How do I get everyone this is Spectra?
Five, I'm fairly sure,
and we're gonna start on part
part three of the of the note.
Up to this point,
we've laid quite a lot of groundwork.
The we have the scaffolding,
or to switch metaphors,
we've acquired some basic tools
and learned how to use them.
The next section is Part 3.
Manifolds, vector vectors,
and differentiation is all about
acquiring some more specialized tools,
specifically a new way of defining
vectors and that one forms and tensors.
And the key thing.
Discovering how to differentiate.
Because the the end the goal here.
Is Einstein's equations which
will come to the next part which
are a differential equation.
On a current manifold.
So we have to learn how to do that
differentiation in order to set up a.
What language do differentiation?
For?
We can set up a differential
equation which we then solve very
briefly at the end of part four and.
In multiple cases in G2 next semester.
Um, anything else I could mention,
the audio notes are in the podcast
that's linked from the Moodle.
The notes are you have to
date with part three and the.
But overview videos of part three
are up and I have been putting
up some of last year's zoom based
lectures up on the the stream.
Site, whatever it's called page,
which you may or may not have found useful.
You're just out of curiosity,
has anyone looked at those and.
I don't look at those.
No pool. A treatment store.
They may or may not be useful to you.
I am interested in on all of these things.
The the version of the the the overviews,
the the audio, the video.
I am clear interested if you
back I think you have been sent
and emphasis questionnaire.
You have right and these are useful.
They are looked at and paid attention to,
so I encourage you to fill
that out in a useful fashion.
Any questions?
OK, let us get going.
So as usual, the aims, objectives,
the high level things and
the more detailed things.
I'm not going to go through them.
You can see them. I I put them
up there in order that I can.
The understands these aims
are all understands XYZ that,
that, that is the goal here,
and there is quite a long list of objectives.
Most of the the the bit of the course
that's most straightforward examinable.
Is this part?
And that's and there's quite a lot
of things you can do in in in this
part I mean it looks for feeling
for bidding thing but if you walk
through the if you walk through the.
The exercise is then you take you take
most of those off so that that look
that look at the most forbidding of the
sets of objectives of the various parts.
But it shouldn't.
I do aim for it to be as
horrible as it looks.
Whether I achieve that aim
is another thing, OK?
So in terms of pacing,
I do plan to get all the way through
section 3.1 in this lecture.
If you do 3.2 and 3.3 in the next lecture,
the 3.4 to the end in the lecture 7,
then I'm doing very well.
I think I won't probably won't manage that,
but we but we have we have we have
about lectures worth of slippage
because I want to be about 3 on Part 4.
So we have we shouldn't panic if
we're not that. That's the plan.
I'm gonna fall behind.
Keep chattering. OK, step one,
we're gonna define vectors. Step 2,
we're gonna learn how to differentiate.
So the first thing is the tangent vector.
Now you learned about vectors in school,
and you learned how to
differentiate them at university.
And.
A lot of the ways in which
you learn how to do,
how to do that differentiation using the
vector calculus that Gibbs developed at the
end of the century was using components.
And you can carry on doing
that in this context.
And that's how GM differential
geometry was introduced for a long
for a large chunk of the 20th century,
you still do that render,
for example, is one of the texts
which takes that approach.
I find it a bit confusing because
focusing on the, on, on, on the.
Um, and the components too early.
Gives them a status that we are at the same
time wanting to deny what we want to see.
The components aren't important and yet
we talked the component all the time.
So the other more modern way
of of doing this,
which was I think first pushed
my missing thought,
Misner,
Thorne and Wheeler in the 70s is to
focus on the geometry and that's
the approach we're taking here.
So I mentioned that just to
highlight that there are two
quite different approaches.
One I think I'd rather old fashioned,
one which is where incidentally the terms
covariant and contravariant come from,
blah blah blah,
we're not going to introduce those terms.
Yeah,
I think I I mentioned somewhere
the the relationship between them,
but they roughly correspond
to vectors in one form.
What we're going to talk about
is we're going to stick with the
that keep things as grim as.
Quota free as long as possible,
which isn't very long,
but we're gonna start there and
talk about the manifold. No.
Is there any pure mathematicians in the room?
You might want to sort of, you know, not,
not not close your eyes at this point,
but I'm going to be.
Playing slightly fast and loose from
the point of view of a mathematician,
and I'm going to be rather
rigorous from the point of view
of of your working physicist,
so I'm going to aim somewhere
in the middle there.
And there are differences,
some differences in terminology between
what's conventional in differential
reused for GR and differential.
Used my petitions.
They're not important differences,
they're just slight differences in.
Path that has curve.
I'll make you three complete.
The manifold is a set of points.
It said with with with very little extra
structure and that petition would would
would want to ask exactly what structure has.
At this point we're not gonna worry about
organization has valid construction.
In particular, there's no notion
of distance in the manifold.
The structure it has is that
locally it's like RN it,
locally it's like a a flat
Euclidean N dimensional space.
What does that mean?
It means that you can define a map.
From any local area of the manifold
to our end to end dimensional space.
And it's, it's, it's like in the
sense of how you move around,
how you'd lay out the different directions.
It's like including space.
And I'll say a little more
about what that means,
what that means.
So the manifold isn't just a set
of points, a ragbag of points.
It has just that much structure that
lets you think of it as locally our own.
And. Our goal is to add little
structure to it in a controlled way.
Now, so that's this is the manifold here.
This this of curvy thing here. And.
What we gonna call a chart is a set of
functions from a point of the manifold.
To the real line.
So I said to functions XI. Which?
Turn a point of the manifold into a number.
And a set of these would call a chart.
And there are any of them.
If this is like like RN, OK.
It might be that these are are
defined for the entire manifold.
It might be the defined for
a subset of the manifold.
The point is that round about
about a point P and our three-point
P there's a set of functions.
So each of these functions is
a map from M the manifold to R.
No, we're all gonna imagine
a path in the manifold.
We're just a set of points in the manifold.
It's there, there, there, there,
there, there, there, there, there,
there, there, there, there. OK.
And we're going to find a curve.
Which is like a path,
except that is it a path which
also has a function which maps the
real lane to a point on that path.
So a curve is a parameterized
path if you like.
So two curves which follow the same path
but with different parameterizations.
Did this curve maps 1 to that point
on the curve and this other curve,
another curve maps 1 to a
different point in the curve.
There are different curves
even though the same path.
OK, you don't have to remember these terms,
these these verse terms long-term.
I need them. I need to define these terms
in order to get through the next section.
So what I meant is if we have a curve.
Which takes the real line or
sectional real line to the manifold?
And our coordinate function,
we're going to call it,
which takes the manifold to the real line.
Then X of Lambda of T.
Is a function which takes
through line through line.
OK.
It was a set of mappings from the curve
parameter to the set of coordinates XI.
Or let's. Out on this one.
What that means is. XI.
Is a function of Lambda.
Which is a function of T.
Or in other words,
XI is our function which maps through line.
Through line. And what that means
is we can differentiate it.
Already we can do.
Already we're talking differentiation.
And we're going to talk about the the
the set of of quadratic functions XI as
a reference framework coordinate frame.
All those various words mean the
same the same thing in the in.
In this context,
we're going to distinguish between.
So the next step is to define a function.
Which maps the manifold to the real line.
So an example of these Xis are an
example of a function which which
maps the the manifold to the line.
But we're gonna pick an arbitrary function.
As the manifold goes to. Through line.
And we're going to talk about
the functions which takes.
The point P through line.
The function which takes.
How do I write this
Lambda of T through line?
The function which takes X.
I of. Lambda of T.
And these are all, if you've been
precise about it, different functions.
But we're going to talk about as if they
were the same function, but we're going
to label them all F as sort of pun.
OK, it's at this point it's, but you
realize that there are different things.
That's a function from from M to R
that's a function from also from MTOR,
because Lambda is the point of the manifold.
This is a function from,
you know, N numbers XI.
You are the different functions,
but they're basically the same function,
just in a way that.
We're going to elide for the moment.
So this function F of Lambda of T.
What happens if we differentiate it with
respect to T or this function F of XI?
Of Lambda of T what happens we
differentiate that with respect to T.
Well, the. Checking the derivative
of F with respect to T.
Is some. Of a DF by DX I.
Txi by. DT. Ohh no.
Yeah, OK.
But that's true of any function F.
So we can write down instead just D by DT.
Either some. Of Dxi by DTD by DX.
And there's two things we can do
at this point. One of the things
we can do is change variables.
So let's instead talk about key A,
which is right way up T. Divided by E.
So. D XI by DT is equal to a DX I by DT.
So that D by DA is going
to end up being a. DX I by.
It's a T.
D. By DX I. OK, that's one thing.
I'm gonna come back to that in a moment.
And. No. Imagine.
There's not just dysfunction
that this curve Lambda.
There's a curve going through the point P.
There's also a curve.
You this so, so, so so here's our point P.
There's our Lambda. Of tea and.
There's a mu of another another curve
which goes through the same point P.
They don't meet anywhere else,
but they both do cross
each other at the point P.
Let's ask what is E? D by D.
DF by DS plus B, DF by D.
T let's not worry about
why we want to do that.
Let, let let's think about that.
Look just. Here look at this.
We'll see that's. Some. Of. XD.
DXIBYD. Yes. Plus BD XI by DT.
ADF by D. To.
Which screen is shown? What?
Because they can't read.
What's going on behind here?
Ohh, sorry, right?
You can't write dranoff. Good idea.
You know, I'll just show both.
Thank you. Yeah. Yeah. If I'm going
off the page or not legible then,
then you do exactly that. Do show.
So we have, we can end up with that.
But. That's a number. I said relative,
so there will be. Another curve.
Called the Tau of what are.
The derivative of which with
respect to R is just that.
Aye of D XXI by Dr.
Would be. And that's
equal to DF by. Yeah. But.
That's a lot of maths, which is the roots.
Don't worry too much about writing it down,
but the point what we've done
here is interesting and important.
Because what we've we've done is
discovered that whatever. DBT is.
Umm. 8 times it. No.
Um. Equals a.
DT8 times it.
It is another thing of the same type.
It's also a derivative.
In other words, DBT is. Think of.
This can be multiplied by the by a scalar
to get another thing of the same type.
We've also discovered.
That you can add.
Some multiple of DDS which is one
of these things and the multiple of
DBT is one of these things and get
DDR with another of these things.
In other words.
These derivatives defined in that way
satisfy the axioms of a vector space.
So This is why we're very genetic
about the the the definition of
a vector space in the last part.
In other words, these objects DBDT,
which you're familiar with and there
isn't anything really exotic there
you're you're you're you're you're
familiar with that sort of thing.
I haven't done,
I haven't pulled any fast one
that's that is nothing more
than what you think it is.
But it obeys the accent of vector
space and so we can see it's a vector.
And the rest of this course,
that's exactly what we're going to do.
We're going to say this is a vector.
And when I talk about a vector from now on,
that's what I mean. OK.
And So what that means is that we
can write do things like write again.
V.
Some I'll provector.
Is. DB DB duty at P Now notice
that vector depends on two things.
It depends on the curve.
To which T is the parameter,
so that vector V.
Is related to the Lambda curve.
Who's prompter is T? OK.
So it's sort of it's it's sort of like
you know what that what that means.
Is that? V is called the tangent vector.
It describes how much the the the function.
So so yeah, therefore V.
Applied to F and that's a fight,
and regularly.
That is a funny thing.
It looks like anything to write is.
DBDT. At P applied to F.
Which is equal to.
So what we would find to
be equal to DF by? DT at.
TOP. This looks like a
strange thing to write.
This V applied to F.
All it is is an A weird way of
writing a differential operator.
OK, it's a differential operator
which apply to F so that debt
PF is defined to have the value.
The value of this operator
applied to F is the derivative.
How fast F is changing when you
change T at the value of T which
corresponds to the point B.
And we call it the tangent
vector because it you can think
of it as lying along the curve.
Lambda of T.
And providing the answer to the question,
how much does this function F
which is applied which is defined
across the whole manifold?
How does it change as you
move along this curve?
Land Lambda of T How does it
change when you change T?
So I'm here.
F is being a scalar field in
the sense that I defined it.
Last part is a field in the sense that it
has a value for each point in the manifold.
So distinguish maps the manifold
to the through lane.
In this case I was wondering.
What? What is?
You have to FB right?
So yeah, sorry I interrupted myself.
So this, this, this,
this V here depends on two things.
It depends on the.
On the curve it's standard vector to a curve.
But but it's defined only.
At the point P on that curve.
So this operator here is is
defined only at one point.
OK, at the point P,
so if we go back to. Umm. Uh.
Or do you? Oh sorry there.
Alright, so that's the the value
of T which the value of T on the
curve which corresponds to point P.
So it's it's sort of the the
backwards map there. So at the so.
This vector here is defined at the
point P at the point on the curve.
The the value of T which
corresponds to that point P.
So as you as you move T along
here you move along this curve.
And you have a function which you go to the
map the manifold to the to to the real line.
As you move along that curve,
the value of that function at the
at the corresponding point changes.
You're asking how much does it change
as you go on that particular curve.
So this V corresponds to that curve.
At that point.
You have to point you to the
local nature of the mind.
We could only have it at a point
because all of the. And the only.
No.
I don't want to resist saying we can
all have it at that point because
that's only defined at that point,
because that's not very illuminating.
We applied only to point at that point.
I think because the yes.
If we're talking about this family of curves.
The, the, The, the A Lambda of T,
the MU of S, the top of our and so on.
The only thing that they have in common
is they all go through the point P.
So we are concerned at this point
with the whole set of curves.
Which go through the point P.
And for each of those curves.
There is one of these
tangent vectors defined.
Which is seeing how much of
the picking up the derivative.
Along that curve at that point.
So for each of these,
did that sort of seem like
an answer to the question?
For each of these curves,
there's a tangent vector.
And so that there's a a
set of tangent vectors,
which we call the tangent plane.
At that. At that point.
So the set of these derivative operators.
V would you define only at point P?
Is a set which is one.
That picture is A2 dimensional flat space,
which is called the tangent plane
at the point P of the manifold M.
And that is called the tangent plane
is seeing that although this looks
like it's laid out on the manifold.
It really touched the manifold at one point,
if you like.
So all the the points that are
all the points in that tangent
plane correspond to vectors V.
But they're not perfect.
But they don't correspond to.
They don't join points in the manifold,
they are purely at on the tangent plane.
They're purely defined at the point P.
And a key thing.
And this might also come back to
what you're thinking is that if
we do all this at a point. Cute.
And what else in the manifold?
Then we get a different tangent plane.
Tangent plane TQM.
Which is a completely different plane.
It's pretty different space the
the the vector is defined in that
different tangent plane at the point
Q somewhere else have nothing to do.
With the vectors in the
tangent plane of the point P.
We don't want that.
Because what we do want to end up with is
be able to ask how do the vectors? How?
How do the rates of change at this point?
How do they relate to rates of change
at that other point in the manifold?
So we do want to talk about how we go
from one tangent plane to the next.
We can't do that yet. We will.
We need to find a way of
connecting this tangent plane,
that tangent plane we'll discover.
The way we do that is through
a thing called the connection.
Conclusion the. But we can't do that yet.
And they're not all going in One Direction,
right? Yeah, yeah, that's right.
So, so, so this, this cover is going
there another cover here whatever.
And a constant each is, is, is the.
The that this vector V gives
the answer to the question.
How much does do things change as you head
along that that that particular curve?
So for all the curves that go through
the point P, each one of them is a.
A vector.
The pending. Yes, look at the curve.
Do they change like with the plane?
And I think the so yes,
you can imagine a number of a number
of of curves here which go all the
way through through the the manifold.
So I think the ones we're interested in
are the ones which go through the point P.
So of all the curves that
go through the point P,
each of those has a corresponding
vector in the tangent plane.
So we're not talking about
all the possible curves,
only the curves which go through.
So I think that is against
circling around your, your,
your question why are we talking about
why we're talking about because we're,
we're picking out what we're
looking at, what happens at P.
Ohh.
They should expand 3.
3. Should also be or.
Yeah, the tangent plane is has the same
dimensionality as the at the manifold.
Yeah, I mean in this picture.
The manifold is 2 dimensional.
And and to the tangent plane,
but and and and so I think that the
the the name I think comes from that.
So, so yes, it's a bad name,
but it has the pick.
The good the picture is of of
a a flat plane just touching
the manager at one point.
Key. Um.
OK. And So what?
But that's so moving on.
And with this in mind. We can.
Look back at this expression here.
And reread this.
With.
And five.
As being. Me too, I thought.
Define defining a set of basis
vectors which are D by DXI. ATP.
And write down DDT. A TP is a sum.
Of.
DXIBYD.
TID by DXI. Which is um,
how do I write this?
Yeah, it's V i.e. Aye.
Jumping back to the tation of our
previous of the previous previous part.
So all I'm doing here is rewriting
something which was what we saw before,
calling it a tangent vector with this
notation here and identifying that as a
basis vector and that as the component VI.
And as you can see the the
index is sort of matched in the
sense that a raised index or the
denominator corresponds to a a a
load index in this so the summation.
Mention still hangs together in that sense.
At which point we can import all
of all of the understanding we
got from the last part of how.
Vectors and components and stuff work.
Which is good.
Next thing is we'll talk with vectors.
So you are asking yourselves,
So what about one forms?
What one forms come in?
One forms come in when we
ask how functions vary.
So again, we're going to consider
a a function. On the manifold
function F arbitrary funct.
And. We're going to define
A1 form field called. DF.
And. That we were going to find that
one form field is by its action on
a vector one formed remember turn.
Vectors into numbers.
So we're going to define this one form.
By asking what its action is when applied to.
DBDT at P.
And we're going to see,
we're going to define.
So remember a function you're
used to seeing a F of X equals
mathematical expression.
All function is is a rule for going from.
Domain to range.
And our rule here is given this picture.
The value of that is DF
by DT at the point P.
And this DF. Is the gradient.
So-called of the function F
the one form which the gradient
one form of the function F.
Which is a wonderful field.
In other words, it is A1 form which is
defined at each point in the manifold.
The value of which it obtained by definition.
By definition by applying the applying
it to a vector in a particular.
Tangent space to give that
to give that result. Um.
Do uh, what does that look like?
I'm also do things like write that as the F.
Thank you. Umm.
What blows right that is that we
picking up this notation that I
briefly mentioned for the contraction
between one form and a vector.
Um, if we write this?
In component form, then we see the DPDT.
Is uh.
This. PDF is applied to.
We could DDT is that that is operator V.
We know from last part that that is it's VI.
DF. EI. Which is VI. Yep.
Yeah, full derivative F by D. XI.
So we we can see a component version
of that same thing. Been that way.
And and similarly if we ask so
each of the coordinate functions.
Remember I said I I I had the picture there
of the coordinate functions which Map XI
which map the manifold to the real line.
Each of those is is is also a function.
So each of those has a gradient
one form associated with it.
Written something like.
The. X. Alright, yes, XI.
And we apply that to one of
the basis vectors DVDX I.
We get. The
XXGXIBYDXJ. Part of relative and if
the the the various quarter function
are independent of each other.
Then that will be one if
I is equal to J and 0.
Otherwise on or the chronicle delta.
So which is reassuring,
that's telling us that our basis.
One forms. The the these gradients,
the gradients of the coordinate functions
are indeed dual to the basis vectors.
As we demanded our last time.
So all the stuff we learned
in the last part about.
Basis vectors basis one forms
component blah blah blah blah
blah can now apply to these.
Species, vectors and these pieces one forms.
So we can import all that
technology and move on.
Any questions about the question?
Yes, I probably am, but so
the point should be up there.
So so yes, that's that's that's a
good point, because all of this,
all of the the the vectors in one
form we're talking about here are
in the tangent plane. Thank you.
So that's all I at a point P.
So so we've sort of we've we've
sort of left manifold behind and
and and most of the stuff we're
going to do immediately is in the
tangent plane at a point P so I
so I I will sometimes miss out the
the the P there we have to remember
that that that that's the space
that we're talking about here the
vector space that we're talking
about here is the tangent plane.
Thank you.
Yeah.
OK, next. We're good progress here.
So one of the things we did last
in last part was change basis,
change of mind, what the basis
functions were the. The.
This function here. Is arbitrary.
We can change our mind about what
that coordinate function should be.
So if we do that.
Then we will write.
E. I bar you go to.
Deep by DXI bar. At peak.
And. This new basis.
Will be related to the old one.
By. Lambda I I bar E. Aye.
And and. I think that there may be an
exercise which allows you to to to to
reassure yourself that Lambda I I bar.
Is be sure to get things right, we up.
Me.
No. Um is.
So we would have to help himself
is the XI by DX. I bought.
Sorry, I got off the boat.
So I I encourage you to go through
that section of the that that last
section is slightly slower to reissue,
which I'm not pulling a fast one here but.
There's nothing surprising
particularly writing here.
The the Lambda that we saw last time
is just the relationship between two of
the of the arbitrary arbitrarily chosen
basis functions coordinate functions.
We are making excellent progress.
We'll move on to section 3.2,
slightly ahead of schedule.
And I think last, I think last
questions about that. So we've.
Yeah. What, what, what,
what, what with them?
Define a useful vector space.
Because the vector space we're able
to pull in all the, all the the.
Vectored at one form tensor manipulation
technology from the last part,
so we now move forward with it. OK.
So we've defined vectors.
We've defined 1 forms.
And now we want to start defining
differentiation of vectors.
In other words,
how given a vector in a?
The electric field vector C.
How does that vector vary?
How does that vector change as
you move around the space? OK.
Because we want to ask things
like how does the argumentum?
Teacher change as you move around the space,
because that's going to tell us.
It turns out how the coverage of
the space it turns out changes
remove around so we've got we've
got we're going to ask yourself the
question how do you take intentions.
How do we manage the to calculate
with the the the the way that tends
to change as they move in space?
We're 2 steps.
We're going to first of all find
out how to do that in flat spaces.
And then discovered that the
step going from there to doing
differentiation in curved spaces.
It's quite a big deal,
but it's a smaller step than you think.
Which is nice.
OK.
So yeah, blah blah.
Well, I ask that question here.
What type of thing is?
Is it is this is this.
Just quickly, who's able to scaler?
A vector. A1 form. Potential.
The matrix had we hang up yet?
Quite a little there. OK.
Talk to each other and I
think that question is?
OK, with a bit of reflection,
then ask the question again.
Who's it with the scaler?
The vector one form.
Tensor. Matrix. And.
It's just scale.
It's just number. Because.
Because. If you look at that,
I think at the end here,
but for me it's just a derivative.
This whole thing is a complicated
way of writing that derivative.
And it's answering.
It's answering the question, how much
does the function F vary as you change T?
Just number F the number.
So as you change T the number changes.
You move from different the manifold.
How much will it change?
OK, and V is a vector.
Is a function actually function so,
but the action of this operator V.
On if.
Is this there is this thing applied to
F which is the derivative of DF by DT?
So if you wanna be picky then used attention.
We could all scale attention too,
but that's slightly.
It's not the simplest answer you can give.
So I see this.
It seems it feels it feels
a bit like a trick question.
But I'm saying that there's slightly
less to this than meets the eye.
That looks a really exotic thing.
You've never seen a vector written
next to a function before you think?
So what this what?
So what this means is that vectors
in this context are slightly dual.
Meaning. They're both pointy things.
And what that's a good thing to
have in your head and they're
also differential operators.
Which should be applied to functions
to give to give numbers. OK.
They're both things you I need
both things you had at one time,
and which what?
Which sense of the of that I
I'm I'm I'm I'm referring to
in a particular context will
depend on what I'm doing with it.
So sometimes I'll be manipulating
these vectors like vectors,
but components, all that stuff.
Sometimes like this there are an
operator which applies to a funk.
And and you've seen that that's
just a summary of other things.
What can you see?
The same question and what sort of thing is?
D XI by DT. In that expression.
Wish it was a scalar. A vector.
A1 form. Tensor. Matrix.
Hadn't read hand but hand up yet,
but still haven't found it yet.
Don't you have a brief chat?
What is DX IDT?
And with that reflection,
who she was the scaler.
Who is he? Was a victor. Who?
She was in one form. Tensor OK.
There's sort of two Oz here.
Strictly it's just it's just a scalar.
It's number. It's the derivative of
this function X of X of I with respect
to T is how much did that that that
coordinate function change as you change T?
So it's just a number again
because XI is just a function.
But of course these are also the
components of function and and.
And as I said last time,
sometimes I'll start the equivocate
between the components of a vector
and the and the vector itself.
So if you said vector, OK, not long.
But strictly, that's just a number.
XI is a function.
It's there's north of them.
That ain't functions,
but in each case the how much that
function changes as you change
T as you go along the curve.
It's just a number.
Um. Yeah, goodbye you abuse of notation.
I was somebody. You think of VI as
being the vector, but but it's not.
Um. There are dual. There's that.
Not sure I'm getting out there.
Um. I think I mean really
tricky tricky there but.
OK. I'll just to start you off. I'm going to.
Mention. That you have seen.
Yeah, don't do that.
No, let's just stop there.
Let's stop a little early.
I think it means we can go right
into a secondary .2 next time,
which I think is next.