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The other thing is the last
lecture before the break ended a

00:00:28.000 --> 00:00:31.000
bit abruptly because I ran out
of time,

00:00:31.000 --> 00:00:34.000
so just to summarize what the
main point was,

00:00:34.000 --> 00:00:37.000
I mean probably you figured
this out if you looked at the

00:00:37.000 --> 00:00:39.000
notes.
It is not that important,

00:00:39.000 --> 00:00:43.000
but anyway.
I just wanted to remind you,

00:00:43.000 --> 00:00:48.000
just to clarify what happened
at the end, we got the diffusion

00:00:48.000 --> 00:00:52.000
equation from two bits of
information.

00:00:52.000 --> 00:00:55.000
I mean the unknown,
in this partial differential

00:00:55.000 --> 00:00:59.000
equation, is a function that we
call u that corresponds to the

00:00:59.000 --> 00:01:04.000
concentration of some substance.
And we used a vector field that

00:01:04.000 --> 00:01:12.000
represents the flow of whatever
the substance is whose diffusion

00:01:12.000 --> 00:01:17.000
we are studying.
And so we got two bits of

00:01:17.000 --> 00:01:25.000
information, one that came from
physics that said the flow goes

00:01:25.000 --> 00:01:31.000
from high concentration to
smaller concentration.

00:01:31.000 --> 00:01:37.000
And that told us that the flow
is proportional to a negative

00:01:37.000 --> 00:01:42.000
gradient of a concentration.
And the second piece of

00:01:42.000 --> 00:01:46.000
information that we got was from
the divergence theorem,

00:01:46.000 --> 00:01:50.000
and that was the one I spent
time trying to explain.

00:01:50.000 --> 00:01:57.000
And that one told us that the
divergence of F is actually

00:01:57.000 --> 00:02:02.000
negative partial u over partial
t.

00:02:02.000 --> 00:02:07.000
When you combine these two
relations together that is how

00:02:07.000 --> 00:02:11.000
you get the diffusion equation.
Sorry, I should say this is not

00:02:11.000 --> 00:02:12.000
the statement of a divergence
theorem.

00:02:12.000 --> 00:02:17.000
This is something that would
derive from it with quite a few

00:02:17.000 --> 00:02:21.000
steps involved.
And so what we got out of that

00:02:21.000 --> 00:02:27.000
is a diffusion equation because
we end up getting that partial u

00:02:27.000 --> 00:02:33.000
over partial t is minus div F,
which is therefore positive k

00:02:33.000 --> 00:02:39.000
times divergence of grad u,
which is what we denoted by del

00:02:39.000 --> 00:02:42.000
square u,
the Laplacian.

00:02:42.000 --> 00:02:59.000
So, that is how we got the
diffusion equation.

00:02:59.000 --> 00:03:04.000
Anyway, I will let you have a
look at the notes that were

00:03:04.000 --> 00:03:08.000
handed out in case you really
want to see more.

00:03:08.000 --> 00:03:16.000
I just wanted to give the
missing part of the last

00:03:16.000 --> 00:03:19.000
lecture.
Let me just switch gears

00:03:19.000 --> 00:03:24.000
completely and switch to today's
topic, which is line integrals

00:03:24.000 --> 00:03:27.000
and work in 3D.
That is going to look a lot

00:03:27.000 --> 00:03:29.000
like what we did in the plane,
except, of course,

00:03:29.000 --> 00:03:32.000
there is a z coordinate.
You will see it doesn't change

00:03:32.000 --> 00:03:35.000
things much when it comes to
computing a line integral.

00:03:35.000 --> 00:03:39.000
It changes things quite a bit,
however, when it comes to

00:03:39.000 --> 00:03:42.000
testing whether a field is a
gradient field.

00:03:42.000 --> 00:03:45.000
That is why we have to be more
careful.

00:03:45.000 --> 00:03:57.000
Let's start right away with
line integrals in space.

00:03:57.000 --> 00:04:08.000
Let's say that we have a vector
field F with components P,

00:04:08.000 --> 00:04:12.000
Q and R.
We should think of it maybe as

00:04:12.000 --> 00:04:18.000
representing a force.
And let's say that we have a

00:04:18.000 --> 00:04:24.000
curve C in space.
Then the work done by the field

00:04:24.000 --> 00:04:29.000
will be the line integral along
C of F dot dr.

00:04:29.000 --> 00:04:34.000
That is a familiar formula.
And what we do with that

00:04:34.000 --> 00:04:37.000
formula is also familiar,
except now, of course,

00:04:37.000 --> 00:04:44.000
we have a z coordinate.
We are going to think of vector

00:04:44.000 --> 00:04:50.000
dr as a space vector with
components dx,

00:04:50.000 --> 00:04:56.000
dy and dz.
When we do the dot product of F

00:04:56.000 --> 00:05:04.000
with dr that will tell us that
we have to integrate Pdx Qdy

00:05:04.000 --> 00:05:07.000
Rdz.
But it is still a line integral

00:05:07.000 --> 00:05:12.000
so it is still going to turn
into a single integral when you

00:05:12.000 --> 00:05:16.000
plug in the correct values.
So the method will be exactly

00:05:16.000 --> 00:05:19.000
the same as in the plane,
namely we will find some way to

00:05:19.000 --> 00:05:22.000
parameterize our curve,
x plus x, y z in terms of a

00:05:22.000 --> 00:05:30.000
single variable,
and then we will integrate with

00:05:30.000 --> 00:05:44.000
respect to that variable.
The way that we evaluate is by

00:05:44.000 --> 00:05:56.000
parameterizing C and express x,
y, z, dx, dy,

00:05:56.000 --> 00:06:07.000
dz in terms of the parameter.
Let's do an example just to

00:06:07.000 --> 00:06:09.000
convince you that you actually
know how to do this,

00:06:09.000 --> 00:06:12.000
or at least you should know how
to do this.

00:06:12.000 --> 00:06:21.000
Let's say that I give you the
vector field with components yz,

00:06:21.000 --> 00:06:27.000
xz and xy.
And let's say that we have a

00:06:27.000 --> 00:06:33.000
curve given by x equals t^3,
y equals t^2,

00:06:33.000 --> 00:06:40.000
z equals t for t going from
zero to one.

00:06:40.000 --> 00:06:45.000
The way we will set up the line
integral for the work done will

00:06:45.000 --> 00:06:49.000
be -- Well, sorry.
Before we actually set up the

00:06:49.000 --> 00:06:51.000
line integral,
we need to know how we will

00:06:51.000 --> 00:06:54.000
express everything in terms of t
and dt.

00:06:54.000 --> 00:06:56.000
x, y and z, in terms of t,
are given here.

00:06:56.000 --> 00:07:01.000
We just need to do also dx,
dy and dz.

00:07:01.000 --> 00:07:05.000
By differentiating you get dx
is 3t^2 dt.

00:07:05.000 --> 00:07:14.000
That is the derivative of t^3,
dy will be 2t dt and dz will

00:07:14.000 --> 00:07:23.000
just be dt.
And we will evaluate the line

00:07:23.000 --> 00:07:36.000
integral for work.
That will be the integral of yz

00:07:36.000 --> 00:07:48.000
dx xz dy xy dz,
which will become -- yz is t^3

00:07:48.000 --> 00:08:04.000
times dx is 3t^2 dt plus xz is
t^4 times dy is 2t dt plus xy is

00:08:04.000 --> 00:08:08.000
t^5 dt.
That just becomes the integral

00:08:08.000 --> 00:08:11.000
from, well, I guess t goes from
zero to one, actually.

00:08:11.000 --> 00:08:14.000
And we are integrating three
plus two plus one.

00:08:14.000 --> 00:08:22.000
That is 6t^5 dt which,
I am sure you know,

00:08:22.000 --> 00:08:30.000
integrates to t^6,
so we will just get one.

00:08:30.000 --> 00:08:33.000
It is the same method as usual.
And if you are being given a

00:08:33.000 --> 00:08:36.000
geometric description of a curve
then, of course,

00:08:36.000 --> 00:08:39.000
you will have to decide for
yourself what the best parameter

00:08:39.000 --> 00:08:43.000
will be.
It might be some time parameter

00:08:43.000 --> 00:08:45.000
t like here.
It might be one of the

00:08:45.000 --> 00:08:49.000
coordinates.
Here we could have used z as

00:08:49.000 --> 00:08:53.000
our parameter because,
in fact, this curve is x equals

00:08:53.000 --> 00:08:57.000
x3 and y equals z2.
And we could also have used

00:08:57.000 --> 00:08:59.000
maybe some angle.
Well, not here,

00:08:59.000 --> 00:09:04.000
but if we had been moving on a
circle or something like that.

00:09:04.000 --> 00:09:11.000
Any questions so far?
No.

00:09:11.000 --> 00:09:26.000
OK.
Well,

00:09:26.000 --> 00:09:31.000
because we can do a bit more
practice,

00:09:31.000 --> 00:09:40.000
let's do another one where we
do the same vector field F but

00:09:40.000 --> 00:09:49.000
our curve C will be going from
the origin to the point (1,0,

00:09:49.000 --> 00:09:54.000
0) along the x-axis.
Let's call that C1.

00:09:54.000 --> 00:10:02.000
Then to (1,1, 0).
Let's call that C2 by moving

00:10:02.000 --> 00:10:08.000
parallel to the y-axis.
And then up to (1,1,

00:10:08.000 --> 00:10:14.000
1) parallel to the z-axis,
let's call that C3.

00:10:14.000 --> 00:10:19.000
I am sure that some of you at
least are suspecting what I am

00:10:19.000 --> 00:10:23.000
getting at here,
but let's not spoil it for

00:10:23.000 --> 00:10:27.000
those who don't see it yet.
OK.

00:10:27.000 --> 00:10:33.000
If we want to compute the line
integral along this guy then we

00:10:33.000 --> 00:10:38.000
have to break it into a sum of
three terms.

00:10:38.000 --> 00:10:42.000
Well, maybe I should call that
C', not C, because that is not

00:10:42.000 --> 00:10:48.000
the same C anymore.
I want to do the sum of the

00:10:48.000 --> 00:10:55.000
line integrals along C1,
C2 and C3.

00:10:55.000 --> 00:11:13.000
And, well, if I look at C1 and
C2 they take place inside the x,

00:11:13.000 --> 00:11:19.000
y plane.
In fact, you know that z will

00:11:19.000 --> 00:11:25.000
be zero and dz will also be zero
on both of these.

00:11:25.000 --> 00:11:30.000
And if you just look at the
formula for line integral in the

00:11:30.000 --> 00:11:34.000
role of yz dx plus xz dy plus xy
dz,

00:11:34.000 --> 00:11:37.000
well, it looks like if you plug
z equals zero and dz equals zero

00:11:37.000 --> 00:11:47.000
you will just get zero.
These are actually very fast.

00:11:47.000 --> 00:12:00.000
Let me write it.
This is going to be zero,

00:12:00.000 --> 00:12:04.000
this is going to be zero,
this is going to zero,

00:12:04.000 --> 00:12:07.000
and we will get zero.
Now, if we do C3,

00:12:07.000 --> 00:12:11.000
well, there we might have to do
some calculation,

00:12:11.000 --> 00:12:18.000
but it won't be all that bad.
C3, well, x and y are both

00:12:18.000 --> 00:12:22.000
equal to one.
And, of course,

00:12:22.000 --> 00:12:26.000
because they are constant that
means dx is zero and dy is zero.

00:12:26.000 --> 00:12:35.000
On the other hand,
z varies from zero to one.

00:12:35.000 --> 00:12:42.000
If I look at the line integral
on C3 -- The first two terms,

00:12:42.000 --> 00:12:48.000
yz, dx and xz dy go away
because the dx and dy are zero,

00:12:48.000 --> 00:12:55.000
so I am just left with xy dz.
But because x and y are one it

00:12:55.000 --> 00:13:01.000
is just the integral of dz from
zero to one, and that will just

00:13:01.000 --> 00:13:07.000
end up being one.
If add these numbers together,

00:13:07.000 --> 00:13:13.000
zero plus zero plus one,
I get one again.

00:13:13.000 --> 00:13:15.000
And, of course,
it is not a coincidence because

00:13:15.000 --> 00:13:18.000
this vector field is a gradient
field.

00:13:18.000 --> 00:13:20.000
I am sure some of you have
already figured out what it is

00:13:20.000 --> 00:13:24.000
the gradient of.
Otherwise, we will figure it

00:13:24.000 --> 00:13:28.000
out together.
And so that is why we get the

00:13:28.000 --> 00:13:33.000
same answer for these two paths
going both from the origin to

00:13:33.000 --> 00:13:36.000
(1,1, 1).
Maybe I should point out,

00:13:36.000 --> 00:13:40.000
to make it clear,
that if you plug t equals zero

00:13:40.000 --> 00:13:42.000
in up there you will get (0,0,
0).

00:13:42.000 --> 00:13:45.000
If you plug t equals one,
you will get (1,1,

00:13:45.000 --> 00:13:56.000
1).
In fact -- F that we have here

00:13:56.000 --> 00:14:11.000
happens to be conservative.
And, if you plug the two curves

00:14:11.000 --> 00:14:20.000
together -- Well,
I am not really sure if I know

00:14:20.000 --> 00:14:28.000
how to plot this correctly.
It is not exactly how it looks.

00:14:28.000 --> 00:14:33.000
Whatever.
The first curve C goes from the

00:14:33.000 --> 00:14:39.000
origin to this point,
and so does C ',

00:14:39.000 --> 00:14:45.000
just in a slightly more
roundabout way.

00:14:45.000 --> 00:14:51.000
They both go from the origin to
(1,1, 1).

00:14:51.000 --> 00:15:00.000
It is not a surprise that you
will get the same answer for

00:15:00.000 --> 00:15:05.000
both line integrals.
And how do we see that?

00:15:05.000 --> 00:15:11.000
Well, actually here it is not
very hard to find a function

00:15:11.000 --> 00:15:15.000
whose gradient is this vector
field.

00:15:15.000 --> 00:15:21.000
Namely, the gradient of x,
y, z looks like it should be

00:15:21.000 --> 00:15:26.000
exactly what we want.
If you take partial of this

00:15:26.000 --> 00:15:29.000
with respect to x,
you will get yz,

00:15:29.000 --> 00:15:33.000
then with respect to y,
xz, and with respect to z,

00:15:33.000 --> 00:15:35.000
xy.
And so, in fact,

00:15:35.000 --> 00:15:38.000
what was the easier way to
compute these line integrals was

00:15:38.000 --> 00:15:41.000
to use the fundamental theorem
of calculus.

00:15:41.000 --> 00:15:45.000
Once we have this remark,
we don't need to compute these

00:15:45.000 --> 00:15:56.000
line integrals anymore.
We can just use the fundamental

00:15:56.000 --> 00:16:08.000
theorem.
If we know this fundamental

00:16:08.000 --> 00:16:20.000
theorem -- -- for line
integrals,

00:16:20.000 --> 00:16:27.000
that tells us that the line
integral of a gradient field is

00:16:27.000 --> 00:16:33.000
equal to the value of the
potential at the final point

00:16:33.000 --> 00:16:40.000
minus the value of the potential
at the starting point.

00:16:40.000 --> 00:16:43.000
And that, of course,
only applies if you have a

00:16:43.000 --> 00:16:46.000
potential.
So, in particular,

00:16:46.000 --> 00:16:52.000
only if you have a conservative
field, a gradient field.

00:16:52.000 --> 00:16:58.000
Here, in our example,
we have to look at,

00:16:58.000 --> 00:17:06.000
let's call little f of x,
y, z that potential xyz,

00:17:06.000 --> 00:17:12.000
then we take f(1,1,
1) - f(0,0, 0).

00:17:12.000 --> 00:17:17.000
And that indeed is one minus
zero which is one.

00:17:17.000 --> 00:17:24.000
Everything is consistent.
All this stuff so far works

00:17:24.000 --> 00:17:32.000
exactly as in the plane.
Any questions?

00:17:32.000 --> 00:17:35.000
No.
OK.

00:17:35.000 --> 00:17:40.000
Let's try to see where things
do get a little bit different.

00:17:40.000 --> 00:17:44.000
And the first such place is
when we try to test whether a

00:17:44.000 --> 00:17:47.000
vector field is a gradient
field.

00:17:47.000 --> 00:17:53.000
Remember when we had a vector
field in the plane,

00:17:53.000 --> 00:17:56.000
to know whether it was a
gradient of a function of two

00:17:56.000 --> 00:17:59.000
variables we just had to check
one condition,

00:17:59.000 --> 00:18:04.000
N sub x equals M sub y.
Now we actually have three

00:18:04.000 --> 00:18:06.000
different conditions to check,
and that means,

00:18:06.000 --> 00:18:07.000
of course, more work.

00:18:23.000 --> 00:18:34.000
OK.
So what is our test for

00:18:34.000 --> 00:18:44.000
gradient fields?
We want to know whether a given

00:18:44.000 --> 00:18:50.000
vector field with components P,
Q and R can be written as f sub

00:18:50.000 --> 00:18:55.000
x, f sub y and f sub z for a
same function F.

00:18:55.000 --> 00:18:59.000
And for that to possibly
happen, well,

00:18:59.000 --> 00:19:03.000
we need certainly some
relations between P,

00:19:03.000 --> 00:19:05.000
Q and R.
And, as before,

00:19:05.000 --> 00:19:11.000
this comes from the fact that
the mixed second derivatives are

00:19:11.000 --> 00:19:16.000
the same, no matter in which
order you take them.

00:19:16.000 --> 00:19:21.000
If that is the case then I can
compute f sub xy,

00:19:21.000 --> 00:19:28.000
which is the same as f sub yx
in two different ways.

00:19:28.000 --> 00:19:35.000
F sub xy should be P sub y.
F sub yx, well,

00:19:35.000 --> 00:19:39.000
since f sub y is Q,
that should be Q sub x.

00:19:39.000 --> 00:19:43.000
That is a part of a criterion
that we already had when we had

00:19:43.000 --> 00:19:46.000
only two variables.
But now, of course,

00:19:46.000 --> 00:19:50.000
we need to do the same thing
when we look at x and z or y and

00:19:50.000 --> 00:19:53.000
z.
That gives us two more

00:19:53.000 --> 00:19:58.000
conditions.
P sub z is f sub xz,

00:19:58.000 --> 00:20:10.000
which is the same as f sub zx,
so it should be the same as R

00:20:10.000 --> 00:20:14.000
sub x.
Finally, Q sub z,

00:20:14.000 --> 00:20:22.000
which is f sub yz,
equals f sub zy equals R sub y.

00:20:22.000 --> 00:20:33.000
We have three conditions,
so our criterion -- Vector

00:20:33.000 --> 00:20:41.000
field F equals <P,
Q, R>.

00:20:41.000 --> 00:20:46.000
And here, to be completely
truthful, I have to say defined

00:20:46.000 --> 00:20:50.000
in a simply connected region.
Otherwise, we might have the

00:20:50.000 --> 00:20:53.000
same kind of strange things
happening as before.

00:20:53.000 --> 00:21:03.000
Let's not worry too much about
it.

00:21:03.000 --> 00:21:07.000
For accuracy we need our vector
field to be defined in a simply

00:21:07.000 --> 00:21:10.000
connected region.
And example is just if it is

00:21:10.000 --> 00:21:14.000
defined everywhere.
If you don't have any evil

00:21:14.000 --> 00:21:21.000
eliminators then you can just go
ahead and there is no problem.

00:21:21.000 --> 00:21:36.000
It is a gradient field.
We need three conditions.

00:21:36.000 --> 00:21:43.000
Let's do it in order.
P sub y equals Q sub x.

00:21:43.000 --> 00:21:52.000
And we have P sub z equals R
sub x and Q sub z equals R sub

00:21:52.000 --> 00:21:55.000
y.
How do you remember these three

00:21:55.000 --> 00:21:56.000
conditions?
Well, it is pretty easy.

00:21:56.000 --> 00:22:01.000
You pick any two components,
say the x and the z component,

00:22:01.000 --> 00:22:04.000
and you take the partial of the
x component with respect to z,

00:22:04.000 --> 00:22:07.000
the partial of the z component
with respect to x and you must

00:22:07.000 --> 00:22:12.000
make them equal.
And the same with every pair of

00:22:12.000 --> 00:22:15.000
variables.
In fact, if you had a function

00:22:15.000 --> 00:22:17.000
of many more variables the
criterion would still look

00:22:17.000 --> 00:22:21.000
exactly like that.
For every pair of components

00:22:21.000 --> 00:22:24.000
the mixed partials must be the
same.

00:22:24.000 --> 00:22:27.000
But we are not going to go
beyond three variables so you

00:22:27.000 --> 00:22:32.000
don't need to know that.
This you need to know so let me

00:22:32.000 --> 00:22:33.000
box it.

00:22:51.000 --> 00:23:02.000
That is pretty straightforward.
Let's do an example just to see

00:23:02.000 --> 00:23:11.000
how it goes.
By the way, we can also think

00:23:11.000 --> 00:23:19.000
of it in terms of differentials.
Before I do the example,

00:23:19.000 --> 00:23:22.000
let me just say in a different
language.

00:23:22.000 --> 00:23:29.000
If we have a differential given
to us of a form Pdx Qdy Rdz is

00:23:29.000 --> 00:23:33.000
going to be an exact
differential,

00:23:33.000 --> 00:23:40.000
which means it is equal to df
for some function F exactly and

00:23:40.000 --> 00:23:43.000
of the same conditions.
That is the same thing.

00:23:43.000 --> 00:23:51.000
Just in the language of
differentials.

00:23:51.000 --> 00:23:57.000
The example that I promised.
Of course, I could do again the

00:23:57.000 --> 00:24:00.000
same one over there and check
that it satisfies the condition,

00:24:00.000 --> 00:24:02.000
but then it wouldn't be much
fun.

00:24:02.000 --> 00:24:10.000
So let's do a better one.
Actually, let's do it in a way

00:24:10.000 --> 00:24:18.000
that looks like an exam problem.
Let's say for which a and b is

00:24:18.000 --> 00:24:29.000
a xy dx plus -- Oh,
it is not going to fit here.

00:24:29.000 --> 00:24:39.000
But it will fit here.
a xy dx ( x^2 z^3) dy (byz^2 -

00:24:39.000 --> 00:24:51.000
4z^3) dz, an exact differential.
Or, if you don't like exact

00:24:51.000 --> 00:24:54.000
differentials,
for which a and b is the

00:24:54.000 --> 00:24:58.000
corresponding vector field with
i, j and k instead,

00:24:58.000 --> 00:25:02.000
a gradient field.
Let's just apply the criterion.

00:25:02.000 --> 00:25:06.000
And, of course,
you can guess that what will

00:25:06.000 --> 00:25:11.000
follow is figuring out how to
find the potential when there is

00:25:11.000 --> 00:25:12.000
one.

00:25:33.000 --> 00:25:40.000
Let's do it one by one.
We want to compare P sub y with

00:25:40.000 --> 00:25:45.000
Q sub x,
we want to compare P sub z with

00:25:45.000 --> 00:25:53.000
R sub x and we want to compare Q
sub z with R sub y where we call

00:25:53.000 --> 00:25:57.000
P, Q and R these guys.
Let's see.

00:25:57.000 --> 00:26:04.000
What is P sub y?
That seems to be ax.

00:26:04.000 --> 00:26:09.000
What is Q sub x?
2x.

00:26:09.000 --> 00:26:12.000
Q is this one.
Actually, let me write them

00:26:12.000 --> 00:26:16.000
down.
Because otherwise I am going to

00:26:16.000 --> 00:26:21.000
get confused myself.
This guy here,

00:26:21.000 --> 00:26:32.000
that is P, this guy here,
that is Q and that guy here,

00:26:32.000 --> 00:26:38.000
that is R.
This one tells us that a should

00:26:38.000 --> 00:26:43.000
be equal to two of the first
product that you hold.

00:26:43.000 --> 00:26:49.000
OK. Let's look at P sub z.
That is just zero.

00:26:49.000 --> 00:26:51.000
R sub x?
Well, R doesn't have any x

00:26:51.000 --> 00:26:56.000
either so that is zero.
This one is not a problem.

00:26:56.000 --> 00:27:02.000
Q sub z?
Well, that seems to be 3z2.

00:27:02.000 --> 00:27:14.000
R sub y seems to be bz2,
so b should be equal to three.

00:27:14.000 --> 00:27:19.000
We need to have a equals two
and, this is an and,

00:27:19.000 --> 00:27:23.000
not or, b equals three for this
to be exact.

00:27:23.000 --> 00:27:27.000
For those values of a and b,
we can look for a potential

00:27:27.000 --> 00:27:30.000
using the method that we are
going to see right now.

00:27:30.000 --> 00:27:33.000
For any other values of a and b
we cannot.

00:27:33.000 --> 00:27:38.000
If we have to compute a line
integral, we have to do it by

00:27:38.000 --> 00:27:42.000
finding a parameter and setting
up everything.

00:27:42.000 --> 00:27:57.000
Any questions at this point?
Yes?

00:27:57.000 --> 00:28:00.000
I see.
Well, if I got the same answer,

00:28:00.000 --> 00:28:05.000
oh, did say bz^2 or 3bz^2?
Well, 3bz^2, for example,

00:28:05.000 --> 00:28:09.000
I would need b to be zero
because the only time that 3bz2

00:28:09.000 --> 00:28:13.000
equals bz2 as not just at one
point but everywhere,

00:28:13.000 --> 00:28:15.000
I need them to be the same
function of x,

00:28:15.000 --> 00:28:18.000
y, z.
Well, if a coefficient of z2 is

00:28:18.000 --> 00:28:22.000
the same that would be give b
equals 3b, that would give me b

00:28:22.000 --> 00:28:25.000
equals zero.
If you got bz2 on both sides

00:28:25.000 --> 00:28:29.000
then it would mean for any value
of b it works,

00:28:29.000 --> 00:28:35.000
and you wouldn't have to worry
about what the value of b is.

00:28:35.000 --> 00:28:41.000
Any other questions?
No.

00:28:41.000 --> 00:28:50.000
OK.
Now, how do we find the

00:28:50.000 --> 00:28:58.000
potential?
Well, there are two methods as

00:28:58.000 --> 00:28:59.000
before.
One of them,

00:28:59.000 --> 00:29:02.000
I don't remember if it was the
first one or the second one last

00:29:02.000 --> 00:29:04.000
time, but it really doesn't
matter.

00:29:04.000 --> 00:29:10.000
One of them was just to say
that the value of F at the

00:29:10.000 --> 00:29:15.000
point,
let me call that x1, y1, z1,

00:29:15.000 --> 00:29:22.000
is equal to the line integral
of my field along a well-chosen

00:29:22.000 --> 00:29:25.000
curve plus,
of course, a constant,

00:29:25.000 --> 00:29:30.000
which is going to be the
integration constant.

00:29:30.000 --> 00:29:39.000
And the kind of curve that I
will take to do this calculation

00:29:39.000 --> 00:29:48.000
will just be my favorite curve
going from the origin to the

00:29:48.000 --> 00:29:51.000
point x1, y1,
z1.

00:29:51.000 --> 00:29:57.000
And so, typically the most
common choice would be to go

00:29:57.000 --> 00:30:04.000
just first along the x-axis,
then parallel to the y-axis and

00:30:04.000 --> 00:30:11.000
then parallel to the z-axis all
the way to my point x1,

00:30:11.000 --> 00:30:14.000
y1, z1.
I would just calculate three

00:30:14.000 --> 00:30:18.000
easy line integrals.
Add them together and that

00:30:18.000 --> 00:30:21.000
would give me the value of my
function.

00:30:21.000 --> 00:30:27.000
That method works exactly the
same way as it did in two

00:30:27.000 --> 00:30:30.000
variables.
Now, I seem to recall that you

00:30:30.000 --> 00:30:32.000
guys mostly preferred the other
method.

00:30:32.000 --> 00:30:34.000
I am going to tell you about
the other method as well,

00:30:34.000 --> 00:30:37.000
but I just want to point out
this one actually doesn't become

00:30:37.000 --> 00:30:40.000
more complicated.
The other one has actually more

00:30:40.000 --> 00:30:42.000
steps.
I mean, of course,

00:30:42.000 --> 00:30:45.000
here there are also a bit more
steps because you have three

00:30:45.000 --> 00:30:47.000
parts to your path instead of
two.

00:30:47.000 --> 00:30:54.000
You have three line integrals
to compute instead of two,

00:30:54.000 --> 00:30:59.000
but conceptually it remains
exactly the same idea.

00:30:59.000 --> 00:31:07.000
I should say it works the same
way as in 2D.

00:31:07.000 --> 00:31:17.000
Not much changes.
Let's look at the other method

00:31:17.000 --> 00:31:24.000
using anti-derivatives.
Remember we want to find a

00:31:24.000 --> 00:31:29.000
function little f whose partials
are exactly the things we have

00:31:29.000 --> 00:31:31.000
been given.
We want to solve,

00:31:31.000 --> 00:31:36.000
well, let me plug in the values
of a and b that will work.

00:31:36.000 --> 00:31:47.000
We said a should be two,
so f sub x should be 2xy,

00:31:47.000 --> 00:31:59.000
f sub y should be x2 plus z3,
and f sub z should be 3yz^2

00:31:59.000 --> 00:32:04.000
minus 4z^3.
We are going to look at them

00:32:04.000 --> 00:32:07.000
one at a time and get partial
information on the function.

00:32:07.000 --> 00:32:11.000
And then we will compare with
the others to get more

00:32:11.000 --> 00:32:15.000
information until we are
completely done.

00:32:15.000 --> 00:32:25.000
The first thing we will do,
we know that f sub x is 2xy.

00:32:25.000 --> 00:32:27.000
That should tell us something
about f.

00:32:27.000 --> 00:32:31.000
Well, let's just integrate that
with respect to x.

00:32:31.000 --> 00:32:36.000
Let me write integral dx next
to that.

00:32:36.000 --> 00:32:41.000
That tells us that f should be,
well, if we integrate that with

00:32:41.000 --> 00:32:44.000
respect to x,
2x integrates to x^2,

00:32:44.000 --> 00:32:47.000
so we should get x2y.
Plus, of course,

00:32:47.000 --> 00:32:50.000
an integration constant.
Now, what do we mean by

00:32:50.000 --> 00:32:53.000
integration constant.
It means that for given values

00:32:53.000 --> 00:32:58.000
of y and z we will get a term
that does not depend on x.

00:32:58.000 --> 00:33:02.000
It still depends on y and z.
In fact, what we get is a

00:33:02.000 --> 00:33:07.000
function of y and z.
See, if you took the derivative

00:33:07.000 --> 00:33:12.000
of this with respect to x you
will get 2xy and this guy will

00:33:12.000 --> 00:33:16.000
go away because there is no x in
it.

00:33:16.000 --> 00:33:19.000
That is the first step.
Now we need to get some

00:33:19.000 --> 00:33:21.000
information on g.
How do we do that?

00:33:21.000 --> 00:33:28.000
Well, we look at the other
partials.

00:33:28.000 --> 00:33:35.000
F sub y, we want that to be x^2
z^3.

00:33:35.000 --> 00:33:41.000
But we have another way to find
it, which is starting from this

00:33:41.000 --> 00:33:50.000
and differentiating.
Let me try to use color for

00:33:50.000 --> 00:33:54.000
this.
Now, if I take the partial of

00:33:54.000 --> 00:33:58.000
this with respect to y,
I am going to get a different

00:33:58.000 --> 00:34:06.000
formula for f sub y.
That will be x^2 plus g sub y.

00:34:06.000 --> 00:34:15.000
Well, if I compare these two
expressions that tells me that g

00:34:15.000 --> 00:34:23.000
sub y should be z3.
Now, if I have this I can

00:34:23.000 --> 00:34:32.000
integrate with respect to y.
That will tell me that g is

00:34:32.000 --> 00:34:39.000
actually yz^3 plus an
integration constant.

00:34:39.000 --> 00:34:42.000
That constant,
again, does not depend on y,

00:34:42.000 --> 00:34:46.000
but it can still depend on z
because we still have not said

00:34:46.000 --> 00:34:49.000
anything about partial with
respect to z.

00:34:49.000 --> 00:34:54.000
In fact, that constant I will
write as a function h of z.

00:34:54.000 --> 00:35:00.000
If I have this function of z
and I take its partial with

00:35:00.000 --> 00:35:04.000
respect to y,
I will still get z^3 no matter

00:35:04.000 --> 00:35:06.000
what h was.
Now, how do I find h?

00:35:06.000 --> 00:35:09.000
Well, obviously,
I have to look at f sub z.

00:35:41.000 --> 00:35:47.000
F sub z.
We know from the given vector

00:35:47.000 --> 00:35:53.000
field that we want it to be
3yz^2 minus 4z^3.

00:35:53.000 --> 00:35:56.000
In case you are wondering where
that came from,

00:35:56.000 --> 00:36:01.000
that was R.
But that is also obtained by

00:36:01.000 --> 00:36:09.000
differentiating with respect to
z what we had so far.

00:36:09.000 --> 00:36:15.000
Sorry.
What did we have so far?

00:36:15.000 --> 00:36:20.000
Well, we had f equals x^2y plus
g.

00:36:20.000 --> 00:36:30.000
And we said g is actually yz^3
plus h of z.

00:36:30.000 --> 00:36:37.000
That is what we have so far.
If we take the derivative of

00:36:37.000 --> 00:36:44.000
that with respect to z,
we will get zero plus 3yz^2

00:36:44.000 --> 00:36:51.000
plus h prime of z,
or dh dz as you want.

00:36:51.000 --> 00:36:59.000
Now, if we compare these two,
we will get the derivative of

00:36:59.000 --> 00:37:03.000
h.
It will tell us that h prime is

00:37:03.000 --> 00:37:08.000
negative for z3.
That means that h is negative

00:37:08.000 --> 00:37:13.000
z^4 plus a constant.
And this it is at last an

00:37:13.000 --> 00:37:17.000
actual constant.
Because it does not depend on z

00:37:17.000 --> 00:37:21.000
and there is nothing else for it
to depend on.

00:37:21.000 --> 00:37:33.000
Now we plug this into what we
had before, and that will give

00:37:33.000 --> 00:37:44.000
us our function f.
We get that f=x^2y yz^3 - z^4

00:37:44.000 --> 00:37:49.000
plus constant.
If you just wanted to find one

00:37:49.000 --> 00:37:51.000
potential, you can just forget
the constant.

00:37:51.000 --> 00:37:55.000
This guy was a potential.
If you want all the potentials

00:37:55.000 --> 00:37:58.000
they differ by this constant.
OK.

00:37:58.000 --> 00:38:01.000
Just to recap the method what
did we do?

00:38:01.000 --> 00:38:04.000
We started with -- And,
of course, you can do it in

00:38:04.000 --> 00:38:07.000
whichever order you prefer,
but you have to still follow

00:38:07.000 --> 00:38:11.000
the systematic method.
You start with f sub x and you

00:38:11.000 --> 00:38:13.000
integrate that with respect to
x.

00:38:13.000 --> 00:38:19.000
That gives you f up to a
function of y and z only.

00:38:19.000 --> 00:38:25.000
Now you compare f sub y as
given to you by the vector field

00:38:25.000 --> 00:38:32.000
with the formula you get from
this expression for f.

00:38:32.000 --> 00:38:36.000
And, of course,
this one will involve g sub y.

00:38:36.000 --> 00:38:41.000
Out of this,
you will get the value of g sub

00:38:41.000 --> 00:38:44.000
y.
When you have g sub y that

00:38:44.000 --> 00:38:48.000
gives you g up to a function of
z only.

00:38:48.000 --> 00:38:52.000
And so now you have f up to a
function of z only.

00:38:52.000 --> 00:38:57.000
And what you will do is look at
the derivative with respect to

00:38:57.000 --> 00:38:59.000
z,
the one you want coming from

00:38:59.000 --> 00:39:02.000
the vector field and the one you
have coming from this formula

00:39:02.000 --> 00:39:05.000
for f,
match them and that will tell

00:39:05.000 --> 00:39:09.000
you h prime.
You will get h and then you

00:39:09.000 --> 00:39:20.000
will get f.
Any questions?

00:39:20.000 --> 00:39:25.000
Who still prefers this method?
OK, still most of you.

00:39:25.000 --> 00:39:29.000
Who is thinking that maybe the
other method was not so bad

00:39:29.000 --> 00:39:33.000
after all?
OK. That is still a minority.

00:39:33.000 --> 00:39:36.000
You can choose whichever one
you prefer.

00:39:36.000 --> 00:39:42.000
I would encourage you to get
some practice by trying both on

00:39:42.000 --> 00:39:47.000
least a couple of examples just
to make sure that you know how

00:39:47.000 --> 00:39:53.000
to do them both and then stick
to whichever one you prefer.

00:39:53.000 --> 00:39:59.000
Any questions on that?
No. I guess I already asked.

00:39:59.000 --> 00:40:07.000
Still no questions?
OK.

00:40:07.000 --> 00:40:13.000
The next logical thing is going
to be curl.

00:40:13.000 --> 00:40:17.000
And the theorem that is going
to replace Green's theorem for

00:40:17.000 --> 00:40:21.000
work in this setting is going to
be called Stokes' theorem.

00:40:21.000 --> 00:40:34.000
Let me start by telling you
about curl in 3D.

00:40:34.000 --> 00:40:39.000
Here is the statement.
The curl is just going to

00:40:39.000 --> 00:40:43.000
measure how much your vector
field fails to be conservative.

00:40:43.000 --> 00:40:47.000
And, if you want to think about
it in terms of motions,

00:40:47.000 --> 00:40:51.000
that also will measure the
rotation part of the motion.

00:40:51.000 --> 00:40:55.000
Well, let me first give a
definition.

00:40:55.000 --> 00:41:00.000
Let's say that my vector field
has components P,

00:41:00.000 --> 00:41:06.000
Q and R.
Then we define the curl of F to

00:41:06.000 --> 00:41:16.000
be R sub y minus Q sub z times i
plus P sub z minus R sub x times

00:41:16.000 --> 00:41:21.000
j plus Q sub x minus P sub y
times k.

00:41:21.000 --> 00:41:25.000
And of course nobody can
remember this formula,

00:41:25.000 --> 00:41:28.000
so what is the structure of
this formula?

00:41:28.000 --> 00:41:35.000
Well, you see,
each of these guys is one of

00:41:35.000 --> 00:41:43.000
the things that have to be zero
for our field to be

00:41:43.000 --> 00:41:52.000
conservative.
If F is defined in a simply

00:41:52.000 --> 00:42:05.000
connected region then we have
that F is conservative and is

00:42:05.000 --> 00:42:15.000
equivalent to if and only if
curl F is zero.

00:42:15.000 --> 00:42:19.000
Now, an important difference
between curl here and curl in

00:42:19.000 --> 00:42:23.000
the plane is that now the curl
of a vector field is again a

00:42:23.000 --> 00:42:27.000
vector field.
These expressions are functions

00:42:27.000 --> 00:42:31.000
of x, y, z and together you form
a vector out of them.

00:42:31.000 --> 00:42:36.000
The curl of a vector field in
space is actually a vector

00:42:36.000 --> 00:42:43.000
field, not a scalar function.
I have delayed the inevitable.

00:42:43.000 --> 00:42:47.000
I have to really tell you how
to remember this evil formula.

00:42:47.000 --> 00:42:55.000
The secret is that,
in fact, you can think of this

00:42:55.000 --> 00:43:01.000
as del cross f.
Maybe you have seen that in

00:43:01.000 --> 00:43:04.000
physics.
This is really where this del

00:43:04.000 --> 00:43:08.000
notation becomes extremely
useful, because that is

00:43:08.000 --> 00:43:13.000
basically the only way to
remember the formula for curl.

00:43:30.000 --> 00:43:34.000
Remember we introduced the dell
operator.

00:43:34.000 --> 00:43:42.000
That was this symbolic vector
operator in which the components

00:43:42.000 --> 00:43:47.000
are the partial derivative
operators.

00:43:47.000 --> 00:43:59.000
We have seen that if you apply
this to a scalar function then

00:43:59.000 --> 00:44:08.000
that will give you the gradient.
And we have seen that if you do

00:44:08.000 --> 00:44:13.000
the dot product between dell and
a vector field,

00:44:13.000 --> 00:44:19.000
maybe I should give it
components P,

00:44:19.000 --> 00:44:24.000
Q and R,
you will get partial P over

00:44:24.000 --> 00:44:30.000
partial x plus partial Q over
partial y plus partial R over

00:44:30.000 --> 00:44:36.000
partial z,
which is the divergence.

00:44:36.000 --> 00:44:47.000
And so now what is new is that
if I try to do dell cross F,

00:44:47.000 --> 00:44:53.000
well, what is dell cross F?
I have to set up a

00:44:53.000 --> 00:44:58.000
cross-product between this
strange thing that is not really

00:44:58.000 --> 00:45:02.000
a vector.
I mean, I cannot really think

00:45:02.000 --> 00:45:05.000
of partial over partial x as a
number.

00:45:05.000 --> 00:45:10.000
And my vector field <P,
Q, R>.

00:45:10.000 --> 00:45:14.000
See, that is really a
completely perverted use of a

00:45:14.000 --> 00:45:18.000
determinant notation.
Initially, determinants were

00:45:18.000 --> 00:45:21.000
just supposed to be you had a
three by three table of numbers

00:45:21.000 --> 00:45:23.000
and you computed a number out of
them.

00:45:23.000 --> 00:45:28.000
These guys are functions so
they count as numbers,

00:45:28.000 --> 00:45:33.000
but these are vectors and these
are partial derivatives.

00:45:33.000 --> 00:45:37.000
It doesn't really make much
sense, except this notation.

00:45:37.000 --> 00:45:43.000
If you try to enter this into a
calculator or computer,

00:45:43.000 --> 00:45:48.000
it will just yell back at you
saying are you crazy.

00:45:48.000 --> 00:45:50.000
[LAUGHTER]
We just use that as a notation

00:45:50.000 --> 00:45:53.000
to remember what is in there.
Let's try and see how that

00:45:53.000 --> 00:45:55.000
works.
The component of i in this

00:45:55.000 --> 00:45:58.000
cross-product,
remember that is this smaller

00:45:58.000 --> 00:46:01.000
determinant,
that smaller determinant is

00:46:01.000 --> 00:46:07.000
partial over partial y of R
minus partial over partial z of

00:46:07.000 --> 00:46:10.000
Q,
the coefficient of i.

00:46:10.000 --> 00:46:14.000
And that seems to be what I had
over there.

00:46:14.000 --> 00:46:18.000
If not then I made a mistake.
Minus the next determinant

00:46:18.000 --> 00:46:20.000
times z.
Remember there is always a

00:46:20.000 --> 00:46:23.000
minus sign in front of a j
component when you do a

00:46:23.000 --> 00:46:27.000
cross-product.
The other one is partial over

00:46:27.000 --> 00:46:33.000
partial x R minus partial over
partial z of P plus the

00:46:33.000 --> 00:46:39.000
component of z which is going to
be partial over partial x Q

00:46:39.000 --> 00:46:46.000
minus partial over partial y P.
And that is indeed going to be

00:46:46.000 --> 00:46:48.000
the curl of F.
In practice,

00:46:48.000 --> 00:46:51.000
if you have to compute the curl
of a vector field,

00:46:51.000 --> 00:46:53.000
you know, don't try to remember
this formula.

00:46:53.000 --> 00:46:59.000
Just set up this cross-product
with whatever formulas you have

00:46:59.000 --> 00:47:04.000
for the components of a field
and then compute it.

00:47:04.000 --> 00:47:15.000
Don't bother to try to remember
the general formula,

00:47:15.000 --> 00:47:25.000
just remember this.
What is the geometric

00:47:25.000 --> 00:47:35.000
interpretation of curl,
just to finish?

00:47:35.000 --> 00:47:48.000
In a way, I will say just curl
measures the rotation component

00:47:48.000 --> 00:47:55.000
in a velocity field.
An exercise that you can do,

00:47:55.000 --> 00:47:59.000
which is actually pretty easy
to check, is say that we have a

00:47:59.000 --> 00:48:04.000
fluid that is just rotating
about the x-axis uniformly.

00:48:04.000 --> 00:48:10.000
Your fluid is just rotating
like that about the z-axis.

00:48:10.000 --> 00:48:15.000
If I take a rotation about the
z-axis.

00:48:15.000 --> 00:48:28.000
That is given by a velocity
field with components at angular

00:48:28.000 --> 00:48:35.000
velocity omega.
That will be negative omega

00:48:35.000 --> 00:48:41.000
times y, then omega x and zero.
And the curl of that you can

00:48:41.000 --> 00:48:45.000
compute, and you will find two
omega times k.

00:48:45.000 --> 00:48:48.000
Concretely,
this curl gives you the angular

00:48:48.000 --> 00:48:51.000
velocity of the rotation,
well, with a factor two but

00:48:51.000 --> 00:48:53.000
that doesn't matter,
and the axis of rotation,

00:48:53.000 --> 00:48:56.000
the direction of the axis of
rotation.

00:48:56.000 --> 00:48:59.000
It tells you it is rotating
about a vertical axis.

00:48:59.000 --> 00:49:01.000
And, in general,
if you have a complicated

00:49:01.000 --> 00:49:03.000
motion some of it might be,
you know, there is a

00:49:03.000 --> 00:49:05.000
translation.
And then within that

00:49:05.000 --> 00:49:08.000
translation there is maybe
expansion and rotation and

00:49:08.000 --> 00:49:11.000
sharing and everything.
And the curl will compute how

00:49:11.000 --> 00:49:14.000
much rotation is taking place.
It will tell you,

00:49:14.000 --> 00:49:16.000
say that you have a very small
solid,

00:49:16.000 --> 00:49:19.000
I don't know like a ping pong
ball in your flow,

00:49:19.000 --> 00:49:21.000
and it is just going with the
flow,

00:49:21.000 --> 00:49:25.000
it tells you how it is going to
start rotating.

00:49:25.000 --> 00:49:32.000
That is what curl measures.
On Thursday we will see Stokes'

00:49:32.000 --> 00:49:37.000
theorem, which will be the last
ingredient before the next exam.

00:49:37.000 --> 00:49:40.000
And then on Friday we will
review stuff.