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Recall that yesterday we saw,
no, two days ago we learned

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00:00:32,000 --> 00:00:37,000
about the curl of a vector field
in space.

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00:00:37,000 --> 00:00:45,000
And we said the curl of F is
defined by taking a cross

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00:00:45,000 --> 00:00:52,000
product between the symbol dell
and the vector F.

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00:00:52,000 --> 00:00:58,000
Concretely, the way we would
compute this would be by putting

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the components of F into this
determinant and expanding and

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00:01:04,000 --> 00:01:09,000
then getting a vector with
components Ry minus Qz,

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00:01:09,000 --> 00:01:21,000
Pz minus Rx and Qx minus Py.
I think I also tried to explain

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00:01:21,000 --> 00:01:25,000
very quickly what the
significance of a curl is.

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00:01:25,000 --> 00:01:28,000
Just to tell you again very
quickly,

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00:01:28,000 --> 00:01:34,000
basically curl measures,
if you mention that your vector

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00:01:34,000 --> 00:01:40,000
field measures the velocity in
some fluid then the curl

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00:01:40,000 --> 00:01:47,000
measures how much rotation is
taking place in that fluid.

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00:01:47,000 --> 00:02:05,000
Measures the rotation part of a
velocity field.

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00:02:05,000 --> 00:02:13,000
More precisely the direction
corresponds to the axis of

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00:02:13,000 --> 00:02:22,000
rotation and the magnitude
corresponds to twice the angular

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00:02:22,000 --> 00:02:24,000
velocity.

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00:02:47,000 --> 00:02:50,000
Just to give you a few quick
examples.

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00:02:50,000 --> 00:02:53,000
If I take a constant vector
field,

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00:02:53,000 --> 00:03:01,000
so everything translates at the
same speed,

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00:03:01,000 --> 00:03:06,000
then obviously when you take
the partial derivatives you will

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00:03:06,000 --> 00:03:10,000
just get a bunch of zeros so the
curl will be zero.

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00:03:10,000 --> 00:03:15,000
If you take a vector field that
stretches things,

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00:03:15,000 --> 00:03:17,000
let's say, for example,
we are going to stretch things

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00:03:17,000 --> 00:03:23,000
along the x-axis,
that would be a vector field

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00:03:23,000 --> 00:03:30,000
that goes parallel to the x
direction but maybe,

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00:03:30,000 --> 00:03:33,000
say, x times i.
So that when you are in front

34
00:03:33,000 --> 00:03:35,000
of a plane of a blackboard you
are moving forward,

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00:03:35,000 --> 00:03:36,000
when you are behind you are
moving backwards,

36
00:03:36,000 --> 00:03:40,000
things are getting expanded in
the x direction.

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00:03:40,000 --> 00:03:48,000
If you compute the curl,
you can check each of these.

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00:03:48,000 --> 00:03:49,000
Again, they are going to be
zero.

39
00:03:49,000 --> 00:03:53,000
There is no curl.
This is not what curl measures.

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00:03:53,000 --> 00:03:58,000
I mean, actually,
what measures expansion,

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00:03:58,000 --> 00:04:03,000
stretching is actually
divergence.

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00:04:03,000 --> 00:04:05,000
If you take the divergence of
this field,

43
00:04:05,000 --> 00:04:07,000
you would get one plus zero
plus zero,

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00:04:07,000 --> 00:04:10,000
it looks like it will be one,
so in case you don't remember,

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00:04:10,000 --> 00:04:15,000
I mean divergence precisely
measures this stretching effect

46
00:04:15,000 --> 00:04:18,000
in your field.
And, on the other hand,

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00:04:18,000 --> 00:04:22,000
if you take something that
corresponds to,

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00:04:22,000 --> 00:04:26,000
say,
rotation about the z-axis at

49
00:04:26,000 --> 00:04:34,000
unit angular velocity -- That
means they are going to moving

50
00:04:34,000 --> 00:04:41,000
in circles around the z-axis.
One way to write down this

51
00:04:41,000 --> 00:04:46,000
field, let's see,
the z component is zero because

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00:04:46,000 --> 00:04:50,000
everything is moving
horizontally.

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00:04:50,000 --> 00:04:54,000
And in the x and y directions,
if you look at it from above,

54
00:04:54,000 --> 00:04:59,000
well, it is just going to be
our good old friend the vector

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00:04:59,000 --> 00:05:02,000
field that rotates everything
[at unit speed?].

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00:05:02,000 --> 00:05:05,000
And we have seen the formula
for this one many times.

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00:05:05,000 --> 00:05:09,000
The first component is minus y,
the second one is x.

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00:05:09,000 --> 00:05:17,000
Now, if you compute the curl of
this guy, you will get zero,

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00:05:17,000 --> 00:05:21,000
zero, two, two k.
And so k is the axis of

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00:05:21,000 --> 00:05:24,000
rotation, two is twice the
angular velocity.

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00:05:24,000 --> 00:05:26,000
And now, of course,
you can imagine much more

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00:05:26,000 --> 00:05:29,000
complicated motions where you
will have -- For example,

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00:05:29,000 --> 00:05:32,000
if you look at the Charles
River very carefully then you

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00:05:32,000 --> 00:05:34,000
will see that water is flowing,
generally speaking,

65
00:05:34,000 --> 00:05:38,000
towards the ocean.
But, at the same time,

66
00:05:38,000 --> 00:05:43,000
there are actually a few eddies
in there and with water

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00:05:43,000 --> 00:05:47,000
swirling.
Those are the places where

68
00:05:47,000 --> 00:05:51,000
there is actually curl in the
flow.

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00:05:51,000 --> 00:05:56,000
Yes.
I don't know how to turn out

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00:05:56,000 --> 00:06:00,000
the lights a bit,
but I'm sure there is a way.

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00:06:00,000 --> 00:06:10,000
Does this do it?
Is it working?

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00:06:10,000 --> 00:06:23,000
OK.
You're welcome.

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00:06:23,000 --> 00:06:31,000
Hopefully it is easier to see
now.

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00:06:31,000 --> 00:06:35,000
That was about curl.
Now, why do we care about curl

75
00:06:35,000 --> 00:06:40,000
besides this motivation of
understanding motions?

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00:06:40,000 --> 00:06:43,000
One place where it comes up is
when we try to understand

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00:06:43,000 --> 00:06:45,000
whether a vector field is
conservative.

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00:06:45,000 --> 00:06:49,000
Remember we have seen that a
vector field is conservative if

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00:06:49,000 --> 00:06:53,000
and only if its curl is zero.
That is the situation in which

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00:06:53,000 --> 00:06:56,000
we are allowed to try to look
for a potential function and

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00:06:56,000 --> 00:06:57,000
then use the fundamental
theorem.

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00:06:57,000 --> 00:07:00,000
But another place where this
comes up,

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00:07:00,000 --> 00:07:02,000
if you remember what we did in
the plane,

84
00:07:02,000 --> 00:07:06,000
curl also came up when we tried
to convert nine integrals into

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00:07:06,000 --> 00:07:10,000
double integrals.
That was Greene's theorem.

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00:07:10,000 --> 00:07:19,000
Well, it turns out we can do
the same thing in space and that

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00:07:19,000 --> 00:07:28,000
is called Stokes' theorem.
What does Stokes' theorem say?

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00:07:28,000 --> 00:07:36,000
It says that the work done by a
vector field along a closed

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00:07:36,000 --> 00:07:44,000
curve can be replaced by a
double integral of curl F.

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00:07:44,000 --> 00:07:47,000
Let me write it using the dell
notation.

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00:07:47,000 --> 00:07:54,000
That is curl F.
Dot ndS on a suitably chosen

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00:07:54,000 --> 00:07:58,000
surface.
That is a very strange kind of

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00:07:58,000 --> 00:08:01,000
statement.
But actually it is not much

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00:08:01,000 --> 00:08:04,000
more strange than things we have
seen before.

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00:08:04,000 --> 00:08:09,000
I should clarify what this
means.

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00:08:09,000 --> 00:08:17,000
C has to be a closed curve in
space.

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00:08:17,000 --> 00:08:32,000
And S can be any surface
bounded by C.

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00:08:32,000 --> 00:08:36,000
For example,
what Stokes' theorem tells me

99
00:08:36,000 --> 00:08:41,000
is that let us say that I have
to compute some line integral on

100
00:08:41,000 --> 00:08:48,000
maybe,
say, the unit circle in the x,

101
00:08:48,000 --> 00:08:52,000
y plane.
Of course I can set a line

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00:08:52,000 --> 00:08:57,000
integral directly and compute it
by setting x equals cosine T,

103
00:08:57,000 --> 00:09:00,000
y equals sine T,
z equals zero.

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00:09:00,000 --> 00:09:03,000
But maybe sometimes I don't
want to do that because my

105
00:09:03,000 --> 00:09:06,000
vector field is really
complicated.

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00:09:06,000 --> 00:09:11,000
And instead I will want to
reduce things to a surface

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00:09:11,000 --> 00:09:13,000
integral.
Now, I know that you guys are

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00:09:13,000 --> 00:09:16,000
not necessarily fond of
computing flux of vector fields

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00:09:16,000 --> 00:09:19,000
for surfaces so maybe you don't
really see the point.

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00:09:19,000 --> 00:09:22,000
But sometimes it is useful.
Sometimes it is also useful

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00:09:22,000 --> 00:09:25,000
backwards because,
actually, you have a surface

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00:09:25,000 --> 00:09:29,000
integral that you would like to
turn into a line integral.

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00:09:29,000 --> 00:09:34,000
What Stokes' theorem says is
that I can choose my favorite

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00:09:34,000 --> 00:09:38,000
surface whose boundary is this
circle.

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00:09:38,000 --> 00:09:42,000
I could choose,
for example,

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00:09:42,000 --> 00:09:50,000
a half sphere if I want or I
can choose, let's call that s1,

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00:09:50,000 --> 00:09:54,000
I don't know,
a pointy thing,

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00:09:54,000 --> 00:09:57,000
s2.
Probably the most logical one,

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00:09:57,000 --> 00:10:00,000
actually, would be just to
choose a disk in the x,

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00:10:00,000 --> 00:10:02,000
y plane.
That would probably be the

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00:10:02,000 --> 00:10:04,000
easiest one to set up for
calculating flux.

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00:10:04,000 --> 00:10:07,000
Anyway,
what Stokes' theorem tells me

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00:10:07,000 --> 00:10:09,000
is I can choose any of these
surfaces,

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00:10:09,000 --> 00:10:14,000
whichever one I want,
and I can compute the flux of

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00:10:14,000 --> 00:10:18,000
curl F through this surface.
Curl F is a new vector field

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00:10:18,000 --> 00:10:22,000
when you have this formula that
gives you a vector field you

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00:10:22,000 --> 00:10:25,000
compute its flux through your
favorite surface,

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00:10:25,000 --> 00:10:31,000
and you should get the same
thing as if you had done the

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00:10:31,000 --> 00:10:37,000
line integral for F.
That is the statement.

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00:10:37,000 --> 00:10:43,000
Now, there is a catch here.
What is the catch?

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00:10:43,000 --> 00:10:47,000
Well, the catch is we have to
figure out what conventions to

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00:10:47,000 --> 00:10:51,000
use because remember when we
have a surface there are two

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00:10:51,000 --> 00:10:54,000
possible orientations.
We have to decide which way we

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00:10:54,000 --> 00:10:58,000
will counter flux positively,
which way we will counter flux

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00:10:58,000 --> 00:11:01,000
negatively.
And, if we change our choice,

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00:11:01,000 --> 00:11:05,000
then of course the flux will
become the opposite.

137
00:11:05,000 --> 00:11:08,000
Well, similarly to define the
work, I need to choose which way

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00:11:08,000 --> 00:11:12,000
I am going to run my curve.
If I change which way I go

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00:11:12,000 --> 00:11:16,000
around the curve then my work
will become the opposite.

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00:11:16,000 --> 00:11:21,000
What happens is I have to
orient the curve C and the

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00:11:21,000 --> 00:11:28,000
surface S in compatible ways.
We have to figure out what the

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00:11:28,000 --> 00:11:36,000
rule is for how the orientation
of S and that of C relate to

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00:11:36,000 --> 00:11:41,000
each other.
What about orientation?

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00:11:41,000 --> 00:11:55,000
Well, we need the orientations
of C and S to be compatible and

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00:11:55,000 --> 00:12:05,000
they have to explain to you what
the rule is.

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00:12:05,000 --> 00:12:15,000
Let me show you a picture.
The rule is if I walk along C

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00:12:15,000 --> 00:12:23,000
with S to my left then the
normal vector is pointing up for

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00:12:23,000 --> 00:12:29,000
me.
Let me write that.

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00:12:29,000 --> 00:12:37,000
If I walk along C,
I should say in the positive

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00:12:37,000 --> 00:12:48,000
direction, in the direction that
I have chosen to orient C.

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00:12:48,000 --> 00:13:06,000
With S to my left then n is
pointing up for me.

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00:13:06,000 --> 00:13:10,000
Here is the example.
If I am walking on this curve,

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00:13:10,000 --> 00:13:12,000
it looks like the surface is to
my left.

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00:13:12,000 --> 00:13:19,000
And so the normal vector is
going towards what is up for me.

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00:13:19,000 --> 00:13:26,000
Any questions about that?
I see some people using their

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00:13:26,000 --> 00:13:28,000
right hands.
That is also right-handable

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00:13:28,000 --> 00:13:31,000
which I am going to say in just
a few moments.

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00:13:31,000 --> 00:13:32,000
That is another way to remember
this.

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00:13:32,000 --> 00:13:38,000
Before I tell you about the
right-handable version,

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00:13:38,000 --> 00:13:43,000
let me just try something.
Actually, I am not happy with

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00:13:43,000 --> 00:13:47,000
this orientation of C and I want
to orient my curve C going

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00:13:47,000 --> 00:13:51,000
clockwise on the picture.
So the other orientation.

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00:13:51,000 --> 00:13:55,000
Then, if I walk on it this way,
the surface would be to my

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00:13:55,000 --> 00:13:56,000
right.
You can remember,

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00:13:56,000 --> 00:13:59,000
if it helps you,
that if a surface is to your

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00:13:59,000 --> 00:14:01,000
right then the normal vector
will go down.

167
00:14:01,000 --> 00:14:04,000
The other way to think about
this rule is enough because if

168
00:14:04,000 --> 00:14:07,000
you are walking clockwise,
well, you can change that to

169
00:14:07,000 --> 00:14:10,000
counterclockwise just by walking
upside down.

170
00:14:10,000 --> 00:14:14,000
This guy is walking clockwise
on C.

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00:14:14,000 --> 00:14:21,000
And while for him,
if you look carefully at the

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00:14:21,000 --> 00:14:31,000
picture, the surface is actually
to his left when you flip upside

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00:14:31,000 --> 00:14:34,000
down.
Yeah, it is kind of confusing.

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00:14:34,000 --> 00:14:38,000
But, anyway,
maybe it's easier if you

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00:14:38,000 --> 00:14:44,000
actually rotate in the picture.
And now it is getting actually

176
00:14:44,000 --> 00:14:50,000
really confusing because his
walking upside up with,

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00:14:50,000 --> 00:14:54,000
actually, the surface is to his
left.

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00:14:54,000 --> 00:14:58,000
I mean where he is at here is
actually at the front and this

179
00:14:58,000 --> 00:15:01,000
is the back, but that is kind of
hard to see.

180
00:15:01,000 --> 00:15:05,000
Anyway, whichever method will
work best for you.

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00:15:05,000 --> 00:15:07,000
Perhaps it is easiest to first
do it with the other

182
00:15:07,000 --> 00:15:09,000
orientation,
this one,

183
00:15:09,000 --> 00:15:13,000
and this side,
if you want the opposite one,

184
00:15:13,000 --> 00:15:21,000
then you will just flip
everything.

185
00:15:21,000 --> 00:15:25,000
Now, what is the other way of
remembering this with the

186
00:15:25,000 --> 00:15:27,000
right-hand rule?
First of all,

187
00:15:27,000 --> 00:15:29,000
take your right hand,
not your left.

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00:15:29,000 --> 00:15:32,000
Even if your right hand is
actually using a pen or

189
00:15:32,000 --> 00:15:34,000
something like that in your
right hand do this.

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00:15:34,000 --> 00:15:37,000
And let's take your fingers in
order.

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00:15:37,000 --> 00:15:40,000
First your thumb.
Let's make your thumb go along

192
00:15:40,000 --> 00:15:42,000
the object that has only one
dimension in there.

193
00:15:42,000 --> 00:15:47,000
That is the curve.
Well, let's look at the top

194
00:15:47,000 --> 00:15:52,000
picture up there.
I want my thumb to go along the

195
00:15:52,000 --> 00:15:56,000
curve so that is kind of towards
the right.

196
00:15:56,000 --> 00:16:06,000
Then I want to make my index
finger point towards the

197
00:16:06,000 --> 00:16:09,000
surface.
Towards the surface I mean

198
00:16:09,000 --> 00:16:12,000
towards the interior of the
surface from the curve.

199
00:16:12,000 --> 00:16:15,000
And when I am on the curve I am
on the boundary of the surface,

200
00:16:15,000 --> 00:16:18,000
so there is a direction along
the surface that is the curve

201
00:16:18,000 --> 00:16:20,000
and the other one is pointing
into the surface.

202
00:16:20,000 --> 00:16:24,000
That one would be pointing kind
of to the back slightly up

203
00:16:24,000 --> 00:16:27,000
maybe, so like that.
And now your middle finger is

204
00:16:27,000 --> 00:16:30,000
going to point in the direction
of the normal vector.

205
00:16:30,000 --> 00:16:37,000
That is up, at least if you
have the same kind of right hand

206
00:16:37,000 --> 00:16:49,000
as I do.
The other way of doing it is

207
00:16:49,000 --> 00:17:08,000
using the right-hand rule along
C positively.

208
00:17:08,000 --> 00:17:18,000
The index finger towards the
interior of S.

209
00:17:18,000 --> 00:17:27,000
Sorry, I shouldn't say interior.
I should say tangent to S

210
00:17:27,000 --> 00:17:34,000
towards the interior of S.
What I mean by that is really

211
00:17:34,000 --> 00:17:39,000
the part of S that is not its
boundary, so the rest of the

212
00:17:39,000 --> 00:17:49,000
surface.
Then the middle finger points

213
00:17:49,000 --> 00:18:00,000
parallel to n.
Let's practice.

214
00:18:00,000 --> 00:18:09,000
Let's say that I gave you this
curve bounding this surface.

215
00:18:09,000 --> 00:18:13,000
Which way do you think the
normal vector will be going?

216
00:18:13,000 --> 00:18:16,000
Up. Yes. Everyone is voting up.
Imaging that I am walking

217
00:18:16,000 --> 00:18:18,000
around C.
That is to my left.

218
00:18:18,000 --> 00:18:24,000
Normal vector points up.
Imagine that you put your thumb

219
00:18:24,000 --> 00:18:32,000
along C, your index towards S
and then your middle finger

220
00:18:32,000 --> 00:18:36,000
points up.
Very good.

221
00:18:36,000 --> 00:18:43,000
N points up.
Another one.

222
00:19:05,000 --> 00:19:07,000
It is interesting to watch you
guys.

223
00:19:07,000 --> 00:19:13,000
I think mostly it is going up.
The correct answer is it goes

224
00:19:13,000 --> 00:19:20,000
up and into the cone.
How do we see that?

225
00:19:20,000 --> 00:19:24,000
Well, one way to think about it
is imagine that you are walking

226
00:19:24,000 --> 00:19:27,000
on C, on the rim of this cone.
You have two options.

227
00:19:27,000 --> 00:19:30,000
Imagine that you are walking
kind of inside or imagine that

228
00:19:30,000 --> 00:19:33,000
you are walking kind of outside.
If you are walking outside then

229
00:19:33,000 --> 00:19:35,000
S is to your right,
but it does not sound good.

230
00:19:35,000 --> 00:19:39,000
Let's say instead that you are
walking on the inside of a cone

231
00:19:39,000 --> 00:19:43,000
following the boundary.
Well, then the surface is to

232
00:19:43,000 --> 00:19:45,000
your left.
And so the normal vector will

233
00:19:45,000 --> 00:19:50,000
be up for you which means it
will be pointing slightly up and

234
00:19:50,000 --> 00:19:52,000
into the cone.
Another way to think about it,

235
00:19:52,000 --> 00:19:56,000
through the right-hand rule,
from this way index going kind

236
00:19:56,000 --> 00:20:00,000
of down because the surface goes
down and a bit to the back.

237
00:20:00,000 --> 00:20:04,000
And then the normal vector
points up and in.

238
00:20:04,000 --> 00:20:08,000
Yet another way,
if you deform continuously your

239
00:20:08,000 --> 00:20:12,000
surface then the conventions
will not change.

240
00:20:12,000 --> 00:20:15,000
See, this is kind of
[UNINTELLIGIBLE]

241
00:20:15,000 --> 00:20:17,000
in a way.
You can deform things and

242
00:20:17,000 --> 00:20:21,000
nothing will change.
So what if we somehow flatten

243
00:20:21,000 --> 00:20:27,000
our cone, push it a bit up so
that it becomes completely flat?

244
00:20:27,000 --> 00:20:30,000
Then, if you had a flat disk
with the curve going

245
00:20:30,000 --> 00:20:33,000
counterclockwise,
the normal vector would go up.

246
00:20:33,000 --> 00:20:36,000
Now take your disk with its
normal vector sticking up.

247
00:20:36,000 --> 00:20:39,000
If you want to paint the face a
different color so that you can

248
00:20:39,000 --> 00:20:43,000
remember that was beside with a
normal vector and then push it

249
00:20:43,000 --> 00:20:45,000
back down to the cone,
you will see that the painted

250
00:20:45,000 --> 00:20:48,000
face,
the one with the normal vector

251
00:20:48,000 --> 00:20:52,000
on that side is the one that is
inside and up.

252
00:20:52,000 --> 00:20:59,000
Does that make sense?
Anyway, I think you have just

253
00:20:59,000 --> 00:21:03,000
to play with these examples for
long enough and get it.

254
00:21:03,000 --> 00:21:07,000
OK. The last one.
Let's say that I have a

255
00:21:07,000 --> 00:21:10,000
cylinder.
So now this guy has actually

256
00:21:10,000 --> 00:21:12,000
two boundary curves,
C and C prime.

257
00:21:12,000 --> 00:21:16,000
And let's say I want to orient
my cylinder so that the normal

258
00:21:16,000 --> 00:21:20,000
vector sticks out.
How should I choose the

259
00:21:20,000 --> 00:21:30,000
orientation of my curves?
Let's start with,

260
00:21:30,000 --> 00:21:40,000
say, the bottom one.
Would the bottom one be going

261
00:21:40,000 --> 00:21:44,000
clockwise or counterclockwise.
Most people seem to say

262
00:21:44,000 --> 00:21:47,000
counterclockwise,
and I agree with that.

263
00:21:47,000 --> 00:21:51,000
Let me write that down and
claim C prime should go

264
00:21:51,000 --> 00:21:55,000
counterclockwise.
One way to think about it,

265
00:21:55,000 --> 00:21:58,000
actually, it's quite easy,
you mentioned that you're

266
00:21:58,000 --> 00:22:02,000
walking on the outside of the
cylinder along C prime.

267
00:22:02,000 --> 00:22:06,000
If you want to walk along C
prime so that the cylinder is to

268
00:22:06,000 --> 00:22:10,000
your left, that means you have
to actually go counterclockwise

269
00:22:10,000 --> 00:22:14,000
around it.
The other way is use your right

270
00:22:14,000 --> 00:22:17,000
hand.
Say when you're at the front of

271
00:22:17,000 --> 00:22:19,000
C prime,
your thumb points to the right,

272
00:22:19,000 --> 00:22:23,000
your index points up because
that's where the surface is,

273
00:22:23,000 --> 00:22:28,000
and then your middle finger
will point out.

274
00:22:28,000 --> 00:22:39,000
What about C?
Well, C I claim we should be

275
00:22:39,000 --> 00:22:43,000
doing clockwise.
I mean think about just walking

276
00:22:43,000 --> 00:22:46,000
again on the surface of the
cylinder along C.

277
00:22:46,000 --> 00:22:52,000
If you walk clockwise,
you will see that the surface

278
00:22:52,000 --> 00:22:57,000
is to your left or use the
right-hand rule.

279
00:22:57,000 --> 00:23:00,000
Now, if a problem gives you
neither the orientation of a

280
00:23:00,000 --> 00:23:04,000
curve nor that of the surface
then it's up to you to make them

281
00:23:04,000 --> 00:23:05,000
up.
But you have to make them up in

282
00:23:05,000 --> 00:23:09,000
a consistent way.
You cannot choose them both at

283
00:23:09,000 --> 00:23:13,000
random.
All right.

284
00:23:13,000 --> 00:23:30,000
Now we're all set to try to use
Stokes' theorem.

285
00:23:30,000 --> 00:23:35,000
Well, let me do an example
first.

286
00:23:35,000 --> 00:23:43,000
The first example that I will
do is actually a comparison.

287
00:23:43,000 --> 00:23:52,000
Stokes' versus Green.
I want to show you how Green's

288
00:23:52,000 --> 00:23:55,000
theorem for work that we saw in
the plane,

289
00:23:55,000 --> 00:23:58,000
but also involved work and curl
and so on,

290
00:23:58,000 --> 00:24:04,000
is actually a special case of
this.

291
00:24:04,000 --> 00:24:11,000
Let's say that we will look at
the special case where our curve

292
00:24:11,000 --> 00:24:16,000
C is actually a curve in the x,
y plane.

293
00:24:16,000 --> 00:24:19,000
And let's make it go
counterclockwise in the x,

294
00:24:19,000 --> 00:24:23,000
y plane because that's what we
did for Green's theorem.

295
00:24:23,000 --> 00:24:25,000
Now let's choose a surface
bounded by this curve.

296
00:24:25,000 --> 00:24:28,000
Well, as I said,
I could make up any surface

297
00:24:28,000 --> 00:24:32,000
that comes to my mind.
But, if I want to relate to

298
00:24:32,000 --> 00:24:35,000
this stuff, I should probably
stay in the x,

299
00:24:35,000 --> 00:24:38,000
y plane.
So I am just going to take my

300
00:24:38,000 --> 00:24:43,000
surface to be the piece of the
x, y plane that is inside my

301
00:24:43,000 --> 00:24:52,000
curve.
So let's say S is going to be a

302
00:24:52,000 --> 00:25:02,000
portion of x,
y plane bounded by a curve C,

303
00:25:02,000 --> 00:25:11,000
and the curve C goes
counterclockwise.

304
00:25:11,000 --> 00:25:17,000
Well, then I should look at
[the table?].

305
00:25:17,000 --> 00:25:24,000
For work along C of my favorite
vector field F dot dr.

306
00:25:24,000 --> 00:25:31,000
So that will be the line
integral of Pdx plus Qdy.

307
00:25:31,000 --> 00:25:34,000
Like I said,
if I call the components of my

308
00:25:34,000 --> 00:25:38,000
field P, Q and R,
it will be Pdx plus Qdy plus

309
00:25:38,000 --> 00:25:42,000
Rdz, but I don't have any Z
here.

310
00:25:42,000 --> 00:25:49,000
Dz is zero on C.
If I evaluate for line

311
00:25:49,000 --> 00:25:53,000
integral, I don't have any term
involving dz.

312
00:25:53,000 --> 00:25:59,000
Z is zero.
Now, let's see what Strokes

313
00:25:59,000 --> 00:26:05,000
says.
Stokes says instead I can

314
00:26:05,000 --> 00:26:13,000
compute for flux through S of
curve F.

315
00:26:13,000 --> 00:26:17,000
But now what's the normal
vector to my surface?

316
00:26:17,000 --> 00:26:19,000
Well, it's going to be either k
or negative k.

317
00:26:19,000 --> 00:26:22,000
I just have to figure out which
one it is.

318
00:26:22,000 --> 00:26:25,000
Well, if you followed what
we've done there,

319
00:26:25,000 --> 00:26:30,000
you know that the normal vector
compatible with this choice for

320
00:26:30,000 --> 00:26:32,000
the curve C is the one that
points up.

321
00:26:32,000 --> 00:26:43,000
My normal vector is just going
to be k hat, so I am going to

322
00:26:43,000 --> 00:26:49,000
replace my normal vector by k
hat.

323
00:26:49,000 --> 00:26:53,000
That means, actually,
I will be integrating curl dot

324
00:26:53,000 --> 00:26:56,000
k.
That means I am integrating the

325
00:26:56,000 --> 00:27:04,000
z component of curl.
Let's look at curl F dot k.

326
00:27:04,000 --> 00:27:14,000
That's the z component of curl
F.

327
00:27:14,000 --> 00:27:18,000
And what's the z component of
curl?

328
00:27:18,000 --> 00:27:21,000
Well, I conveniently still have
the values up there.

329
00:27:21,000 --> 00:27:36,000
It's Q sub x minus P sub y.
My double integral becomes

330
00:27:36,000 --> 00:27:42,000
double integral of Q sub x minus
P sub y.

331
00:27:42,000 --> 00:27:46,000
What about dS?
Well, I am in a piece of the x,

332
00:27:46,000 --> 00:27:52,000
y plane, so dS is just dxdy or
your favorite combination that

333
00:27:52,000 --> 00:27:56,000
does the same thing.
Now, see, if you look at this

334
00:27:56,000 --> 00:28:02,000
equality, integral of Pdx plus
Qdy along a closed curve equals

335
00:28:02,000 --> 00:28:05,000
double integral of Qx minus Py
dxdy.

336
00:28:05,000 --> 00:28:11,000
That is exactly the statement
of Green's theorem.

337
00:28:11,000 --> 00:28:17,000
I mean except at that time we
called things m and n,

338
00:28:17,000 --> 00:28:20,000
but really that shouldn't
matter.

339
00:28:20,000 --> 00:28:35,000
This tells you that,
in fact, Green's theorem is

340
00:28:35,000 --> 00:28:51,000
just a special case of Stokes'
in the x, y plane.

341
00:28:51,000 --> 00:28:55,000
Now, another small remark I
want to make right away before I

342
00:28:55,000 --> 00:28:57,000
forget,
you might think that these

343
00:28:57,000 --> 00:29:01,000
rules that we've made up about
compatibility of orientations

344
00:29:01,000 --> 00:29:05,000
are completely arbitrary.
Well, they are literally in the

345
00:29:05,000 --> 00:29:10,000
same way as our convention for
which we guy curl is arbitrary.

346
00:29:10,000 --> 00:29:14,000
We chose to make the curl be
this thing and not the opposite

347
00:29:14,000 --> 00:29:17,000
which would have been pretty
much just as sensible.

348
00:29:17,000 --> 00:29:21,000
And, ultimately,
that comes from our choice of

349
00:29:21,000 --> 00:29:25,000
making the cross-product be what
it is but of the opposite.

350
00:29:25,000 --> 00:29:30,000
Ultimately, it all comes from
our preference for right-handed

351
00:29:30,000 --> 00:29:33,000
coordinate systems.
If we had been on the planet

352
00:29:33,000 --> 00:29:36,000
with left-handed coordinate
systems then actually our

353
00:29:36,000 --> 00:29:40,000
conventions would be all the
other way around,

354
00:29:40,000 --> 00:30:05,000
but they are this way.
Any other questions?

355
00:30:05,000 --> 00:30:08,000
A surface that you use in
Stokes' theorem is usually not

356
00:30:08,000 --> 00:30:12,000
going to be closed because its
boundary needs to be the curve

357
00:30:12,000 --> 00:30:14,000
C.
So if you had a closed surface

358
00:30:14,000 --> 00:30:17,000
you wouldn't know where to put
your curve.

359
00:30:17,000 --> 00:30:20,000
I mean of course you could make
a tiny hole in it and get a tiny

360
00:30:20,000 --> 00:30:22,000
curve.
Actually, what that would say,

361
00:30:22,000 --> 00:30:26,000
and we are going to see more
about that so not very important

362
00:30:26,000 --> 00:30:28,000
right now,
but what we would see is that

363
00:30:28,000 --> 00:30:31,000
for a close surface we would end
up getting zero for the flux.

364
00:30:31,000 --> 00:30:34,000
And that is actually because
divergence of curl is zero,

365
00:30:34,000 --> 00:30:36,000
but I am getting ahead of
myself.

366
00:30:36,000 --> 00:30:45,000
We are going to see that
probably tomorrow in more

367
00:30:45,000 --> 00:30:49,000
detail.
Stokes' theorem only works if

368
00:30:49,000 --> 00:30:53,000
you can make sense of this.
That means you need your vector

369
00:30:53,000 --> 00:30:58,000
field to be continuous and
differentiable everywhere on the

370
00:30:58,000 --> 00:31:03,000
surface S.
Now, why is that relevant?

371
00:31:03,000 --> 00:31:05,000
Well, say that your vector
field was not defined at the

372
00:31:05,000 --> 00:31:07,000
origin and say that you wanted
to do,

373
00:31:07,000 --> 00:31:11,000
you know, the example that I
had first with the unit circling

374
00:31:11,000 --> 00:31:14,000
the x, y plane.
Normally, the most sensible

375
00:31:14,000 --> 00:31:17,000
choice of surface to apply
Stokes' theorem to would be just

376
00:31:17,000 --> 00:31:19,000
the flat disk in the x,
y plane.

377
00:31:19,000 --> 00:31:23,000
But that assumes that your
vector field is well-defined

378
00:31:23,000 --> 00:31:24,000
there.
If your vector field is not

379
00:31:24,000 --> 00:31:27,000
defined at the origin but
defined everywhere else you

380
00:31:27,000 --> 00:31:29,000
cannot use this guy,
but maybe you can still use,

381
00:31:29,000 --> 00:31:31,000
say, the half-sphere,
for example.

382
00:31:31,000 --> 00:31:35,000
Or, you could use a piece of
cylinder plus a flat top or

383
00:31:35,000 --> 00:31:38,000
whatever you want but not
pressing for the origin.

384
00:31:38,000 --> 00:31:41,000
So you could still use Stokes
but you'd have to be careful

385
00:31:41,000 --> 00:31:44,000
about which surface you choose.
Now, if instead your vector

386
00:31:44,000 --> 00:31:49,000
field is not defined anywhere on
the z-axis then you're out of

387
00:31:49,000 --> 00:31:54,000
luck because there is no way to
find a surface bounded by this

388
00:31:54,000 --> 00:31:59,000
unit circle without crossing the
z-axis somewhere.

389
00:31:59,000 --> 00:32:07,000
Then you wouldn't be able to
Stokes' theorem at all or at

390
00:32:07,000 --> 00:32:16,000
least not directly.
Maybe I should write it F

391
00:32:16,000 --> 00:32:25,000
defines a differentiable
everywhere on this.

392
00:32:25,000 --> 00:32:27,000
But we don't care about what
happens outside of this.

393
00:32:27,000 --> 00:32:35,000
It's really only on the surface
that we need it to be OK.

394
00:32:35,000 --> 00:32:39,000
I mean, again,
99% of the vector fields that

395
00:32:39,000 --> 00:32:44,000
we see in this class are defined
everywhere so that's not an

396
00:32:44,000 --> 00:32:47,000
urgent concern,
but still.

397
00:32:47,000 --> 00:32:49,000
OK.
Should we move on?

398
00:32:49,000 --> 00:33:01,000
Yes. I have a yes.
Let me explain to you quickly

399
00:33:01,000 --> 00:33:08,000
why Stokes is true.
How do we prove a theorem like

400
00:33:08,000 --> 00:33:10,000
that?
Well,

401
00:33:10,000 --> 00:33:12,000
the strategy,
I mean there are other ways,

402
00:33:12,000 --> 00:33:16,000
but the least painful strategy
at this point is to observe what

403
00:33:16,000 --> 00:33:19,000
we already know is a special
case of Stokes's theorem.

404
00:33:19,000 --> 00:33:22,000
Namely we know the case where
the curve is actually in the x,

405
00:33:22,000 --> 00:33:24,000
y plane and the surface is a
flat piece of the x,

406
00:33:24,000 --> 00:33:34,000
y plane because that's Green's
theorem which we proved a while

407
00:33:34,000 --> 00:33:42,000
ago.
We know it for C and S in the

408
00:33:42,000 --> 00:33:47,000
x, y plane.
Now, what if C and S were,

409
00:33:47,000 --> 00:33:49,000
say, in the y,
z plane instead of the x,

410
00:33:49,000 --> 00:33:51,000
y plane?
Well, then it will not quite

411
00:33:51,000 --> 00:33:55,000
give the same picture because
the normal vector would be i hat

412
00:33:55,000 --> 00:33:58,000
instead of k hat and they would
be having different notations

413
00:33:58,000 --> 00:34:01,000
and it would be integrating with
y and z.

414
00:34:01,000 --> 00:34:02,000
But you see that it would
become, again,

415
00:34:02,000 --> 00:34:05,000
exactly the same formula.
We'd know it for any of the

416
00:34:05,000 --> 00:34:08,000
coordinate planes.
In fact, I claim we know it for

417
00:34:08,000 --> 00:34:13,000
absolutely any plane.
And the reason for that is,

418
00:34:13,000 --> 00:34:15,000
sure, when we write it in
coordinates,

419
00:34:15,000 --> 00:34:19,000
when we write that this line
integral is integral of Pdx plus

420
00:34:19,000 --> 00:34:24,000
Qdy plus Rdz or when we write
that the curl is given by this

421
00:34:24,000 --> 00:34:28,000
formula we use the x,
y, z coordinate system.

422
00:34:28,000 --> 00:34:30,000
But there is something I
haven't quite told you about.

423
00:34:30,000 --> 00:34:33,000
Which is if I switch to any
other right-handed coordinate

424
00:34:33,000 --> 00:34:35,000
system,
so I do some sort of rotation

425
00:34:35,000 --> 00:34:40,000
of my space coordinates,
then somehow the line integral,

426
00:34:40,000 --> 00:34:44,000
the flux integral,
the notion of curl makes sense

427
00:34:44,000 --> 00:34:47,000
in coordinates.
And the reason is that they all

428
00:34:47,000 --> 00:34:50,000
have geometric interpretations.
For example,

429
00:34:50,000 --> 00:34:52,000
when I think of this as the
work done by a force,

430
00:34:52,000 --> 00:34:55,000
well, the force doesn't care
whether it's being put in x,

431
00:34:55,000 --> 00:34:56,000
y coordinates this way or that
way.

432
00:34:56,000 --> 00:35:00,000
It still does the same work
because it's the same force.

433
00:35:00,000 --> 00:35:03,000
And when I say that the curl
measures the rotation in a

434
00:35:03,000 --> 00:35:06,000
motion, well,
that depends on which

435
00:35:06,000 --> 00:35:09,000
coordinates you use.
And the same for interpretation

436
00:35:09,000 --> 00:35:12,000
of flux.
In fact, if I rotated my

437
00:35:12,000 --> 00:35:17,000
coordinates to fit with any
other plane, I could still do

438
00:35:17,000 --> 00:35:23,000
the same things.
What I'm trying to say is,

439
00:35:23,000 --> 00:35:31,000
in fact, if C and S are in any
plane then we can still claim

440
00:35:31,000 --> 00:35:37,000
that it reduces to Green's
theorem.

441
00:35:37,000 --> 00:35:45,000
It will be Green's theorem not
in x, y, z coordinates but in

442
00:35:45,000 --> 00:35:50,000
some funny rotated coordinate
systems.

443
00:35:50,000 --> 00:35:56,000
What I'm saying is that work,
flux and curl makes sense

444
00:35:56,000 --> 00:35:59,000
independently of coordinates.

445
00:36:20,000 --> 00:36:23,000
Now, this has to stop somewhere.
I can start claiming that I can

446
00:36:23,000 --> 00:36:26,000
somehow bend my coordinates to a
plane, any surface is flat.

447
00:36:26,000 --> 00:36:29,000
That doesn't really work.
But what I can say is if I have

448
00:36:29,000 --> 00:36:31,000
any surface I can cut it into
tiny pieces.

449
00:36:31,000 --> 00:36:35,000
And these tiny pieces are
basically flat.

450
00:36:35,000 --> 00:36:39,000
So that's basically the idea of
a proof.

451
00:36:39,000 --> 00:36:47,000
I am going to decompose my
surface into very small flat

452
00:36:47,000 --> 00:36:56,000
pieces.
Given any S we are just going

453
00:36:56,000 --> 00:37:08,000
to decompose it into tiny almost
flat pieces.

454
00:37:08,000 --> 00:37:15,000
For example,
if I have my surface like this,

455
00:37:15,000 --> 00:37:23,000
what I will do is I will just
cut it into tiles.

456
00:37:23,000 --> 00:37:28,000
I mean a good example of that
is if you look at

457
00:37:28,000 --> 00:37:31,000
[UNINTELLIGIBLE],
for example,

458
00:37:31,000 --> 00:37:36,000
it's made of all these hexagons
and pentagons.

459
00:37:36,000 --> 00:37:38,000
Well, actually,
they're not quite flat in the

460
00:37:38,000 --> 00:37:41,000
usual rule, but you could make
them flat and it would still

461
00:37:41,000 --> 00:37:45,000
look pretty much like a sphere.
Anyway, you're going to cut

462
00:37:45,000 --> 00:37:49,000
your surface into lots of tiny
pieces.

463
00:37:49,000 --> 00:37:53,000
And then you can use Stokes'
theorem on each small piece.

464
00:37:53,000 --> 00:38:00,000
What it says on each small flat
piece -- It says that the line

465
00:38:00,000 --> 00:38:04,000
integral along say,
for example,

466
00:38:04,000 --> 00:38:08,000
this curve is equal to the flux
of a curl through this tiny

467
00:38:08,000 --> 00:38:12,000
piece of surface.
And now I will add all of these

468
00:38:12,000 --> 00:38:14,000
terms together.
If I add all of the small

469
00:38:14,000 --> 00:38:17,000
contributions to flux I get the
total flux.

470
00:38:17,000 --> 00:38:19,000
What if I add all of the small
line integrals?

471
00:38:19,000 --> 00:38:23,000
Well, I get lots of extra junk
because I never asked to compute

472
00:38:23,000 --> 00:38:26,000
the line integral along this.
But this guy will come in twice

473
00:38:26,000 --> 00:38:30,000
when I do this little plate and
when I do that little plate with

474
00:38:30,000 --> 00:38:34,000
opposite orientations.
When I sum all of the little

475
00:38:34,000 --> 00:38:38,000
line integrals together,
all of the inner things cancel

476
00:38:38,000 --> 00:38:40,000
out,
and the only ones that I go

477
00:38:40,000 --> 00:38:44,000
through only once are those that
are at the outer most edges.

478
00:38:44,000 --> 00:38:50,000
So, when I sum all of my works
together, I will get the work

479
00:38:50,000 --> 00:38:54,000
done just along the outer
boundary C.

480
00:38:54,000 --> 00:39:12,000
Sum of work around each little
piece is just actually the work

481
00:39:12,000 --> 00:39:27,000
along C, the outer curve.
And the sum of the flux for

482
00:39:27,000 --> 00:39:39,000
each piece is going to be the
flux through S.

483
00:39:39,000 --> 00:39:45,000
From Stokes' theorem for flat
surfaces, I can get it for any

484
00:39:45,000 --> 00:39:47,000
surface.
I am cheating a little bit

485
00:39:47,000 --> 00:39:50,000
because you would actually have
to check carefully that this

486
00:39:50,000 --> 00:39:53,000
approximately where you flatten
the little pieces that are

487
00:39:53,000 --> 00:39:56,000
almost flat is [UNINTELLIGIBLE].
But, trust me,

488
00:39:56,000 --> 00:39:56,000
it actually works.

489
00:40:13,000 --> 00:40:15,000
Let's do an actual example.
I mean I said example,

490
00:40:15,000 --> 00:40:19,000
but that was more like getting
us ready for the proof so

491
00:40:19,000 --> 00:40:22,000
probably that doesn't count as
an actual example.

492
00:40:22,000 --> 00:40:25,000
I should probably keep these
statements for now so I am not

493
00:40:25,000 --> 00:40:26,000
going to erase this side.

494
00:41:08,000 --> 00:41:21,000
Let's do an example.
Let's try to find the work of

495
00:41:21,000 --> 00:41:40,000
vector field zi plus xj plus yk
around the unit circle in the x,

496
00:41:40,000 --> 00:41:58,000
y plane counterclockwise.
The picture is conveniently

497
00:41:58,000 --> 00:42:05,000
already there.
Just as a quick review,

498
00:42:05,000 --> 00:42:08,000
let's see how we do that
directly.

499
00:42:08,000 --> 00:42:14,000
If we do that directly,
I have to find the integral

500
00:42:14,000 --> 00:42:21,000
along C.
So F dot dr becomes zdx plus

501
00:42:21,000 --> 00:42:28,000
xdy plus ydz.
But now we actually know that

502
00:42:28,000 --> 00:42:33,000
on this circle,
well, z is zero.

503
00:42:33,000 --> 00:42:39,000
And we can parameterize x and
y, the unit circle in the x,

504
00:42:39,000 --> 00:42:44,000
y plane, so we can take x
equals cosine t,

505
00:42:44,000 --> 00:42:49,000
y equals sine t.
That will just become the

506
00:42:49,000 --> 00:42:54,000
integral over C.
Well, z times dx,

507
00:42:54,000 --> 00:43:05,000
z is zero so we have nothing,
plus x is cosine t times dy is

508
00:43:05,000 --> 00:43:17,000
-- Well, if y is sine t then dy
is cosine tdt plus ydz but z is

509
00:43:17,000 --> 00:43:22,000
zero.
Now, the range of values for t,

510
00:43:22,000 --> 00:43:26,000
well, we are going
counterclockwise around the

511
00:43:26,000 --> 00:43:31,000
entire circle so that should go
from zero to 2pi.

512
00:43:31,000 --> 00:43:39,000
We will get integral from zero
to 2pi of cosine square tdt

513
00:43:39,000 --> 00:43:45,000
which, if you do the
calculation, turns out to be

514
00:43:45,000 --> 00:43:50,000
just pi.
Now, let's instead try to use

515
00:43:50,000 --> 00:43:55,000
Stokes' theorem to do the
calculation.

516
00:43:55,000 --> 00:44:00,000
Now, of course the smart choice
would be to just take the flat

517
00:44:00,000 --> 00:44:02,000
unit disk.
I am not going to do that.

518
00:44:02,000 --> 00:44:06,000
That would be too boring.
Plus we have already kind of

519
00:44:06,000 --> 00:44:09,000
checked it because we already
trust Green's theorem.

520
00:44:09,000 --> 00:44:11,000
Instead, just to convince you
that,

521
00:44:11,000 --> 00:44:14,000
yes, I can choose really any
surface I want,

522
00:44:14,000 --> 00:44:23,000
let's say that I'm going to
choose a piece of paraboloid z

523
00:44:23,000 --> 00:44:30,000
equals one minus x squared minus
y squared.

524
00:44:30,000 --> 00:44:36,000
Well, to get our conventions
straight, we should take the

525
00:44:36,000 --> 00:44:43,000
normal vector pointing up for
compatibility with our choice.

526
00:44:43,000 --> 00:44:48,000
Well, we will have to compute
the flux through S.

527
00:44:48,000 --> 00:44:50,000
We don't really have to because
we could have chosen the disk,

528
00:44:50,000 --> 00:44:54,000
it would be easier,
but if we want to do it this

529
00:44:54,000 --> 00:45:00,000
way we will compute the flux of
curl F through our paraboloid.

530
00:45:00,000 --> 00:45:03,000
How do we do that?
Well, we need to find the curl

531
00:45:03,000 --> 00:45:10,000
and we need to find ndS.
Let's start with the curl.

532
00:45:10,000 --> 00:45:23,000
Curl F let's take the
cross-product between dell and F

533
00:45:23,000 --> 00:45:28,000
which is zxy.
If we compute this,

534
00:45:28,000 --> 00:45:31,000
the i component will be one
minus zero.

535
00:45:31,000 --> 00:45:37,000
It looks like it is one i.
Minus the j component is zero

536
00:45:37,000 --> 00:45:41,000
minus one.
Plus the k component is one

537
00:45:41,000 --> 00:45:52,000
minus zero.
In fact, the curl of the field

538
00:45:52,000 --> 00:45:59,000
is one, one, one.
Now, what about ndS?

539
00:45:59,000 --> 00:46:03,000
Well, this is a surface for
which we know z is a function of

540
00:46:03,000 --> 00:46:08,000
x and y.
ndS we can write as,

541
00:46:08,000 --> 00:46:14,000
let's call this F of xy,
then we can use the formula

542
00:46:14,000 --> 00:46:19,000
that says ndS equals negative F
sub x, negative F sub y,

543
00:46:19,000 --> 00:46:26,000
one dxdy,
which here gives us 2x,

544
00:46:26,000 --> 00:46:34,000
2y, one dxdy.
Now, when we want to compute

545
00:46:34,000 --> 00:46:41,000
the flux, we will have to do
double integral over S of one,

546
00:46:41,000 --> 00:46:47,000
one, one dot product with 2x,
2y, one dxdy.

547
00:46:47,000 --> 00:46:55,000
It will become the double
integral of 2x plus 2y plus one

548
00:46:55,000 --> 00:46:58,000
dxdy.
And, of course,

549
00:46:58,000 --> 00:47:01,000
the region which we are
integrating, the range of values

550
00:47:01,000 --> 00:47:04,000
of x and y will be the shadow of
our surface.

551
00:47:04,000 --> 00:47:07,000
That is just going to be,
if you look at this paraboloid

552
00:47:07,000 --> 00:47:11,000
from above,
all you will see is the unit

553
00:47:11,000 --> 00:47:17,000
disk so it will be a double
integral of the unit disk.

554
00:47:17,000 --> 00:47:23,000
And the way we will do that,
one way is to switch to polar

555
00:47:23,000 --> 00:47:28,000
coordinates and do the
calculation and then you will

556
00:47:28,000 --> 00:47:31,000
end up with pi.
The other way is to try to do

557
00:47:31,000 --> 00:47:34,000
it by symmetry.
Observe, when you integrate x

558
00:47:34,000 --> 00:47:37,000
above this, x is as negative on
the left as it is positive on

559
00:47:37,000 --> 00:47:40,000
the right.
So the integral of x will be

560
00:47:40,000 --> 00:47:42,000
zero.
The integral of y will be zero

561
00:47:42,000 --> 00:47:46,000
also by symmetry.
Then the integral of one dxdy

562
00:47:46,000 --> 00:47:52,000
will just be the area of this
unit disk which is pi.

563
00:47:52,000 --> 00:47:54,000
That was our first example.
And, of course,

564
00:47:54,000 --> 00:47:57,000
if you're actually free to
choose your favorite surface,

565
00:47:57,000 --> 00:48:01,000
there is absolutely no reason
why you would actually choose

566
00:48:01,000 --> 00:48:04,000
this paraboloid in this example.
I mean it would be much easier

567
00:48:04,000 --> 00:48:05,000
to choose a flat disk.
OK.

568
00:48:05,000 --> 00:48:09,000
Tomorrow I will tell you a few
more things about curl fits in

569
00:48:09,000 --> 00:48:13,000
with conservativeness and with
the divergence theorem,

570
00:48:13,000 --> 00:48:17,000
Stokes all together,
and we will look at Practice

571
00:48:17,000 --> 00:48:20,000
Exam 4B so please bring the exam
with you.

572
00:48:20,000 --> 00:48:25,000
with you.