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Today I am going to tell you
about flux of a vector field for
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a curve.
In case you have seen flux in
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physics,
probably you have seen flux in
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space,
and we are going to come to
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that in a couple of weeks,
but for now we are still doing
00:00:41.000 --> 00:00:44.000
everything in the plane.
So bear with me if you have
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seen a more complicated version
of flux.
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We are going to do the easy one
first.
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What is flux?
Well, flux is actually another
00:00:59.000 --> 00:01:10.000
kind of line integral.
Let's say that I have a plane
00:01:10.000 --> 00:01:18.000
curve and a vector field in the
plane.
00:01:18.000 --> 00:01:27.000
Then the flux of F across a
curve C is, by definition,
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a line integral,
but I will use notation F dot n
00:01:35.000 --> 00:01:38.000
ds.
I have to explain to you what
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it means, but let me first box
that because that is the
00:01:43.000 --> 00:01:50.000
important formula to remember.
That is the definition.
00:01:50.000 --> 00:01:55.000
What does that mean?
First, mostly I have to tell
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you what this little n is.
The notation suggests it is a
00:02:01.000 --> 00:02:05.000
normal vector,
so what does that mean?
00:02:05.000 --> 00:02:16.000
I have a curve in the plane and
I have a vector field.
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Let's see.
The vector field will be yellow
00:02:24.000 --> 00:02:28.000
today.
And I will want to integrate
00:02:28.000 --> 00:02:33.000
along the curve the dot product
of F with the normal vector to
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the curve, a unit normal vector
to the curve.
00:02:37.000 --> 00:02:44.000
That means a vector that is at
every point of the curve
00:02:44.000 --> 00:02:51.000
perpendicular to the curve and
has length one.
00:02:51.000 --> 00:03:06.000
N everywhere will be the unit
normal vector to the curve C
00:03:06.000 --> 00:03:17.000
pointing 90 degrees clockwise
from T.
00:03:17.000 --> 00:03:19.000
What does that mean?
That means I have two normal
00:03:19.000 --> 00:03:22.000
vectors, one that is pointing
this way, one that is pointing
00:03:22.000 --> 00:03:24.000
that way.
I have to choose a convention.
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And the convention is that the
normal vector that I take goes
00:03:28.000 --> 00:03:32.000
to the right of the curve as I
am traveling along the curve.
00:03:32.000 --> 00:03:36.000
You mentioned that you were
walking along this curve,
00:03:36.000 --> 00:03:40.000
then you look to your right,
that is that direction.
00:03:40.000 --> 00:03:44.000
What we will do is just,
at every point along the curve,
00:03:44.000 --> 00:03:48.000
the dot product between the
vector field and the normal
00:03:48.000 --> 00:03:51.000
vector.
And we will sum that along the
00:03:51.000 --> 00:03:58.000
various pieces of the curve.
What this notation means is
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that if we actually break C into
small pieces of length delta(s)
00:04:09.000 --> 00:04:17.000
then the flux will be the limit,
as the pieces become smaller
00:04:17.000 --> 00:04:25.000
and smaller,
of the sum of F dot n delta S.
00:04:25.000 --> 00:04:29.000
I take each small piece of my
curve, I do the dot product
00:04:29.000 --> 00:04:32.000
between F and n and I multiply
by the length of a piece.
00:04:32.000 --> 00:04:35.000
And then I add these together.
That is what the line integral
00:04:35.000 --> 00:04:38.000
means.
Of course that is,
00:04:38.000 --> 00:04:41.000
again, not how I will compute
it.
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Just to compare this with work,
conceptually it is similar to
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the line integral we did for
work except the line integral
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for work -- Work is the line
integral of F dot dr,
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which is also the line integral
of F dot T ds.
00:05:15.000 --> 00:05:20.000
That is how we reformulated it.
That means we take our curve
00:05:20.000 --> 00:05:26.000
and we figure out at each point
how big the tangent component --
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I guess I should probably take
the same vector field as before.
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Let's see.
My field was pointing more like
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that way.
What I do at any point is
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project F to the tangent
direction, I figure out how much
00:05:45.000 --> 00:05:50.000
F is going along my curve and
then I sum these things
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together.
I am actually summing -- -- the
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tangential component of my field
F.
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Roughly-speaking the work
measures, you know,
00:06:17.000 --> 00:06:21.000
when I move along my curve,
how much I am going with or
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against F.
Flux, on the other hand,
00:06:23.000 --> 00:06:26.000
measures, when I go along the
curve, roughly how much the
00:06:26.000 --> 00:06:28.000
field is going to across the
curve.
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Counting positively what goes
to the right,
00:06:31.000 --> 00:06:34.000
negatively what goes to the
left.
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Flux is integral F dot n ds,
and that one corresponds to
00:06:45.000 --> 00:06:54.000
summing the normal component of
a vector field.
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But apart from that
conceptually it is the same kind
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of thing.
Just the physical
00:06:59.000 --> 00:07:01.000
interpretations will be very
different,
00:07:01.000 --> 00:07:10.000
but for a mathematician these
are two line integrals that you
00:07:10.000 --> 00:07:17.000
set up and compute in pretty
much the same way.
00:07:17.000 --> 00:07:21.000
Let's see.
I should probably tell you what
00:07:21.000 --> 00:07:22.000
it means.
Why do we make this definition?
00:07:22.000 --> 00:07:27.000
What does it correspond to?
Well, the interpretation for
00:07:27.000 --> 00:07:31.000
work made a lot of sense when F
was representing a force.
00:07:31.000 --> 00:07:36.000
The line integral was actually
the work done by the force.
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The interpretation for flux
makes more sense if you think of
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F as a velocity field.
What is the interpretation?
00:07:50.000 --> 00:07:55.000
Let's say that for F is a
velocity field.
00:07:55.000 --> 00:08:00.000
That means I am thinking of
some fluid that is moving,
00:08:00.000 --> 00:08:04.000
maybe water or something,
and it is moving at a certain
00:08:04.000 --> 00:08:08.000
speed.
And my vector field represents
00:08:08.000 --> 00:08:14.000
how things are moving at every
point of the plane.
00:08:14.000 --> 00:08:31.000
I claim that flux measures how
much fluid passes through -- --
00:08:31.000 --> 00:08:41.000
the curve C per unit time.
If you imagine that maybe you
00:08:41.000 --> 00:08:45.000
have a river and you are somehow
building a damn here,
00:08:45.000 --> 00:08:48.000
a damn with holes in it so that
the water still passes through,
00:08:48.000 --> 00:08:53.000
then this measures how much
water passes through your
00:08:53.000 --> 00:08:57.000
membrane per unit time.
Let's try to figure out why
00:08:57.000 --> 00:09:00.000
this is true.
Why does this make sense?
00:09:00.000 --> 00:09:07.000
Let's look at what happens on a
small portion of our curve C.
00:09:07.000 --> 00:09:21.000
I am zooming in on my curve C.
I guess I need to zoom further.
00:09:21.000 --> 00:09:26.000
That is a little piece of my
curve, of length delta S,
00:09:26.000 --> 00:09:30.000
and there is a fluid flow.
On my picture things are
00:09:30.000 --> 00:09:33.000
flowing to the right.
Here I am drawing a constant
00:09:33.000 --> 00:09:38.000
vector field because if you zoom
in enough then your vectors will
00:09:38.000 --> 00:09:40.000
pretty much be the same
everywhere.
00:09:40.000 --> 00:09:44.000
If you enlarge the picture
enough then things will be
00:09:44.000 --> 00:09:48.000
pretty much a uniform flow.
Now, how much stuff goes
00:09:48.000 --> 00:09:51.000
through this little piece of
curve per unit time?
00:09:51.000 --> 00:09:57.000
Well, what happens over time is
the fluid is moving while my
00:09:57.000 --> 00:10:04.000
curve is staying the same place
so it corresponds to something
00:10:04.000 --> 00:10:09.000
like this.
I claim that what goes through
00:10:09.000 --> 00:10:16.000
C in unit time is actually going
to be a parallelogram.
00:10:16.000 --> 00:10:21.000
Here is a better picture.
I claim that what will be going
00:10:21.000 --> 00:10:24.000
through C is this shaded
parallelogram to the left of C.
00:10:24.000 --> 00:10:32.000
Let's see.
If I move for unit time it
00:10:32.000 --> 00:10:35.000
works.
That is the stuff that goes
00:10:35.000 --> 00:10:38.000
through my curve,
for a small portion of curve in
00:10:38.000 --> 00:10:40.000
unit time.
And, of course,
00:10:40.000 --> 00:10:43.000
I would need to add all of
these together to get the entire
00:10:43.000 --> 00:10:47.000
curve.
Let's try to understand how big
00:10:47.000 --> 00:10:50.000
this parallelogram is.
To know how big this
00:10:50.000 --> 00:10:53.000
parallelogram is I would like to
use base times height or
00:10:53.000 --> 00:10:56.000
something like that.
And maybe I want to actually
00:10:56.000 --> 00:10:59.000
flip my picture so that the base
and the height make more sense
00:10:59.000 --> 00:11:04.000
to me.
Let me actually turn it this
00:11:04.000 --> 00:11:10.000
way.
And, in case you have trouble
00:11:10.000 --> 00:11:21.000
reading the rotated picture,
let me redo it on the board.
00:11:21.000 --> 00:11:31.000
What passes through a portion
of C in unit time is the
00:11:31.000 --> 00:11:40.000
contents of a parallelogram
whose base is on C.
00:11:40.000 --> 00:11:49.000
So it has length delta s.
That is a piece of C.
00:11:49.000 --> 00:12:06.000
And the other side is going to
be given by my velocity vector
00:12:06.000 --> 00:12:11.000
F.
And to find the height of this
00:12:11.000 --> 00:12:17.000
thing, I need to know what
actually the normal component of
00:12:17.000 --> 00:12:24.000
this vector is.
If I call n the unit normal
00:12:24.000 --> 00:12:35.000
vector to the curve then the
area is base times height.
00:12:35.000 --> 00:12:42.000
The base is delta S and the
height is the normal component
00:12:42.000 --> 00:12:48.000
of F, so it is F dot n.
And so you see that when you
00:12:48.000 --> 00:12:54.000
sum these things together you
get, what I said,
00:12:54.000 --> 00:12:56.000
flux.
Now, if you are worried about
00:12:56.000 --> 00:13:00.000
the fact that actually -- If
your unit time is too long then
00:13:00.000 --> 00:13:03.000
of course things might start
changing as it flows.
00:13:03.000 --> 00:13:07.000
You have to take the time unit
and the length unit that are
00:13:07.000 --> 00:13:11.000
sufficiently small so that
really this approximation where
00:13:11.000 --> 00:13:15.000
C is a straight line and where
flow is at constant speed are
00:13:15.000 --> 00:13:17.000
valid.
You want to take maybe a
00:13:17.000 --> 00:13:20.000
segment here that is a few
micrometers.
00:13:20.000 --> 00:13:24.000
And the time unit might be a
few nanoseconds or whatever,
00:13:24.000 --> 00:13:28.000
and then it is a good
approximation.
00:13:28.000 --> 00:13:31.000
What I mean by per unit time
is, well, actually,
00:13:31.000 --> 00:13:35.000
that works, but you want to
think of a really,
00:13:35.000 --> 00:13:39.000
really small time.
And then the amount of matter
00:13:39.000 --> 00:13:44.000
that passes in that really,
really small time is the flux
00:13:44.000 --> 00:13:48.000
times the amount of time.
Let's be a tiny bit more
00:13:48.000 --> 00:13:50.000
careful.
And what I am saying is the
00:13:50.000 --> 00:13:53.000
amount of stuff that passes
through C depends actually on
00:13:53.000 --> 00:13:56.000
whether n is going this way or
the opposite way.
00:13:56.000 --> 00:14:00.000
Actually,
what is implicit in this
00:14:00.000 --> 00:14:05.000
explanation is that I am
counting positively all the
00:14:05.000 --> 00:14:11.000
stuff that flows across C in the
direction of n and negatively
00:14:11.000 --> 00:14:15.000
what flows in the opposite
direction.
00:14:15.000 --> 00:14:32.000
What flows to the right of C,
well, across C from left to
00:14:32.000 --> 00:14:47.000
right is counted positively.
While what flows right to left
00:14:47.000 --> 00:14:53.000
is counted negatively.
So, in fact,
00:14:53.000 --> 00:15:00.000
it is the net flow through C
per unit time.
00:15:00.000 --> 00:15:06.000
Any questions about the
definition or the interpretation
00:15:06.000 --> 00:15:16.000
or things like that?
Yes?
00:15:16.000 --> 00:15:19.000
Well, you can have both not in
the same small segment.
00:15:19.000 --> 00:15:24.000
But it could be that,
well, imagine that my vector
00:15:24.000 --> 00:15:28.000
field accidentally goes in the
opposite direction then this
00:15:28.000 --> 00:15:32.000
part of the curve,
while things are flowing to the
00:15:32.000 --> 00:15:35.000
left,
contributes negatively to flux.
00:15:35.000 --> 00:15:39.000
And here maybe the field is
tangent so the normal component
00:15:39.000 --> 00:15:42.000
becomes zero.
And then it becomes positive
00:15:42.000 --> 00:15:47.000
and this part of the curve
contributes positively.
00:15:47.000 --> 00:15:51.000
For example,
if you imagine that you have a
00:15:51.000 --> 00:15:54.000
round tank in which the fluid is
rotating and you put your dam
00:15:54.000 --> 00:15:57.000
just on a diameter across then
things are going one way on one
00:15:57.000 --> 00:15:59.000
side,
the other way on the other
00:15:59.000 --> 00:16:04.000
side,
and actually it just evens out.
00:16:04.000 --> 00:16:06.000
We don't have complete
information.
00:16:06.000 --> 00:16:13.000
It is just the total net flux.
OK.
00:16:13.000 --> 00:16:19.000
If there are no other questions
then I guess we will need to
00:16:19.000 --> 00:16:25.000
figure out how to compute this
guy and how to actually do this
00:16:25.000 --> 00:16:33.000
line integral.
Well, let's start with a couple
00:16:33.000 --> 00:16:43.000
of easy examples.
Let's say that C is a circle of
00:16:43.000 --> 00:16:55.000
radius (a) centered at the
origin going counterclockwise.
00:16:55.000 --> 00:17:02.000
And let's say that our vector
field is xi yj.
00:17:02.000 --> 00:17:09.000
What does that look like?
Remember, xi plus yj is a
00:17:09.000 --> 00:17:15.000
vector field that is pointing
radially away from the origin.
00:17:15.000 --> 00:17:19.000
Because at every point it is
equal to the vector from the
00:17:19.000 --> 00:17:25.000
origin to that point.
Now, if we have a circle and
00:17:25.000 --> 00:17:30.000
let's say we are going
counterclockwise.
00:17:30.000 --> 00:17:32.000
Actually, I have a nicer
picture.
00:17:32.000 --> 00:17:48.000
Let me do it here.
That is my curve and my vector
00:17:48.000 --> 00:17:55.000
field.
And the normal vector, see,
00:17:55.000 --> 00:17:57.000
when you go counterclockwise in
a closed curve,
00:17:57.000 --> 00:18:01.000
this convention that a normal
vector points to the right of
00:18:01.000 --> 00:18:04.000
curve makes it point out.
The usual convention,
00:18:04.000 --> 00:18:08.000
when you take flux for a closed
curve, is that you are counting
00:18:08.000 --> 00:18:11.000
the flux going out of the region
enclosed by the curve.
00:18:11.000 --> 00:18:13.000
And, of course,
if you went clockwise it would
00:18:13.000 --> 00:18:18.000
be the other way around.
You choose to do it the way you
00:18:18.000 --> 00:18:27.000
want, but the most common one is
to count flux going out of the
00:18:27.000 --> 00:18:31.000
region.
Let's see what happens.
00:18:31.000 --> 00:18:35.000
Well, if I am anywhere on my
circle, see, the normal vector
00:18:35.000 --> 00:18:38.000
is sticking straight out of the
circle.
00:18:38.000 --> 00:18:43.000
That is a property of the
circle that the radial direction
00:18:43.000 --> 00:18:49.000
is perpendicular to the circle.
Actually, let me complete this
00:18:49.000 --> 00:18:52.000
picture.
If I take a point on the
00:18:52.000 --> 00:18:59.000
circle, I have my normal vector
that is pointing straight out so
00:18:59.000 --> 00:19:05.000
it is parallel to F.
Along C we know that F is
00:19:05.000 --> 00:19:10.000
parallel to n,
so F dot n will be equal to the
00:19:10.000 --> 00:19:16.000
magnitude of F times,
well, the magnitude of n,
00:19:16.000 --> 00:19:20.000
but that is one.
Let me put it anywhere,
00:19:20.000 --> 00:19:23.000
but that is the unit normal
vector.
00:19:23.000 --> 00:19:27.000
Now, what is the magnitude of
this vector field if I am at a
00:19:27.000 --> 00:19:29.000
point x, y?
Well, it is square root of x
00:19:29.000 --> 00:19:32.000
squared plus y squared,
which is the same as the
00:19:32.000 --> 00:19:36.000
distance from the origin.
So if this distance,
00:19:36.000 --> 00:19:46.000
if this radius is a then the
magnitude of F will just be a.
00:19:46.000 --> 00:19:51.000
In fact, F dot n is constant,
always equal to a.
00:19:51.000 --> 00:19:57.000
So the line integral will be
pretty easy because all I have
00:19:57.000 --> 00:20:04.000
to do is the integral of F dot n
ds becomes the integral of a ds.
00:20:04.000 --> 00:20:07.000
(a) is a constant so I can take
it out.
00:20:07.000 --> 00:20:16.000
And integral ds is just a
length of C which is 2pi a,
00:20:16.000 --> 00:20:24.000
so I will get 2pi a squared.
And that is positive,
00:20:24.000 --> 00:20:28.000
as we expected,
because stuff is flowing out of
00:20:28.000 --> 00:20:36.000
the circle.
Any questions about that?
00:20:36.000 --> 00:20:41.000
No.
OK.
00:20:41.000 --> 00:20:45.000
Just out of curiosity,
let's say that we had taken our
00:20:45.000 --> 00:20:52.000
other favorite vector field.
Let's say that we had the same
00:20:52.000 --> 00:20:57.000
C, but now the vector field
.
00:20:57.000 --> 00:21:05.000
Remember, that one goes
counterclockwise around the
00:21:05.000 --> 00:21:09.000
origin.
If you remember what we did
00:21:09.000 --> 00:21:12.000
several times,
well, along the circle that
00:21:12.000 --> 00:21:16.000
vector field now is tangent to
the circle.
00:21:16.000 --> 00:21:19.000
If it is tangent to the circle
it doesn't have any normal
00:21:19.000 --> 00:21:22.000
component.
The normal component is zero.
00:21:22.000 --> 00:21:25.000
Things are not flowing into the
circle or out of it.
00:21:25.000 --> 00:21:30.000
They are just flowing along the
circle around and around so the
00:21:30.000 --> 00:21:38.000
flux will be zero.
F now is tangent to C.
00:21:38.000 --> 00:21:51.000
F dot n is zero and,
therefore, the flux will be
00:21:51.000 --> 00:21:55.000
zero.
These are examples where you
00:21:55.000 --> 00:21:57.000
can compute things
geometrically.
00:21:57.000 --> 00:22:00.000
And I would say,
generally speaking,
00:22:00.000 --> 00:22:03.000
with flux, well,
if it is a very complicated
00:22:03.000 --> 00:22:06.000
field then you cannot.
But, if a field is fairly
00:22:06.000 --> 00:22:08.000
simple,
you should be able to get some
00:22:08.000 --> 00:22:11.000
general feeling for whether your
answer should be positive,
00:22:11.000 --> 00:22:15.000
negative or zero just by
thinking about which way is my
00:22:15.000 --> 00:22:20.000
flow going.
Is it going across the curve
00:22:20.000 --> 00:22:32.000
one way or the other way?
Still no questions about these
00:22:32.000 --> 00:22:36.000
examples?
The next thing we need to know
00:22:36.000 --> 00:22:40.000
is how we will actually compute
these things because here,
00:22:40.000 --> 00:22:43.000
yeah, it works pretty well,
but what if you don't have a
00:22:43.000 --> 00:22:47.000
simple geometric interpretation.
What if I give you a really
00:22:47.000 --> 00:22:50.000
complicated curve and then you
have trouble finding the normal
00:22:50.000 --> 00:22:53.000
vector?
It is going to be annoying to
00:22:53.000 --> 00:22:56.000
set up things this way.
Actually, there is a better way
00:22:56.000 --> 00:22:59.000
to do it in coordinates.
Just as we do work,
00:22:59.000 --> 00:23:04.000
when we compute this line
integral, usually we don't do it
00:23:04.000 --> 00:23:08.000
geometrically like this.
Most of the time we just
00:23:08.000 --> 00:23:12.000
integrate M dx plus N dy in
coordinates.
00:23:12.000 --> 00:23:16.000
That is a similar way to do it
because it is,
00:23:16.000 --> 00:23:20.000
again, a line integral so it
should work the same way.
00:23:20.000 --> 00:23:21.000
Let's try to figure that out.
00:24:05.000 --> 00:24:22.000
How do we do the calculation in
coordinates, or I should say
00:24:22.000 --> 00:24:29.000
using components?
That is the general method of
00:24:29.000 --> 00:24:33.000
calculation when we don't have
something geometric to do.
00:24:33.000 --> 00:24:41.000
Remember, when we were doing
things for work we said this
00:24:41.000 --> 00:24:49.000
vector dr, or if you prefer T
ds, we said just becomes
00:24:49.000 --> 00:24:56.000
symbolically dx and dy.
When you do the line integral
00:24:56.000 --> 00:25:01.000
of F dot dr you get line
integral of n dx plus n dy.
00:25:01.000 --> 00:25:07.000
Now let's think for a second
about how we would express n ds.
00:25:07.000 --> 00:25:11.000
Well, what is n ds compared to
T ds?
00:25:11.000 --> 00:25:15.000
Well, M is just T rotated by 90
degrees, so n ds is T ds rotated
00:25:15.000 --> 00:25:19.000
by 90 degrees.
That might sound a little bit
00:25:19.000 --> 00:25:23.000
outrageous because these are
really symbolic notations but it
00:25:23.000 --> 00:25:25.000
works.
I am not going to spend too
00:25:25.000 --> 00:25:28.000
much time trying to convince you
carefully.
00:25:28.000 --> 00:25:33.000
But if you go back to where we
wrote this and how we tried to
00:25:33.000 --> 00:25:36.000
justify this and you work your
way through it,
00:25:36.000 --> 00:25:42.000
you will see that n ds can be
analyzed the same way.
00:25:42.000 --> 00:25:51.000
N is T rotated 90 degrees
clockwise.
00:25:51.000 --> 00:25:57.000
That tells us that n ds is --
How do we rotate a vector by 90
00:25:57.000 --> 00:26:00.000
degrees?
Well, we swept the two
00:26:00.000 --> 00:26:05.000
components and we put a minus
sign.
00:26:05.000 --> 00:26:07.000
You have dy and dx.
And you have to be careful
00:26:07.000 --> 00:26:11.000
where to put the minus sign.
Well, if you are doing it
00:26:11.000 --> 00:26:13.000
clockwise, it is in front of dx.
00:26:26.000 --> 00:26:29.000
Well, actually,
let me just convince you
00:26:29.000 --> 00:26:32.000
quickly.
Let's say we have a small piece
00:26:32.000 --> 00:26:36.000
of C.
If we do T delta S,
00:26:36.000 --> 00:26:44.000
that is also vector delta r.
That is going to be just the
00:26:44.000 --> 00:26:48.000
vector that goes along the curve
given by this.
00:26:48.000 --> 00:26:54.000
Its components will be indeed
the change in x,
00:26:54.000 --> 00:27:00.000
delta x, and the change in y,
delta y.
00:27:00.000 --> 00:27:07.000
And now, if I want to get n
delta S, well,
00:27:07.000 --> 00:27:15.000
I claim now that it is
perfectly valid and rigorous to
00:27:15.000 --> 00:27:24.000
just rotate that by 90 degrees.
If I want to rotate this by 90
00:27:24.000 --> 00:27:31.000
degrees clockwise then the x
component will become the same
00:27:31.000 --> 00:27:36.000
as the old y component.
And the y component will be
00:27:36.000 --> 00:27:40.000
minus delta x.
Then you take the limit when
00:27:40.000 --> 00:27:44.000
the segment becomes shorter and
shorter, and that is how you can
00:27:44.000 --> 00:27:47.000
justify this.
That is the key to computing
00:27:47.000 --> 00:27:50.000
things in practice.
It means, actually,
00:27:50.000 --> 00:27:55.000
you already know how to compute
line integrals for flux.
00:27:55.000 --> 00:28:05.000
Let me just write it explicitly.
Let's say that our vector field
00:28:05.000 --> 00:28:08.000
has two components.
And let me just confuse you a
00:28:08.000 --> 00:28:12.000
little bit and not call them M
and N for this time just to
00:28:12.000 --> 00:28:16.000
stress the fact that we are
doing a different line integral.
00:28:16.000 --> 00:28:22.000
Let me call them P and Q for
now.
00:28:22.000 --> 00:28:31.000
Then the line integral of F dot
n ds will be the line integral
00:28:31.000 --> 00:28:39.000
of
dot product .
00:28:39.000 --> 00:28:46.000
That will be the integral of -
Q dx P dy.
00:28:46.000 --> 00:28:50.000
Well, I am running out of space
here.
00:28:50.000 --> 00:29:01.000
It is integral along C of
negative Q dx plus P dy.
00:29:01.000 --> 00:29:04.000
And from that point onwards you
just do it the usual way.
00:29:04.000 --> 00:29:10.000
Remember, here you have two
variables x and y but you are
00:29:10.000 --> 00:29:14.000
integrating along a curve.
If you are integrating along a
00:29:14.000 --> 00:29:18.000
curve x and y are related.
They depend on each other or
00:29:18.000 --> 00:29:21.000
maybe on some other parameter
like T or theta or whatever.
00:29:21.000 --> 00:29:28.000
You express everything in terms
of a single variable and then
00:29:28.000 --> 00:29:36.000
you do a usual single integral.
Any questions about that?
00:29:36.000 --> 00:29:39.000
I see a lot of confused faces
so maybe I shouldn't have called
00:29:39.000 --> 00:29:41.000
my component P and Q.
00:30:04.000 --> 00:30:14.000
If you prefer,
if you are really sentimentally
00:30:14.000 --> 00:30:27.000
attached to M and N then this
new line integral becomes the
00:30:27.000 --> 00:30:35.000
integral of - N dx M dy.
If a problem tells you compute
00:30:35.000 --> 00:30:37.000
flux instead of saying compute
work,
00:30:37.000 --> 00:30:41.000
the only thing you change is
instead of doing M dx plus N dy
00:30:41.000 --> 00:30:45.000
you do minus N dx plus M dy.
And I am sorry to say that I
00:30:45.000 --> 00:30:49.000
don't have any good way of
helping you remember which one
00:30:49.000 --> 00:30:52.000
of the two gets the minus sign,
so you just have to remember
00:30:52.000 --> 00:30:58.000
this formula by heart.
That is the only way I know.
00:30:58.000 --> 00:31:04.000
Well, you can try to go through
this argument again,
00:31:04.000 --> 00:31:10.000
but it is really best if you
just remember that formula.
00:31:10.000 --> 00:31:15.000
I am not going to do an example
because we already know how to
00:31:15.000 --> 00:31:19.000
do line integrals.
Hopefully you will get to see
00:31:19.000 --> 00:31:23.000
one at least in recitation on
Monday.
00:31:23.000 --> 00:31:29.000
That is all pretty good.
Let me tell you now what if I
00:31:29.000 --> 00:31:35.000
have to compute flux along a
closed curve and I don't want to
00:31:35.000 --> 00:31:39.000
compute it?
Well, remember in the case of
00:31:39.000 --> 00:31:43.000
work we had Green's theorem.
We saw yesterday Green's
00:31:43.000 --> 00:31:45.000
theorem.
Let's us replace a line
00:31:45.000 --> 00:31:48.000
integral along a closed curve by
a double integral.
00:31:48.000 --> 00:31:51.000
Well, here it is the same.
We have a line integral along a
00:31:51.000 --> 00:31:53.000
curve.
If it is a closed curve,
00:31:53.000 --> 00:31:57.000
we should be able to replace it
by a double integral.
00:31:57.000 --> 00:32:09.000
There is a version of Green's
theorem for flux.
00:32:09.000 --> 00:32:13.000
And you will see it is not
scarier than the other one.
00:32:13.000 --> 00:32:18.000
It is perhaps less scarier or
perhaps just as scary or just
00:32:18.000 --> 00:32:22.000
not as scary,
depending on how you feel about
00:32:22.000 --> 00:32:26.000
it, but it works pretty much the
same way.
00:32:26.000 --> 00:32:30.000
What does Green's theorem for
flux say?
00:32:30.000 --> 00:32:39.000
It says if C is a curve that
encloses a region R
00:32:39.000 --> 00:32:51.000
counterclockwise and if I have a
vector field that is defined
00:32:51.000 --> 00:32:56.000
everywhere,
not just on C but also inside,
00:32:56.000 --> 00:33:11.000
so also on R.
Well, maybe I should give names
00:33:11.000 --> 00:33:14.000
to the components.
If you will forgive me for a
00:33:14.000 --> 00:33:16.000
second, I will still use P and Q
for now.
00:33:16.000 --> 00:33:23.000
You will see why.
It is defined and
00:33:23.000 --> 00:33:30.000
differentiable in R.
Then I can actually -- --
00:33:30.000 --> 00:33:40.000
replace the line integral for
flux by a double integral over R
00:33:40.000 --> 00:33:47.000
of some function.
And that function is called the
00:33:47.000 --> 00:33:58.000
divergence of F dA.
This is the divergence of F.
00:33:58.000 --> 00:34:08.000
And I have to define for you
what this guy is.
00:34:08.000 --> 00:34:15.000
The divergence of a vector
field with components P and Q is
00:34:15.000 --> 00:34:20.000
just P sub x Q sub y.
This one is actually easier to
00:34:20.000 --> 00:34:23.000
remember than curl because you
just take the x component,
00:34:23.000 --> 00:34:26.000
take its partial with respect
to x,
00:34:26.000 --> 00:34:29.000
take the y component,
take its partial with respect
00:34:29.000 --> 00:34:31.000
to y and add them together.
No signs.
00:34:31.000 --> 00:34:36.000
No switching things around.
This one is pretty
00:34:36.000 --> 00:34:43.000
straightforward.
The picture again is if I have
00:34:43.000 --> 00:34:50.000
my curve C going
counterclockwise around a region
00:34:50.000 --> 00:34:58.000
R and I want to find the flux of
some vector field F that is
00:34:58.000 --> 00:35:03.000
everywhere in here.
Maybe some parts of C will
00:35:03.000 --> 00:35:06.000
contribute positively and some
parts will contribute
00:35:06.000 --> 00:35:10.000
negatively.
Just to reiterate what I said,
00:35:10.000 --> 00:35:14.000
positively here means,
because we are going
00:35:14.000 --> 00:35:19.000
counterclockwise,
the normal vector points out of
00:35:19.000 --> 00:35:31.000
the region.
This guy here is the flux out
00:35:31.000 --> 00:35:39.000
of R through C.
That is the formula.
00:35:39.000 --> 00:35:45.000
Any questions about what the
statement says or how to use it
00:35:45.000 --> 00:35:48.000
concretely?
No.
00:35:48.000 --> 00:35:51.000
OK.
It is pretty similar to Green's
00:35:51.000 --> 00:35:58.000
theorem for work.
Actually, I should say -- This
00:35:58.000 --> 00:36:07.000
is called Green's theorem in
normal form also.
00:36:07.000 --> 00:36:17.000
Not that the other one is
abnormal, but just that the old
00:36:17.000 --> 00:36:23.000
one for work was,
you could say,
00:36:23.000 --> 00:36:28.000
in tangential form.
That just means,
00:36:28.000 --> 00:36:32.000
well, Green's theorem,
as seen yesterday was for the
00:36:32.000 --> 00:36:36.000
line integral F dot T ds,
integrating the tangent
00:36:36.000 --> 00:36:39.000
component of F.
The one today is for
00:36:39.000 --> 00:36:43.000
integrating the normal component
of F.
00:36:43.000 --> 00:36:47.000
OK. Let's prove this.
Good news.
00:36:47.000 --> 00:36:50.000
It is much easier to prove than
the one we did yesterday because
00:36:50.000 --> 00:36:53.000
we are just going to show that
it is the same thing just using
00:36:53.000 --> 00:36:54.000
different notations.
00:37:23.000 --> 00:37:29.000
How do I prove it?
Well, maybe actually it would
00:37:29.000 --> 00:37:33.000
help if first,
before proving it,
00:37:33.000 --> 00:37:38.000
I actually rewrite what it
means in components.
00:37:38.000 --> 00:37:46.000
We said the line integral of F
dot n ds is actually the line
00:37:46.000 --> 00:37:54.000
integral of - Q dx P dy.
And we want to show that this
00:37:54.000 --> 00:38:03.000
is equal to the double integral
of P sub x Q sub y dA.
00:38:03.000 --> 00:38:09.000
This is really one of the
features of Green's theorem.
00:38:09.000 --> 00:38:12.000
No matter which form it is,
it relates a line integral to a
00:38:12.000 --> 00:38:16.000
double integral.
Let's just try to see if we can
00:38:16.000 --> 00:38:19.000
reduce it to the one we had
yesterday.
00:38:19.000 --> 00:38:24.000
Let me forget what these things
mean physically and just focus
00:38:24.000 --> 00:38:26.000
on the math.
On the math it is a line
00:38:26.000 --> 00:38:29.000
integral of something dx plus
something dy.
00:38:29.000 --> 00:38:36.000
Let's call this guy M and let's
call this guy N.
00:38:36.000 --> 00:38:42.000
Let M equal negative Q and N
equal P.
00:38:42.000 --> 00:38:53.000
Then this guy here becomes
integral of M dx plus N dy.
00:38:53.000 --> 00:38:57.000
And I know from yesterday what
this is equal to,
00:38:57.000 --> 00:39:01.000
namely using the tangential
form of Green's theorem.
00:39:01.000 --> 00:39:05.000
Green for work.
This is the double integral of
00:39:05.000 --> 00:39:11.000
curl of this guy.
That is Nx minus My dA.
00:39:11.000 --> 00:39:15.000
But now let's think about what
this is in terms of M and N.
00:39:15.000 --> 00:39:24.000
Well, we said that M is
negative Q so this is negative
00:39:24.000 --> 00:39:29.000
My.
And we said P is the same as N,
00:39:29.000 --> 00:39:33.000
so this is Nx.
Just by renaming the
00:39:33.000 --> 00:39:37.000
components, I go from one form
to the other one.
00:39:37.000 --> 00:39:38.000
So it is really the same
theorem.
00:39:38.000 --> 00:39:41.000
That's why it is also called
Green's theorem.
00:39:41.000 --> 00:39:45.000
But the way we think about it
when we use it is different,
00:39:45.000 --> 00:39:48.000
because one of them computes
the work done by a force along a
00:39:48.000 --> 00:39:53.000
closed curve,
the other one computes the flux
00:39:53.000 --> 00:39:59.000
maybe of a velocity field out of
region.
00:39:59.000 --> 00:40:10.000
Questions?
Yes?
00:40:10.000 --> 00:40:14.000
That is correct.
If you are trying to compute a
00:40:14.000 --> 00:40:18.000
line integral for flux,
wait, where did I put it?
00:40:18.000 --> 00:40:20.000
A line integral for flux just
becomes this.
00:40:20.000 --> 00:40:25.000
And once you are here you know
how to compute that kind of
00:40:25.000 --> 00:40:27.000
thing.
The double integral side does
00:40:27.000 --> 00:40:29.000
not even have any kind of
renaming to do.
00:40:29.000 --> 00:40:31.000
You know how to compute a
double integral of a function.
00:40:31.000 --> 00:40:35.000
This is just a particular kind
of function that you get out of
00:40:35.000 --> 00:40:38.000
a vector field,
but it is like any function.
00:40:38.000 --> 00:40:41.000
The way you would evaluate
these double integrals is just
00:40:41.000 --> 00:40:46.000
the usual way.
Namely, you have a function of
00:40:46.000 --> 00:40:54.000
x and y, you have a region and
you set up the bounds for the
00:40:54.000 --> 00:40:57.000
isolated integral.
The way you would evaluate the
00:40:57.000 --> 00:40:59.000
double integrals is really the
usual way,
00:40:59.000 --> 00:41:02.000
by slicing the region and
setting up the bounds for
00:41:02.000 --> 00:41:06.000
iterated integrals in dx,
dy or dydx or maybe rd,
00:41:06.000 --> 00:41:12.000
rd theta or whatever you want.
In fact, in terms of computing
00:41:12.000 --> 00:41:14.000
integrals, we just have two sets
of skills.
00:41:14.000 --> 00:41:18.000
One is setting up and
evaluating double integrals.
00:41:18.000 --> 00:41:21.000
The other one is setting up and
evaluating line integrals.
00:41:21.000 --> 00:41:25.000
And whether these line
integrals or double integrals
00:41:25.000 --> 00:41:29.000
are representing work,
flux, integral of a curve,
00:41:29.000 --> 00:41:34.000
whatever,
the way that we actually
00:41:34.000 --> 00:41:40.000
compute them is the same.
Let's do an example.
00:41:40.000 --> 00:41:47.000
Oh, first. Sorry.
This renaming here, see,
00:41:47.000 --> 00:41:51.000
that is why actually I call my
components P and Q because the
00:41:51.000 --> 00:41:54.000
argument would have gotten very
messy if I had told you now I
00:41:54.000 --> 00:41:57.000
call M ,N and I call N minus M
and so on.
00:41:57.000 --> 00:42:00.000
But, now that we are through
with this,
00:42:00.000 --> 00:42:03.000
if you still like M and N
better,
00:42:03.000 --> 00:42:15.000
you know, what this says -- The
formulation of Green's theorem
00:42:15.000 --> 00:42:27.000
in this language is just
integral of minus N dx plus M dy
00:42:27.000 --> 00:42:37.000
is the double integral over R of
Mx plus Ny dA.
00:42:37.000 --> 00:42:42.000
Now let's do an example.
Let's look at this picture
00:42:42.000 --> 00:42:51.000
again, the flux of xi plus yj
out of the circle of radius A.
00:42:51.000 --> 00:42:53.000
We did the calculation directly
using geometry,
00:42:53.000 --> 00:42:57.000
and it wasn't all that bad.
But let's see what Green's
00:42:57.000 --> 00:42:58.000
theorem does for us here.
00:43:19.000 --> 00:43:22.000
Example.
Let's take the same example as
00:43:22.000 --> 00:43:28.000
last time.
F equals xi yj.
00:43:28.000 --> 00:43:43.000
C equals circle of radius a
counterclockwise.
00:43:43.000 --> 00:43:46.000
How do we set up Green's
theorem.
00:43:46.000 --> 00:43:57.000
Well, let's first figure out
the divergence of F.
00:43:57.000 --> 00:44:00.000
The divergence of this field,
I take the x component,
00:44:00.000 --> 00:44:03.000
which is x, and I take its
partial respect to x.
00:44:03.000 --> 00:44:08.000
And then I do the same with the
y component, and I will get one
00:44:08.000 --> 00:44:12.000
plus one equals two.
So, the divergence of this
00:44:12.000 --> 00:44:17.000
field is two.
Now, Green's theorem tells us
00:44:17.000 --> 00:44:25.000
that the flux out of this region
is going to be the double
00:44:25.000 --> 00:44:29.000
integral of 2 dA.
What is R now?
00:44:29.000 --> 00:44:31.000
Well, R is the region enclosed
by C.
00:44:31.000 --> 00:44:38.000
So if C is the circle,
R is the disk of radius A.
00:44:38.000 --> 00:44:42.000
Of course, we can compute it,
but we don't have to because
00:44:42.000 --> 00:44:46.000
double integral of 2dA is just
twice the double integral of dA
00:44:46.000 --> 00:44:51.000
so it is twice the area of R.
And we know the area of a
00:44:51.000 --> 00:44:54.000
circle of radius A.
That is piA2.
00:44:54.000 --> 00:45:01.000
So, it is 2piA2.
That is the same answer that we
00:45:01.000 --> 00:45:04.000
got directly,
which is good news.
00:45:04.000 --> 00:45:08.000
Now we can even do better.
Let's say that my circle is not
00:45:08.000 --> 00:45:12.000
at the origin.
Let's say that it is out here.
00:45:12.000 --> 00:45:17.000
Well, then it becomes harder to
calculate the flux directly.
00:45:17.000 --> 00:45:21.000
And it is harder even to guess
exactly what will happen because
00:45:21.000 --> 00:45:24.000
on this side here the vector
field will go into the region so
00:45:24.000 --> 00:45:27.000
the contribution to flux will be
negative here.
00:45:27.000 --> 00:45:31.000
Here it will be positive
because it is going out of the
00:45:31.000 --> 00:45:33.000
region.
There are positive and negative
00:45:33.000 --> 00:45:35.000
terms.
Well, it looks like positive
00:45:35.000 --> 00:45:38.000
should win because here the
vector field is much larger than
00:45:38.000 --> 00:45:41.000
over there.
But, short of computing it,
00:45:41.000 --> 00:45:45.000
we won't actually know what it
is.
00:45:45.000 --> 00:45:48.000
If you want to do it by direct
calculation then you have to
00:45:48.000 --> 00:45:51.000
parametize this circle and
figure out what the line
00:45:51.000 --> 00:45:55.000
integral will be.
But if you use Green's theorem,
00:45:55.000 --> 00:46:00.000
well, we never used the fact
that it is the circle of radius
00:46:00.000 --> 00:46:03.000
A at the origin.
It is true actually for any
00:46:03.000 --> 00:46:08.000
closed curve that the flux out
of it is going to be twice the
00:46:08.000 --> 00:46:12.000
area of the region inside.
It still will be 2piA2 even if
00:46:12.000 --> 00:46:16.000
my circle is anywhere else in
the plane.
00:46:16.000 --> 00:46:18.000
If I had asked you a trick
question where do you want to
00:46:18.000 --> 00:46:21.000
place this circle so that that
the flux is the largest?
00:46:21.000 --> 00:46:28.000
Well, the answer is it doesn't
matter.
00:46:28.000 --> 00:46:33.000
Now, let's just finish quickly
by answering a question that
00:46:33.000 --> 00:46:36.000
some of you,
I am sure, must have,
00:46:36.000 --> 00:46:40.000
which is what does divergence
mean and what does it measure?
00:46:40.000 --> 00:46:44.000
I mean, we said for curl,
curl measures how much things
00:46:44.000 --> 00:46:48.000
are rotating somehow.
What does divergence mean?
00:46:48.000 --> 00:46:53.000
Well, the answer is divergence
measures how much things are
00:46:53.000 --> 00:47:08.000
diverging.
Let's be more explicit.
00:47:08.000 --> 00:47:20.000
Interpretation of divergence.
You can think of it,
00:47:20.000 --> 00:47:23.000
you know, what do I want to say
first?
00:47:23.000 --> 00:47:28.000
If you take a vector field that
is a constant vector field where
00:47:28.000 --> 00:47:32.000
everything just translates then
there is no divergence involved
00:47:32.000 --> 00:47:34.000
because the derivatives will be
zero.
00:47:34.000 --> 00:47:37.000
If you take the guy that
rotates things around you will
00:47:37.000 --> 00:47:40.000
also compute and find zero for
divergence.
00:47:40.000 --> 00:47:43.000
This is not sensitive to
translation motions where
00:47:43.000 --> 00:47:46.000
everything moves together or to
rotation motions,
00:47:46.000 --> 00:47:51.000
but instead it is sensitive to
explaining motions.
00:47:51.000 --> 00:48:04.000
A possible answer is that it
measures how much the flow is
00:48:04.000 --> 00:48:10.000
expanding areas.
If you imagine this flow that
00:48:10.000 --> 00:48:14.000
we have here on the picture,
things are moving away from the
00:48:14.000 --> 00:48:16.000
origin and they fill out the
plane.
00:48:16.000 --> 00:48:19.000
If we mention this fluid
flowing out there,
00:48:19.000 --> 00:48:21.000
it is occupying more and more
space.
00:48:21.000 --> 00:48:24.000
And so that is what it means to
have positive divergence.
00:48:24.000 --> 00:48:28.000
If you took the opposite vector
field that contracts everything
00:48:28.000 --> 00:48:31.000
to the origin that will have
negative divergence.
00:48:31.000 --> 00:48:34.000
That is a good way to think
about it if you are thinking of
00:48:34.000 --> 00:48:37.000
a gas maybe that can expand to
fill out more volume.
00:48:37.000 --> 00:48:41.000
If you thinking of water,
well, water doesn't really
00:48:41.000 --> 00:48:43.000
shrink or expand.
The fact that it is taking more
00:48:43.000 --> 00:48:46.000
and more space actually means
that there is more and more
00:48:46.000 --> 00:48:51.000
water.
The other way to think about it
00:48:51.000 --> 00:48:56.000
is divergence is the source
rate,
00:48:56.000 --> 00:49:00.000
it is the amount of fluid that
is being inserted into the
00:49:00.000 --> 00:49:12.000
system,
that is being pumped into the
00:49:12.000 --> 00:49:27.000
system per unit time per unit
area.
00:49:27.000 --> 00:49:31.000
What div F equals two here
means is that here you actually
00:49:31.000 --> 00:49:35.000
have matter being created or
being pumped into the system so
00:49:35.000 --> 00:49:39.000
that you have more and more
water filling more and more
00:49:39.000 --> 00:49:41.000
space as it flows.
But, actually,
00:49:41.000 --> 00:49:43.000
divergence is not two just at
the origin.
00:49:43.000 --> 00:49:46.000
It is two everywhere.
So, in fact,
00:49:46.000 --> 00:49:49.000
to have this you need to have a
system of pumps that actually is
00:49:49.000 --> 00:49:52.000
in something water absolutely
everywhere uniformly.
00:49:52.000 --> 00:49:55.000
That is the only way to do this.
I mean if you imagine that you
00:49:55.000 --> 00:49:57.000
just have one spring at the
origin then,
00:49:57.000 --> 00:50:00.000
sure, water will flow out,
but as you go further and
00:50:00.000 --> 00:50:02.000
further away it will do so more
and more slowly.
00:50:02.000 --> 00:50:04.000
Well, here it is flowing away
faster and faster.
00:50:04.000 --> 00:50:09.000
And that means everywhere you
are still pumping more water
00:50:09.000 --> 00:50:11.000
into it.
So, that is what divergence
00:50:11.000 --> 00:50:13.000
measures.