when you go through your notes
afterwards it it will be illuminating.
This this is.
Quick mathematical defense that that
we are laying on the definitions here.
I think it's worth stressing that.
This definition is doing a lot of work.
All the things that I've said.
Your last 10 minutes so are just
immediate consequences of this.
Attention to function is linear. Which
takes these arguments about a real number.
The only thing I've said there isn't.
An immediate consequence of that is this
last point here, which is an extra. OK.
And all that depends on the statement that
the tensors are illness of vector space.
So you get all those other properties
you're being scooped up being provided
for free in a sense by that definition.
OK. So at this point,
we don't mean you do not have a picture.
I don't expect to have a picture
of what tensors are yet.
But you really know a lot about them.
OK. And that's that,
that's the the I think the the the sort of.
When you come back to think through it again,
that's with the clarity of a fully
activated approach comes from.
You already know a lot about them,
even though you have a picture.
OK.
The so these definitions
are doing a lot of work.
Key. Um and?
Umm. We have, and I'll mention just
a bit of terminology. This is all.
I will at this point freely,
freely refer to contractions
between vectors in one forms.
And I say they were contraction.
I mean that.
I mean applying of extra one form
that's contracting the vector P
the the vector A and the one form
one form P so you contraction.
That's what I mean.
There's more we can say about that,
but we're not going to. Ohh question.
In the example you're given for the.
Yeah, yeah, this is.
So the Q, is that the one form or is
that it's a one form with child over?
But that's also considered
as an argument for M.
So we considered as MN, so that's M&amp;N.
And if and if I'm if.
I'm not sure that's a A21 tensor.
It's telling us that there are two
wonderful arguments, and one vector.
Is that what you meant?
OK, yeah, so that's that's an
example of a true 1 tensor, and.
And take your tea in each of both
cases is A21 tensor by partial
application which we turn into
A-11 tensor and A10 tensor. OK?
Sorry, what are the thoughts?
Which bracket? And in
practice. Throw some dots.
Ohh the brackets here.
And that alright, that I an important
question and that's just picking
up that that we've said T is a is a
is a is a a function. So just like.
Like here we're seeing F
of X is a function symbol,
brackets argument is a function,
and that's that's why we're
seeing that the we're big,
that's the same rotation.
So we're seeing the tensors are functions.
They are machines which turn
one thing into another.
In this, in this case,
it's a real number into
another real number.
In this case,
it's a function which turns
a number of 1 forms of
vectors into a real number.
The dot or the dots or?
I think that's similar question here.
That's just a fairly informal notation.
Just to sort of guide the eyes
what to to to to see that this is
a thing which has three arguments.
And these are, remember I said that the
idea of a function as being a machine
which has multiple holes in the top.
You put things in the top, turn the handle,
and now comes something else.
So in this case the that that that
tension is a machine which has
two one form shaped arguments.
One vector shaped argument.
When you put things in,
turn handle oakum sausage.
Ohh yeah, because one one is A1 form
shaped argument and one is a a vector
shape argument and the notation there
and informal notation intend to show
that what we have with they're so empty.
So so we'll we'll pre fills in one
of the one of the arguments and
we'll have two empty arguments.
Is it about to see?
Speaking. The two behind?
Yeah, they're empty.
They're 22 empty slots.
Two empty slots.
Yes you can, because remember
that these attentions as well.
That's just A10 tensor.
That's a 01 tensor.
So I think that with this
update donors comes from so.
A10 tensor. Takes 0 vectors as and one
one form as input and a 01 tensor takes.
1. One form and 0101 forms and
one-on-one vector as argument,
so, but that's that's.
Hence that that's all the answer to the
upside downness there appeared to be
present in the definition of of tensors.
OK. And the question there, I'll call
you next and then we better move on.
21 forms and the vector, yeah.
I'm assuming, yeah, that.
That could be anything you can have.
Exactly, yeah, yeah,
that, that, that, that.
That's just an attempt to keep
consistency between in each case.
It's sort of sort of notion of the same.
The same tensor.
Yeah, here's a question.
When we say so.
Why is it not?
So T is A21 tensor because it has two.
Slots two holes which are which are one form
shaped and one which is vector shaped. Um.
Ohh, but S yes, true.
So S with that,
you know you've got machine.
You've got one whole filled in.
You then have two.
A whole open one one form and one vector,
and so S is A-11.
Attention.
Because it S with that whole
field in has one one form
and one vector as argument.
But I should pressure. And.
So. A quick question.
Given an arbitrary tensor,
an arbitrary 11 tensor T.
What's the value of
T2P3A?
How many folks it was just just
quickly would say it was one.
2. 3. Possible to see?
Haven't given up yet.
Have a chat to your neighbor.
And and why and why you yeah before
going and what are the rank of the
object to T what the rank of that
object and have a think and if S is
A02 sensor an SAB is 5, what is?
Have a chance to each of those questions.
OK, so give an operator one tensor.
Books.
So given an average 11 tensor,
what's the value of T2P3A?
Was it with one? 2. 3. 4.
Can't see. I think we didn't get
everybody putting their hands up,
but I'll, I'll, I'll make a,
I'll make, I'll system that later.
It's it's that. Because.
Since T is linear in each argument.
TF2P. Is 2 to FP.
TO of O P3 is 3 times. TV.
So two and three is 6. That's just
because I was entertained, Sir.
Therefore it's linear in each argument.
What's the rank of the object 2TP
empty space? And everyone shout out.
Yes, it it it takes it.
It takes 1 vector at argument.
Yeah, so it's, so it's it's just A1 form.
And if this is a.
And as you 2 tensor,
an SAB is 5 for a given A&amp;B.
What is a SBA, BBA?
Do arguments live around?
Who said it was five?
Which it was minus 5.
Who's impossible to see?
Who had brand up yet?
Who was it was five.
Who was it minus 5? Who seemed policy?
Is it possible to see?
Because you don't know what the function is.
How many? What the function is?
It's a function.
Which will turn 2 vectors into a number.
But I think, I think the tension
I've told you that much.
It will be the answer when both
things are filled in a number.
But I've said nothing about what.
What this what this machine
does to the two vectors?
Could they ignores the second argument?
OK. So if it is the case that.
SAB and SBA are equal for all A&amp;B.
Then the tension is known as symmetric.
It is the case that SAB is equal
to minus SBA and the tensor
is not anti symmetric.
But most sensors,
it's whatever it is.
So we won't see many tensors which.
Are so important to see we're mostly
dealing with symmetrical metric tensors.
But the distinction is important and
and the fact that this machine does
stuff to effect to it to its arguments
doesn't tell you what what it does.
OK, it. Uncertain about that.
You will have noticed I put these slides up.
Generally, after a little while after the
fact in the lecture notes thing I think
I don't put the slides up beforehand,
but after the the the part is finished,
I put the the size up in the
lecture notes folder on the Moodle.
Ohh, and just because it's absolutely say.
And I. Aspired to get the recordings
audio recordings up before now.
Hasn't happened. I indicate this
week so it should be up and I know.
OK, so I said that. Right.
I last thing about, um,
vectors is the notion of the and there's
also just another another definition.
Examples for me in the moment one last
definition, the idea of the outer product.
This circled cross. So given 2.
Two tensors where of whatever rank you want,
but let's in this example use 2 vectors.
The outer product.
Of these two two vectors this V cross W.
Is it tender?
Which tells you a lot about it.
Which has two arguments.
And and and the action of this outer
product on those two arguments.
Is they actually the first
one on the 1st argument?
Action second one of the second argument.
Those are just real numbers
multiplied together.
OK.
So that that that times there just an
ordinary times that that's the times
that you learned about in primary school,
OK, that's not an exotic times,
that is a company that's an exotic times.
I think other product that's just
the the the real product that's
that's that's the apple themes.
Oranges, right.
I'm not going to mention outer
products again for some weeks,
but this is the place to introduce that term.
So finally some examples.
Umm.
This is an example of a. Victor.
Nice simple vector A2 in in the plane it's
a vector which has a direction and length.
And it has it composed.
The space is spanned by two
basis vectors E1 and E2.
That's E1 and E2. And as you know,
I'm I'm stressing this just to
reassure you that stuff you know
is still true that vector A.
Can be broken down into components.
It's some number,
a real number times one piece of vector plus.
You know, cause they're both vectors.
That's a vector addition plus some
number times the other basis vector.
Nothing exotic there.
Those things are slightly exotic.
Is the placement of these
indexes A1 and A2 OK,
and they are there because these
are the components of a vector.
We'll see why that annotation
matters in a moment.
So that's just a vector.
OK. And we'll also. So there.
And we can write that as a equal to A1.
And that's how you remember I say in
school writing a column vector. OK,
we can also imagine in this context A1 form.
Being a rule vector.
And you you feel like familiar with the idea
of one form of vector turning of all vectors
called vectors and and all that stuff.
I'm in this context distinguishing
vectors and one forms. They can.
These two things are exist
in different spaces.
You have to, you have to.
You have to do something to turn
one of those into one of those.
We can contract these, we can track the
straight forwardly and the contraction.
Of P. And E. You would be fine.
That being P1U2. One trick I do apologize.
Ah, you NP21. I'm going to be,
yes, so apologies for that.
The components of 1 forms are are labeled
with the A1A2 and that as you know
will is going to be P1A1 plus P2. 2.
And since these components are numbers.
And that should ordinary,
you know, simple numerical.
That is a number. In other words,
the contraction of this one form
and this vector is a number.
And we've are our choice here.
Is our choice to define the
contraction in this way?
Now as you can see here.
We ask, OK, that's P applied to a.
What's a applied to P?
Well, we can't really use our experience
of of of row vectors and column
vectors to answer that question,
so we're just going to say ohh.
But of course we don't have to
because we know it's. HP print A. OK.
Through the contract, so we define. That.
To be the thing that we've just defined here,
the contraction, OK, so that's an
example of all the things that were
mentioned up to this point. And. This is.
These are both elements of vector space.
You can add two column vectors together,
a column vector, you can add two
row vectors together or row vector,
etcetera, etcetera.
All the other I can supply and
we have defined a contraction.
And so on.
And we can also define hierarchy question.
In this case, well, the.
That is because it when you
learn about about vectors,
I convectors vector stuff, it's.
It's cool you are told are all vector
just a convection outside. Yeah, yeah,
your tools are very straightforward.
We are going from one to the other.
And in this case we could do that,
but we're keeping them as separate things.
So we have, so I haven't said how you,
how you get, how you get.
So so so this.
I have said nothing about how you get that.
From that, OK,
that's just completely different thing.
OK, because I have, I've chosen not to.
OK, so the difference is that they are just.
The difference is we haven't is is that
we have abstained from the thing that
we learn about in school, which is,
which is how do you go from one to the other?
So I I wish we implicitly said
that the two things are,
are are the same sort of thing.
So we are we are holding you
back from that and saying these
separate things and and.
But but I I've linked them.
By this process of defining the
contraction between them and.
I can also define.
Our attention. Um.
Which in this example is square matrix.
And now I can apply a tensor.
To a vector. And get.
The.
Et cetera. I mean,
in usual things, super.
So the point here is that this
tensor is definable in this way.
It's a, a thing which takes a.
A single one form.
Of course could be we could left
apply a road matrix to this and
we can right apply column matrix to this.
And if we? To.
P1P2.
T1A2, we could let's apply
the row vector, right,
apply the column vector and
get a number. So this works.
We would that that is equal to.
Is A is a thing.
And on your line, if we partially apply it,
if if we give this tensor,
there's 11 tensor, just one thing,
what we get is another vector.
Which is a is a I think, which takes
one form of argument and gives a number.
And similarly,
if we're left applied at one form,
we'd get also get another
one form which is a I think,
which is a vector vector as argument, so.
So. The thing this this stuff real vectors,
column vectors,
matrices you're very familiar with.
It has been an example of of of
three different types of tensors,
which you didn't realize when
you learn about the school.
OK, so these ideas are not
the the terminology is exotic.
And we're going to use the
power of that terminology,
and the power of those debased definitions,
fairly freely hereafter.
But the things are examples of them.
Are not exotic.
There are things you you you you,
you you are familiar with.
And um.
Another point here is that I'm going to.
I want you to end up with.
And this will matter a lot.
In the next part,
we're going to hold on to the idea
of a vector as being two things.
I would hold on to the idea.
There's a pointy thing, OK?
Cause that's a good thing to hold on to you.
That's a clear idea in your head,
and you could hold on to that as a as a
as a notion to hold on to that thought.
I also want to stress that you can.
That you must be also be able to think of um.
I've lost an example of it.
To think of vectors as functions.
Think of a vector as a function
which takes one form as argument.
And turned a number. So yeah,
you have these two pictures in your head,
swapping in and out as we go on,
because they're both right. OK.
OK.
And I and I I I I jumped ahead of myself.
So so this will have to apply A1
form to to to to to our tensor.
We get a I think,
which is another victory.
So it's another another one form.
In this example. OK.
And.
In general, we'll also talk about fields.
In this context, a field is a.
This is under any mathematicians in the
audience you might want to close your ears.
Now I feel is a thing which takes
different values over the space.
So a scalar field is something
which takes different values in say,
the 3D space of this room.
So the pressure in the room
is a 3D is a scalar field.
At different points of the room,
the pressure, air pressure is different.
It's just a number.
On Vector field is something
which different vector values at
different points in the space.
So the magnetic field of the
earth is a vector field,
the different points in the space.
In the Space 3 space it takes different
values and each at each of those
points it it has a a vector value.
It has a middle of the earth has
a a direction and a strength.
OK, that's what field means in this context.
Um.
And we can then go on to
visualize these things. Were we?
Victors.
There's nothing complicated
about visualizing a vector.
It's a pointy thing.
OK, but we're going to also,
but we're going to visualize.
One forms.
I'm saying this because you
know to help you, to help you.
A need to thought rather it's a
movement actually calculated with
good would visualize 1 forms as.
Well, in 2D lines they have. A direction.
That this has a direction and length.
This set of planes has a direction.
And it has a sort of pitch. OK.
So there there's a magnitude to this, so.
That's in this in this picture.
That's a one form which is the same
direction but has a higher magnitude.
They're closer together.
Why closer together?
Because are we in this picture?
So this is not another example of of danger.
We we can imagine the contraction
of a vector in one form in this
picture is by laying the. Vector.
Across this one forms and saying.
This vector so goes crosses 2 lines.
It goes from one cross cross.
And it gives an answer of two
so that the contraction of that
one form and that vector is 2.
The direction of this one form 11234 is 4,
so that the contraction of that the same
vector with with with with the that the
larger one form gives a different number.
Tell the direction of one
form without using the right,
because it could be, it could be, yes.
That's not a very mathematically
very precise picture,
so I think that I think that.
We think of vectors is useful to have
this notion of a point of of an arrow.
You're just trying to get
your things in your head.
When I took one form useful to
have a a picture like that,
maybe you could color one side
of them once, one side the other.
Somewhere in between 21, yes, so.
Yeah, so the field, obviously,
I mean, I think we we don't want to
overstress the precision of this,
that, that, that.
That's just a picture. That's useful.
And also it means that things like.
Like this big sense.
So there's a A1 form field time stop.
There's one form field.
And like like the gradient lines on the map.
So so the gradient it turns
out to be modeled as A1 form.
In this case, the, you know, up uphill.
Has a direction,
and the closer these lines are together,
the steeper the slope is and if you.
Move, move like that.
Or like that, or like that.
In each case, you've crossed the same number
of grid lines or of of contour lines, sorry.
And so you've you've risen the same
amount of, you've claimed the same distance.
So in that sense the contraction
of those three different vectors.
With the one form field.
There's the same number.
So when I talk about of of vectors,
think of pointer pointy things.
We're talking of A1 form fields.
Think of the gradient lines of the gradient
lines on the country lines on a map.
And with that picture,
I'll let you go.
We should.
I think we've we've done not too
badly in getting just about right
just through this this part.
I'll see you again on Friday, I think.
Not in this room.
OK, all the other elected in
this room and other words,
the Davidson building.