Books. Hello again and welcome
to lecture six of the GRE course,
and before we get going,
a couple of things about notes and things.
One is that thank you for those who filled
in things about the the emphasis things,
some very useful feedback there.
One of the things that was mentioned
was which I think it's very easy
to just think about audibility.
And I think the rumours against us there
because there's quite a low ceiling here
of soft material fabric at the back,
so there's not A and also there's
think because a shelf there a
bit of a poor impedance match.
So I think it is quite possible
that I'm yelling down there.
You can't necessarily be very well here,
so I should aim to project.
But if I'm failing.
At the back,
just shout out volume or something.
Just yell.
And I think we're going to handle that.
No, another thing that was mentioned was
the issue of just the volume of stuff.
And I have a sleep and and and how
how one navigates around those and so
I'll recap a bit and and remind you.
They believe each chapters aims and
objectives aims at the high level things.
The you know the point.
The objectives are the things which
are also useful but are less exciting,
but are accessible in the sense that
there are things which you can I
can ask you to do and in an exam,
and those structure the you should have
those in mind when reading through
reprocessing the notes after the lecture.
If you look at the exercises
at the back of the chapters,
a lot of those are keyed to
fairly specific objectives.
So you you,
you,
you can see all this exercise is in
the service of that objective and look
through things with that in mind.
Another issue is that it's
sometimes difficult to.
It is. It is a defect of the
notes that there are a lot of it.
It's something a little hard to
see what the the key thing is.
So what I've spent some time last
weekend doing is adding to the notes
some sort of key points in selective
sections and in the slides that
you'll see now at the end of these
sections that are point, point,
point of just the highlights of that.
Now having said that,
I'd encourage you not to look at
those because I encourage you.
You go and look at and and and and
do the same exercise for yourself
because picking a section and going
what am I supposed to be getting out
of this and writing that down is by
itself a very useful exercise because
seeing standing back a bit and seeing
what was what was the point of that,
I get it is a useful exercise
but you see my version there as
well in in the in the slides.
I've also decided to put the slides
up ahead of time rather behind.
There's barely,
there's there's essentially nothing.
The slides that are.
That isn't in the notes.
But it might be, depending how you
want to do what you would scribble on.
You may want to use that.
And I've also put up the as
I said in the model posting.
I've also put up the compendium
of all the exercises.
Usual thing is not useful to do these
to look at the answers too quickly.
But your Honor, students know.
You know that and you can do
what you like with those.
You can undermine yourself if you want,
and I'm sure you won't.
And so that I I put that up just to
remind me that those things are there.
Not entirely obvious.
You have to click on that
tiny little arrow to see the.
The contents are rearranged a bit
just to make it look less huge.
There, there, there are still there.
They're just basically three-part
three sets of of things in there,
but in a couple of different formats.
Any questions?
OK, there there were other useful
points in the the feedback.
Which I will. Moreover,
and I think of at various points,
so that's not the only feedback.
Other things you can think of to
see mail me or it's all good. OK,
where we got your last time was the end of.
Section of of three one and
we're just encroaching on.
The differentiation we basically
ran out of time, so if I can find.
Five slides few. You moved.
And we got.
But so far there so there's a you
know one of those key point things
that that that that I thought the
the key point from that section.
So what I'm going to do this time is talk
about how you do how do differentiation.
In our of bases, in a case where the
basis vectors that that were using
to create our coordinates where
they are changing over the space,
we're first going to do that in flat space.
And then we're going to do
it in curved space.
And the surprising thing is
that that second step turns out
to be the easier of the two.
You think it was the other way around.
So you have done this.
This is to some extent another
case where you've done this before,
but not in this notation.
So there's a slight notational issue.
And there's also slight stepping back
and seeing what it was you did before,
because what you did before was
things like you've seen the Laplacian
in spherical porous bit of a mess.
But it's a bit of a mess because R,
Theta and Phi basis vectors are
changing as you move over the
the space and so the when you
differentiate the components of a
vector in those coordinates you have
to do all sorts of stuff to get the.
The the coordinate independent
change in this vector as you move
as it moves around the space.
So, so, so this is about
differentiating a vector field.
You've got a vector field of
for example the electric field.
Are you asking how does that
change as I move around the space?
And that and and the Laplacian
is part of the.
It's part of the answer.
It was simplified set of things
a bit and talk instead just
of plain polar coordinates.
Let's start off simple and work up
so the plane polar coordinates.
Are um.
Near Project source camera.
Not that far. Not that far.
Playing public coordinates are.
ER. And E Theta. And we want
to ask and and those are.
Derived from the.
The basis vectors?
The Cartesian basis vectors.
ER. Is. Costita. Having this
done just to get the signs right.
Ex plus. Sine Theta EY&E,
Theta is equal to minus R.
Sign. Theta EX plus.
Are Cos Theta EY? No. Uh-huh.
But you may say I don't remember
that being there. And you don't.
That's because the basis vectors of
the the the plane polar basis vector
you're used to are orthonormal vectors,
where specifically the R is removed
in order that these both be squared.
However, this is in some sense the more
natural one without that correction.
And we can go into what?
Into why that, why there is,
but that are as they're
deliberately and inconsequentially.
It doesn't really matter how I define my,
my, my, my, my, my basis vectors,
just in this case,
these ones are orthogonal but
not orthonormal. For reasons.
Which we can talk about.
If need be.
Now we want to ask how do
these change as they go,
as we move around the space?
And that's not hard.
We do things we we can say D by DROFER is.
Well.
Yard doesn't vary.
It doesn't depend on our at all.
D Theta is this honest with me?
Either. You know, funny echo,
I'm not sure it's just.
Whose speech anyway D by D Theta.
Of, ER, it's going to be.
Yeah, it's going to be. A minus costs.
Theta EX plus sine Theta E. Why?
The other way. You have school.
Thank you my sine Theta EX plus Cos
Theta EY which is just 1 / r E.
Teacher. And so on.
So we we we can just just walk
through those four possibilities,
differentiating the the radial and
tangential basis vectors with respect to R
and Theta and get expressions which are.
The remaining two are D by Dre. Theta.
Equals 1 / r, E Theta and D by D Theta.
E Theta is equal to.
Made our ER and and and one of the
reasons why orthonormal coordinates
are good is because you your your void
these extra factors of factors of R.
So no surprises there.
Now what happens if we pick?
You pick a vector V and ask how does that?
Change as we move around,
as we move radially.
So as you move away from the origin,
how does how does the vector V change?
Um, that's uh. Ebitdar.
Of VR plus. Breathe easy.
Peter. And which is? The DVR by
Dre R + V RDERBYDR plus and and so on.
And notice, by the way,
that I'm I'm slipping in a
notation here that. I'll use.
Occasionally I'm indexing the basis vector.
And the component with the symbol
R rather than an index 123.
So that's sort of a slightly
slangy way of of indexing the arc,
the arc component and the Theta
component of of the basis vector of
put that on on that slide as well.
OK. And?
Or we can just write that
in in index notation as a
DDIBYDREI plus.
The IDE I by Dr. And. Ohh question.
With respect to our like in this case,
because it could be. Or or or or Theta.
So what we can do is maybe more
generic and exactly I think
you suggest and say DV by D XI.
Is going to be the VP of
called JD VI by DXJE I + V.
ITE.
Uh. You know.
Right. Aye, aye. By DXG.
Yeah. So is that what you meant?
Yeah. And we could have written
that down from the outset.
I'm just sort of easing us into
into that expression from me
because I just written that down.
OK, but notice. That's a victor.
But asking how does that vector
change as you move around and and if
the vector starts off here in this
position and here in this position,
then there's there's there's
a change in that vector.
So that change in the vector is a vector.
Which is a number. Times a vector.
A number times a vector.
In other words, this DEIBYDXJ is,
not entirely surprisingly,
also a vector.
Which vector is it? It's a vector.
Then we can express that vector
in terms of the basis vectors.
So if you're right, DE.
I by DX J. It's some some.
Components.
In multiplying the basis vectors and
we write the components. Gamma K. GI.
I think I've written IG. Sorry IG.
And gamma here is called the Christoffel
symbol or christophel symbols.
No one seems to be quite clear
whether a symbol or symbols,
the crystal symbols are nothing more than.
The one second, nothing more than the
components of that vector in that basis.
Basis the same as. Yes it is.
So. So we're just,
there's just we're picking a
different, a different index.
Yeah, so it's it's, it seems,
a bit of vectors. This, this,
this is a vector in that space.
And so we're seeing if it's
a vector in that space,
then it's expressible in terms of
the basis vectors in that space.
And we could also have just written
that down from scratch with the there.
There's nothing was stopping us doing that.
But this is a this is a motivating.
This is mogamma. 50%.
So good. It's. It's a, yes.
So this is a number. Exactly.
It's a set of N by N by N numbers. OK.
Is that number, and yeah,
yeah, so this isn't a tensor.
It's it's not the components
of a geometrical thing,
it's just a set of N by N
by N numbers. Thank you.
So we can write so, so if we.
If we go back a bit and and ask.
DERBYD. Theta I wrote. Will make that
EDEDEE 1 by DX. Two calling RX1 and Theta.
And Theta equals X2. That will be. Um.
Gamma one. 12 E 1 plus. Gamma 212 E 2.
I'm just illustrating what the
what what that some looks like.
And from above and and and we could
look back a page discovered that.
The the the the ER by the Theta is 0 * E.
1ER plus 1 / R.
He 2IN other words, gamma 1/2.
Is equal to 0 gamma 212 is equal
to 1 / R and that's how you you
you we we calculate what the these
Christoffel symbols are for a
particular set of basis vectors.
It's a turning the handle thing.
A bit tedious, but it's the sort of
thing it's very easy to test, like,
very easy to do, and several of the the the,
the question of of the exercises are
encourage you to just turn that handle and.
There are no thrills in turning that handle.
It's just a matter of of practicing
doing so and not getting lost.
OK.
And and.
Uh. So that.
That means that in, for,
for, for plane pollers.
Gamma One woman two is 0 Gamma 2 and two.
Is it same as gamma 2 to one and gamma?
122 If you could monitor and we'll
also write that sometimes as gamma.
Are. R Theta equals 0 gamma Theta
R Theta equals gamma Theta Theta
r = 1 / R and gamma R Theta Theta
equals minus R and again this is
a slightly slangy notation which I
hope is is clear but by our I mean.
These would match up.
So I'll, I'll just.
So does that mean so any questions on that?
That's mostly notational.
Another notational section
telling you something you.
Again, the idea is this is
telling something you do know,
but in different notation.
But it allows us to define
the covariant derivative.
Because what we have,
if we if we go back a bit we we have the.
V by DX J is equal to D
VI by DX JEI plus. The.
IDEI by DX J but we know
that that is equal to gamma
KIJK.
And so if then we decide to
renew, relabel that as. Uh.
DVI by DXJ. He. Aye. And.
And instead of eyes right keys instead
of keys right eyes so VK. Gamma. I.
KGEI. No, all I've done there.
Is that these?
Dummy these repeated indexes are
dummy indexes, the eye and the key,
so there can be anything.
So I've decided to rewrite them
simply with different letters.
There's no that that's exactly
equivalent expression,
but what it means is I can take
this EI out of there and get
DVIBYDXJ. Plus Ek gamma
IKJEI. And discover. That.
Right, right. So.
So what we have there is this is a.
A vector. With components. That.
And I'm going to write those
components in a particular way.
Would write those components as VI,
semi colon, J. EI.
And equal to VI comma J.
Plus VK Gamma I KKJJ.
EI where this notation VI semi
colon subscript semi colon G refers
specifically to that expression.
And the the the notation of
just introduced the VI comma.
J refers to the the plane.
Usual derivative of DVI by the exchange,
so that's DVI by DX J.
And like that.
That's the last notational bad surprise.
OK, punctuation in subscripts.
I'm sorry,
I didn't make it up.
And. G is a little tricky to write neatly.
I I I agree. My handwriting improved
massively when I started doing this.
No.
Umm.
Uh, you know, I have a quick question here.
This illustration of that as that.
And. So the the key point of the of
of previous section were that the
basis vectors vary over the space.
We knew that. And that variation can
be characterised using this notation
using the Christoffel symbols.
Um, so. A quick question.
What sort of thing is
determined brackets in the
expression and that expression?
Who's he with scaler?
Who said it was a vector?
Who said it was a one form?
Tensure. A matrix.
Have a brief chat about.
OK. With that reflection in mind.
Who would say that was a scalar?
OK, a vector. One form. Tensor. Amatrix.
Two of those answers sort of are correct.
In one sense, yes.
This this this is a scalar because there
is the for pick and I pick an IG and key.
Pick an iron key, and yes,
there's a number which corresponds to this,
so yes. But at the same time.
For a reason which I'm about to elaborate on,
this turns out to be a tensor as well,
because or the components of a tensor.
Because.
What we have here is our. Thing
which linearly depends on the. Um.
Right.
I'm going to go through the, the, the.
That's actually in the way I expressed on
in my notes because rather than busk are
possibly confusing answer, but it is. And.
So um.
The key thing is.
That this this vector, this vector here.
Is. Proportional to the.
Basis vector. Each egg.
If you made EG twice as big,
you make all the components half the size,
and that vector would would would change
inside inside the cornely accordingly.
So there's a a proportional
relationship between those things.
So this is a a thing which
depends on the variation of.
V around the plane and the size of the
depends on the size of the vector,
which corresponds to D by the XJ.
Which is the basis vector.
So what we can do?
Is we can define.
Are A11 tinger.
Nabla V.
And we'll define that by saying that
the action of that on the vector.
East. How do I call it EG so the
11 tensor so it takes a vector
argument and A1 form argument?
And the action of it on on there. Yeah.
Right.
I'm, I'm, I'm, I'm, I'm, I'm gonna
write it in a way other way around,
in a way to just make sure
it's consistent with my notes.
Just just the the the
distinction is important but.
The action of that one one
tensor when we give it when we
give it 1 vector as argument.
One basis vector as argument is DV by DX J.
As a victor. So.
We're seeing let there be a tensor.
Which is related to V which we're going
to call the covariant derivative tensor.
And our definition of that tensor is through
is through the slightly indirect way.
Remember that the tensors A11 tensor
takes A1 form and a vector as argument.
And if we give the vector argument to
that covariant tensor, I'll get a moment.
If we give a as the vector
argument to that tensor,
one of the basis vectors,
then by Fiat we say the value of
that tensor is this vector here,
and being a vector is something
which takes A1 form of argument.
So there is a missing one form
argument in both in both places.
Yep.
The acoustics in this room are not good.
Right. And I it's we could
pick any basis vector here,
So what we've said. And and.
If you remember the basis vectors. Are.
EI equal to D by DX I.
And so we'll just pick a random
basis vector in this case,
EG and we're seeing if we apply
EG could be anyone picked, EG,
then we get the derivative of
that vector V with respect to
the corresponding coordinate.
Coordinate function. OK, right. And so.
What we have here then?
That's as I've said this this notation
with the dots and and and and missing
arguments is that your conventional,
the there isn't really a completely
conventional way of writing of writing these.
But one Commissioner,
we are writing these is to
write that as nabla, EG. V.
Where we're rate the.
This this vector argument
as a subscript there.
And. We that we write this visually often,
that of course we want to abbreviate
it so we end up writing we now blog.
V. And that is it. It tends to be a
short version of that, and that is.
Just means that it means that this tensor
nabla V with one argument filled in.
In a in a slightly funny place.
Written down is like money. Please. So.
So if we add so we have a tensor nabla V.
Want to ask what other?
So stepping back a bit, what other
components IG of that tensor?
And. What we do is we now
have a V and fill in Omega.
I egg. But we know that that is
also written as nabla, EG um.
V and the ith component of
that by filling in the. And.
So so by by filling in this Omega I we're
extracting the ith component of this.
Vector here of this vector here.
So again the ith component of the of this.
Napa V EEG, which we also
write as an Apple ID. P.
So G. Aye, I'm and that and in
particular by comparing it with
on the previous sheet that is.
VI. G. So it's important to look
at to to to be clear what we're
looking at the very stages here.
So Navisa is a tensor.
We're asking what are the IG
components of that tensor?
How do we do that?
We plug in Omega and and and and egg.
You know the two are the two one form
and vector arguments of that thing.
But we know what that is.
We can we just notationally change.
That and there. To this.
The Omega I is extracting the
ith component of the result,
so it's the ith component of this
vector because so this is by now.
This by now is a vector.
We rewrite that as.
For convenience, not nabla EG,
but just nabla G.
And we know from from this.
That the other way we write that
is with this VI semi colon J.
Which expands. To this.
So there's quite a lot
packed into that line.
It's we'll we'll take a
couple of goals through it,
but I think it's important that you
understand what different thing,
what different things are happening
in each of those equal signs.
So make sure that you're that
you have a story in your mind for
what that equals sign is doing.
That equal sign is doing,
that's equal sign is doing,
and that equal sign is.
Question.
It's it's a semi colon at the end, yes.
Yeah. What the meaning of
ohh that that semi colon is.
Have this. Yeah. So this.
the I semi colon G.
Is this expression here?
So is VI comma G where we've
introduced the rotation?
Iconology is just that, it's the derivative
of the ith component of the vector.
With respect to the XJ, nothing, nothing.
Covariant just goes straight forward.
That's real four thing plus the other term.
So that's and and that is the.
The the components of the
11 tensor which is nabla V.
How is it the one one?
If you vote, you buy the vector space,
the vector position, so so.
With both of those empty.
But there's a on one format and and
a vector thing. If we fill one in,
then that thing there becomes a vector.
Because that thing had it has
a single one form argument,
so shouldn't it be a 0?
So, so so that is a 01 tenger.
That is a. But.
It's a it's A10 tensor.
It takes a single one form of argument,
so that takes up one form of
argument that takes A1 form
and a vector as argument.
So, so, so the nabla.
The the tension. Is it?
That thing is 11. It won't.
Whole thing is 10 once we
fill in one of its arguments.
Then what we what we're left
with is 1 unfilled argument.
And therefore it's a vector.
So remember that if you have a, a.
A11 tensor. It's a move.
Takes 1 vector one, one form
and one vector as argument.
So sitting there with two holes at the top,
ready to have things dropped in
handle turned a number come out.
If we put me into one of those holes.
There's only one hole left over.
In this case this this thing here. Nabavi.
Is the thing with two holes at the top.
Had one form and a vector hole one,
but we have filled one of those holes.
So what we're left overall?
Is a thing with one form whole on.
Open as it were.
In other words, a vector.
Which vector? We're seeing it's this, victor.
And I'm carefully writing here
the argument of this vector.
It being a vector,
it has a single one form hole. OK.
So. This is, it's like we've sort of
reversed into this definition of what?
Of what the tensor is.
We've said there shall be a tensor. And with.
Attention with attention with two slots,
yes, one form and and and and and a vector.
It is empty, yes.
In other words, this thing here is a vector.
This thing here is a vector,
which we've illustrated by showing
that there's a single one form shape
slot which which we filled in.
And we know what that is.
And so this whole business here is.
It is doing a bit of a notational dance to
to discover what the components of this.
Tensor.
There's one tensor.
It corresponds to this thing that
we've just that we worked out earlier.
OK. That might need going back
back through again, but a bit of
digestion you will have a little bit.
Well, it's another sort of thing that
you can sit back in the bathroom,
think, think, think your way through.
I think you have to stare at it for a
while and get some illumination that way.
Just to clarify implementation,
so if you ever give us a nobler, yeah,
then something after the Nova and then you
either superscript or subscript something,
that means that it's you have added
a basis something onto the and the
thing that comes after the number.
If that makes sense.
I know where you're going,
but I think that in this case.
You should think of that.
I think here Nabila isn't
really being an operator.
I mean, it's not with an operator and isn't,
but in this case there isn't a
nabla operator that acts on things.
We we ask what is the
covariant derivative of of V?
How can we talk about how V changes
as we move around the space?
Then. There's a tension there.
We've said there's a tensor which gives
that which which gives that, that,
that, that, that, that, that answer.
We could call it Q, we could call it Omega.
We call it anything we're like,
but the way we the the name we give it.
It's a 2 character name if you like.
There's a number of.
OK, so I can have like A1 form,
that's nabla A1 form,
and then I have an I substitute
because I have added the one form.
We're going to complete the command
derivatives of 1 forms and momently.
But I think right now the useful
thing to think about is is is is
not to think of this as an operator.
This is A2 symbol name for the tensor.
Which corresponds to V.
Which is the,
which describes how that tensor varies.
How how how that vector varies.
That vector field varies as
you move around the space.
So I think the next bit may
answer your question, I I think.
Keep this is moving less quickly
than I'd rather had hoped, but.
Another thing that you might see is no
might about it you will see is if you ask.
Specifically annotation like. And.
Nabla X number subscript X or V?
And what does that mean?
It. It means as we can see here.
This tensor nabla V.
With this X argument.
X is a vector is important X
i.e I so that is going to be.
Excuse me? XI, Tableau V.
Till the E. Aye. Which is. XI.
VI semi colon GE.
G. To.
And is X IVG semi colon IE.
Gee. I would have done there.
So here this this tensor what
character is is it's. It's.
It's vector argument is written
conventionally subscript.
In other words,
it's the it's the vector argument
we supplied to this one one tensor.
Then you need your argument so we can
take the XXI out and and have this,
but we know what to do with that.
You're turning the handle here
and we get X IVJ semi colon I
e.g so this is is the.
So this is the comment derivative of.
Of of of V with with argument X
rather than one of the basis of
vectors is asking. How does the?
The house how what this is doing
is is is essentially asking how
does the the vector field V vary
as you move in the EEG direction.
This is asking how does the vector feel
very as you move in a more general,
more general, different direction.
So we started off talking
about the support terms.
That's just a slightly more
general remark based on that.
I want to get into this section um.
So.
What we've done it remains to define
a tensor field associated with the V.
So V is a as a vector field,
it takes a different value at
at each point in the space.
We've managed to find a tensor field,
and there was a 11 tensor which takes a
different value at different points of space.
We've called that tends to field
the covariant derivative and would
define it in such a way that the
answer gives us when we put in its
two arguments is the amount that the
speed at which the vector field V
varies at that point in the space.
Remember vectors you put in
a vector about tensors,
you put in a vector in one
form and you get a number out.
In this case,
the answer to the question when you
put these two things in is how fast
is what's the ith component of the
variation of V as you change X.
G.
So so when I when I talk to the beginning
about this idea that Victor was our
attention was a a real valued function
of vectors in one and one forms.
This sort of thing is the thing I'm
saying attention is the put vector.
Wonderful. Then turn the hand it
comes in a number and answer.
In this case that's what the answer is.
Now we can also in the last five minutes. Uh.
Umm.
The point of a covariant um.
A covariant derivative is that
its coordinate independent.
We've described this.
Using arbitrary coordinates.
But the answer is not a
coordinate dependent thing.
That's the point of what's
the point of all this?
We're we're we're able to
answer question how do vary in
a coordinate independent way.
But if you remember.
If we had a function.
That's a scalar field. Then when we.
Obtaining the gradient of
that scalar field. We got.
A1 form answer which was was also
because a scaler is coordinate
independent which is also the the thing.
Also here is coordinate independent
so for functions. The coordinate?
The covariant derivative.
Of a function is just that,
it has it been just that. Great.
Operator, we we've seen before.
And um.
I'm not going to go through.
I invite you to look at. Equation.
The discussion just don't really
have time to go through in detail.
Equation 3242526 where we discover that
the covariant derivative of one form.
Is Pi semi? G is equal to Pi comma G minus.
Gamma. And. Key IGP.
OK, which looks just like the
expression for the derivative
of the derivative of a vector,
except there's a A minus sign.
Here where there's a plus sign for
the for the vector and there's a
quick derivation of that there which
I'm not going to explain effort
on and and similarly with with.
With the sensitive higher rank you you
end up with one plus and 1 minus sign
for each of the one form and vector.
Are they? Are they measuring
on the on on the next page?
And the last thing to to remark is
that the Leibnitz rule. Does apply?
Well, the key to make use of this PK, PK.
That's that's the commander
of a vector of a scalar,
which is a contraction of those things,
and it does end up being indeed P.
He semi colon GVK. Plus PKV.
The key thing called G.
Actually the library's rule in that
case and I I mentioned that just to
reassure that still is is the case,
it would be a strange and disturbing
derivative operator which
didn't where that wasn't true.
So the key thing here I think.
Blah blah blah.
Not over that.
Measuring why the work
very is used is useful.
Which will put over this and
ohh yeah that's the point.
When we take the derivative
of a higher rank tensor,
we get one plus and 1 minus for each of the.
Of the indexes,
I think it is useful to to just quickly
go through these key points because
we've covered a lot in this section.
It all, it is a bit of a hair dryer
lecture this one which is disturbing,
but there are.
All the time the question being
answered had been had been
afraid of straight forward one.
Given a vector field, how does it vary?
How can you talk about how it
varies as you move around the space
in a coordinate independent way?
And we'll talk about those defined
what the components of that,
we'll talk how that and there is
a tensor A11 tensor which gives
you the answer to that question.
The question then comes how do we
find the components of that tensor,
the IJ components of that tensor?
We discovered by slightly reversing
into it that we could do that
by simply considering the, the,
the, the, the,
the steps involved in differentiating
the components and then the basis.
Picture of our vector.
And we had this interesting rather
strange notation with semicolons,
discovered the covariant derivatives of
vectors of scalars and one forms to be
something we had sort of seen before.
And although I had rather hoped to
get on to the to the next section,
I have managed to get through this
section without rebellion on your part.