Double integrals 5

WEBVTT
Kind: captions
Language: en

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In all of the double
integrals we've done so

00:00:03.320 --> 00:00:06.820
far, the boundaries on
x and y were fixed.

00:00:06.820 --> 00:00:09.490
Now we'll see what happens
when the boundaries on

00:00:09.490 --> 00:00:12.690
x and y are variables.

00:00:12.690 --> 00:00:15.580
So let's say I have the same
surface, and I'm not going to

00:00:15.580 --> 00:00:16.910
draw it the way it looks, I'll
just kind of draw

00:00:16.910 --> 00:00:17.580
it figuratively.

00:00:17.580 --> 00:00:21.310
But the problem we're actually
going to do is z, and this is

00:00:21.310 --> 00:00:23.820
the exact same one we've
been doing all along.

00:00:23.820 --> 00:00:25.900
The point of here isn't to show
you how to integrate, the point

00:00:25.900 --> 00:00:28.020
of here is to show you how
to visualize and think

00:00:28.020 --> 00:00:29.100
about these problems.

00:00:29.100 --> 00:00:31.430
And frankly, in double integral
problems the hardest part is

00:00:31.430 --> 00:00:32.690
figuring out the boundaries.

00:00:32.690 --> 00:00:34.560
Once you do that, the
integration is pretty

00:00:34.560 --> 00:00:35.130
straightforward.

00:00:35.130 --> 00:00:39.260
It's really not any harder then
single variable integration.

00:00:39.260 --> 00:00:41.130
So let's say that's our
surface: z is equal

00:00:41.130 --> 00:00:42.990
to xy squared.

00:00:42.990 --> 00:00:47.150
Let me draw the axes again.

00:00:47.150 --> 00:00:51.140
So that's my x-axis.

00:00:51.140 --> 00:00:53.950
That's my z-axis.

00:00:53.950 --> 00:00:55.310
That's my y-axis.

00:00:58.155 --> 00:01:02.350
x, y, and z.

00:01:02.350 --> 00:01:05.290
And you saw what this graph
looked like several videos ago.

00:01:05.290 --> 00:01:08.340
I took out the whole grapher
and we rotated and things.

00:01:08.340 --> 00:01:10.190
I'm not going to draw the graph
the way it looks; I'm just

00:01:10.190 --> 00:01:13.020
going to brought fairly
abstractly as just an

00:01:13.020 --> 00:01:14.140
abstract surface.

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Because the point here it's
really to figure out the

00:01:16.210 --> 00:01:18.420
boundaries of integration.

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Before I actually even draw
the surface, I'm going

00:01:20.033 --> 00:01:21.690
to draw the boundary.

00:01:21.690 --> 00:01:23.630
The first time we did this
problem we said, OK, x goes

00:01:23.630 --> 00:01:27.600
from 0 to 2, y goes from 0 to
1, and then we figured out the

00:01:27.600 --> 00:01:31.240
volume above that
bounded domain.

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Now let's do something else.

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Let's say that x
goes from 0 to 1.

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And let's say that the volume
that we want to figure out

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under the surface, it's not
from a fixed y to

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an upper-bound y.

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I'll show you: it's
actually a curve.

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So this is all on the xy plane,
everything I'm drawing here.

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And this curve, we could view
it two ways: we could say y is

00:02:10.920 --> 00:02:13.450
a function of x, y is
equal to x squared.

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Or we could write is equal
to square root of y.

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We don't have to write plus or
minus or anything like that

00:02:18.480 --> 00:02:20.880
because we're in the
first quadrant.

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So this is the area
above which we want to

00:02:25.150 --> 00:02:26.010
figure out the volume.

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Let me, yeah, it doesn't hurt
to color it in just so we

00:02:32.070 --> 00:02:36.070
can really hone in on
what we care about.

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So that's the area
above which we want to

00:02:38.170 --> 00:02:39.490
figure out the volume.

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You could kind of say,
that's our bounded domain.

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And so x goes from 0 to
1, and then this point

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is going to be what?

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That point's going to
be 1 comma 1, right?

00:02:48.260 --> 00:02:51.210
1 is equal to 1 squared, 1 is
equal to the square root of 1.

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So this point is
y is equal to 1.

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And then I'm not going to
draw this surface exactly.

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I'm just trying to give you a
sense of what the volume of

00:03:01.530 --> 00:03:04.700
the figure we're trying
to calculate is.

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If this is just some arbitrary
surface-- let me do it in a

00:03:07.210 --> 00:03:12.910
different color --so
this is the top.

00:03:12.910 --> 00:03:14.770
This line is going vertical
in the z-direction.

00:03:20.640 --> 00:03:23.340
Actually, I could draw it like
this, like it's a curve.

00:03:23.340 --> 00:03:26.865
And then this curve back here
is going to be like a wall.

00:03:33.090 --> 00:03:37.460
And maybe I'll paint this side
of the wall just so you can see

00:03:37.460 --> 00:03:40.270
what it kind of looks like.

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Trying my best.

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Think you get an idea.

00:03:43.960 --> 00:03:46.890
Let me make it a little darker;
this is actually more of an

00:03:46.890 --> 00:03:53.220
exercise in art than in
math, in many ways.

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You get the idea.

00:03:54.090 --> 00:03:57.460
And then the boundary
here is like this.

00:03:57.460 --> 00:03:59.260
And this top isn't flat,
you know, it could

00:03:59.260 --> 00:04:00.890
be curved surface.

00:04:00.890 --> 00:04:03.420
I do a little like that,
but it's a curved surface.

00:04:03.420 --> 00:04:06.340
And we know in the example
we're about to do that the

00:04:06.340 --> 00:04:08.770
surface right here is z
is equal to x squared.

00:04:08.770 --> 00:04:12.430
So we want to figure out
the volume under this.

00:04:12.430 --> 00:04:14.400
So how do we do it?

00:04:14.400 --> 00:04:16.840
Well, let's think about it.

00:04:16.840 --> 00:04:21.790
We could actually use the
intuition that I just gave you.

00:04:21.790 --> 00:04:25.030
We're essentially just going to
take a da, which is a little

00:04:25.030 --> 00:04:31.440
small square down here, and
that little area, that's the

00:04:31.440 --> 00:04:38.490
same thing as the dx-- let me
use a darker color --as a dx

00:04:38.490 --> 00:04:42.320
times a dy, and then we just
have to multiply it times f of

00:04:42.320 --> 00:04:46.480
xy, which is this, for
each area, and then

00:04:46.480 --> 00:04:48.110
some them all up.

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And then we could take a sum
in the x-direction first

00:04:50.610 --> 00:04:52.540
or the y-direction first.

00:04:52.540 --> 00:04:55.440
Now before doing that, just
to make sure that you have

00:04:55.440 --> 00:04:57.480
the intuition because the
boundaries are the hard part,

00:04:57.480 --> 00:04:59.270
let me just draw our xy plane.

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So let me rotate
it up like that.

00:05:01.860 --> 00:05:04.610
I'm just going to
draw our xy plane.

00:05:04.610 --> 00:05:07.020
Because that's what matters.

00:05:07.020 --> 00:05:08.790
Because the hard part here
is just figuring out our

00:05:08.790 --> 00:05:09.745
bounds of integration.

00:05:14.160 --> 00:05:17.600
So the curve is just y is
equal to x squared, look

00:05:17.600 --> 00:05:20.730
something like that.

00:05:20.730 --> 00:05:22.400
This is the point
y is equal to 1.

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This is y-axis, this is
the x-axis, this is the

00:05:25.840 --> 00:05:28.050
point x is equal to 1.

00:05:31.720 --> 00:05:32.780
That's not an x, that's a 1.

00:05:32.780 --> 00:05:34.070
This is the x.

00:05:34.070 --> 00:05:38.770
Anyway, so we want to figure
out, how do we sum up this dx

00:05:38.770 --> 00:05:44.100
times dy, or this da,
along this domain?

00:05:44.100 --> 00:05:44.760
So let's draw it.

00:05:44.760 --> 00:05:46.590
Let's visually draw it and it
doesn't hurt to do this when

00:05:46.590 --> 00:05:47.670
you actually have to do
the problem because this

00:05:47.670 --> 00:05:49.200
frankly is the hard part.

00:05:49.200 --> 00:05:52.300
A lot of calculus teachers will
just have you set up the

00:05:52.300 --> 00:05:54.670
integral and then say, OK,
well the rest is easy.

00:05:54.670 --> 00:05:57.360
Or the rest is Calc 1.

00:05:57.360 --> 00:06:00.810
OK, so this area, this area
here is the same thing

00:06:00.810 --> 00:06:02.540
as this area here.

00:06:02.540 --> 00:06:07.730
So its base is dx and
its height is dy.

00:06:07.730 --> 00:06:10.020
And then you could imagine
that we're looking at

00:06:10.020 --> 00:06:11.320
this thing from above.

00:06:11.320 --> 00:06:13.430
So the surface is up here some
place and we're looking

00:06:13.430 --> 00:06:17.260
straight down on it, and so
this is just this area.

00:06:17.260 --> 00:06:20.580
So let's say we wanted to
take the integral with

00:06:20.580 --> 00:06:21.850
respect to x first.

00:06:21.850 --> 00:06:28.250
So we want to sum up, so if we
want the volume above this

00:06:28.250 --> 00:06:33.270
column, first of all, is this
area times dx, dy, right?

00:06:33.270 --> 00:06:35.470
So let's write the volume
above that column.

00:06:35.470 --> 00:06:39.700
It's going to be the value of
the function, the height at

00:06:39.700 --> 00:06:48.830
that point, which is xy
squared times dx, dy.

00:06:52.360 --> 00:06:55.170
This expression gives us the
volume above this area, or

00:06:55.170 --> 00:06:56.540
this column right here.

00:06:56.540 --> 00:06:58.730
And let's say we want the sum
in the x direction first.

00:06:58.730 --> 00:07:02.980
So we want to sum that dx,
sum one here, sum here,

00:07:02.980 --> 00:07:04.290
et cetera, et cetera.

00:07:04.290 --> 00:07:05.910
So we're going to sum
in the x-direction.

00:07:05.910 --> 00:07:08.540
So my question to you
is, what is our lower

00:07:08.540 --> 00:07:09.330
bound of integration?

00:07:15.170 --> 00:07:18.620
Well, we're kind of holding
our y constant, right?

00:07:18.620 --> 00:07:21.700
And so if we go to the left, if
we go lower and lower x's we

00:07:21.700 --> 00:07:24.040
kind of bump into
the curve here.

00:07:24.040 --> 00:07:25.600
So the lower bound
of integration is

00:07:25.600 --> 00:07:27.530
actually the curve.

00:07:27.530 --> 00:07:29.780
And what is this curve
if we were to write x

00:07:29.780 --> 00:07:31.040
is a function of y?

00:07:31.040 --> 00:07:34.260
This curve is y is equal to
x squared, or x is equal

00:07:34.260 --> 00:07:36.230
to the square root of y.

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So if we're integrating with
respect to x for a fixed y

00:07:39.320 --> 00:07:42.130
right here-- we're integrating
in the horizontal direction

00:07:42.130 --> 00:07:46.010
first --our lower bound is x is
equal to the square root of y.

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That's interesting.

00:07:51.550 --> 00:07:53.140
I think it's the first time
you've probably seen a

00:07:53.140 --> 00:07:54.420
variable bound integral.

00:07:54.420 --> 00:07:57.510
But it makes sense because for
this row that we're adding up

00:07:57.510 --> 00:07:59.210
right here, the upper
bound is easy.

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The upper bound is
x is equal to 1.

00:08:02.330 --> 00:08:05.860
The upper bound is x is equal
to 1, but the lower bound is

00:08:05.860 --> 00:08:07.810
x is equal to the
square root of y.

00:08:07.810 --> 00:08:10.220
Because you go back like,
oh, I bump into the curve.

00:08:10.220 --> 00:08:10.800
And what's the curve?

00:08:10.800 --> 00:08:12.760
Well the curve is x is equal to
the square root of y because we

00:08:12.760 --> 00:08:14.720
don't know which y we picked.

00:08:14.720 --> 00:08:15.210
Fair enough.

00:08:15.210 --> 00:08:19.320
So once we've figured out the
volume-- so that'll give us the

00:08:19.320 --> 00:08:23.470
volume above this rectangle
right here --and then we

00:08:23.470 --> 00:08:24.510
want to add up the dy's.

00:08:27.020 --> 00:08:29.030
And remember, there's a
whole volume above what

00:08:29.030 --> 00:08:30.330
I'm drawing right here.

00:08:30.330 --> 00:08:35.370
I'm just drawing this
part in the xy plane.

00:08:35.370 --> 00:08:37.590
So what we've done just now,
this expression, as it's

00:08:37.590 --> 00:08:41.840
written right now, figures
out the volume above

00:08:41.840 --> 00:08:43.520
that rectangle.

00:08:43.520 --> 00:08:49.300
Now if we want to figure out
the entire volume of the solid,

00:08:49.300 --> 00:08:51.485
we integrate along the y-axis.

00:08:51.485 --> 00:08:53.710
Or we add up all the dy's.

00:08:53.710 --> 00:08:55.870
This was a dy right
here, not a dx.

00:08:55.870 --> 00:08:58.780
My dx's and dy's
look too similar.

00:08:58.780 --> 00:09:04.860
So now what is the lower bound
on the y-axis if I'm summing

00:09:04.860 --> 00:09:05.560
up these rectangles?

00:09:08.180 --> 00:09:11.270
Well, the lower bound
is y is equal to 0.

00:09:11.270 --> 00:09:14.580
So we're going to go from
y is equal to 0 to what--

00:09:14.580 --> 00:09:15.910
what is the upper bound?

00:09:15.910 --> 00:09:18.630
--to y is equal to 1.

00:09:18.630 --> 00:09:19.400
And there you have it.

00:09:19.400 --> 00:09:21.590
Let me rewrite that integral.

00:09:21.590 --> 00:09:26.250
So the double integral is going
to be from x is equal to square

00:09:26.250 --> 00:09:35.810
root of y to x is equal to
1, xy squared, dx, dy.

00:09:35.810 --> 00:09:43.090
And then the y bound, y
goes from 0 to y to 1.

00:09:43.090 --> 00:09:44.580
I've just realized
I've run out of time.

00:09:44.580 --> 00:09:47.020
In the next video we'll
evaluate this, and then we'll

00:09:47.020 --> 00:09:48.155
do it in the other order.

00:09:48.155 --> 00:09:49.770
See you soon.


Double integrals 2

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:02.980
You hopefully have a little
intuition now on what a double

00:00:02.980 --> 00:00:06.920
integral is or how we go about
figuring out the volume

00:00:06.920 --> 00:00:07.490
under a surface.

00:00:07.490 --> 00:00:09.910
So let's actually compute it
and I think it'll all become

00:00:09.910 --> 00:00:10.910
a lot more concrete.

00:00:10.910 --> 00:00:14.220
So let's say I have the
surface, z, and it's a

00:00:14.220 --> 00:00:15.530
function of x and y.

00:00:15.530 --> 00:00:20.670
And it equals xy squared.

00:00:20.670 --> 00:00:22.850
It's a surface in
three-dimensional space.

00:00:22.850 --> 00:00:26.020
And I want to know the
volume between this

00:00:26.020 --> 00:00:28.660
surface and the xy-plane.

00:00:28.660 --> 00:00:33.320
And the domain in the xy-plane
that I care about is x is

00:00:33.320 --> 00:00:38.380
greater than or equal to 0,
and less than or equal to 2.

00:00:38.380 --> 00:00:42.450
And y is greater than or
equal to 0, and less

00:00:42.450 --> 00:00:43.740
than or equal to 1.

00:00:43.740 --> 00:00:45.370
Let's see what that looks
like just so we have a

00:00:45.370 --> 00:00:47.960
good visualization of it.

00:00:47.960 --> 00:00:50.260
So I graphed it here and
we can rotate it around.

00:00:50.260 --> 00:00:52.750
This is z equals xy squared.

00:00:52.750 --> 00:00:56.240
This is the bounding box,
right? x goes from 0 to

00:00:56.240 --> 00:00:58.300
2; y goes from 0 to 1.

00:00:58.300 --> 00:01:00.720
We literally want this-- you
could almost view it the

00:01:00.720 --> 00:01:02.710
volume-- well, not almost.

00:01:02.710 --> 00:01:05.590
Exactly view it as the
volume under this surface.

00:01:05.590 --> 00:01:08.530
Between this surface, the top
surface, and the xy-plane.

00:01:08.530 --> 00:01:11.580
And I'll rotate it around so
you can get a little bit better

00:01:11.580 --> 00:01:14.210
sense of the actual volume.

00:01:14.210 --> 00:01:16.250
Let me rotate.

00:01:16.250 --> 00:01:19.330
Now I should use the
mouse for this.

00:01:19.330 --> 00:01:21.380
So it's this space,
underneath here.

00:01:21.380 --> 00:01:23.975
It's like a makeshift
shelter or something.

00:01:23.975 --> 00:01:27.060
I could rotate it a little bit.

00:01:27.060 --> 00:01:29.340
Whatever's under this,
between the two surfaces--

00:01:29.340 --> 00:01:30.920
that's the volume.

00:01:30.920 --> 00:01:32.550
Whoops, I've flipped it.

00:01:32.550 --> 00:01:33.500
There you go.

00:01:33.500 --> 00:01:35.690
So that's the volume
that we care about.

00:01:35.690 --> 00:01:38.490
Let's figure out how to do and
we'll try to gather a little

00:01:38.490 --> 00:01:41.480
bit of the intuition
as we go along.

00:01:41.480 --> 00:01:44.850
So I'm going to draw a not as
impressive version of that

00:01:44.850 --> 00:01:49.026
graph, but I think it'll
do the job for now.

00:01:49.026 --> 00:01:50.185
Let me draw my axes.

00:01:52.710 --> 00:02:01.030
That's my x-axis, that's my
y-axis, and that's my z-axis.

00:02:04.552 --> 00:02:08.814
x, y, z.

00:02:08.814 --> 00:02:10.870
x is going from 0 to 2.

00:02:10.870 --> 00:02:12.300
Let's say that's 2.

00:02:12.300 --> 00:02:16.160
y is going from 0 to 1.

00:02:16.160 --> 00:02:20.796
So we're taking the volume
above this rectangle

00:02:20.796 --> 00:02:23.570
in the xy-plane.

00:02:23.570 --> 00:02:25.740
And then the surface, I'm going
to try my best to draw it.

00:02:25.740 --> 00:02:27.660
I'll draw it in a
different color.

00:02:27.660 --> 00:02:30.680
I'm looking at the picture.

00:02:30.680 --> 00:02:32.600
At this end it looks
something like this.

00:02:36.300 --> 00:02:37.743
And then it has a
straight line.

00:02:37.743 --> 00:02:43.580
Let's see if I can draw this
surface going down like that.

00:02:43.580 --> 00:02:47.176
And then if I was really
good I could shade it.

00:02:47.176 --> 00:02:50.695
It looks something like this.

00:02:50.695 --> 00:02:55.740
If I were to shade it,
the surface looks

00:02:55.740 --> 00:02:57.020
something like that.

00:02:57.020 --> 00:02:59.780
And this right here
is above this.

00:02:59.780 --> 00:03:04.380
This is the bottom left corner,
you can almost view it.

00:03:04.380 --> 00:03:08.700
So let me just say the yellow
is the top of the surface.

00:03:08.700 --> 00:03:09.830
That's the top of the surface.

00:03:09.830 --> 00:03:11.830
And then this is
under the surface.

00:03:11.830 --> 00:03:15.260
So we care about this
volume underneath here.

00:03:15.260 --> 00:03:17.840
Let me show you the
actual surface.

00:03:17.840 --> 00:03:20.280
So this volume underneath here.

00:03:20.280 --> 00:03:21.060
I think you get the idea.

00:03:21.060 --> 00:03:22.560
So how do we do that?

00:03:22.560 --> 00:03:26.590
Well, in the last example we
said, well, let's pick an

00:03:26.590 --> 00:03:29.920
arbitrary y and for that
y, let's figure out the

00:03:29.920 --> 00:03:31.250
area under the curve.

00:03:31.250 --> 00:03:36.280
So if we fix some y-- when you
actually do the problem, you

00:03:36.280 --> 00:03:39.550
don't have to think about it in
this much detail, but I want

00:03:39.550 --> 00:03:40.410
to give you the intuition.

00:03:40.410 --> 00:03:43.810
So if we pick just an
arbitrary y here.

00:03:43.810 --> 00:03:48.250
So on that y, we could think of
it-- if we have a fixed y, then

00:03:48.250 --> 00:03:51.480
the function of x and y you can
almost view it as a function

00:03:51.480 --> 00:03:56.620
of just x for that given y.

00:03:56.620 --> 00:04:02.610
And so, we're kind of figuring
out the value of this, of

00:04:02.610 --> 00:04:04.470
the area under this curve.

00:04:08.430 --> 00:04:11.820
You should view this as kind of
an up down curve for a given y.

00:04:11.820 --> 00:04:15.870
So if we know a y we can figure
out then-- for example, if y

00:04:15.870 --> 00:04:20.200
was 5, this function would
become z equals 25x.

00:04:20.200 --> 00:04:22.570
And then that's easy to
figure out the value

00:04:22.570 --> 00:04:23.350
of the curve under.

00:04:23.350 --> 00:04:26.070
So we'll make the value under
the curve as a function of y.

00:04:26.070 --> 00:04:27.500
We'll pretend like
it's just a constant.

00:04:27.500 --> 00:04:28.770
So let's do that.

00:04:28.770 --> 00:04:33.680
So if we have a dx
that's our change in x.

00:04:33.680 --> 00:04:36.710
And then our height of each of
our rectangles is going to be a

00:04:36.710 --> 00:04:40.010
function-- it's going to be z.

00:04:40.010 --> 00:04:42.660
The height is z, which is
a function of x and y.

00:04:42.660 --> 00:04:45.190
So we can take the integral.

00:04:45.190 --> 00:04:50.020
So the area of each of these is
going to be our function, xy

00:04:50.020 --> 00:04:54.760
squared-- I'll do it here
because I'll run out of space.

00:04:54.760 --> 00:04:59.015
xy squared times the
width, which is dx.

00:04:59.015 --> 00:05:05.710
And if we want the area of this
slice for a given y, we just

00:05:05.710 --> 00:05:08.030
integrate along the x-axis.

00:05:08.030 --> 00:05:10.095
We're going to integrate
from x is equal to 0

00:05:10.095 --> 00:05:12.230
to x is equal to 2.

00:05:12.230 --> 00:05:15.210
From x is equal to 0 to 2.

00:05:15.210 --> 00:05:16.790
Fair enough.

00:05:16.790 --> 00:05:21.050
Now, but we just don't want to
figure out the area under the

00:05:21.050 --> 00:05:23.600
curve at one slice, for one
particular y, we want to

00:05:23.600 --> 00:05:25.830
figure out the entire
area of the curve.

00:05:25.830 --> 00:05:27.570
So what we do is
we say, OK, fine.

00:05:27.570 --> 00:05:33.370
The area under the curve, not
the surface-- under this curve

00:05:33.370 --> 00:05:37.050
for a particular y,
is this expression.

00:05:37.050 --> 00:05:40.550
Well, what if I gave it
a little bit of depth?

00:05:40.550 --> 00:05:45.540
If I multiplied this area times
dy then it would give me a

00:05:45.540 --> 00:05:46.850
little bit of depth, right?

00:05:46.850 --> 00:05:50.140
We'd kind of have a
three-dimensional slice of the

00:05:50.140 --> 00:05:51.240
volume that we care about.

00:05:51.240 --> 00:05:52.870
I know it's hard to imagine.

00:05:52.870 --> 00:05:54.350
Let me bring that here.

00:05:54.350 --> 00:05:58.560
So if I had a slice here, we
just figured out the area of a

00:05:58.560 --> 00:06:01.400
slice and then I'm multiplying
it by dy to give it a

00:06:01.400 --> 00:06:04.200
little bit of depth.

00:06:04.200 --> 00:06:08.000
So you multiply it by dy to
give it a little bit of depth,

00:06:08.000 --> 00:06:11.550
and then if we want the entire
volume under the curve we add

00:06:11.550 --> 00:06:14.070
all the dy's together, take the
infinite sum of these

00:06:14.070 --> 00:06:17.300
infinitely small
volumes really now.

00:06:17.300 --> 00:06:21.450
And so we will integrate
from y is equal to 0

00:06:21.450 --> 00:06:22.570
to y is equal to 1.

00:06:22.570 --> 00:06:24.290
I know this graph is a little
hard to understand, but you

00:06:24.290 --> 00:06:27.180
might want to re-watch
the first video.

00:06:27.180 --> 00:06:30.540
I had a slightly easier
to understand surface.

00:06:30.540 --> 00:06:33.590
So now, how do we
evaluate this?

00:06:33.590 --> 00:06:36.510
Well, like we said,
you evaluate from the

00:06:36.510 --> 00:06:37.500
inside and go outward.

00:06:40.480 --> 00:06:43.510
It's taking a partial
derivative in reverse.

00:06:43.510 --> 00:06:47.540
So we're integrating here with
respect to x, so we can treat

00:06:47.540 --> 00:06:49.420
y just like a constant.

00:06:49.420 --> 00:06:51.670
Like it's like the number
5 or something like that.

00:06:51.670 --> 00:06:53.620
So it really doesn't
change the integral.

00:06:53.620 --> 00:06:57.060
So what's the antiderivative
of xy squared?

00:06:57.060 --> 00:07:00.160
Well, the antiderivative of
xy squared-- I want to make

00:07:00.160 --> 00:07:02.280
sure I'm color consistent.

00:07:02.280 --> 00:07:05.720
Well, the antiderivative
of x is x to the 1/2--

00:07:05.720 --> 00:07:09.080
sorry. x squared over 2.

00:07:09.080 --> 00:07:12.180
And then y squared is
just a constant, right?

00:07:12.180 --> 00:07:14.580
And then we don't have to
worry about plus c since

00:07:14.580 --> 00:07:15.960
this is a definite integral.

00:07:15.960 --> 00:07:18.990
And we're going to
evaluate that at 2 and 0.

00:07:18.990 --> 00:07:21.190
And then we still have
the outside integral

00:07:21.190 --> 00:07:22.650
with respect to y.

00:07:22.650 --> 00:07:25.190
So once we figure that out
we're going to integrate it

00:07:25.190 --> 00:07:29.800
from 0 to 1 with respect to dy.

00:07:29.800 --> 00:07:31.470
Now what does this evaluate?

00:07:31.470 --> 00:07:32.912
We put a 2 in here.

00:07:32.912 --> 00:07:36.276
If you put a 2 in there
you get 2 squared over 2.

00:07:39.230 --> 00:07:41.740
That's just 4 over 2.

00:07:41.740 --> 00:07:43.565
So it's 2 y squared.

00:07:47.670 --> 00:07:51.210
Minus 0 squared over
2 times y squared.

00:07:51.210 --> 00:07:52.080
Well, that's just
going to be 0.

00:07:52.080 --> 00:07:52.950
So it's minus 0.

00:07:52.950 --> 00:07:55.220
I won't write that down because
hopefully that's a little

00:07:55.220 --> 00:07:56.190
bit of second nature to you.

00:07:56.190 --> 00:07:58.510
We just evaluated this
at the 2 endpoints and

00:07:58.510 --> 00:08:00.660
I'm short for space.

00:08:00.660 --> 00:08:03.710
So this evaluated at 2y
squared and now we evaluate

00:08:03.710 --> 00:08:05.580
the outside integral.

00:08:05.580 --> 00:08:08.910
0, 1 dy.

00:08:08.910 --> 00:08:10.230
And this is an important
thing to realize.

00:08:10.230 --> 00:08:13.120
When we evaluated this
inside integral, remember

00:08:13.120 --> 00:08:13.820
what we were doing?

00:08:13.820 --> 00:08:16.950
We were trying to figure out
for a given y, what the

00:08:16.950 --> 00:08:19.180
area of this surface was.

00:08:19.180 --> 00:08:23.070
Well, not this surface, the
area under the surface

00:08:23.070 --> 00:08:24.380
for a given y.

00:08:24.380 --> 00:08:27.190
For a given y that surface
kind of turns into a curve.

00:08:27.190 --> 00:08:30.110
And we tried to figure out
the area under that curve

00:08:30.110 --> 00:08:33.540
in the traditional sense.

00:08:33.540 --> 00:08:36.870
This ended up being
a function of y.

00:08:36.870 --> 00:08:40.500
And that makes sense because
depending on which y we pick

00:08:40.500 --> 00:08:44.390
we're going to get a
different area here.

00:08:44.390 --> 00:08:47.810
Obviously, depending on which y
we pick, the area-- kind of a

00:08:47.810 --> 00:08:52.620
wall dropped straight down--
that area's going to change.

00:08:52.620 --> 00:08:55.760
So we got a function of y when
we evaluated this and now we

00:08:55.760 --> 00:08:58.330
just integrate with respect to
y and this is just plain old

00:08:58.330 --> 00:09:00.810
vanilla definite integration.

00:09:00.810 --> 00:09:03.350
What's the antiderivative
of 2y squared?

00:09:03.350 --> 00:09:08.140
Well, that equals 2 times
y to the third over 3,

00:09:08.140 --> 00:09:11.510
or 2/3 y to the third.

00:09:11.510 --> 00:09:14.740
We're going to evaluate
that at 1 and 0, which

00:09:14.740 --> 00:09:16.100
is equal to-- let's see.

00:09:16.100 --> 00:09:17.480
1 to the third times 2/3.

00:09:17.480 --> 00:09:18.870
That's 2/3.

00:09:18.870 --> 00:09:20.460
Minus 0 to the third times 2/3.

00:09:20.460 --> 00:09:21.580
Well, that's just 0.

00:09:21.580 --> 00:09:25.270
So it equals 2/3.

00:09:25.270 --> 00:09:29.620
If our units were meters
these would be 2/3 meters

00:09:29.620 --> 00:09:31.230
cubed or cubic meters.

00:09:31.230 --> 00:09:32.280
But there you go.

00:09:32.280 --> 00:09:34.890
That's how you evaluate
a double integral.

00:09:34.890 --> 00:09:36.450
There really isn't
a new skill here.

00:09:36.450 --> 00:09:38.650
You just have to make sure to
keep track of the variables.

00:09:38.650 --> 00:09:39.760
Treat them constant.

00:09:39.760 --> 00:09:41.620
They need to be treated
constant, and then treat them

00:09:41.620 --> 00:09:44.710
as a variable of integration
when it's appropriate.

00:09:44.710 --> 00:09:49.090
Anyway, I will see you
in the next video.


Double integrals 3

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:04.390
In the last a video we figured
out the volume between this

00:00:04.390 --> 00:00:08.570
surface, which was xy squared
and the xy-plane when x went

00:00:08.570 --> 00:00:11.540
from 0 to 2 and y
went from 0 to 1.

00:00:11.540 --> 00:00:13.390
And the way we did it
is we integrated with

00:00:13.390 --> 00:00:15.890
respect to x first.

00:00:15.890 --> 00:00:20.680
We said, pick a y, and
let's just figure out the

00:00:20.680 --> 00:00:21.860
area under the curve.

00:00:21.860 --> 00:00:24.640
And so we integrated with
respect to x first, and then we

00:00:24.640 --> 00:00:25.850
integrated with respect to y.

00:00:25.850 --> 00:00:27.390
But we could have done it
the other way around.

00:00:27.390 --> 00:00:31.080
So let's do that and just make
sure we got the right answer.

00:00:31.080 --> 00:00:34.716
So let me erase a lot of this.

00:00:34.716 --> 00:00:37.636
So remember, our answer was 2/3
when we integrated with respect

00:00:37.636 --> 00:00:40.440
to x first, and then
with respect to y.

00:00:40.440 --> 00:00:44.560
But I will show you that we can
integrate the other way around.

00:00:44.560 --> 00:00:46.610
That's good when you can
get the same answer in

00:00:46.610 --> 00:00:48.160
two different ways.

00:00:48.160 --> 00:00:52.400
So let me redraw that graph
because I want to give

00:00:52.400 --> 00:00:54.570
you the intuition again.

00:00:54.570 --> 00:01:03.626
So that's my x-axis,
y-axis, z-axis.

00:01:03.626 --> 00:01:10.850
x, y, z.

00:01:10.850 --> 00:01:13.540
Then this is my xy-plane.

00:01:13.540 --> 00:01:14.180
down here.

00:01:16.990 --> 00:01:25.370
y goes from 0 to 1; x goes from
0 to 2, This is x equals 1,

00:01:25.370 --> 00:01:29.650
this is equals 2, this is y
equals 1, and then the graph--

00:01:29.650 --> 00:01:33.290
I will do my best to draw it.

00:01:33.290 --> 00:01:39.060
Looks something-- let me get
some contrast going here.

00:01:39.060 --> 00:01:41.310
So the graph looks
something like this.

00:01:41.310 --> 00:01:45.380
Let me see if I can draw it.

00:01:45.380 --> 00:01:49.260
On this side it looks something
like that and then it comes

00:01:49.260 --> 00:01:54.070
down like that, straight.

00:01:54.070 --> 00:01:55.910
And then, the volume
we care about.

00:01:55.910 --> 00:01:57.925
It's actually this volume
underneath the graph.

00:02:01.680 --> 00:02:04.690
This is the top of the
surface on that side.

00:02:04.690 --> 00:02:07.150
We care about this volume
underneath the surface.

00:02:07.150 --> 00:02:09.730
And then when we draw the
bottom of the surface-- let

00:02:09.730 --> 00:02:13.050
me do it in a darker color--
looks something like this.

00:02:13.050 --> 00:02:17.060
This is the bottom
underneath the surface.

00:02:17.060 --> 00:02:19.060
I can even shade it a little
bit just to show you that

00:02:19.060 --> 00:02:20.750
it's the underneath.

00:02:24.010 --> 00:02:26.970
Hopefully that's a
decent rendering of it.

00:02:26.970 --> 00:02:28.670
Let's look back at
what we had before.

00:02:28.670 --> 00:02:30.840
It's like a page that I just
flipped up at this point, and

00:02:30.840 --> 00:02:32.560
we care about this volume,
kind of the colored

00:02:32.560 --> 00:02:35.120
area under there.

00:02:35.120 --> 00:02:36.250
So let's figure
out how to do it.

00:02:36.250 --> 00:02:37.900
Last time we integrated
with respect to x first.

00:02:37.900 --> 00:02:39.330
Let's integrate with
respect to y first.

00:02:39.330 --> 00:02:41.520
So let's hold x constant.

00:02:41.520 --> 00:02:44.600
So if we hold x constant what
we could do is for a given

00:02:44.600 --> 00:02:45.560
x-- let's pick an x.

00:02:48.820 --> 00:02:55.170
So if we pick a given x,
let's pick the x here.

00:02:57.880 --> 00:03:01.900
Then what we can do, for a
given x, you can view that

00:03:01.900 --> 00:03:03.310
function of x and y.

00:03:03.310 --> 00:03:07.070
If x is a constant, let's say
if x is 1 then z is just

00:03:07.070 --> 00:03:08.490
equal to y squared.

00:03:08.490 --> 00:03:11.710
That's easy to figure out the
area under because we can see

00:03:11.710 --> 00:03:13.650
that x isn't the constant, but
we can treat it as a constant.

00:03:13.650 --> 00:03:16.640
So for example, for any
given x, we would have

00:03:16.640 --> 00:03:17.730
a curve like this.

00:03:22.560 --> 00:03:24.200
What we could do is we could
try to figure out the

00:03:24.200 --> 00:03:27.610
area of this curve first.

00:03:27.610 --> 00:03:29.300
So how do we do that?

00:03:29.300 --> 00:03:32.650
Well, we just said, we could
kind of view this function up

00:03:32.650 --> 00:03:35.860
here as z is equal to xy
squared because that's

00:03:35.860 --> 00:03:36.830
exactly what it is.

00:03:36.830 --> 00:03:38.610
But we're holding x constant.

00:03:38.610 --> 00:03:41.010
We're treating it
like a constant.

00:03:41.010 --> 00:03:44.360
To figure out that area we
could take a dy, a change

00:03:44.360 --> 00:03:49.665
in y, multiply it by the
height, which is xy squared.

00:03:54.130 --> 00:04:02.830
So we take xy squared, multiply
it by dy, and then if we want

00:04:02.830 --> 00:04:05.020
this entire area we integrate
it from y is equal to

00:04:05.020 --> 00:04:06.910
0 to y is equal to 1.

00:04:11.080 --> 00:04:11.600
Fair enough.

00:04:11.600 --> 00:04:14.570
Now once we have that area, if
you want the volume underneath

00:04:14.570 --> 00:04:18.210
this entire surface what we
could do is we can multiply

00:04:18.210 --> 00:04:22.450
this area times dx and
get some depth going.

00:04:22.450 --> 00:04:24.780
Let me pick a nice
color, that's green.

00:04:24.780 --> 00:04:27.120
So that's our dx.

00:04:27.120 --> 00:04:29.640
So if we multiply that times
dx we would get some depth.

00:04:29.640 --> 00:04:32.950
Let me do a darker color,
get some contrast.

00:04:32.950 --> 00:04:37.080
Sometimes I feel like that
guy who paints on PBS.

00:04:37.080 --> 00:04:39.820
So now we have the volume of
this, you kind of view it-- the

00:04:39.820 --> 00:04:42.950
area under the curve times a
dx, so we have some depth here.

00:04:42.950 --> 00:04:44.750
So it's time dx.

00:04:44.750 --> 00:04:47.540
And if we want to figure out
the entire volume under this

00:04:47.540 --> 00:04:51.030
surface-- between the surface
and the xy-plane given this

00:04:51.030 --> 00:04:55.370
constraint to our domain-- we
just integrate from x

00:04:55.370 --> 00:04:56.930
is equal to 0 to 2.

00:05:02.120 --> 00:05:05.050
All right, so let's
think about it.

00:05:05.050 --> 00:05:07.870
This area in green here
that we started with, that

00:05:07.870 --> 00:05:09.100
should be a function of x.

00:05:09.100 --> 00:05:11.600
We held x constant, but
depending on which x you pick

00:05:11.600 --> 00:05:13.780
this area is going to change.

00:05:13.780 --> 00:05:16.820
So when we evaluate this
magenta inner integral with

00:05:16.820 --> 00:05:19.330
respect to y we should
get a function of x.

00:05:19.330 --> 00:05:20.470
And then when you evaluate
the whole thing we'll

00:05:20.470 --> 00:05:21.220
get our volumes.

00:05:21.220 --> 00:05:22.320
So let's do it.

00:05:22.320 --> 00:05:24.700
Let's evaluate this
inner integral.

00:05:24.700 --> 00:05:26.420
Hold x constant.

00:05:26.420 --> 00:05:28.090
What's the antiderivative
of y squared?

00:05:28.090 --> 00:05:30.020
It's y of the third over 3.

00:05:35.530 --> 00:05:37.586
The x is a constant, right?

00:05:37.586 --> 00:05:42.290
We're going to evaluate
that at 1 and at 0.

00:05:42.290 --> 00:05:47.670
The outer integral is still
with respect to x dx.

00:05:47.670 --> 00:05:49.730
This is equal to-- let's see.

00:05:49.730 --> 00:05:52.860
When you evaluate y is equal
to 1 you get 1 to the third.

00:05:52.860 --> 00:05:53.370
That's 1.

00:05:53.370 --> 00:05:58.050
So it's x/3 minus when
y is 0 then that whole

00:05:58.050 --> 00:06:00.560
thing just becomes 0.

00:06:00.560 --> 00:06:02.870
This purple expression
is just x/3.

00:06:06.350 --> 00:06:11.120
And then we still have
the outside integral

00:06:11.120 --> 00:06:13.560
from 0 to 2 dx.

00:06:13.560 --> 00:06:18.290
So given what x we have, the
area of this green surface--

00:06:18.290 --> 00:06:20.410
that was where we started.

00:06:20.410 --> 00:06:25.430
Given any given x, that
area-- I wanted something

00:06:25.430 --> 00:06:26.700
with some contrast.

00:06:26.700 --> 00:06:32.550
This area is x/3 depending
on which x you pick.

00:06:32.550 --> 00:06:35.830
If x is 1, this area
right here is 1/3.

00:06:35.830 --> 00:06:38.580
But now we're going to
integrate underneath the entire

00:06:38.580 --> 00:06:40.550
surface and get our volume.

00:06:40.550 --> 00:06:42.160
And like I said, when
you integrate it,

00:06:42.160 --> 00:06:43.870
it's a function of x.

00:06:43.870 --> 00:06:45.220
So let's do that.

00:06:45.220 --> 00:06:51.440
And this is just plain old
vanilla, standard integral.

00:06:51.440 --> 00:06:52.930
So what's the
antiderivative of x?

00:06:52.930 --> 00:06:54.930
It's x squared over 2.

00:06:54.930 --> 00:06:58.890
We have a 1/3 there so
it equals x squared

00:06:58.890 --> 00:07:01.180
over 2 times 3.

00:07:01.180 --> 00:07:04.410
So x squared over 6.

00:07:04.410 --> 00:07:07.410
And we're going to evaluate
it at 2 and at 0.

00:07:07.410 --> 00:07:11.030
2 squared over 6 is 4/6.

00:07:11.030 --> 00:07:14.110
Minus 0/6, which is equal to 0.

00:07:14.110 --> 00:07:15.640
Equals 4/6.

00:07:15.640 --> 00:07:16.560
What is 4/6?

00:07:16.560 --> 00:07:19.850
Well, that's just the
same thing as 2/3.

00:07:19.850 --> 00:07:23.360
So the volume under the surface
is 2/3, and if you watched

00:07:23.360 --> 00:07:29.150
the previous video you will
appreciate the fact that when

00:07:29.150 --> 00:07:32.590
we integrated the other way
around, when we did it with

00:07:32.590 --> 00:07:35.540
respect to x first and then y,
we got the exact same answer.

00:07:35.540 --> 00:07:38.950
So the universe is in
proper working order.

00:07:38.950 --> 00:07:41.580
And I've surprisingly,
actually finished this

00:07:41.580 --> 00:07:43.090
video with extra time.

00:07:43.090 --> 00:07:47.400
So for fun, we can just spin
this graph and just appreciate

00:07:47.400 --> 00:07:52.290
the fact that we have figured
out the volume between this

00:07:52.290 --> 00:07:56.710
surface, xy squared
and the xy-plane.

00:07:56.710 --> 00:07:58.950
Pretty neat.

00:07:58.950 --> 00:08:02.360
Anyway, I'll see you
in the next video.


Double integral 1

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.020
So far, we've used integrals
to figure out the

00:00:03.020 --> 00:00:03.950
area under a curve.

00:00:03.950 --> 00:00:06.320
And let's just review a little
bit of the intuition, although

00:00:06.320 --> 00:00:08.830
this should hopefully be second
nature to you at this point.

00:00:08.830 --> 00:00:12.720
If it's not, you might want
to review the definite

00:00:12.720 --> 00:00:15.360
integration videos.

00:00:15.360 --> 00:00:18.280
But if I have some function--
this is the xy plane, that's

00:00:18.280 --> 00:00:22.300
the x-axis, that's the y-axis--
and I have some function.

00:00:22.300 --> 00:00:26.430
Let's call that, you know,
this is y is equal to

00:00:26.430 --> 00:00:27.660
some function of x.

00:00:27.660 --> 00:00:29.750
Give me an x and
I'll give you a y.

00:00:29.750 --> 00:00:33.740
If I wanted to figure out the
area under this curve, between,

00:00:33.740 --> 00:00:40.910
let's say, x is equal to
a and x is equal to b.

00:00:40.910 --> 00:00:42.980
So this is the area I
want to figure out.

00:00:46.870 --> 00:00:50.870
What I do is, I split it up
into a bunch of columns

00:00:50.870 --> 00:00:52.890
or a bunch of rectangles.

00:00:52.890 --> 00:00:56.510
Where-- let me draw one of
those rectangles-- where you

00:00:56.510 --> 00:00:58.610
could view-- and there's
different ways to do this,

00:00:58.610 --> 00:01:00.330
but this is just a review.

00:01:00.330 --> 00:01:04.080
Where you could review-- that's
maybe 1 of the rectangles.

00:01:04.080 --> 00:01:05.720
Well, the area of the
rectangle is just base

00:01:05.720 --> 00:01:07.530
times height, right?

00:01:07.530 --> 00:01:09.360
Well, we're going to make these
rectangles really skinny and

00:01:09.360 --> 00:01:11.100
just sum up an infinite
number of them.

00:01:11.100 --> 00:01:12.340
So we want to make them
infinitely small.

00:01:12.340 --> 00:01:17.060
But let's just call the
base of this rectangle dx.

00:01:17.060 --> 00:01:19.590
And then the height of this
rectangle is going to be

00:01:19.590 --> 00:01:21.620
f of x, at that point.

00:01:21.620 --> 00:01:24.545
It's going to be f of-- if this
is x0, or whatever, you can

00:01:24.545 --> 00:01:25.690
just call it f of x, right?

00:01:25.690 --> 00:01:28.170
That's the height
of that rectangle.

00:01:28.170 --> 00:01:30.470
And if we wanted to take
the sum of all of these

00:01:30.470 --> 00:01:31.090
rectangles-- right?

00:01:31.090 --> 00:01:32.210
There's just going to
be a bunch of them.

00:01:32.210 --> 00:01:34.160
One there, one there.

00:01:34.160 --> 00:01:36.910
Then we'll get the area, and if
we have infinite number of

00:01:36.910 --> 00:01:38.860
these rectangles, and they're
infinitely skinny, we

00:01:38.860 --> 00:01:40.840
have exactly the area
under that curve.

00:01:40.840 --> 00:01:43.160
That's the intuition behind
the definite integral.

00:01:43.160 --> 00:01:47.140
And the way we write that--
it's the definite integral.

00:01:47.140 --> 00:01:50.300
We're going to take the sums of
these rectangles, from x is

00:01:50.300 --> 00:01:53.490
equal to a, to x is equal to b.

00:01:53.490 --> 00:01:57.120
And the sum, or the areas that
we're summing up, are going to

00:01:57.120 --> 00:02:00.770
be-- the height is f of x,
and the width is d of x.

00:02:00.770 --> 00:02:04.580
It's going to be f
of x times d of x.

00:02:04.580 --> 00:02:06.970
This is equal to the
area under the curve.

00:02:06.970 --> 00:02:09.630
f of x, y is equal to f
of x, from x is equal to

00:02:09.630 --> 00:02:11.040
a to x is equal to b.

00:02:11.040 --> 00:02:12.440
And that's just a
little bit of review.

00:02:12.440 --> 00:02:14.720
But hopefully, you'll now
see the parallel of how we

00:02:14.720 --> 00:02:19.250
extend this to taking the
volume under a surface.

00:02:19.250 --> 00:02:20.670
So first of all,
what is a surface?

00:02:20.670 --> 00:02:23.130
Well, if we're thinking
in three dimensions, a

00:02:23.130 --> 00:02:27.540
surface is going to be
a function of x and y.

00:02:27.540 --> 00:02:31.060
So we can write a surface as,
instead of y is a function

00:02:31.060 --> 00:02:32.930
of f and x-- I'm sorry.

00:02:32.930 --> 00:02:36.060
Instead of saying that y is a
function of x, we can write a

00:02:36.060 --> 00:02:42.810
surface as z is equal to
a function of x and y.

00:02:42.810 --> 00:02:45.270
So you can kind of view
it as the domain.

00:02:45.270 --> 00:02:45.460
Right?

00:02:45.460 --> 00:02:48.030
The domain is all of the set
of valid things that you

00:02:48.030 --> 00:02:49.200
can input into a function.

00:02:49.200 --> 00:02:52.690
So now, before, our domain was
just-- at least, you know, for

00:02:52.690 --> 00:02:57.070
most of what we dealt with--
was just the x-axis, or kind of

00:02:57.070 --> 00:02:59.640
the real number line
in the x direction.

00:02:59.640 --> 00:03:02.580
Now our domain is the xy plane.

00:03:02.580 --> 00:03:06.040
We can give any x and any y--
and we'll just deal with the

00:03:06.040 --> 00:03:07.740
reals right now, I don't
want to get too technical.

00:03:07.740 --> 00:03:10.480
And then it'll pop out another
number, and if we wanted to

00:03:10.480 --> 00:03:11.950
graph it, it'll be our height.

00:03:11.950 --> 00:03:13.970
And so that could be the
height of a surface.

00:03:13.970 --> 00:03:15.960
So let me just show you what
a surface looks like, in

00:03:15.960 --> 00:03:16.800
case you don't remember.

00:03:16.800 --> 00:03:20.360
And we'll actually figure out
the volume under this surface.

00:03:20.360 --> 00:03:22.080
So this is a surface.

00:03:22.080 --> 00:03:24.140
I'll tell you its function
in a second, but it's

00:03:24.140 --> 00:03:25.370
pretty neat to look at.

00:03:25.370 --> 00:03:26.400
But as you can see,
it's a server.

00:03:26.400 --> 00:03:30.760
It's like a piece of
paper that's bent.

00:03:30.760 --> 00:03:33.870
Let's see, let me rotate it
to its traditional form.

00:03:33.870 --> 00:03:37.485
So this is the x direction,
this is the y direction.

00:03:37.485 --> 00:03:43.120
And the height is a function of
where we are in the xy plane.

00:03:43.120 --> 00:03:45.880
So how do we figure out
the volume under a

00:03:45.880 --> 00:03:48.970
surface like this?

00:03:48.970 --> 00:03:50.540
How do we figure
out the volume?

00:03:50.540 --> 00:03:53.810
It seems like a bit of a
stretch, given what we've

00:03:53.810 --> 00:03:55.500
learned from this.

00:03:55.500 --> 00:03:58.300
But what if-- and I'm just
going to draw an abstract

00:03:58.300 --> 00:04:02.186
surface here-- let
me draw some axis.

00:04:02.186 --> 00:04:03.730
Let's see, that's my x-axis.

00:04:06.390 --> 00:04:09.072
That's my y-axis.

00:04:09.072 --> 00:04:11.650
That's my z-axis.

00:04:11.650 --> 00:04:14.970
I don't practice these videos
ahead of time, so I'm often

00:04:14.970 --> 00:04:17.590
wondering what I'm
about to draw.

00:04:17.590 --> 00:04:17.960
OK.

00:04:17.960 --> 00:04:22.420
So that's x, that's
y, and that's z.

00:04:22.420 --> 00:04:23.990
And let's say I
have some surface.

00:04:27.350 --> 00:04:30.330
I'll just draw something.

00:04:30.330 --> 00:04:33.380
I don't know what it is.

00:04:33.380 --> 00:04:35.050
Some surface.

00:04:35.050 --> 00:04:36.090
This is our surface.

00:04:36.090 --> 00:04:38.300
z is a function of x and y.

00:04:38.300 --> 00:04:41.470
So, for example, you give me a
coordinate in the xy plane.

00:04:41.470 --> 00:04:44.360
Say, here, I'll put it into
the function and it'll

00:04:44.360 --> 00:04:45.630
give us a z value now.

00:04:45.630 --> 00:04:47.290
And I'll plot it there
and it'll be a point

00:04:47.290 --> 00:04:48.440
on the surface.

00:04:48.440 --> 00:04:50.250
So what we want to figure
out is the volume

00:04:50.250 --> 00:04:50.990
under the surface.

00:04:50.990 --> 00:04:52.670
And we have to specify
bounds, right?

00:04:52.670 --> 00:04:56.330
From here, we said x is equal
to a, to x is equal to b.

00:04:56.330 --> 00:04:59.030
So let's make a square
bound first, because this

00:04:59.030 --> 00:05:00.610
keeps it a lot simpler.

00:05:00.610 --> 00:05:05.150
So let's say that the domain or
the region-- not the domain--

00:05:05.150 --> 00:05:09.770
the region of-- the x and y
region of this part of the

00:05:09.770 --> 00:05:12.000
surface under which we want
to calculate the volume.

00:05:12.000 --> 00:05:15.650
Let's say, the shadow-- if
the sun was right above

00:05:15.650 --> 00:05:19.800
the surface, the shadow
would be right there.

00:05:19.800 --> 00:05:24.530
Let me try my best to
draw this neatly.

00:05:24.530 --> 00:05:26.110
So this is what we're
going to try to figure

00:05:26.110 --> 00:05:27.240
out the volume of.

00:05:30.490 --> 00:05:34.210
And let's say-- so, if we
wanted to draw it in the xy

00:05:34.210 --> 00:05:36.950
plane, like you can kind of
view the projection of the

00:05:36.950 --> 00:05:38.980
surface of the xy plane,
or the shadow of the

00:05:38.980 --> 00:05:40.240
surface of the xy plane.

00:05:40.240 --> 00:05:41.450
What are the bounds?

00:05:41.450 --> 00:05:44.470
You can almost view-- what are
the bounds of the domain?

00:05:44.470 --> 00:05:47.650
Well, let's say that this
point-- let's say that this

00:05:47.650 --> 00:05:51.340
right here, that's 0,
0 in the xy plane.

00:05:51.340 --> 00:05:54.470
Let's say that this is y is
equal to-- I don't know,

00:05:54.470 --> 00:05:57.780
that's y is equal to a.

00:05:57.780 --> 00:05:59.010
That's this line right here.

00:05:59.010 --> 00:06:00.370
Y is equal to a.

00:06:00.370 --> 00:06:04.940
And let's say that this line
right here is x is equal to b.

00:06:04.940 --> 00:06:05.910
Hope you get that, right?

00:06:05.910 --> 00:06:07.230
This is the xy plane.

00:06:07.230 --> 00:06:10.080
If we have a constant x, it
would be a line like that.

00:06:10.080 --> 00:06:11.740
A constant y, a line like that.

00:06:11.740 --> 00:06:13.370
And then we have the
area in between it.

00:06:13.370 --> 00:06:17.080
So how do we figure out
the volume under this?

00:06:17.080 --> 00:06:23.130
Well, if I just wanted to
figure out the area of--

00:06:23.130 --> 00:06:24.310
let's just say, this sliver.

00:06:24.310 --> 00:06:26.560
Let's say we had a--
well, actually let

00:06:26.560 --> 00:06:26.950
me go the other way.

00:06:26.950 --> 00:06:28.570
Let's say we had a constant y.

00:06:28.570 --> 00:06:32.060
So let's say I had some sliver.

00:06:32.060 --> 00:06:33.100
I don't want to confuse you.

00:06:33.100 --> 00:06:36.980
Let's say that I had
some constant y.

00:06:36.980 --> 00:06:39.020
I just want to give
you the intuition.

00:06:39.020 --> 00:06:39.810
You know, let's say.

00:06:39.810 --> 00:06:40.620
I don't know what that is.

00:06:40.620 --> 00:06:42.665
It's an arbitrary y.

00:06:42.665 --> 00:06:47.820
But for some constant y, what
if I could just figure out the

00:06:47.820 --> 00:06:50.810
area under the curve there?

00:06:54.130 --> 00:06:58.600
How would I figure out just
the area under that curve?

00:06:58.600 --> 00:07:01.210
It'll be a function of
which y I pick, right?

00:07:01.210 --> 00:07:03.050
Because if I pick a y here,
it'll be a different area.

00:07:03.050 --> 00:07:05.280
If I pick a y there, it'll
be a different area.

00:07:05.280 --> 00:07:07.790
But I could view this now
as a very similar problem

00:07:07.790 --> 00:07:09.560
to this one up here.

00:07:09.560 --> 00:07:12.265
I could have my dx's-- let
me pick a vibrant color

00:07:12.265 --> 00:07:13.530
so you can see it.

00:07:13.530 --> 00:07:14.930
Let's say that's dx, right?

00:07:14.930 --> 00:07:16.760
That's a change in x.

00:07:16.760 --> 00:07:27.940
And then the height is going
to be a function of the x

00:07:27.940 --> 00:07:33.440
I have and the y I picked.

00:07:33.440 --> 00:07:35.000
Although I'm assuming,
to some degree, that

00:07:35.000 --> 00:07:36.660
that's a constant y.

00:07:36.660 --> 00:07:42.990
So what would be the area
of this sheet of paper?

00:07:42.990 --> 00:07:45.100
It's kind of a constant y.

00:07:45.100 --> 00:07:48.390
It's part of-- it's a sheet
of paper within this volume,

00:07:48.390 --> 00:07:49.990
you can kind of view it.

00:07:49.990 --> 00:07:53.750
Well, it would be-- we said
the height of each of these

00:07:53.750 --> 00:07:59.030
rectangles is f of xy, right?

00:07:59.030 --> 00:07:59.515
That's the height.

00:07:59.515 --> 00:08:03.270
It depends which x and
y we pick down here.

00:08:03.270 --> 00:08:06.000
And then its width is
going to be d of x.

00:08:06.000 --> 00:08:08.830
Not d of x, dx.

00:08:08.830 --> 00:08:12.120
And then if we integrated it,
from x is equal to 0, which was

00:08:12.120 --> 00:08:14.530
back here, all the way to x
is equal to b, what

00:08:14.530 --> 00:08:16.006
would it look like?

00:08:16.006 --> 00:08:20.600
It would look like that.
x is going from 0 to b.

00:08:20.600 --> 00:08:21.120
Fair enough.

00:08:21.120 --> 00:08:26.220
And this would actually
give us a function of y.

00:08:26.220 --> 00:08:30.290
This would give us an
expression so that I would know

00:08:30.290 --> 00:08:35.670
the area of this kind of sliver
of the volume, for any

00:08:35.670 --> 00:08:36.690
given value of y.

00:08:36.690 --> 00:08:39.300
If you give me a y, I can tell
you the area of the sliver

00:08:39.300 --> 00:08:42.680
that corresponds to that y.

00:08:42.680 --> 00:08:43.720
Now what could I do?

00:08:43.720 --> 00:08:47.110
If I know the area of any given
sliver, what if I multiply the

00:08:47.110 --> 00:08:51.680
area of that sliver times dy?

00:08:51.680 --> 00:08:52.660
This is a dy.

00:08:52.660 --> 00:08:54.650
Let me do it in a
vibrant color.

00:08:54.650 --> 00:08:59.180
So dy, a very small
change in y.

00:08:59.180 --> 00:09:02.430
If I multiplied this area
times a small dy, then

00:09:02.430 --> 00:09:06.970
all of a sudden I have
a sliver of volume.

00:09:06.970 --> 00:09:09.840
Hopefully that
makes some sense.

00:09:09.840 --> 00:09:12.970
I'm making that-- that little
cut that I took the area of--

00:09:12.970 --> 00:09:14.800
by making it three dimensional.

00:09:14.800 --> 00:09:17.490
So what would be the
volume of that sliver?

00:09:17.490 --> 00:09:20.820
The volume of that sliver will
be this function of y times dy,

00:09:20.820 --> 00:09:23.280
or this whole thing times dy.

00:09:23.280 --> 00:09:31.910
So it would be the integral
from 0 to b of f of xy dx.

00:09:31.910 --> 00:09:34.670
That gives us the area
of this blue sheet.

00:09:34.670 --> 00:09:39.130
Now if I multiply this
whole thing times dy,

00:09:39.130 --> 00:09:40.710
I get this volume.

00:09:40.710 --> 00:09:42.150
It gets some depth.

00:09:42.150 --> 00:09:44.150
This little area that I'm
shading right here gets

00:09:44.150 --> 00:09:46.010
depth of that sheet.

00:09:46.010 --> 00:09:51.650
Now if I added all of those
sheets that now have depth, if

00:09:51.650 --> 00:09:55.130
I took the infinite sum-- so if
I took the integral of this

00:09:55.130 --> 00:10:04.120
from my lower y bound-- from 0
to my upper y bound, a, then--

00:10:04.120 --> 00:10:08.860
at least based on our intuition
here-- maybe I will have

00:10:08.860 --> 00:10:12.110
figured out the volume
under this surface.

00:10:12.110 --> 00:10:13.690
But anyway, I didn't
want to confuse you.

00:10:13.690 --> 00:10:15.670
But that's the intuition of
what we're going to do.

00:10:15.670 --> 00:10:18.140
And I think you're going to
find out that actually

00:10:18.140 --> 00:10:20.840
calculating the volumes are
pretty straightforward,

00:10:20.840 --> 00:10:23.660
especially when you have
fixed x and y bounds.

00:10:23.660 --> 00:10:26.510
And that's what we're going
to do in the next video.

00:10:26.510 --> 00:10:28.012
See you soon.


Curl 3

WEBVTT
Kind: captions
Language: en

00:00:00.890 --> 00:00:05.730
Let's say I have the
vector field v.

00:00:05.730 --> 00:00:09.920
The vector at any given point
is equal to minus-- or the

00:00:09.920 --> 00:00:14.570
magnitude of my x direction
is actually dependent on y.

00:00:14.570 --> 00:00:19.280
So where we are in the y
coordinate and the xy plane.

00:00:19.280 --> 00:00:21.385
Plus-- and then my magnitude
in the y direction

00:00:21.385 --> 00:00:22.470
is dependent on x.

00:00:25.690 --> 00:00:26.920
Fair enough.

00:00:26.920 --> 00:00:29.210
So first, let's just
chuck through this, and

00:00:29.210 --> 00:00:31.090
figure out it's curl.

00:00:31.090 --> 00:00:32.680
So the curl of v.

00:00:36.450 --> 00:00:43.160
It is equal to our del
vector operator, cross v.

00:00:43.160 --> 00:00:45.780
Which is nothing but this.

00:00:45.780 --> 00:00:49.200
And even though this looks like
a two dimensional vector field,

00:00:49.200 --> 00:00:51.480
we actually have to take the
cross product in 3 dimensions.

00:00:51.480 --> 00:00:55.420
Because a curl is just like
torque, and when you-- like we

00:00:55.420 --> 00:00:57.410
did the right hand rule when we
studied-- well, I hope you

00:00:57.410 --> 00:01:00.750
watched some of the videos on
magnetism and torque-- but the

00:01:00.750 --> 00:01:03.780
torque actually goes in a
direction that is perpendicular

00:01:03.780 --> 00:01:08.060
to both of the vectors
in your cross product.

00:01:08.060 --> 00:01:11.550
If both of these only have x
and y components, your actual

00:01:11.550 --> 00:01:13.640
result is going to be
in the z direction.

00:01:13.640 --> 00:01:16.210
It's going to be perpendicular
to both of these vectors.

00:01:16.210 --> 00:01:18.750
So when you take the cross
product, you still have to

00:01:18.750 --> 00:01:20.730
do it in 3 dimensions.

00:01:20.730 --> 00:01:31.660
So i j k, partial with respect
to x, partial with respect to

00:01:31.660 --> 00:01:35.400
y, partial with respect to z.

00:01:35.400 --> 00:01:38.690
The x component is minus y.

00:01:38.690 --> 00:01:40.500
The j component is x.

00:01:40.500 --> 00:01:42.920
And we have no k component.

00:01:42.920 --> 00:01:45.110
This one should be a little
bit cleaner to calculate

00:01:45.110 --> 00:01:47.370
than the last example.

00:01:47.370 --> 00:01:50.150
So this is going to be equal
to-- well, see the i

00:01:50.150 --> 00:01:53.195
component-- let's cross out
its column and its row.

00:01:53.195 --> 00:02:00.180
So it's going to be the partial
with respect to y of 0

00:02:00.180 --> 00:02:06.980
minus the partial with
respect to z of x.

00:02:06.980 --> 00:02:08.030
Minus that.

00:02:08.030 --> 00:02:13.140
And all of that times i
minus-- and now it's going

00:02:13.140 --> 00:02:13.870
to be the j component.

00:02:13.870 --> 00:02:17.180
Remember, you do
plus minus plus.

00:02:17.180 --> 00:02:18.960
So minus and this is
the j component.

00:02:18.960 --> 00:02:20.880
Cross out its row and column,
the partial derivative

00:02:20.880 --> 00:02:23.300
of x with respect to 0.

00:02:23.300 --> 00:02:26.550
Or partial derivative of 0
with respect to x, actually.

00:02:26.550 --> 00:02:32.450
Minus the partial
derivative of z.

00:02:32.450 --> 00:02:33.840
Or the partial derivative
of the [? vector ?]

00:02:33.840 --> 00:02:37.540
z of minus y.

00:02:37.540 --> 00:02:39.860
That's its j component.

00:02:39.860 --> 00:02:41.820
And finally, plus
its k component.

00:02:41.820 --> 00:02:42.630
Row and column.

00:02:42.630 --> 00:02:50.740
So the partial derivative with
respect to x minus the partial

00:02:50.740 --> 00:02:55.280
derivative of y with
respect to minus y.

00:02:55.280 --> 00:02:57.360
And I know you can't read it,
but that's that minus y there,

00:02:57.360 --> 00:02:59.900
and that's the k component.

00:02:59.900 --> 00:03:00.990
And now let's simplify it.

00:03:00.990 --> 00:03:03.810
I'll simplify it above it.

00:03:03.810 --> 00:03:06.220
So this term-- let's see,
partial derivative of 0.

00:03:06.220 --> 00:03:07.950
Well, that's 0.

00:03:07.950 --> 00:03:10.360
Partial derivative of
x with respect to z.

00:03:10.360 --> 00:03:14.310
Well, as far as z's concerned,
x is a constant, so that's 0.

00:03:14.310 --> 00:03:17.280
Partial derivative of 0
with respect to x, 0.

00:03:17.280 --> 00:03:20.550
Partial derivative of minus
y with respect to z.

00:03:20.550 --> 00:03:22.740
As far as z's concerned,
y is also constant.

00:03:22.740 --> 00:03:24.010
So that's 0.

00:03:24.010 --> 00:03:25.530
So all we're left is
with this last term.

00:03:25.530 --> 00:03:26.270
That was pretty
straightforward.

00:03:26.270 --> 00:03:28.320
Why don't we just cross
all of that out?

00:03:28.320 --> 00:03:30.420
And that makes intuitive
sense, too, right?

00:03:30.420 --> 00:03:37.000
Because at least from x's point
of view, the rotation is--

00:03:37.000 --> 00:03:39.990
although, the rotation, if you
think about it, is going to be

00:03:39.990 --> 00:03:47.080
in the direction perpendicular
to both the x direction and

00:03:47.080 --> 00:03:51.380
perpendicular to the direction
in which it is changing.

00:03:51.380 --> 00:03:54.370
You could kind of view it
orthogonally, to its direction

00:03:54.370 --> 00:03:55.650
of motion, which is y.

00:03:55.650 --> 00:03:57.310
So it would be the z direction.

00:03:57.310 --> 00:03:58.860
If that confuses you,
don't worry about it.

00:03:58.860 --> 00:04:01.420
But if it doesn't, then
you could apply the same

00:04:01.420 --> 00:04:03.350
argument to the j vector.

00:04:03.350 --> 00:04:05.270
But anyway, lets simplify this.

00:04:05.270 --> 00:04:07.890
So this is equal to the
partial derivative of

00:04:07.890 --> 00:04:09.280
x with respect to x.

00:04:09.280 --> 00:04:11.510
Well, that's just 1.

00:04:11.510 --> 00:04:15.270
Minus the partial derivative
of y-- of minus y

00:04:15.270 --> 00:04:16.130
with respect to y.

00:04:16.130 --> 00:04:18.750
Well, that's just minus 1.

00:04:18.750 --> 00:04:20.560
So it equals 2.

00:04:20.560 --> 00:04:26.330
So the curl, at any point of
this vector field, is 2.

00:04:26.330 --> 00:04:28.380
Let's see what this vector
field looks like, and let's

00:04:28.380 --> 00:04:33.590
see if that gives us-- if
our intuition holds

00:04:33.590 --> 00:04:34.610
in this example.

00:04:34.610 --> 00:04:38.360
And let me try to make
it a little bit bigger.

00:04:38.360 --> 00:04:41.650
Make the window bigger.

00:04:41.650 --> 00:04:42.510
There you go.

00:04:42.510 --> 00:04:45.150
Well, I think it's clear, you
know, right when you look at

00:04:45.150 --> 00:04:47.010
it, that this vector field
looks like it's spinning.

00:04:47.010 --> 00:04:49.760
If you were to stick something,
especially in the middle, it's

00:04:49.760 --> 00:04:51.450
very clear that it would spin.

00:04:51.450 --> 00:04:53.450
But what might be a little
unintuitive-- you might think,

00:04:53.450 --> 00:04:55.870
wow, well, wouldn't something
spin faster near the center

00:04:55.870 --> 00:04:57.280
than it would here?

00:04:57.280 --> 00:04:59.660
Why is the-- you know,
the curl we got is 2.

00:04:59.660 --> 00:05:00.450
It's a constant.

00:05:00.450 --> 00:05:04.150
The curl is the same throughout
this entire vector field.

00:05:04.150 --> 00:05:06.680
So you'll be like, whoa, that's
kind of implying that the field

00:05:06.680 --> 00:05:09.440
is making something spin
equally, no matter where

00:05:09.440 --> 00:05:10.210
you are in the field.

00:05:10.210 --> 00:05:11.950
Let's see if that makes sense.

00:05:11.950 --> 00:05:13.730
Well, in the middle, it
definitely makes sense that

00:05:13.730 --> 00:05:14.970
something is spinning.

00:05:14.970 --> 00:05:17.360
If I had a little stick here,
I'd be pushing in this

00:05:17.360 --> 00:05:19.890
direction, with not that much
of a magnitude in

00:05:19.890 --> 00:05:20.790
that direction.

00:05:20.790 --> 00:05:23.720
And then I'd be pushing down to
the right in this direction,

00:05:23.720 --> 00:05:25.570
so it would cause it to spin.

00:05:25.570 --> 00:05:29.900
But what if I had that
same stick here?

00:05:29.900 --> 00:05:31.810
You'd say, well, on the
top right I'm pushing

00:05:31.810 --> 00:05:34.020
up, up and to the left.

00:05:34.020 --> 00:05:37.180
And the bottom left, I'm also
pushing up and to the left.

00:05:37.180 --> 00:05:38.270
You know, it wouldn't
spin as much.

00:05:38.270 --> 00:05:38.880
But it would.

00:05:38.880 --> 00:05:42.390
Because the difference in
magnitudes of these 2-- you

00:05:42.390 --> 00:05:45.530
could almost view them as the
torque producing forces-- the

00:05:45.530 --> 00:05:49.390
difference in magnitude is
enough that you'd still have

00:05:49.390 --> 00:05:53.030
the same counterclockwise
rotation here as you

00:05:53.030 --> 00:05:53.800
would have here.

00:05:53.800 --> 00:05:58.110
So because the curl is a
constant positive number, when

00:05:58.110 --> 00:06:01.410
we look at the xy plane like
this, if you put a twig-- that

00:06:01.410 --> 00:06:04.390
same twig anywhere where you
put it on this plane--

00:06:04.390 --> 00:06:07.370
it'll have the same
counterclockwise rotation.

00:06:07.370 --> 00:06:09.370
I think that's pretty neat.

00:06:09.370 --> 00:06:12.150
Now let's do a
little experiment.

00:06:12.150 --> 00:06:18.600
What would have happened
if this was plus y.

00:06:18.600 --> 00:06:20.900
So let's just do
that experiment.

00:06:20.900 --> 00:06:29.930
If this was plus y then this
would have been plus y, then

00:06:29.930 --> 00:06:32.060
this final term-- we would have
taken the partial derivative

00:06:32.060 --> 00:06:35.110
with respect to y of plus y--
and so this would have

00:06:35.110 --> 00:06:37.260
been 1 minus plus 1.

00:06:37.260 --> 00:06:40.050
And then our curl
would have been 0.

00:06:40.050 --> 00:06:42.710
Which would have meant that we
would have had no rotation.

00:06:42.710 --> 00:06:44.030
And what's the
intuition of that?

00:06:44.030 --> 00:06:46.070
Well, if we just look at it
mathematically, if we have 0

00:06:46.070 --> 00:06:50.770
curl, somehow, the rotation in
our x direction must be being

00:06:50.770 --> 00:06:53.230
offset by a rotation
in the y direction.

00:06:53.230 --> 00:06:55.290
That the torques must
be just perfectly

00:06:55.290 --> 00:06:56.210
offsetting each other.

00:06:56.210 --> 00:06:58.350
Let's see what happens
if I were to change.

00:06:58.350 --> 00:06:59.740
So this was our old graph.

00:06:59.740 --> 00:07:03.000
Let me actually change it
to my new vector field.

00:07:06.310 --> 00:07:08.160
So that's our new vector field.

00:07:08.160 --> 00:07:12.700
This is our vector
field plus yi plus xj.

00:07:12.700 --> 00:07:14.890
And now the curl
is 0 everywhere.

00:07:14.890 --> 00:07:18.540
Which implies, or which means,
that I could put a twig

00:07:18.540 --> 00:07:23.310
anywhere here and I'm not going
to get any kind of rotation.

00:07:23.310 --> 00:07:24.160
Let's see if that makes sense.

00:07:24.160 --> 00:07:29.070
If I put a twig in the center,
or some kind of stick in

00:07:29.070 --> 00:07:30.740
the center, let's see.

00:07:30.740 --> 00:07:34.330
I would have the forces or the
fluids pushing in in this

00:07:34.330 --> 00:07:36.860
direction, but they're not
helping to rotate, and pushing

00:07:36.860 --> 00:07:38.280
out in that direction.

00:07:38.280 --> 00:07:40.870
Well, that's not going
to help me either.

00:07:40.870 --> 00:07:42.110
So I'm definitely not
going to rotate there.

00:07:42.110 --> 00:07:44.070
And actually, you could
put a twig anywhere.

00:07:44.070 --> 00:07:46.240
And maybe a twig might be
pushed in a direction.

00:07:46.240 --> 00:07:48.520
For example, if I put a twig
here, it's going to be pushed

00:07:48.520 --> 00:07:51.450
outward by the flow of the
water, by the velocity

00:07:51.450 --> 00:07:52.350
of the water.

00:07:52.350 --> 00:07:53.920
But it's not going to rotate.

00:07:53.920 --> 00:07:54.720
So the [? ink ?]

00:07:54.720 --> 00:07:55.230
kind of holds.

00:07:55.230 --> 00:07:59.740
That's even though I kind of
have curl in the x direction,

00:07:59.740 --> 00:08:01.300
or I have curl in the
y direction, they're

00:08:01.300 --> 00:08:02.590
offsetting each other.

00:08:02.590 --> 00:08:04.720
So that in 2 dimensions,
I actually end up

00:08:04.720 --> 00:08:06.990
having no rotation.

00:08:06.990 --> 00:08:10.070
And actually, this is called an
irrotational-- I think that's

00:08:10.070 --> 00:08:11.410
the word-- vector field.

00:08:11.410 --> 00:08:13.510
Where you're not going to
have any rotation here.

00:08:13.510 --> 00:08:17.520
All of the-- if you think of it
as force or velocity of the

00:08:17.520 --> 00:08:21.350
vector field-- is going to be
applying translation to

00:08:21.350 --> 00:08:23.870
objects in that field.

00:08:23.870 --> 00:08:25.520
And actually, just for fun,
let's think about the

00:08:25.520 --> 00:08:26.630
divergence of that field.

00:08:26.630 --> 00:08:27.900
Just to, I don't know.

00:08:27.900 --> 00:08:30.250
Just because I have 2 minutes.

00:08:30.250 --> 00:08:33.180
So if I were to think about the
divergence of that field, it's

00:08:33.180 --> 00:08:35.340
fairly easy to calculate.

00:08:35.340 --> 00:08:37.920
Let me erase this real fast.

00:08:37.920 --> 00:08:40.240
It's always fun to just
interpret a vector

00:08:40.240 --> 00:08:41.400
field to death.

00:08:41.400 --> 00:08:44.320
Let's do the divergence
of this one.

00:08:44.320 --> 00:08:47.000
So the divergence of that
vector field, is just a

00:08:47.000 --> 00:08:50.210
partial derivative of
this with respect to x.

00:08:50.210 --> 00:08:58.350
So div of v is the same thing
as our del operator dot

00:08:58.350 --> 00:09:00.230
our vector field, v.

00:09:00.230 --> 00:09:01.810
And that's the partial
derivative of this

00:09:01.810 --> 00:09:03.260
with respect to x.

00:09:03.260 --> 00:09:06.290
Well, the partial derivative of
y with respect to x is just 0.

00:09:06.290 --> 00:09:11.140
Plus the partial derivative
of x with respect to y.

00:09:11.140 --> 00:09:12.860
So this is 0.

00:09:12.860 --> 00:09:14.520
So the divergence of
this field is 0.

00:09:14.520 --> 00:09:16.430
And even in the other case,
when this was a minus, it

00:09:16.430 --> 00:09:18.240
still would have been 0.

00:09:18.240 --> 00:09:20.590
And does that make sense?

00:09:20.590 --> 00:09:23.110
Well, if we take a circle
anywhere here-- let's take it

00:09:23.110 --> 00:09:25.050
in the center, because the
center's the most interesting.

00:09:25.050 --> 00:09:26.300
Let's take a circle.

00:09:26.300 --> 00:09:29.690
So we do have some fluid or
particles coming in at a

00:09:29.690 --> 00:09:32.940
certain velocity from the
bottom right and the top left.

00:09:32.940 --> 00:09:35.410
But just as much is coming
out through the top right

00:09:35.410 --> 00:09:38.100
and the bottom left.

00:09:38.100 --> 00:09:40.650
Whatever's coming in through
here and here is leaving

00:09:40.650 --> 00:09:41.490
through there and there.

00:09:41.490 --> 00:09:46.330
So if I had an infinitesimally
small circle, or sphere, in

00:09:46.330 --> 00:09:50.650
this vector field, I would have
no net density increasing.

00:09:50.650 --> 00:09:53.500
Or nothing would be entering
into that circle, or in any

00:09:53.500 --> 00:09:55.880
given amount of time the
concentration of that

00:09:55.880 --> 00:09:57.060
circle wouldn't change.

00:09:57.060 --> 00:09:58.660
And that's true pretty
much anywhere.

00:09:58.660 --> 00:10:03.060
If I were to draw a circle
here, because the divergence

00:10:03.060 --> 00:10:05.630
is 0, it's telling us that
whatever's coming in in 1

00:10:05.630 --> 00:10:08.060
direction is coming out
in the other directions.

00:10:08.060 --> 00:10:11.540
So I'm not getting any
denser, or any less dense.

00:10:11.540 --> 00:10:15.080
I have no divergence
or convergence.

00:10:15.080 --> 00:10:16.040
So that's just interesting.

00:10:16.040 --> 00:10:17.120
Well, now I've run of time.

00:10:17.120 --> 00:10:19.150
So we're done analyzing
this vector field.

00:10:19.150 --> 00:10:21.300
See you in the next video.


Curl 2

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:01.820
Welcome back.

00:00:01.820 --> 00:00:03.690
Hopefully you have a
little intuition now

00:00:03.690 --> 00:00:04.770
of what the curl is.

00:00:04.770 --> 00:00:08.070
Now let's actually compute it
because if your sole goal is to

00:00:08.070 --> 00:00:11.990
pass a test and not understand
the nature of the universe,

00:00:11.990 --> 00:00:14.820
which I think would be sad, but
if that is your goal you at

00:00:14.820 --> 00:00:16.880
least need to know how to
calculate these things.

00:00:16.880 --> 00:00:21.700
But it's even more fun when
you have the intuition.

00:00:21.700 --> 00:00:23.420
And then you'll hopefully
never forget it.

00:00:23.420 --> 00:00:26.480
We'll take the curl of a
fairly fancy vector field.

00:00:26.480 --> 00:00:29.010
One that I have trouble
visualizing but that we can

00:00:29.010 --> 00:00:31.020
mathematically chug through.

00:00:31.020 --> 00:00:35.460
So let's say our vector field--
and I'll do a three dimensional

00:00:35.460 --> 00:00:39.480
vector field just to do a
fairly complicated example; I'm

00:00:39.480 --> 00:00:41.505
just going to make it up on the
fly --so let's say in the

00:00:41.505 --> 00:00:45.545
x-direction the magnitude of
the field is, I don't know,

00:00:45.545 --> 00:00:57.670
let's say it's x squared, y,
sine, z, in the x-direction,

00:00:57.670 --> 00:01:06.790
plus-- I don't know --let's
make it x, y squared, z in

00:01:06.790 --> 00:01:09.420
the j-direction, or
the y-direction.

00:01:09.420 --> 00:01:13.410
And in the z-direction, I don't
know, let's make it cosine of

00:01:13.410 --> 00:01:19.880
x times cosine of y,
in the z-direction.

00:01:19.880 --> 00:01:25.460
Now we said that you can view
the curl of this vector field--

00:01:25.460 --> 00:01:27.640
and I have no intuition of
what this vector field looks

00:01:27.640 --> 00:01:30.180
like; I just made this up.

00:01:30.180 --> 00:01:33.870
Maybe we'll graph it for fun
just to see how messed up it

00:01:33.870 --> 00:01:38.440
looks --but we said this curl,
you could view it as a cross

00:01:38.440 --> 00:01:45.230
product of our del operator
and the vector field.

00:01:45.230 --> 00:01:47.990
Well, when you were using this
engineering notation, when you

00:01:47.990 --> 00:01:53.620
have a vector broken down into
it x, y, and z components, or

00:01:53.620 --> 00:01:57.130
it's i, j, and k components,
you can take the determinant of

00:01:57.130 --> 00:01:59.490
that matrix-- when I showed you
how to compute the cross

00:01:59.490 --> 00:02:01.620
product --to figure out
the cross product.

00:02:01.620 --> 00:02:02.590
So how do we do this?

00:02:02.590 --> 00:02:11.440
So the cross product is going
to be equal to-- I didn't have

00:02:11.440 --> 00:02:15.970
to draw a straight line --so
how you take the cross product

00:02:15.970 --> 00:02:19.680
of this vector field and
the gradient operator?

00:02:19.680 --> 00:02:23.990
Well, you write i, j, k on top
like you're taking the cross

00:02:23.990 --> 00:02:29.680
product of any two three
dimensional vectors, and then

00:02:29.680 --> 00:02:34.100
you take the first vector-- but
it's really a vector operator,

00:02:34.100 --> 00:02:38.410
but it's this del operator.

00:02:38.410 --> 00:02:40.250
And what are the components
of the del operator?

00:02:40.250 --> 00:02:44.270
It's the partial derivative
with respect to x, the partial

00:02:44.270 --> 00:02:48.600
derivative with respect to y,
partial derivative with

00:02:48.600 --> 00:02:50.160
respect to z, right?

00:02:50.160 --> 00:02:52.750
Let me just rewrite
the del operator.

00:02:52.750 --> 00:02:55.350
You can view it as being equal
to the partial with respect to

00:02:55.350 --> 00:03:01.750
x, i plus the partial with
respect to y, j plus the

00:03:01.750 --> 00:03:05.630
partial with respect to z, k.

00:03:05.630 --> 00:03:07.900
So its x, y, and z components
are the partials with respect

00:03:07.900 --> 00:03:10.390
to x, with respect to
y, with respect to z.

00:03:10.390 --> 00:03:13.160
And then the second, where
we're taking this operator

00:03:13.160 --> 00:03:14.375
cross the vector field.

00:03:14.375 --> 00:03:16.500
So what are the components
of the vector field?

00:03:16.500 --> 00:03:25.130
I'll probably run out of space,
but it's x squared, y, sine, z.

00:03:25.130 --> 00:03:29.940
Then here it's xy squared, z--
I should written all of this

00:03:29.940 --> 00:03:33.040
bigger --and then the third
column, the z component is

00:03:33.040 --> 00:03:36.980
cosine of x times cosine of y.

00:03:36.980 --> 00:03:39.800
Just the x, y, and
z components.

00:03:39.800 --> 00:03:43.350
And now we are ready to take
the determinant, which will

00:03:43.350 --> 00:03:46.260
probably, well, it'll
probably get pretty

00:03:46.260 --> 00:03:47.990
messy, but let's try it.

00:03:47.990 --> 00:03:55.710
So this is equal to the i-unit
vector-- let me use a more

00:03:55.710 --> 00:03:59.245
vibrant color --the i-unit
vector times it's

00:03:59.245 --> 00:03:59.500
subdeterminant.

00:03:59.500 --> 00:04:02.740
So you cross out it's row and
column, and so you take the

00:04:02.740 --> 00:04:06.540
determinant of this expression,
so it's going to be times--

00:04:06.540 --> 00:04:09.130
this times this, but it's
really the partial.

00:04:09.130 --> 00:04:12.400
If you multiply the partial
with respect to y operator

00:04:12.400 --> 00:04:14.986
times that expression, you're
really just taking, since it's

00:04:14.986 --> 00:04:17.240
an operator and not an
expression, you're really just

00:04:17.240 --> 00:04:19.150
going to take the partial of
this with respect to y,

00:04:19.150 --> 00:04:19.970
but I'll write it down.

00:04:19.970 --> 00:04:26.750
So it's going to be the partial
with respect to y of cosine x,

00:04:26.750 --> 00:04:37.020
cosine y minus the partial with
respect to z times xy squared,

00:04:37.020 --> 00:04:45.490
z-- and now we're on to
our j component --plus j.

00:04:45.490 --> 00:04:48.800
So now what's the magnitude of
our curl in the j-direction?

00:04:48.800 --> 00:04:50.860
Let's cross out the row
in the column of j.

00:04:50.860 --> 00:04:54.580
So it's a partial with
respect to x of this.

00:04:54.580 --> 00:04:58.130
So this is maybe messier
than I originally intended.

00:04:58.130 --> 00:05:05.700
Cosine of x, cosine of y, cross
these columns there, minus

00:05:05.700 --> 00:05:20.540
the partial with respect to z
of x squared, y sine of z.

00:05:20.540 --> 00:05:23.370
And then finally
our k component.

00:05:23.370 --> 00:05:28.100
Oh, and sorry, when you take
the determinant, you use that--

00:05:28.100 --> 00:05:31.180
and this is all kind of a bit
of voodoo --but you put a plus

00:05:31.180 --> 00:05:33.550
here, a minus here, a plus
here, so it's kind of

00:05:33.550 --> 00:05:34.350
this checkered pattern.

00:05:34.350 --> 00:05:37.630
To this is plus i, this
should actually be minus j.

00:05:37.630 --> 00:05:40.420
Don't want to make that
mistake; this is minus j.

00:05:40.420 --> 00:05:46.340
This is just kind of the
algorithm of how do you

00:05:46.340 --> 00:05:48.180
take a determinant.

00:05:48.180 --> 00:05:53.730
OK, then finally we have plus
k times the determinant

00:05:53.730 --> 00:05:57.110
of its submatrix.

00:05:57.110 --> 00:06:06.660
So the partial with respect to
x of this, sorry, we take out

00:06:06.660 --> 00:06:13.040
it's row and column, so xy
squared, z minus the partial

00:06:13.040 --> 00:06:19.190
with respect to y of this.

00:06:19.190 --> 00:06:20.990
Why don't we take this row and
columns, and this is the

00:06:20.990 --> 00:06:28.780
submatrix of x squared,
y, sine of z.

00:06:28.780 --> 00:06:30.870
All right, now let me try
to simplify it, and I'll

00:06:30.870 --> 00:06:31.860
have to get some space.

00:06:31.860 --> 00:06:34.960
Hopefully you understood what I
did here and now we got this.

00:06:34.960 --> 00:06:36.930
And now I think I can erase
all of this and just so I

00:06:36.930 --> 00:06:39.840
can have some room to
simplify things in.

00:06:43.090 --> 00:06:46.540
No, that's not what I wanted
to do; I wanted to do

00:06:46.540 --> 00:06:49.570
it in a darker color.

00:06:49.570 --> 00:06:51.910
That's what I wanted to do.

00:06:51.910 --> 00:06:54.320
Erase that.

00:06:54.320 --> 00:06:55.030
Erase that.

00:06:55.030 --> 00:06:57.750
Now we just have to simplify
it, taking a bunch of

00:06:57.750 --> 00:06:58.260
partial derivatives.

00:06:58.260 --> 00:07:02.630
What's the partial derivative
of this with respect to y?

00:07:02.630 --> 00:07:08.320
Well, x is just a constant, so
it's going to be-- well, we can

00:07:08.320 --> 00:07:10.480
just put the i out front, but
eventually we want to write our

00:07:10.480 --> 00:07:14.350
magnitude before the vector
--so it's i times the partial

00:07:14.350 --> 00:07:17.720
of this with respect to
y with our constant.

00:07:17.720 --> 00:07:19.850
Cosine of x is just a constant.

00:07:19.850 --> 00:07:21.800
And then what's the derivative
of this with respect to y?

00:07:21.800 --> 00:07:24.090
It's minus sine of y.

00:07:24.090 --> 00:07:30.550
So we'll write sine of y, and
let's put the minus out front;

00:07:30.550 --> 00:07:32.170
these are what's multiplied.

00:07:32.170 --> 00:07:39.310
OK, and then we have minus, now
we stick the partial with

00:07:39.310 --> 00:07:42.180
respect-- sorry, actually,
I forgot to do this part.

00:07:42.180 --> 00:07:46.510
Let me start over, actually.

00:07:46.510 --> 00:07:49.050
So let me just take this
expression and I'll

00:07:49.050 --> 00:07:49.635
multiply it by i.

00:07:49.635 --> 00:07:55.180
The partial of this with
respect to y is cosine of x

00:07:55.180 --> 00:07:58.590
times minus sine of y-- let's
put the minus out front

00:07:58.590 --> 00:07:59.760
--minus sine of y.

00:08:02.290 --> 00:08:05.080
Now minus the partial of
this with respect to z.

00:08:05.080 --> 00:08:08.780
Well the partial of this with
respect to z, xy squared

00:08:08.780 --> 00:08:10.110
is just a constant, right?

00:08:10.110 --> 00:08:11.560
So the partial of this
with respect to z

00:08:11.560 --> 00:08:13.530
is just xy squared.

00:08:13.530 --> 00:08:18.360
So minus xy squared, and then
we're going to have all of

00:08:18.360 --> 00:08:23.270
that; that's the magnitude
in our i-direction.

00:08:23.270 --> 00:08:27.170
And now we have minus-- because
minus in the j-direction

00:08:27.170 --> 00:08:30.490
--what's the partial derivative
of this with respect to x?

00:08:30.490 --> 00:08:32.483
Well, the partial of cosine
of x with respect to

00:08:32.483 --> 00:08:33.779
x is minus sine of x.

00:08:37.470 --> 00:08:39.970
And cosine of y is just
a constant, so it just

00:08:39.970 --> 00:08:43.080
carries over, cosine of y.

00:08:43.080 --> 00:08:47.240
And then that should be-- oh,
there we go --minus this

00:08:47.240 --> 00:08:50.350
expression, the partial of
this with respect to z.

00:08:50.350 --> 00:08:52.780
Well, the derivative of
sine of z with respect

00:08:52.780 --> 00:08:53.960
to z is cosine of z.

00:08:53.960 --> 00:08:57.660
This is just a constant,
so it's minus x squared,

00:08:57.660 --> 00:09:00.670
y, cosine of z.

00:09:00.670 --> 00:09:03.160
And that's the magnitude
in the j-direction.

00:09:03.160 --> 00:09:04.500
We're almost there.

00:09:04.500 --> 00:09:08.030
And now finally, plus,
what's the partial of

00:09:08.030 --> 00:09:09.990
this with respect to x?

00:09:09.990 --> 00:09:17.480
Well, these are just constants,
so it's y squared, z minus--

00:09:17.480 --> 00:09:19.600
once again, we just have a y
term; everything else is a

00:09:19.600 --> 00:09:22.680
constant --so the partial
with respect to y is

00:09:22.680 --> 00:09:27.250
x squared, sine of z.

00:09:27.250 --> 00:09:31.962
And that's the magnitude
in our k-direction.

00:09:31.962 --> 00:09:33.300
And we're pretty much done.

00:09:33.300 --> 00:09:36.570
I mean we can simplify
it a little bit just

00:09:36.570 --> 00:09:38.280
to make it clean.

00:09:38.280 --> 00:09:40.420
And essentially we could just,
I don't have to rewrite it,

00:09:40.420 --> 00:09:43.130
we can just multiply
this by negative 1.

00:09:43.130 --> 00:09:46.940
So then this becomes
plus, plus, plus.

00:09:46.940 --> 00:09:48.310
Yeah, that's pretty much it.

00:09:48.310 --> 00:09:52.050
This is the curl of the
vector field v at any

00:09:52.050 --> 00:09:53.450
point x, y, and z.

00:09:53.450 --> 00:09:54.590
So that's how you calculate it.

00:09:54.590 --> 00:09:59.810
You just literally take the
cross product of that del

00:09:59.810 --> 00:10:02.020
operator and your vector field.

00:10:02.020 --> 00:10:04.350
And you'll get something fairly
hairy, although this was, I

00:10:04.350 --> 00:10:06.700
think, a hairier than
average problem.

00:10:06.700 --> 00:10:09.320
In the next video, we'll do a
little bit of this, but I think

00:10:09.320 --> 00:10:13.120
it'll give you more intuition
and less of just the algorithm

00:10:13.120 --> 00:10:15.020
and the computation
of how do you do it.

00:10:15.020 --> 00:10:17.100
So you in the next video.


Curl 1

WEBVTT
Kind: captions
Language: en

00:00:01.210 --> 00:00:04.500
Before I actually show you the
mechanics of what the curl of a

00:00:04.500 --> 00:00:06.370
vector field really is, let's
try to get a little

00:00:06.370 --> 00:00:07.600
bit of intuition.

00:00:07.600 --> 00:00:11.110
So here I've drawn, I'm going
to just draw a two-dimensional

00:00:11.110 --> 00:00:11.760
vector field.

00:00:11.760 --> 00:00:13.525
You can extrapolate to 3,
but when we're getting

00:00:13.525 --> 00:00:15.770
the intuition, it's
good to do it in 2.

00:00:15.770 --> 00:00:17.110
And so, let's see.

00:00:17.110 --> 00:00:19.290
I didn't even label
the x and y axis.

00:00:19.290 --> 00:00:21.720
This is x, this is y.

00:00:21.720 --> 00:00:25.300
So when y is relatively low,
our magnitude vector goes in

00:00:25.300 --> 00:00:27.830
the x direction, when it
increases a little bit, it

00:00:27.830 --> 00:00:29.380
gets a little bit longer.

00:00:29.380 --> 00:00:33.150
So as we can see, as our change
in the y-direction, as we go

00:00:33.150 --> 00:00:36.200
in the y-direction, the
x-component of our vectors

00:00:36.200 --> 00:00:38.020
get larger and larger.

00:00:38.020 --> 00:00:41.130
And maybe in the x-direction
they're constant, regardless

00:00:41.130 --> 00:00:45.980
of your level of x,
the magnitude stays.

00:00:45.980 --> 00:00:53.230
So for given y, the magnitude
of your x-component vector

00:00:53.230 --> 00:00:53.810
might stay the same.

00:00:53.810 --> 00:00:55.700
So I mean, this vector field
might look something like this.

00:00:55.700 --> 00:00:56.710
I'm just making up numbers.

00:00:56.710 --> 00:01:02.870
Maybe it's just, I don't
know, y squared i.

00:01:02.870 --> 00:01:05.860
So the magnitude of the
x-direction is just a

00:01:05.860 --> 00:01:07.840
function of your y-value.

00:01:07.840 --> 00:01:10.810
And as your y-values get bigger
and bigger, the magnitude in

00:01:10.810 --> 00:01:13.660
your x-direction will get
bigger and bigger, proportional

00:01:13.660 --> 00:01:16.870
to the square of the magnitude
of the y direction.

00:01:16.870 --> 00:01:20.950
But for any given x, it's
always going to be the same.

00:01:20.950 --> 00:01:21.920
It's only dependent on y.

00:01:21.920 --> 00:01:25.100
So here, even if we make
x larger, we still get

00:01:25.100 --> 00:01:25.750
the same magnitude.

00:01:28.390 --> 00:01:29.640
And remember, these are
just sample points

00:01:29.640 --> 00:01:31.250
on our vector field.

00:01:31.250 --> 00:01:31.570
But anyway.

00:01:31.570 --> 00:01:34.000
That's enough of just getting
the intuition behind

00:01:34.000 --> 00:01:34.580
that vector field.

00:01:34.580 --> 00:01:36.630
But let me ask you a question.

00:01:36.630 --> 00:01:39.850
If I were to, let's say that
this vector field shows the

00:01:39.850 --> 00:01:44.020
velocity of a fluid
at various points.

00:01:44.020 --> 00:01:47.610
And so you can view this, we're
looking down on a river, maybe.

00:01:47.610 --> 00:01:50.880
If I were to take a little twig
or something, and I were to

00:01:50.880 --> 00:01:57.520
place it in this fluid, so let
me place the twig right here.

00:01:57.520 --> 00:02:00.520
Let me draw my twig.

00:02:00.520 --> 00:02:04.480
So let's say I place a twig,
it's a funny-looking twig,

00:02:04.480 --> 00:02:06.690
but that's good enough.

00:02:06.690 --> 00:02:07.920
Let's say I place a
twig right there.

00:02:07.920 --> 00:02:10.010
What's going to
happen to the twig?

00:02:10.010 --> 00:02:13.230
Well, at this point on the
twig, the water's moving to the

00:02:13.230 --> 00:02:17.260
right, so it'll push this part
of the twig to the right.

00:02:17.260 --> 00:02:20.660
At the top of the twig, the
water is also moving to the

00:02:20.660 --> 00:02:22.720
right, maybe with a faster
velocity, but it's also going

00:02:22.720 --> 00:02:24.570
to push the top of the
twig to the right.

00:02:24.570 --> 00:02:27.090
But the top of the twig is
going to be being pushed to

00:02:27.090 --> 00:02:30.970
the right faster than the
bottom of the twig, right?

00:02:30.970 --> 00:02:32.480
So what's going to happen?

00:02:32.480 --> 00:02:34.190
The twig's going
to rotate, right?

00:02:34.190 --> 00:02:36.680
After, I don't know, some
period of time, the

00:02:36.680 --> 00:02:38.950
twig's going to look
something like this.

00:02:38.950 --> 00:02:40.960
The bottom will move a little
bit to the right, but the

00:02:40.960 --> 00:02:44.520
top will move a lot
more to the right.

00:02:44.520 --> 00:02:44.810
Right?

00:02:44.810 --> 00:02:46.650
And the whole thing would have
been shifted to the right.

00:02:46.650 --> 00:02:47.650
But it's going to
rotate a little bit.

00:02:47.650 --> 00:02:50.370
And maybe after a little bit
further, maybe it looks

00:02:50.370 --> 00:02:52.940
something like this.

00:02:52.940 --> 00:02:59.880
So you can see that because the
vectors increasing in a

00:02:59.880 --> 00:03:02.650
direction that is perpendicular
to our direction

00:03:02.650 --> 00:03:04.790
of motion, right?

00:03:04.790 --> 00:03:08.560
This fairly simple example,
all of the vectors point

00:03:08.560 --> 00:03:09.960
in the x-direction.

00:03:09.960 --> 00:03:15.300
But the magnitude of the
vectors increase, they increase

00:03:15.300 --> 00:03:18.160
perpendicular, they increase
in the y-dimension, right?

00:03:18.160 --> 00:03:21.700
And when this happens, when the
flow is going in the same

00:03:21.700 --> 00:03:24.700
direction, but it's going at a
different magnitude, you see

00:03:24.700 --> 00:03:27.460
that any object in it
will rotate, right?

00:03:27.460 --> 00:03:28.200
So let's think about that.

00:03:28.200 --> 00:03:32.010
So if the derivative, the
partial derivative, of this

00:03:32.010 --> 00:03:36.490
vector field with respect to y
is increasing or decreasing, if

00:03:36.490 --> 00:03:40.230
it's just changing, that means
as we increase in y, or as we

00:03:40.230 --> 00:03:44.460
decrease in y, the magnitude of
the x-component of our vectors,

00:03:44.460 --> 00:03:46.860
right, the x-direction
of our vectors changes.

00:03:46.860 --> 00:03:49.230
And so if you have a different
speed for different levels of

00:03:49.230 --> 00:03:53.250
y, as something moves in the
x-direction, it's going

00:03:53.250 --> 00:03:55.520
to be rotated, right?

00:03:55.520 --> 00:03:59.920
You could almost view it as if
there's a net torque on an

00:03:59.920 --> 00:04:02.700
object that sits in
the water here.

00:04:02.700 --> 00:04:06.320
And the ultimate would be, let
me draw another vector field,

00:04:06.320 --> 00:04:11.560
the ultimate would be if
I had this situation.

00:04:11.560 --> 00:04:15.180
Let me draw another
vector field.

00:04:15.180 --> 00:04:20.850
If I had this situation, where
maybe down here it's like this,

00:04:20.850 --> 00:04:24.200
then maybe it's like this, and
then maybe it gets really

00:04:24.200 --> 00:04:26.960
small, then maybe it switches
directions, up here, and then

00:04:26.960 --> 00:04:28.570
the vector field
goes like this.

00:04:28.570 --> 00:04:32.080
So you could imagine up here
that's going to the left, with

00:04:32.080 --> 00:04:33.150
a fairly large magnitude.

00:04:33.150 --> 00:04:36.440
So if you put a twig here, you
would definitely hopefully see

00:04:36.440 --> 00:04:39.490
that the twig, not only will it
not be shifted to the right,

00:04:39.490 --> 00:04:41.650
this side is going to be moved
to the left, this side is going

00:04:41.650 --> 00:04:43.410
to be the right, it's
going to be rotated.

00:04:43.410 --> 00:04:47.150
And you'll see that there's
a net torque on the twig.

00:04:47.150 --> 00:04:48.950
So what's the intuition there?

00:04:48.950 --> 00:04:52.810
All of a sudden, we care about
how much is the magnitude of a

00:04:52.810 --> 00:04:56.505
vector changing, not in its
direction of motion, like in

00:04:56.505 --> 00:05:00.640
the divergence example, but we
care how much is the magnitude

00:05:00.640 --> 00:05:04.040
of a vector changing as we go
perpendicular to its

00:05:04.040 --> 00:05:05.570
direction of motion.

00:05:05.570 --> 00:05:07.520
So when we learned about
dot and cross product,

00:05:07.520 --> 00:05:08.070
what did we learn?

00:05:08.070 --> 00:05:11.480
We learned that the dot product
of 2 vectors tells you how much

00:05:11.480 --> 00:05:15.820
2 vectors move together, and
the cross product tells you how

00:05:15.820 --> 00:05:19.300
much the perpendicular, it's
kind of the multiplication

00:05:19.300 --> 00:05:22.260
of the perpendicular
components of a vector.

00:05:22.260 --> 00:05:25.720
So this might give you a little
intuition of what is the curl.

00:05:25.720 --> 00:05:31.510
Because the curl essentially
measures what is the rotational

00:05:31.510 --> 00:05:35.800
effect, or I guess you could
say, what is the curl of a

00:05:35.800 --> 00:05:37.240
vector field at a given point?

00:05:37.240 --> 00:05:38.710
And you can you
can visualize it.

00:05:38.710 --> 00:05:40.870
You put a twig there, what
would happen to the twig?

00:05:40.870 --> 00:05:44.460
If the twig rotates and there's
some curl, if the magnitude

00:05:44.460 --> 00:05:46.730
of the rotation is larger,
then the curl is larger.

00:05:46.730 --> 00:05:48.640
If it rotates in the other
direction, you'll have the

00:05:48.640 --> 00:05:50.300
negative direction of curl.

00:05:50.300 --> 00:05:53.820
And so just like what we did
with torque, we now care

00:05:53.820 --> 00:05:54.420
about the direction.

00:05:54.420 --> 00:05:56.435
Because we care whether it's
going counterclockwise or

00:05:56.435 --> 00:05:58.620
clockwise, so we're going to
end up with a vector

00:05:58.620 --> 00:05:59.710
quantity, right?

00:05:59.710 --> 00:06:05.270
So, and all of this should
hopefully start fitting

00:06:05.270 --> 00:06:07.210
together at this point.

00:06:07.210 --> 00:06:10.430
We've been dealing
with this Dell

00:06:10.430 --> 00:06:12.830
vector or this, you know, we
could call this abusive

00:06:12.830 --> 00:06:16.550
notation, but it kind of is
intuitive, although it really

00:06:16.550 --> 00:06:18.690
doesn't have any meaning when
I describe it like this.

00:06:18.690 --> 00:06:21.520
You can kind of write it as a
vector operator, and then it

00:06:21.520 --> 00:06:23.140
has a little bit more meeting.

00:06:23.140 --> 00:06:23.800
But this Dell

00:06:23.800 --> 00:06:26.000
operator, we use it
a bunch of times.

00:06:26.000 --> 00:06:30.190
You know, if the partial
derivative of something in the

00:06:30.190 --> 00:06:34.180
i-direction, plus the partial
derivative, something with

00:06:34.180 --> 00:06:37.080
respect to y in the
j-direction, plus the partial

00:06:37.080 --> 00:06:40.270
derivative, well, this is if we
do it in three dimensions

00:06:40.270 --> 00:06:43.460
with respect to z
in the k-direction.

00:06:43.460 --> 00:06:46.660
When we applied it to just a
scalar or vector field, you

00:06:46.660 --> 00:06:49.670
know, like a three-dimensional
function, we just multiplied

00:06:49.670 --> 00:06:52.430
this times that scalar
function, we got the gradient.

00:06:52.430 --> 00:07:00.890
When we took the dot product of
this with a vector field, we

00:07:00.890 --> 00:07:05.710
got the divergence of
the vector field.

00:07:05.710 --> 00:07:07.090
And this should be a
little bit intuitive

00:07:07.090 --> 00:07:08.110
to you, at this point.

00:07:08.110 --> 00:07:11.300
Because when we, you might want
to review our original videos

00:07:11.300 --> 00:07:16.380
where we compared the dot
product to the cross product.

00:07:16.380 --> 00:07:18.830
Because the dot product
was, how much do two

00:07:18.830 --> 00:07:20.840
vectors move together?

00:07:20.840 --> 00:07:26.070
So when you're taking this Dell
operator and dotting it with a

00:07:26.070 --> 00:07:29.070
vector field, you're saying,
how much is the vector

00:07:29.070 --> 00:07:29.840
field changing, right?

00:07:29.840 --> 00:07:32.200
All a derivative is, a partial
derivative or a normal

00:07:32.200 --> 00:07:34.020
derivative, it's just
a rate of change.

00:07:34.020 --> 00:07:36.195
Partial derivative with respect
to x is rate of change

00:07:36.195 --> 00:07:37.440
in the x-direction.

00:07:37.440 --> 00:07:40.170
So all you're saying is, when
you're taking a dot product,

00:07:40.170 --> 00:07:44.500
how much is my rate of
change increasing in my

00:07:44.500 --> 00:07:45.890
direction of movement?

00:07:45.890 --> 00:07:48.700
How much is my rate of change
in the y-direction increasing

00:07:48.700 --> 00:07:49.755
in the y-direction?

00:07:49.755 --> 00:07:52.410
And so it makes sense that it
helps us with divergence.

00:07:52.410 --> 00:07:56.825
Because remember, if this is a
vector, and then as we increase

00:07:56.825 --> 00:08:00.480
this in the x-direction, the
vectors increase, we took a

00:08:00.480 --> 00:08:04.280
little point, and we said, oh,
at this point we're going to

00:08:04.280 --> 00:08:06.100
have more leaving than
entering, so we have a

00:08:06.100 --> 00:08:07.840
positive divergence.

00:08:07.840 --> 00:08:10.000
But that makes sense, also,
because as you go in the

00:08:10.000 --> 00:08:12.790
x-direction, the magnitudes
of the vectors increase.

00:08:12.790 --> 00:08:14.310
Anyway, I don't want to
confuse you too much.

00:08:14.310 --> 00:08:17.380
So now, the intuition, because
now we don't care about the

00:08:17.380 --> 00:08:20.270
rate of change along with the
direction of the vector.

00:08:20.270 --> 00:08:22.420
We care about the rate of
change of the magnitudes of

00:08:22.420 --> 00:08:25.560
the vectors perpendicular
the direction of the vector.

00:08:25.560 --> 00:08:34.460
So the curl, you might guess,
is equal to the cross product

00:08:34.460 --> 00:08:39.620
of our Dell operator
and the vector field.

00:08:39.620 --> 00:08:43.210
And if that was where your
intuition led you, and that

00:08:43.210 --> 00:08:45.780
is what your guess is,
you would be correct.

00:08:45.780 --> 00:08:49.950
That is the curl of
the vector field.

00:08:49.950 --> 00:08:56.490
And it is a measure of how much
is that field rotating, or

00:08:56.490 --> 00:08:58.890
maybe if you imagine an object
in the field, how much is the

00:08:58.890 --> 00:09:01.720
field causing something
to rotate because it's

00:09:01.720 --> 00:09:03.220
exerting a net torque?

00:09:03.220 --> 00:09:07.510
Because at different points
in the object, you have a

00:09:07.510 --> 00:09:13.230
different magnitude of a
field in the same direction.

00:09:13.230 --> 00:09:14.410
Anyway, I don't want to
confuse you too much.

00:09:14.410 --> 00:09:15.910
Hopefully that example I
just showed you will make

00:09:15.910 --> 00:09:17.700
a little bit of sense.

00:09:17.700 --> 00:09:20.530
Anyway, I realize I've
already pushed 9 minutes.

00:09:20.530 --> 00:09:24.970
In the next video, I'll
actually compute curl, and

00:09:24.970 --> 00:09:27.840
maybe we'll try to draw
a couple more to hit

00:09:27.840 --> 00:09:29.320
the intuition home.

00:09:29.320 --> 00:09:31.380
See you in the next video.