Divergence theorem proof (part 3)

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:03.900
Let's now evaluate this green
surface integral right over

00:00:03.900 --> 00:00:04.660
here.

00:00:04.660 --> 00:00:07.460
And to do that, we need
to parameterize surface 2.

00:00:07.460 --> 00:00:10.080
So let's say surface
2 can be represented

00:00:10.080 --> 00:00:12.810
by the vector position function.

00:00:12.810 --> 00:00:18.280
I'll just call it t-- t
for two-- so t-- which

00:00:18.280 --> 00:00:20.610
is a vector-- it's going to
be a function of x and y.

00:00:20.610 --> 00:00:23.680
Those are going are going
to be our parameters.

00:00:23.680 --> 00:00:25.430
And it's going to
be equal to-- and

00:00:25.430 --> 00:00:28.530
we can do this, once again,
because our surface is

00:00:28.530 --> 00:00:30.730
a function of x and y.

00:00:30.730 --> 00:00:36.150
So it's going to be equal
to x times i plus y times j

00:00:36.150 --> 00:00:47.980
plus f2 of xy times
k for all the xy's

00:00:47.980 --> 00:00:51.555
that are a member of our domain.

00:00:51.555 --> 00:00:52.930
Now, with that
out of the way, we

00:00:52.930 --> 00:00:56.400
can re-express
what k dot n ds is.

00:00:56.400 --> 00:00:59.130
So let me write this over here.

00:00:59.130 --> 00:01:08.431
k dot n ds is equal to-- and
we could put parentheses here

00:01:08.431 --> 00:01:10.180
that we're going to
take this dot product.

00:01:10.180 --> 00:01:12.138
That's, at least, how I
like to think about it.

00:01:12.138 --> 00:01:19.080
This is the exact
same thing as k

00:01:19.080 --> 00:01:27.140
dotted with the cross
product of the partial of t--

00:01:27.140 --> 00:01:29.394
let me make this clear.

00:01:29.394 --> 00:01:30.810
I'm going to do
the magenta color.

00:01:30.810 --> 00:01:35.870
The partial of t with respect
to x crossed with the partial

00:01:35.870 --> 00:01:42.270
of t with respect to y times
a little chunk of our area--

00:01:42.270 --> 00:01:43.360
times da.

00:01:43.360 --> 00:01:45.900
A little chunk of our
area in the xy domain.

00:01:45.900 --> 00:01:47.570
We've done this
multiple times as we

00:01:47.570 --> 00:01:49.250
evaluated surface integrals.

00:01:49.250 --> 00:01:51.230
And we got the intuition
for why this works.

00:01:51.230 --> 00:01:53.165
And so then, we're
essentially just evaluating

00:01:53.165 --> 00:01:54.040
the surface integral.

00:01:54.040 --> 00:01:55.360
And the one thing
we want to make sure

00:01:55.360 --> 00:01:57.130
is make sure this has
the right orientation.

00:01:57.130 --> 00:01:59.629
Because, remember, in order for
the divergence theorem to be

00:01:59.629 --> 00:02:01.080
true, the way we've
defined it is,

00:02:01.080 --> 00:02:03.400
all the normal vectors
have to be outward-facing.

00:02:03.400 --> 00:02:05.600
So for this top surface,
the normal vector

00:02:05.600 --> 00:02:07.830
has to be pointing straight up.

00:02:10.844 --> 00:02:12.010
Not necessarily straight up.

00:02:12.010 --> 00:02:12.842
At least, upwards.

00:02:12.842 --> 00:02:15.300
If this is a curve, it wouldn't
be necessarily straight up.

00:02:15.300 --> 00:02:17.990
But it needs to be up kind
of outward-facing, like that.

00:02:17.990 --> 00:02:20.290
On the sides, it would be
outward-facing like that.

00:02:20.290 --> 00:02:22.890
And down here, it
would be outward-facing

00:02:22.890 --> 00:02:25.550
going in the general
downwards direction.

00:02:25.550 --> 00:02:29.090
So let's just make sure
that this is upwards-facing.

00:02:29.090 --> 00:02:30.630
If we're changing
with respect to x,

00:02:30.630 --> 00:02:32.030
we're going in this direction.

00:02:32.030 --> 00:02:34.540
Changing with respect to y,
we're going in that direction.

00:02:34.540 --> 00:02:37.580
Take the right-hand rule with
the cross product-- index

00:02:37.580 --> 00:02:42.080
finger there, middle finger
there, your right thumb,

00:02:42.080 --> 00:02:43.910
I should say, will
go straight up.

00:02:43.910 --> 00:02:45.710
So it goes in the
right direction.

00:02:45.710 --> 00:02:48.610
So this would be an
upward-pointing vector.

00:02:48.610 --> 00:02:53.301
So we got the right
orientation for our surface.

00:02:53.301 --> 00:02:54.800
Now, let's think
about what this is.

00:02:54.800 --> 00:02:56.995
And it's important to
realize we could calculate

00:02:56.995 --> 00:02:58.870
all of the components
of this, but then we're

00:02:58.870 --> 00:03:01.100
just going to take the dot
product of that with k.

00:03:01.100 --> 00:03:03.280
So really, we care about
the k component only,

00:03:03.280 --> 00:03:06.250
but I'll work it out.

00:03:06.250 --> 00:03:17.290
This is equal to k
times a matrix i, j,

00:03:17.290 --> 00:03:22.180
k of the partial of
t with respect to x.

00:03:22.180 --> 00:03:24.200
Well, the partial
of t with respect

00:03:24.200 --> 00:03:26.190
to x-- I'll do this
in blue-- is going

00:03:26.190 --> 00:03:33.800
to be 1, 0, and the partial
of f2 with respect to x.

00:03:33.800 --> 00:03:36.580
And then the partial
of t with respect to y

00:03:36.580 --> 00:03:40.410
is just going to be
0, 1, and the partial

00:03:40.410 --> 00:03:44.420
of f2 with respect to y.

00:03:44.420 --> 00:03:46.720
And then, of course, we
have to multiply times da.

00:03:49.250 --> 00:03:57.460
And this is all going to be
equal to unit vector k dotted

00:03:57.460 --> 00:04:01.090
with-- and I don't even
have to work it all out.

00:04:01.090 --> 00:04:04.730
It's going to be
something times the i unit

00:04:04.730 --> 00:04:06.760
vector minus
something-- checkerboard

00:04:06.760 --> 00:04:10.790
pattern-- minus something times
something else necessarily--

00:04:10.790 --> 00:04:13.010
times our j unit vector.

00:04:13.010 --> 00:04:14.980
And now, we can think
about the k unit vector.

00:04:14.980 --> 00:04:17.630
So the k unit vector is going
to be 1 times 1 minus 0.

00:04:17.630 --> 00:04:20.300
So it's just going to be
plus the k unit vector.

00:04:20.300 --> 00:04:23.690
We know it's 1 times
the k unit vector.

00:04:23.690 --> 00:04:28.220
And of course, we have
our da out right here.

00:04:28.220 --> 00:04:29.920
But when you take
this dot product,

00:04:29.920 --> 00:04:32.310
you only are left
with the k components.

00:04:32.310 --> 00:04:33.845
And it's essentially
just 1 times 1.

00:04:33.845 --> 00:04:35.470
You end up with a
scalar quantity of 1.

00:04:38.660 --> 00:04:43.930
All of this business
just simplified to da.

00:04:43.930 --> 00:04:46.260
So now, we can rewrite
our surface integral.

00:04:46.260 --> 00:04:49.010
And we're going to rewrite
it in the xy domain

00:04:49.010 --> 00:04:51.170
now-- in our parameters domain.

00:04:51.170 --> 00:04:53.941
So our surface integral
right up here--

00:04:53.941 --> 00:04:55.440
so this will be
good for this video.

00:04:55.440 --> 00:04:57.160
And then, we'll do the same
thing with this surface here.

00:04:57.160 --> 00:04:59.800
Just making sure that we
get the orientation right.

00:04:59.800 --> 00:05:04.500
So this surface
integral, s2-- and I'll

00:05:04.500 --> 00:05:07.520
even rewrite a little bit--
s2, which is a function.

00:05:07.520 --> 00:05:18.550
r is a function of x, y,
and z times k dot n ds--

00:05:18.550 --> 00:05:20.540
I just rewrote all of
this right over here--

00:05:20.540 --> 00:05:23.750
is equivalent to
the double integral

00:05:23.750 --> 00:05:28.953
over our parameters domain--
which is just D-- of r of x, y,

00:05:28.953 --> 00:05:31.000
z times all of this business.

00:05:31.000 --> 00:05:33.600
All of this business
just simplified to da.

00:05:33.600 --> 00:05:36.830
And since I want to write it
in terms of my parameters,

00:05:36.830 --> 00:05:41.070
I'll write it as R of xy.

00:05:41.070 --> 00:05:44.500
And while we're on that
surface, z is equal to f2.

00:05:44.500 --> 00:05:49.550
So it's xy and f2 of xy.

00:05:49.550 --> 00:05:54.635
And then, all of this business
we just saw simplified to da.

00:05:54.635 --> 00:05:56.010
So you might be
saying, hey, Sal.

00:05:56.010 --> 00:05:58.300
It didn't look like you
simplified it a lot.

00:05:58.300 --> 00:06:01.600
But at least now put it in terms
of a double integral instead

00:06:01.600 --> 00:06:03.020
of a surface integral.

00:06:03.020 --> 00:06:05.311
So at least, in my mind,
that is a simplification.

00:06:05.311 --> 00:06:07.560
In the next video, we're
going to the exact same thing

00:06:07.560 --> 00:06:09.810
with this-- just making sure
that our vectors are oriented

00:06:09.810 --> 00:06:10.310
properly.

00:06:10.310 --> 00:06:12.140
And we could just
introduce a negative sign

00:06:12.140 --> 00:06:15.080
to make sure that they are.

00:06:15.080 --> 00:06:17.800
And then, we're going to think
about the triple integrals

00:06:17.800 --> 00:06:20.329
and try to simplify those.


Divergence theorem proof (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.244 --> 00:00:02.660
I think we're now ready to get
into the meat of the proof.

00:00:02.660 --> 00:00:04.160
In the last video, we
said if we can just

00:00:04.160 --> 00:00:06.580
prove that each of these
parts are equal to each other,

00:00:06.580 --> 00:00:08.850
we essentially have proven
that that is equal to that.

00:00:08.850 --> 00:00:10.830
Because here in
yellow is another way

00:00:10.830 --> 00:00:14.623
of writing the flux across
the surface and here in green

00:00:14.623 --> 00:00:17.220
is another way of writing
the triple integral

00:00:17.220 --> 00:00:19.827
over our region of
the divergence of f.

00:00:19.827 --> 00:00:21.410
And what I'm going
to do in this video

00:00:21.410 --> 00:00:23.260
and probably in
the next video is

00:00:23.260 --> 00:00:26.580
prove that these two are
equivalent to each other.

00:00:26.580 --> 00:00:28.830
And I'm going to prove
it using the fact

00:00:28.830 --> 00:00:32.520
that our original region
is a simple solid region,

00:00:32.520 --> 00:00:37.240
in particular is
a Type 1 region.

00:00:37.240 --> 00:00:39.030
And then that's
essentially going

00:00:39.030 --> 00:00:41.500
to be it because you can
use the exact same argument

00:00:41.500 --> 00:00:49.260
and the fact that it is a
Type 2 region to prove this,

00:00:49.260 --> 00:00:51.210
and the exact same
argument and the fact

00:00:51.210 --> 00:00:56.720
that it is a Type 3
region to prove that.

00:00:56.720 --> 00:01:00.140
So I'm going to assume
it's a Type 1, which I can.

00:01:00.140 --> 00:01:02.130
It's a Type 1, 2, and 3 region.

00:01:02.130 --> 00:01:03.790
So given the fact
that it's Type 1,

00:01:03.790 --> 00:01:06.777
I'm now going to prove this
relationship right over here.

00:01:06.777 --> 00:01:09.360
And then I'll leave it up to you
to do the exact same argument

00:01:09.360 --> 00:01:12.490
with a Type 2 and
a Type 3 region.

00:01:12.490 --> 00:01:13.890
So let's get going.

00:01:13.890 --> 00:01:17.410
So Type 1 region, just
to remind ourselves,

00:01:17.410 --> 00:01:26.350
a Type 1 region is
a region that is

00:01:26.350 --> 00:01:33.770
equal to the set of
all x, y's and z's such

00:01:33.770 --> 00:01:41.060
that the xy pairs are a member
of a domain in the xy plane,

00:01:41.060 --> 00:01:43.640
and z is bounded
by two functions.

00:01:43.640 --> 00:01:47.940
z's lower bound
is f1 of x and y.

00:01:47.940 --> 00:01:51.220
And that's going to be
less than or equal to z.

00:01:51.220 --> 00:01:59.190
And z's upper bound, we
can call it f2 of xy.

00:01:59.190 --> 00:02:02.100
And then let me close the
set notation right over here.

00:02:02.100 --> 00:02:07.390
And let me just draw a general
version of a Type 1 region.

00:02:07.390 --> 00:02:11.710
So let me draw my
x, y, and z-axes.

00:02:11.710 --> 00:02:14.940
So this is my z-axis.

00:02:14.940 --> 00:02:17.930
This is my x-axis.

00:02:17.930 --> 00:02:21.470
And there is my y-axis.

00:02:21.470 --> 00:02:28.160
And so we might have a
region D. So our region,

00:02:28.160 --> 00:02:32.340
I'll draw it as a little
circle right over here.

00:02:32.340 --> 00:02:38.370
This is our region D. And
for any xy in our region D,

00:02:38.370 --> 00:02:40.050
you can evaluate the function f.

00:02:40.050 --> 00:02:42.420
You can figure out
an f1, I should say.

00:02:42.420 --> 00:02:45.760
So this might define an f1,
which we can kind of imagine

00:02:45.760 --> 00:02:47.500
as a surface or a
little thing that's

00:02:47.500 --> 00:02:49.460
at the bottom of a
cylinder if you want.

00:02:49.460 --> 00:02:51.913
So every xy there,
when you evaluate

00:02:51.913 --> 00:02:55.150
or when you figure out
what the corresponding f1

00:02:55.150 --> 00:02:57.155
of those points in
this domain would be,

00:02:57.155 --> 00:02:59.640
you might get a surface that
looks something like this.

00:02:59.640 --> 00:03:03.490
It doesn't have to be flat, but
hopefully this gives the idea.

00:03:03.490 --> 00:03:05.190
It doesn't have to
be completely flat.

00:03:05.190 --> 00:03:07.330
It can be curved
or whatever else.

00:03:07.330 --> 00:03:10.130
But this just shows you that
every xy, when you evaluate it

00:03:10.130 --> 00:03:12.010
right over here,
it gets associated

00:03:12.010 --> 00:03:14.650
with a point, this lower
bound surface right over here.

00:03:14.650 --> 00:03:16.520
And I'll draw a
dotted line to show

00:03:16.520 --> 00:03:19.020
that it's only for the
xy's in this domain.

00:03:19.020 --> 00:03:22.790
And then we have an upper bound
surface that might be up here.

00:03:22.790 --> 00:03:23.840
Give me any xy.

00:03:23.840 --> 00:03:28.849
When I evaluate f2, I
get this surface up here.

00:03:28.849 --> 00:03:30.890
And once again, they don't
have to look the same.

00:03:30.890 --> 00:03:33.000
This could be like a dome
or it could be slanted

00:03:33.000 --> 00:03:34.250
or who knows what it might be.

00:03:34.250 --> 00:03:36.110
But this will give
you the general idea.

00:03:36.110 --> 00:03:38.174
And then z fills up the region.

00:03:38.174 --> 00:03:40.590
Remember, the region isn't
just the surface of the figure,

00:03:40.590 --> 00:03:43.070
it's the entire
volume inside of it.

00:03:43.070 --> 00:03:46.200
So when z varies between that
surface and that surface,

00:03:46.200 --> 00:03:52.890
for any given xy in our domain,
we fill up the entire region.

00:03:52.890 --> 00:03:56.500
So we can fill up
this entire region.

00:03:56.500 --> 00:03:57.750
And this is the way I drew it.

00:03:57.750 --> 00:03:59.416
It looks like a
cylinder, but it doesn't

00:03:59.416 --> 00:04:01.030
have to be a cylinder like this.

00:04:01.030 --> 00:04:03.620
And these two surfaces might
touch each other, in which case

00:04:03.620 --> 00:04:05.260
there would be no
side of the cylinder.

00:04:05.260 --> 00:04:07.610
They could be lumpier than this.

00:04:07.610 --> 00:04:09.470
They might be
inclined in some way.

00:04:09.470 --> 00:04:12.020
But hopefully, this is
a good generalization

00:04:12.020 --> 00:04:14.440
of a Type 1 region.

00:04:14.440 --> 00:04:16.870
Now, a Type 1 region, you
can kind of think of it,

00:04:16.870 --> 00:04:19.660
it can be broken up
into three parts.

00:04:19.660 --> 00:04:22.240
It can be broken
up into surface--

00:04:22.240 --> 00:04:24.990
or the surfaces of a Type
1 region, I should say,

00:04:24.990 --> 00:04:27.385
can be broken up
into three parts.

00:04:32.550 --> 00:04:35.780
Let's call that surface one.

00:04:35.780 --> 00:04:38.490
Let's call this right
over here surface

00:04:38.490 --> 00:04:41.770
two, the top of the cylinder
or whatever kind of lumpy top

00:04:41.770 --> 00:04:42.730
it might be.

00:04:42.730 --> 00:04:46.170
And let's call the side, if
these two surfaces don't touch

00:04:46.170 --> 00:04:48.090
each other, let's call
that surface three.

00:04:48.090 --> 00:04:50.390
There might not necessarily
even be a surface three

00:04:50.390 --> 00:04:53.150
if these two touch each other
as in the case of the sphere.

00:04:53.150 --> 00:04:57.040
But let's just assume that there
actually is a surface three.

00:04:57.040 --> 00:04:59.030
So if we're evaluating
the surface integral--

00:04:59.030 --> 00:05:02.290
we'll think about this
triple integral in a second.

00:05:02.290 --> 00:05:04.690
But let's think about how
we can rewrite this surface

00:05:04.690 --> 00:05:06.690
integral right over here.

00:05:06.690 --> 00:05:10.802
This entire surface
is S1 plus S2 plus S3.

00:05:10.802 --> 00:05:12.260
So we can essentially
break this up

00:05:12.260 --> 00:05:14.490
into three separate
surface integrals.

00:05:14.490 --> 00:05:15.276
So let's do that.

00:05:15.276 --> 00:05:16.650
Just remember
we're just focusing

00:05:16.650 --> 00:05:18.230
on this part right over here.

00:05:18.230 --> 00:05:24.000
So the surface
integral of R times

00:05:24.000 --> 00:05:28.058
k dot n, the dot
product of k and n,

00:05:28.058 --> 00:05:33.410
ds can be rewritten as the-- let
me write it this way-- can be

00:05:33.410 --> 00:05:37.840
rewritten as the
surface integral

00:05:37.840 --> 00:05:49.202
over S2 of R times k dot n ds,
plus the surface integral-- I'm

00:05:49.202 --> 00:05:50.660
just breaking up
the surface here--

00:05:50.660 --> 00:05:57.134
plus the surface integral
over S1 of R times k

00:05:57.134 --> 00:06:04.940
dot n ds plus the surface
integral over surface three

00:06:04.940 --> 00:06:09.760
of the same thing,
R times k dot n ds.

00:06:09.760 --> 00:06:12.450
Now, the way I've drawn it,
and this is actually the case,

00:06:12.450 --> 00:06:15.060
S3 is if those surfaces
don't touch each other.

00:06:15.060 --> 00:06:18.320
And for Type 1 situation
right over here,

00:06:18.320 --> 00:06:20.290
the normal vector
at any given point

00:06:20.290 --> 00:06:23.660
on the side of the cylinder
for this Type 1 region--

00:06:23.660 --> 00:06:25.149
if there is this
in between region.

00:06:25.149 --> 00:06:25.940
There always isn't.

00:06:25.940 --> 00:06:28.230
In a sphere, there wouldn't
be this surface, in which case

00:06:28.230 --> 00:06:28.896
this would be 0.

00:06:28.896 --> 00:06:31.760
But if there is this surface in
a Type 1 region, the one that

00:06:31.760 --> 00:06:34.590
essentially connects the
boundaries of the top

00:06:34.590 --> 00:06:36.540
and the bottom, then
the normal vector

00:06:36.540 --> 00:06:38.040
will never have a k component.

00:06:38.040 --> 00:06:40.870
The normal vector will
always be pointing flat out.

00:06:40.870 --> 00:06:42.950
It will only have an
i and j component.

00:06:42.950 --> 00:06:45.820
So if you take this normal
vector right over here

00:06:45.820 --> 00:06:48.450
does not have a k component
and you're dotting it

00:06:48.450 --> 00:06:51.760
with a k vector, then the dot
product of two things that

00:06:51.760 --> 00:06:54.250
are orthogonal, the k
vector goes like that,

00:06:54.250 --> 00:06:55.520
you're going to get 0.

00:06:55.520 --> 00:06:59.690
So this thing is going
to be 0 because k

00:06:59.690 --> 00:07:02.830
dot n is going to be 0 in this
situation for this surface.

00:07:02.830 --> 00:07:08.480
k dot n is going
to be equal to 0.

00:07:08.480 --> 00:07:12.380
So this part right
over here simplifies

00:07:12.380 --> 00:07:15.250
to this right over here.

00:07:15.250 --> 00:07:19.740
Now in the next video what we
can do is essentially express

00:07:19.740 --> 00:07:21.770
these in terms of
surface integrals,

00:07:21.770 --> 00:07:25.414
but in terms of double
integrals over this domain

00:07:25.414 --> 00:07:26.080
right over here.

00:07:26.080 --> 00:07:29.810
We'll kind of evaluate
these surface integrals.


Divergence theorem proof (part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:03.580
Let's now prove the divergence
theorem, which tells us

00:00:03.580 --> 00:00:06.602
that the flux across the
surface of a vector field--

00:00:06.602 --> 00:00:08.560
and our vector field
we're going to think about

00:00:08.560 --> 00:00:11.060
is F. So the flux
across that surface,

00:00:11.060 --> 00:00:13.800
and I could call
that F dot n, where

00:00:13.800 --> 00:00:15.640
n is a normal vector
of the surface--

00:00:15.640 --> 00:00:18.050
and I can multiply
that times ds--

00:00:18.050 --> 00:00:23.720
so this is equal to the
trip integral summing up

00:00:23.720 --> 00:00:27.410
throughout the volume
of that region,

00:00:27.410 --> 00:00:34.121
summing up that volume
of the divergence of F.

00:00:34.121 --> 00:00:36.620
And we've done several videos
explaining the intuition here,

00:00:36.620 --> 00:00:38.490
but now we are actually
going to prove it.

00:00:38.490 --> 00:00:42.560
And of course, we're times
each little differential cube

00:00:42.560 --> 00:00:43.329
of volume.

00:00:43.329 --> 00:00:45.120
And we're going to make
an assumption here.

00:00:45.120 --> 00:00:46.960
We're going to assume
that we're dealing

00:00:46.960 --> 00:00:49.525
with a simple solid region.

00:00:58.010 --> 00:00:59.815
And what this means,
more formally,

00:00:59.815 --> 00:01:01.440
is that the region
we're thinking about

00:01:01.440 --> 00:01:04.680
can be a type I, type
II, and type III region.

00:01:04.680 --> 00:01:06.890
I should say it is
all of the three.

00:01:06.890 --> 00:01:12.630
So it is types I, II, and III.

00:01:12.630 --> 00:01:15.540
And there are videos that go
into what each of these regions

00:01:15.540 --> 00:01:18.790
are, but a lot of
the basic shapes

00:01:18.790 --> 00:01:20.630
fall into this
simple solid region.

00:01:20.630 --> 00:01:24.620
Like a sphere or a
cylinder of some kind,

00:01:24.620 --> 00:01:28.630
they can be types I,
type II, and type III.

00:01:28.630 --> 00:01:31.370
And for a lot of situations that
aren't simple solid regions,

00:01:31.370 --> 00:01:34.500
you can break them up
into simple solid regions.

00:01:34.500 --> 00:01:38.780
But let's just prove it for
this case right over here.

00:01:38.780 --> 00:01:40.610
So let's just assume
that our vector field

00:01:40.610 --> 00:01:46.440
F can be written as P, which is
a function of x, y, and z times

00:01:46.440 --> 00:01:49.520
i, plus Q, which is
a function of x, y,

00:01:49.520 --> 00:01:54.690
and z times j, plus R,
which is a function of x, y,

00:01:54.690 --> 00:01:58.590
and z times k.

00:01:58.590 --> 00:02:01.460
So let's think about what each
of these sides of the equation

00:02:01.460 --> 00:02:02.790
would come out to be.

00:02:02.790 --> 00:02:06.600
Well, first of all, what
is going to be F dot n?

00:02:06.600 --> 00:02:08.449
So let's think about
that a little bit.

00:02:08.449 --> 00:02:19.270
F dot n is going to be
equal to this component

00:02:19.270 --> 00:02:23.050
right over here times
n's i-component,

00:02:23.050 --> 00:02:26.310
plus this component right over
here times n's j-component,

00:02:26.310 --> 00:02:28.900
plus this component here
times n's k-component.

00:02:28.900 --> 00:02:34.960
So we could write
it as P times--

00:02:34.960 --> 00:02:38.270
or I'll just write P
open parentheses, the dot

00:02:38.270 --> 00:02:47.470
product of i and
n, and let me make

00:02:47.470 --> 00:02:51.010
sure I write i as a unit vector.

00:02:51.010 --> 00:02:52.050
Now, I want to be clear.

00:02:52.050 --> 00:02:53.674
What's going to happen
right over here?

00:02:53.674 --> 00:02:55.620
If you take the dot
product of i and n,

00:02:55.620 --> 00:02:59.500
you're just going to get
the i-component, the scaling

00:02:59.500 --> 00:03:02.024
factor of the i-component
of the n normal vector,

00:03:02.024 --> 00:03:03.940
and we're just going to
multiply that times P.

00:03:03.940 --> 00:03:07.070
So that's essentially the
product of the x-components,

00:03:07.070 --> 00:03:09.660
or I guess you could say the
magnitude of the x-components.

00:03:09.660 --> 00:03:17.660
And then to that, we are going
to add Q times j dotted with n.

00:03:17.660 --> 00:03:19.460
And once again, when
you dot j with n,

00:03:19.460 --> 00:03:22.010
you get the magnitude
of the j-component

00:03:22.010 --> 00:03:25.710
of the normal vector
right over there,

00:03:25.710 --> 00:03:29.750
and then times-- or
plus, I should say,

00:03:29.750 --> 00:03:35.070
plus R times k dotted with n.

00:03:35.070 --> 00:03:37.140
This isn't how we
normally see it,

00:03:37.140 --> 00:03:39.960
but I think it's reasonable to
say that this is actually true.

00:03:39.960 --> 00:03:41.910
This right over
here is going to be

00:03:41.910 --> 00:03:46.517
equal to P times the magnitude
of the i-component of n's

00:03:46.517 --> 00:03:49.100
normal vector, which is exactly
what we want in a dot product.

00:03:49.100 --> 00:03:50.891
This is the same thing
for the j-component.

00:03:50.891 --> 00:03:53.990
This is the same thing
for the k-component.

00:03:53.990 --> 00:03:58.390
And you can try it out Define
n as equal to m times i plus n

00:03:58.390 --> 00:04:01.660
times j plus o times k,
or something like that,

00:04:01.660 --> 00:04:04.710
and, you'll see that this
actually does work out fine.

00:04:04.710 --> 00:04:08.170
So how could we simplify this
expression right up here?

00:04:08.170 --> 00:04:12.410
Well, we could rewrite
the left-hand side

00:04:12.410 --> 00:04:16.970
as-- so the surface
integral of F--

00:04:16.970 --> 00:04:18.550
now let me write
it multiple ways.

00:04:18.550 --> 00:04:25.570
F dot ds, which is equal to
the surface integral of F dot n

00:04:25.570 --> 00:04:30.409
times the scalar ds is
equal to the double integral

00:04:30.409 --> 00:04:32.700
of the surface of all of this
business right over here,

00:04:32.700 --> 00:04:37.070
is equal to the double integral
over the surface of-- let me

00:04:37.070 --> 00:04:40.430
just copy and paste that--
of all of that business.

00:04:48.350 --> 00:04:51.920
And I just noticed that I forgot
to put the little unit vector

00:04:51.920 --> 00:04:55.360
symbol, a little caret
right over there.

00:04:55.360 --> 00:05:01.310
Put some parentheses, and
then we are left with our ds.

00:05:01.310 --> 00:05:03.160
And then this, all
of this, can be

00:05:03.160 --> 00:05:08.270
rewritten as the
surface integral

00:05:08.270 --> 00:05:09.900
of P times this business.

00:05:09.900 --> 00:05:12.520
And I'll just do it in
the same color-- of P

00:05:12.520 --> 00:05:17.700
times the dot product
of i and n ds,

00:05:17.700 --> 00:05:23.910
plus the surface integral of
Q times the dot product of j

00:05:23.910 --> 00:05:30.800
and n ds, plus the surface
integral of R times the dot

00:05:30.800 --> 00:05:37.630
product of k and n-- I
Forgot a caret-- k and n ds.

00:05:37.630 --> 00:05:38.740
So I just broke it up.

00:05:38.740 --> 00:05:40.720
We were taking the
integral of this sum,

00:05:40.720 --> 00:05:43.840
and so I just rewrote it as
the sum of the integrals.

00:05:43.840 --> 00:05:46.400
So that's the left-hand
side right over here.

00:05:46.400 --> 00:05:48.490
Now let's think about
the right-hand side.

00:05:48.490 --> 00:05:49.947
What is the divergence of F?

00:05:49.947 --> 00:05:52.030
And actually I'm going to
take some space up here.

00:05:52.030 --> 00:05:53.960
What is the divergence of F?

00:05:53.960 --> 00:05:56.670
Well, the divergence of F
based on this expression of F

00:05:56.670 --> 00:05:58.490
is just going to
be-- let me just

00:05:58.490 --> 00:06:00.180
write it over here real small.

00:06:00.180 --> 00:06:03.520
The divergence of
F is going to be

00:06:03.520 --> 00:06:05.630
the partial of P
with respect to-- let

00:06:05.630 --> 00:06:07.770
me do this in a new
color, because I'm

00:06:07.770 --> 00:06:10.070
using that yellow too much.

00:06:10.070 --> 00:06:13.730
The divergence of
F is going to be

00:06:13.730 --> 00:06:17.360
the partial of P with respect
to x, plus the partial of Q

00:06:17.360 --> 00:06:24.020
with respect to y, plus the
partial of R with respect to z.

00:06:24.020 --> 00:06:27.780
So this triple integral
right over here

00:06:27.780 --> 00:06:30.810
could be written as the triple
integral of the partial of P

00:06:30.810 --> 00:06:33.890
with respect to x, plus the
partial of Q with respect to y,

00:06:33.890 --> 00:06:37.370
plus the partial of
R with respect to z.

00:06:37.370 --> 00:06:39.545
Well, this thing, once
again, instead of writing it

00:06:39.545 --> 00:06:41.362
as the triple
integral of this sum,

00:06:41.362 --> 00:06:43.320
we could write it as a
sum of triple integrals.

00:06:43.320 --> 00:06:45.300
So this thing
right over here can

00:06:45.300 --> 00:06:50.820
be rewritten as
the triple integral

00:06:50.820 --> 00:06:53.107
over our
three-dimensional region.

00:06:53.107 --> 00:06:54.815
Actually, let me copy
and paste that so I

00:06:54.815 --> 00:06:56.148
don't have to keep rewriting it.

00:06:59.010 --> 00:07:02.350
So it's going to be equal to the
triple integral of the partial

00:07:02.350 --> 00:07:11.610
of P with respect to x dv
plus the triple integral

00:07:11.610 --> 00:07:18.990
of the partial of Q with respect
to y dv plus, once again,

00:07:18.990 --> 00:07:29.531
triple integral of the partial
of R with respect to z dv.

00:07:32.070 --> 00:07:36.630
So we've essentially restated
our divergence theorem.

00:07:36.630 --> 00:07:39.000
This is our surface integral,
and the divergence theorem

00:07:39.000 --> 00:07:42.150
says that this needs to
be equal to this business

00:07:42.150 --> 00:07:42.840
right over here.

00:07:42.840 --> 00:07:44.570
We've just written
it a different way.

00:07:44.570 --> 00:07:47.190
And so what I'm going to
do, in order to prove it,

00:07:47.190 --> 00:07:49.970
is just show that each of
these corresponding terms

00:07:49.970 --> 00:07:53.860
are equal to each other, that
these are equal to each other,

00:07:53.860 --> 00:07:58.750
that these are
equal to each other,

00:07:58.750 --> 00:08:04.704
and that these are
equal to each other.

00:08:04.704 --> 00:08:07.120
And in particular, we're going
to focus the proof on this.

00:08:07.120 --> 00:08:09.780
And we're going to use the fact
that our region is a type I

00:08:09.780 --> 00:08:10.280
region.

00:08:10.280 --> 00:08:12.130
It's a type I, type
II, and type II.

00:08:12.130 --> 00:08:14.350
But we're going to use the
fact that it's a type I

00:08:14.350 --> 00:08:16.737
region to prove that these
two things are equivalent.

00:08:16.737 --> 00:08:18.570
And then you can use
the fact that it's also

00:08:18.570 --> 00:08:22.820
a type II and type III region to
make the exact same argument as

00:08:22.820 --> 00:08:28.410
to why this is equal to this
and why this is equal to that.


Type III regions in three dimensions

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.600
After going through type 1
and type 2 region definitions,

00:00:03.600 --> 00:00:07.550
you can probably guess what a
type 3 region is going to be.

00:00:07.550 --> 00:00:14.995
So a type 3 is a region
in three dimensions.

00:00:18.200 --> 00:00:21.130
Since we called the other
the type 2 region R sub 2

00:00:21.130 --> 00:00:23.500
and the type 1
region R sub 1, I'll

00:00:23.500 --> 00:00:27.200
call this region R sub
3-- R with a subscript 3.

00:00:27.200 --> 00:00:31.490
It's going to be the set of
all points in three dimensions.

00:00:31.490 --> 00:00:39.230
The set of all x, y and z's
such that the x, z pairs

00:00:39.230 --> 00:00:45.730
are member of a domain, I'll
call this domain D sub 3.

00:00:45.730 --> 00:00:51.600
And y is going to vary
between two surfaces that

00:00:51.600 --> 00:00:54.420
are functions of x and z.

00:00:54.420 --> 00:00:58.860
So y is going to be greater than
or equal to the surface-- I'll

00:00:58.860 --> 00:01:03.550
call it h1 of x, z is going
to be less than or equal to y.

00:01:07.910 --> 00:01:10.540
y is going to be bounded
from above by the surface

00:01:10.540 --> 00:01:14.240
h2 of x, z.

00:01:14.240 --> 00:01:17.550
And once again, let's
close our set notation.

00:01:17.550 --> 00:01:20.700
So let's think about whether
some of these regions that we

00:01:20.700 --> 00:01:23.350
already saw were type 1 and
type 2, whether they're type 3.

00:01:23.350 --> 00:01:26.490
And then think about what
would not be a type 3 region.

00:01:26.490 --> 00:01:28.140
So let's go to this sphere.

00:01:28.140 --> 00:01:29.310
What could be our domain?

00:01:29.310 --> 00:01:31.640
Well the domain
is a set of x, z.

00:01:31.640 --> 00:01:33.340
It's just going to
be the in x, z plane.

00:01:33.340 --> 00:01:37.180
So over here our domain
could be this region right

00:01:37.180 --> 00:01:40.180
over here in the x, z plane.

00:01:40.180 --> 00:01:42.340
So I'll color it in.

00:01:42.340 --> 00:01:45.320
It could be that region right
over there in the x, z plane.

00:01:45.320 --> 00:01:47.720
And then the lower
bound on y will

00:01:47.720 --> 00:01:51.580
be the part that is behind
the sphere in this direction

00:01:51.580 --> 00:01:53.134
right over here.

00:01:53.134 --> 00:01:55.550
It's the stuff-- actually,
this might be a little bit hard

00:01:55.550 --> 00:01:58.210
to visualize since I'm
redrawing on top of it.

00:01:58.210 --> 00:02:00.050
And then the upper
bound on y is going

00:02:00.050 --> 00:02:02.150
to be this side right over here.

00:02:05.950 --> 00:02:11.060
So all of this is now
going to be green.

00:02:11.060 --> 00:02:14.570
Let me redraw this sphere
just to make it clear.

00:02:14.570 --> 00:02:18.010
Let me just draw
another coordinate axes.

00:02:18.010 --> 00:02:20.130
So another coordinate axes.

00:02:20.130 --> 00:02:22.640
The backside of this
sphere in the y direction.

00:02:22.640 --> 00:02:24.710
So I guess let's
think of it this way.

00:02:24.710 --> 00:02:28.610
This is the
hemisphere mark, this

00:02:28.610 --> 00:02:32.240
is the halfway
point for my sphere.

00:02:32.240 --> 00:02:35.610
And once again, that's kind
of the boundary of our domain.

00:02:35.610 --> 00:02:39.070
And then the backside-- let
me do the front side first.

00:02:39.070 --> 00:02:42.330
The front side y is upper
bound, that would be h2,

00:02:42.330 --> 00:02:47.750
would be all of this
business right over here.

00:02:47.750 --> 00:02:50.910
So this would be h2
that I'm coloring in.

00:02:50.910 --> 00:02:55.310
h2 would be the side that
is facing in that direction.

00:02:55.310 --> 00:02:57.230
Let me see how well
I can color it.

00:02:57.230 --> 00:02:59.000
That didn't do a good thing.

00:02:59.000 --> 00:03:03.340
So h2 is all of this stuff out
on this side of the sphere.

00:03:03.340 --> 00:03:06.090
And then h1 was the
lower bound on y.

00:03:06.090 --> 00:03:08.499
So it's going to be that
side right over there.

00:03:08.499 --> 00:03:10.540
And I could draw it probably
a little bit neater.

00:03:10.540 --> 00:03:13.123
But hopefully you get the point,
it would be all of that side.

00:03:13.123 --> 00:03:15.160
And then y can vary
between those two

00:03:15.160 --> 00:03:17.460
and essentially
fill up the region.

00:03:17.460 --> 00:03:20.060
Make the exact same
argument with the cylinder.

00:03:20.060 --> 00:03:21.940
The cylinder can be
defined the same way.

00:03:21.940 --> 00:03:24.260
So first of all, the
sphere is a type 1, type 2,

00:03:24.260 --> 00:03:25.950
and type 3 region.

00:03:25.950 --> 00:03:28.166
It meets all of the constraints.

00:03:28.166 --> 00:03:29.540
The cylinder, at
least the way it

00:03:29.540 --> 00:03:31.710
was oriented there,
actually any cylinder,

00:03:31.710 --> 00:03:33.760
will also be a type 3 region.

00:03:33.760 --> 00:03:35.980
Exact same argument.

00:03:35.980 --> 00:03:39.340
So let me draw my axes again.

00:03:42.810 --> 00:03:45.680
And here to make the cylinder
that we've been making,

00:03:45.680 --> 00:03:49.720
our domain could be a
rectangle in the x, z plane.

00:03:49.720 --> 00:03:54.460
So our domain can be
a rectangular region

00:03:54.460 --> 00:03:57.480
in the x, z plane,
just like that.

00:03:57.480 --> 00:03:59.960
And then the lower
bound on y could

00:03:59.960 --> 00:04:04.950
be that side of the cylinder.

00:04:04.950 --> 00:04:07.460
The side facing in that
direction right over there,

00:04:07.460 --> 00:04:08.882
that is the lower bound.

00:04:08.882 --> 00:04:10.340
And then the upper
bound on y could

00:04:10.340 --> 00:04:12.000
be the side facing
in that direction.

00:04:18.860 --> 00:04:21.740
So once again, this is
also a type 3 region.

00:04:21.740 --> 00:04:26.620
By the same logic, this
one right over here,

00:04:26.620 --> 00:04:30.580
this hourglass, can
be a type 3 region.

00:04:30.580 --> 00:04:37.960
The front side would be this
front side right over here,

00:04:37.960 --> 00:04:41.540
all of this, including
that stuff that I--

00:04:41.540 --> 00:04:46.640
And then the backside when
you think of it in terms of y,

00:04:46.640 --> 00:04:52.290
the backside would be that
part right over there.

00:04:52.290 --> 00:04:59.240
And once again, this could
also be a type 3 region.

00:05:04.441 --> 00:05:06.690
The boundary of the domain
would be this cross section

00:05:06.690 --> 00:05:08.610
right over here.

00:05:08.610 --> 00:05:10.170
So the boundary of
our domain could

00:05:10.170 --> 00:05:12.160
be that cross section
right over there.

00:05:12.160 --> 00:05:15.010
The lower bound on y
would be the back half,

00:05:15.010 --> 00:05:17.840
the back half of
this hour glass.

00:05:17.840 --> 00:05:20.790
And the upper bound on y
would be the front half.

00:05:23.544 --> 00:05:25.960
Let me do that magenta color
because I've been using that.

00:05:25.960 --> 00:05:28.170
Or actually that green color.

00:05:28.170 --> 00:05:32.390
So the upper bound on y
would be this right half

00:05:32.390 --> 00:05:33.570
right over here.

00:05:33.570 --> 00:05:35.580
So what would not be
a type three region?

00:05:35.580 --> 00:05:38.040
Well, if we just rotated
this around like that.

00:05:38.040 --> 00:05:41.410
So let me just draw something
that is not a type 3,

00:05:41.410 --> 00:05:43.930
just to show you that
this definition does not

00:05:43.930 --> 00:05:45.401
include everything.

00:05:45.401 --> 00:05:47.400
So something that would
not be a type 3 region--

00:05:47.400 --> 00:05:50.520
for the same argument as
we've seen before for a type

00:05:50.520 --> 00:05:54.000
2 and type 1 region
is an hourglass

00:05:54.000 --> 00:05:56.304
where it's along the
y-axis or at least

00:05:56.304 --> 00:05:57.262
it's oriented this way.

00:05:57.262 --> 00:06:00.270
It actually does not have to be
right centered on the y-axis.

00:06:00.270 --> 00:06:07.770
But an hour glass that looks
like this, now all of a sudden

00:06:07.770 --> 00:06:11.970
the y can't be just expressed as
being between two surfaces that

00:06:11.970 --> 00:06:13.630
are functions of x and z.

00:06:13.630 --> 00:06:16.020
You would have to break
this up in order to do that.

00:06:16.020 --> 00:06:18.040
But you could break
up this region

00:06:18.040 --> 00:06:19.830
into two types 3 regions.

00:06:19.830 --> 00:06:24.000
But the whole thing itself
is not a type 3 region.

00:06:24.000 --> 00:06:29.770
So not type 3.


Type II regions in three dimensions

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:02.655
Let's now think
about Type 2 regions.

00:00:02.655 --> 00:00:05.155
And you'll see that they're
kind of very similar definitions

00:00:05.155 --> 00:00:07.075
and it's really a
question of orientation.

00:00:11.220 --> 00:00:15.950
Type 2 region is a
region-- I'll call it

00:00:15.950 --> 00:00:21.400
R2-- that's the set of all
x, y's, and z's in three

00:00:21.400 --> 00:00:24.310
dimensions such
that-- and now instead

00:00:24.310 --> 00:00:29.917
of thinking of our domain
in terms of xy-coordinates,

00:00:29.917 --> 00:00:32.250
we're going to think of them
in terms of yz coordinates,

00:00:32.250 --> 00:00:36.920
such that our yz pairs are
a member of some domain.

00:00:36.920 --> 00:00:40.260
I'll call it D2 since we're
talking about Type 2 regions.

00:00:40.260 --> 00:00:45.740
And x is bounded below
by some function of yz.

00:00:45.740 --> 00:00:58.160
So I'll call it g1 of yz is
less than or equal to x, which

00:00:58.160 --> 00:01:06.200
is less than or equal to some
other function of yz, g2 of yz.

00:01:06.200 --> 00:01:08.890
And so you'll immediately see
a very similar way of thinking

00:01:08.890 --> 00:01:12.750
about it, but instead of
having z vary between two

00:01:12.750 --> 00:01:15.670
functions of x and y as
we had in a Type 1 region,

00:01:15.670 --> 00:01:20.412
we now have x varying between
two functions of y and z.

00:01:20.412 --> 00:01:22.620
Now let's think about some
of the shapes we explored.

00:01:22.620 --> 00:01:24.590
We saw that these
two right up here,

00:01:24.590 --> 00:01:27.550
this sphere and the cylinder,
were Type 1 regions,

00:01:27.550 --> 00:01:30.080
but this dumbbell, the way
that I oriented it here,

00:01:30.080 --> 00:01:31.810
was not a Type 1 region.

00:01:31.810 --> 00:01:34.220
Let's think about which of
these are Type 2 regions

00:01:34.220 --> 00:01:36.600
and what might not
be a Type 2 region.

00:01:36.600 --> 00:01:39.280
So first let's think
about the sphere.

00:01:39.280 --> 00:01:41.780
So I have my axes
right over here.

00:01:41.780 --> 00:01:43.350
Let me scroll down a little bit.

00:01:43.350 --> 00:01:45.500
So I got my axes.

00:01:45.500 --> 00:01:47.610
And so over here, our
domain, we could still

00:01:47.610 --> 00:01:49.470
construct our sphere,
but our domain

00:01:49.470 --> 00:01:52.250
is now going to be
in the yz plane.

00:01:52.250 --> 00:01:58.960
So yz plane is this
business right over here.

00:01:58.960 --> 00:02:01.000
So this will be our domain.

00:02:01.000 --> 00:02:03.740
I want to make it more
spherical than that.

00:02:03.740 --> 00:02:08.650
So our domain is this right
over here in the yz plane.

00:02:08.650 --> 00:02:11.666
That is our D2.

00:02:11.666 --> 00:02:13.040
And now the lower
bound, in order

00:02:13.040 --> 00:02:17.510
to construct the solid region
of the sphere or the globe

00:02:17.510 --> 00:02:20.310
or whatever you want to call
it, the lower bound on x

00:02:20.310 --> 00:02:23.850
would be kind of the back half
of the sphere, the one that's

00:02:23.850 --> 00:02:25.760
away from us right over here.

00:02:25.760 --> 00:02:27.710
So the lower bound--
so let me see

00:02:27.710 --> 00:02:32.670
how well I can wire
frame it at first.

00:02:32.670 --> 00:02:35.240
I can do a better job than that.

00:02:35.240 --> 00:02:37.780
So with my ghost do
something like that,

00:02:37.780 --> 00:02:40.490
then do something like that.

00:02:40.490 --> 00:02:43.120
But this is if the domain
right over here is transparent.

00:02:43.120 --> 00:02:46.100
But all we might catch-- we'll
just catch a glimpse of it

00:02:46.100 --> 00:02:47.960
in the back right over here.

00:02:47.960 --> 00:02:50.980
So it's the side of the sphere
that's facing away from us.

00:02:50.980 --> 00:02:53.420
And then the upper
bound on x would

00:02:53.420 --> 00:02:55.660
be the side of the
sphere that's facing us.

00:02:55.660 --> 00:02:58.340
So if I were to
do some contours,

00:02:58.340 --> 00:03:00.690
it might look
something like this

00:03:00.690 --> 00:03:03.940
and then look
something like this.

00:03:03.940 --> 00:03:10.770
And then we would color in this
entire region right over here.

00:03:10.770 --> 00:03:14.520
And x can take on all of the
values above that magenta

00:03:14.520 --> 00:03:16.486
surface and below
this green surface.

00:03:16.486 --> 00:03:17.860
And essentially,
it would fill up

00:03:17.860 --> 00:03:24.020
the globe for every yz
point in our domain.

00:03:24.020 --> 00:03:26.295
So a sphere is both a Type
1 and a Type 2 region.

00:03:26.295 --> 00:03:27.670
Actually, we're
going to see it's

00:03:27.670 --> 00:03:30.180
going to be a Type
3 region as well.

00:03:30.180 --> 00:03:32.110
What about this cylinder
right over here?

00:03:32.110 --> 00:03:34.360
Can we construct it or
think about it in a way

00:03:34.360 --> 00:03:36.330
that it would actually
be a Type 2 region?

00:03:36.330 --> 00:03:38.000
So let's try to do that.

00:03:38.000 --> 00:03:41.360
So let me paste it.

00:03:41.360 --> 00:03:44.560
So what if we had
a domain-- what

00:03:44.560 --> 00:03:46.550
if our domain was
something like this?

00:03:46.550 --> 00:03:48.380
It was a rectangle
in the yz plane.

00:03:51.740 --> 00:03:57.310
So this is our domain, a
rectangle in the yz plane.

00:03:57.310 --> 00:03:59.630
So that would be my D2.

00:03:59.630 --> 00:04:03.280
And what if the
lower bound was kind

00:04:03.280 --> 00:04:05.870
of the back side
of the cylinder?

00:04:05.870 --> 00:04:08.030
So the backside of
the cylinder, try

00:04:08.030 --> 00:04:11.930
to draw it as good as I can.

00:04:11.930 --> 00:04:13.710
And so if we just saw
the outside of it,

00:04:13.710 --> 00:04:15.130
it would look
something like that.

00:04:15.130 --> 00:04:17.060
It's facing away from
us so we barely see it.

00:04:17.060 --> 00:04:19.220
If we could see
through the cylinder

00:04:19.220 --> 00:04:21.850
or see through the little
flat cut of the cylinder,

00:04:21.850 --> 00:04:23.780
it would look
something like that.

00:04:23.780 --> 00:04:26.490
So that over there
would be our g1.

00:04:26.490 --> 00:04:31.020
And then our g2 would be the
front side of the cylinder.

00:04:31.020 --> 00:04:33.160
The g2 could be the front
side of the cylinder.

00:04:37.780 --> 00:04:40.590
So let me color it
in as best as I can.

00:04:40.590 --> 00:04:42.700
So the g2 would be the
front side of cylinder.

00:04:42.700 --> 00:04:46.260
And x can vary above
g1 and below g2,

00:04:46.260 --> 00:04:48.400
and it would fill up
this entire cylinder.

00:04:48.400 --> 00:04:51.170
So we see that this same
cylinder that we also saw

00:04:51.170 --> 00:04:54.810
was a Type 1 region can
also be a Type 2 region.

00:04:54.810 --> 00:04:57.430
Now what about this
hourglass thing that we saw

00:04:57.430 --> 00:04:59.010
could not be a Type 1 region?

00:04:59.010 --> 00:05:01.290
Can this be a Type 2 region?

00:05:01.290 --> 00:05:04.370
Well, let's think about it.

00:05:04.370 --> 00:05:05.770
I'll do it the same way.

00:05:05.770 --> 00:05:07.910
We can construct a domain.

00:05:07.910 --> 00:05:09.684
So maybe our domain,
it's in the y-- well,

00:05:09.684 --> 00:05:12.100
it should be in the yz plane
if we're talking about Type 2

00:05:12.100 --> 00:05:14.910
regions or if we want to think
of it as a Type 2 region.

00:05:14.910 --> 00:05:19.030
So our domain could be this kind
of flat hourglass shape that's

00:05:19.030 --> 00:05:21.670
in the yz plane.

00:05:21.670 --> 00:05:25.090
So our domain could be a
region that looks something

00:05:25.090 --> 00:05:28.170
like this in the yz plane.

00:05:28.170 --> 00:05:30.300
So this is kind
of flattened out.

00:05:30.300 --> 00:05:33.860
So this is our domain
right over there.

00:05:33.860 --> 00:05:36.110
And then the lower
bound on x, g1,

00:05:36.110 --> 00:05:39.230
could be a surface,
the function of y and z

00:05:39.230 --> 00:05:43.700
that is kind of the
backside of our hourglass.

00:05:43.700 --> 00:05:46.990
The backside of our
hourglass you could see.

00:05:46.990 --> 00:05:49.020
I'll try to show the
contours from the underside

00:05:49.020 --> 00:05:49.728
right over there.

00:05:49.728 --> 00:05:51.481
So that could be our g1.

00:05:51.481 --> 00:05:53.855
And then our g2 could be the
front side of the hourglass.

00:05:56.780 --> 00:06:01.860
So my best attempt to draw the
front side of the hourglass.

00:06:01.860 --> 00:06:03.200
And I could color it in.

00:06:03.200 --> 00:06:06.370
And so the way I somewhat
confusingly drew it just now,

00:06:06.370 --> 00:06:09.740
you see that this hourglass
oriented the way it is

00:06:09.740 --> 00:06:12.640
would actually be
a Type 2 region.

00:06:12.640 --> 00:06:17.510
Now if we were to rotate
it like this-- so let

00:06:17.510 --> 00:06:19.070
me draw it like this.

00:06:19.070 --> 00:06:21.050
Edit.

00:06:21.050 --> 00:06:24.760
So if we were to
make it like this so

00:06:24.760 --> 00:06:26.760
that the top of my
hourglass is facing

00:06:26.760 --> 00:06:29.650
us-- try my best to draw it.

00:06:29.650 --> 00:06:34.990
So let's say the top intersects
the x-axis right over there.

00:06:34.990 --> 00:06:38.370
This is the bottom of my
hourglass right over there.

00:06:38.370 --> 00:06:42.680
And then it bends in and then
comes back out like that.

00:06:46.110 --> 00:06:49.180
For the same reasons that
this was not a Type 1 region,

00:06:49.180 --> 00:06:51.360
this now would not
be a Type 2 region.

00:06:56.100 --> 00:06:59.450
For any zy, you can see there
could be multiple x points that

00:06:59.450 --> 00:07:01.320
are associated with
the different points

00:07:01.320 --> 00:07:02.350
of this hourglass.

00:07:02.350 --> 00:07:05.330
You can't just have a
simple lower and upper round

00:07:05.330 --> 00:07:06.530
functions right over here.

00:07:06.530 --> 00:07:13.450
So this right over here
is not a Type 2 region.

00:07:13.450 --> 00:07:15.810
You could show a
rationale or this

00:07:15.810 --> 00:07:18.450
is going to be a Type 1 region.

00:07:18.450 --> 00:07:21.680
You could create a region
over here in the xy plane

00:07:21.680 --> 00:07:24.380
and have an upper and lower
bound functions for z.

00:07:24.380 --> 00:07:28.680
So you could be Type 1, but
this will not be Type 2.


Type I regions in three dimensions

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:02.220
In this and the
next few videos, I

00:00:02.220 --> 00:00:05.880
hope to explore different types
of regions in three dimensions.

00:00:05.880 --> 00:00:07.580
And these will be
useful for thinking

00:00:07.580 --> 00:00:10.860
about how to evaluate different
double and triple integrals

00:00:10.860 --> 00:00:14.660
and also some interesting proofs
in multivariable calculus.

00:00:14.660 --> 00:00:17.340
So the first type of region,
and it's appropriately named,

00:00:17.340 --> 00:00:18.955
we will call a type 1 region.

00:00:23.499 --> 00:00:25.165
At first, I'll give
a formal definition.

00:00:25.165 --> 00:00:26.665
And hopefully, the
formal definition

00:00:26.665 --> 00:00:27.910
makes some intuitive sense.

00:00:27.910 --> 00:00:30.329
But then I'll draw a
couple of type 1 regions,

00:00:30.329 --> 00:00:31.870
and then I'll show
you what would not

00:00:31.870 --> 00:00:33.840
be a type 1 region
because sometimes that's

00:00:33.840 --> 00:00:35.600
the more important question.

00:00:35.600 --> 00:00:39.520
So type 1 region, maybe a type
one region R, is the set--

00:00:39.520 --> 00:00:41.420
and these little curly
brackets means set--

00:00:41.420 --> 00:00:44.500
is the set of all
x, y's, and z's.

00:00:44.500 --> 00:00:46.870
It's the set of all
points in three dimensions

00:00:46.870 --> 00:00:54.120
such that the x and y's
are part of some domain,

00:00:54.120 --> 00:00:57.270
are a member-- that's what
this little symbol represents--

00:00:57.270 --> 00:00:59.960
are a member of some domain.

00:00:59.960 --> 00:01:04.400
And z can-- essentially
varies between two functions

00:01:04.400 --> 00:01:05.890
of x and y.

00:01:05.890 --> 00:01:07.150
So let me write it over here.

00:01:07.150 --> 00:01:13.910
So f1 of x, y is kind
of the lower bound on z.

00:01:13.910 --> 00:01:18.320
So this is going to be less
than or equal to z, which

00:01:18.320 --> 00:01:21.950
is less than or equal to
another function of x and y,

00:01:21.950 --> 00:01:27.790
which is going to be less than
or equal to f2 of x and y.

00:01:27.790 --> 00:01:29.540
And let me close
the curly brackets

00:01:29.540 --> 00:01:31.200
to show that this was all a set.

00:01:31.200 --> 00:01:32.980
This is a set of
x, y's, and z's.

00:01:32.980 --> 00:01:35.430
And right here, we
are defining that set.

00:01:35.430 --> 00:01:37.940
So what would be a
reasonable type 1 region?

00:01:37.940 --> 00:01:40.980
Well, a very simple type
one region is a sphere.

00:01:40.980 --> 00:01:43.400
So let me draw a
sphere right over here.

00:01:43.400 --> 00:01:46.230
So in a sphere, where it
intersects the x, y plane--

00:01:46.230 --> 00:01:49.090
that's essentially this
domain D right over here.

00:01:49.090 --> 00:01:50.510
So I'll do it in blue.

00:01:50.510 --> 00:01:55.620
So let me draw my best
attempt at drawing that domain

00:01:55.620 --> 00:02:00.086
so that this is the domain D
right over here for a sphere.

00:02:00.086 --> 00:02:01.710
This is a sphere
centered at 0, but you

00:02:01.710 --> 00:02:05.010
could make the same argument
for a sphere anywhere else.

00:02:05.010 --> 00:02:07.650
So that is my domain.

00:02:07.650 --> 00:02:11.650
And then f1 of x, y, which
is a lower bound of z,

00:02:11.650 --> 00:02:13.430
will be the bottom
half of the sphere.

00:02:13.430 --> 00:02:16.160
So you really can't see
it well right over here,

00:02:16.160 --> 00:02:19.034
but it would be-- these
contours right over here

00:02:19.034 --> 00:02:20.200
would be on the bottom half.

00:02:20.200 --> 00:02:23.020
And I can even color in
this part right over here.

00:02:23.020 --> 00:02:27.770
The bottom surface of our
sphere would be f1 of x, y,

00:02:27.770 --> 00:02:29.470
and f2 of x, y as
you could imagine,

00:02:29.470 --> 00:02:32.470
will be the top half of the
sphere, the top hemisphere.

00:02:32.470 --> 00:02:40.637
So it'll look
something like that.

00:02:40.637 --> 00:02:42.220
This thing that I'm
drawing right over

00:02:42.220 --> 00:02:44.620
here is definitely
a type 1 region.

00:02:44.620 --> 00:02:46.470
As we'll see, this could
be a type 1 type 2,

00:02:46.470 --> 00:02:47.530
or a type 3 region.

00:02:47.530 --> 00:02:52.480
But it's definitely
a type one region.

00:02:52.480 --> 00:02:54.730
Another example of a type 1
region-- and actually this

00:02:54.730 --> 00:02:56.640
might even be more obvious.

00:02:56.640 --> 00:03:00.500
So let me draw some
axes again, and let

00:03:00.500 --> 00:03:04.440
me draw some type of a cylinder.

00:03:04.440 --> 00:03:07.620
Just to make it clear that
our domain, where the x, y

00:03:07.620 --> 00:03:10.420
plane does not have to
be inside of our region--

00:03:10.420 --> 00:03:15.190
let's imagine a cylinder
that is below-- well,

00:03:15.190 --> 00:03:18.450
actually I'll draw it above--
that is above the x, y plane.

00:03:18.450 --> 00:03:20.226
So this is the bottom
of the cylinder.

00:03:20.226 --> 00:03:21.100
It's right over here.

00:03:21.100 --> 00:03:22.641
And once again, it
doesn't have to be

00:03:22.641 --> 00:03:24.080
centered around the z-axis.

00:03:24.080 --> 00:03:26.400
But I'll do it that way
just for this video.

00:03:26.400 --> 00:03:28.780
Actually, I could draw it a
little bit better than that.

00:03:28.780 --> 00:03:32.940
So this is the bottom
surface of our cylinder,

00:03:32.940 --> 00:03:35.040
and then the top
surface of our cylinder

00:03:35.040 --> 00:03:37.560
might be right over here.

00:03:37.560 --> 00:03:41.220
And these things actually
don't even have to be flat.

00:03:41.220 --> 00:03:43.530
They could actually
be curvy in some way.

00:03:43.530 --> 00:03:45.650
And in this situation,
so in this cylinder--

00:03:45.650 --> 00:03:47.940
let me draw it a
little bit neater.

00:03:47.940 --> 00:03:53.270
In this cylinder
right over here,

00:03:53.270 --> 00:03:58.344
our domain are all of the values
that the x and y's can take on.

00:03:58.344 --> 00:04:00.010
So our domain is going
to be this region

00:04:00.010 --> 00:04:02.080
right over here
in the x, y plane.

00:04:02.080 --> 00:04:06.100
And then for each of that,
those x, y pairs, f1 of x, y

00:04:06.100 --> 00:04:08.540
defines the bottom
boundary of our region.

00:04:08.540 --> 00:04:11.960
So f1 of x, y is going to
be this right over here.

00:04:11.960 --> 00:04:14.890
So you give me any of these
x, y's in this domain D,

00:04:14.890 --> 00:04:18.060
and then you evaluate the
function at those points,

00:04:18.060 --> 00:04:21.620
and it will correspond to
this surface right over here.

00:04:21.620 --> 00:04:23.820
And then f2 of x, y,
once again, give me

00:04:23.820 --> 00:04:26.290
any one of those x, y
points in our domain,

00:04:26.290 --> 00:04:29.900
and you evaluate
f2 at those points,

00:04:29.900 --> 00:04:33.030
and it will give you
this surface up here.

00:04:33.030 --> 00:04:36.080
And we're saying
that z will take

00:04:36.080 --> 00:04:39.160
on all the values in between,
and so it is really this whole

00:04:39.160 --> 00:04:44.210
solid-- it's really
this entire solid area.

00:04:44.210 --> 00:04:45.880
Likewise, over
here, z could take

00:04:45.880 --> 00:04:48.150
on any value
between this magenta

00:04:48.150 --> 00:04:50.200
surface and this green surface.

00:04:50.200 --> 00:04:53.490
So it would essentially
fill up our entire volume

00:04:53.490 --> 00:04:56.860
so it would become
a solid region.

00:04:56.860 --> 00:05:01.030
Now, you might be wondering what
would not be a type 1 region?

00:05:01.030 --> 00:05:03.191
So let's think about that.

00:05:03.191 --> 00:05:04.690
So it would essentially
be something

00:05:04.690 --> 00:05:06.520
that we could not
define in this way,

00:05:06.520 --> 00:05:08.630
and I'll try my best to draw it.

00:05:08.630 --> 00:05:11.940
But you could
imagine a shape that

00:05:11.940 --> 00:05:13.670
does something funky like this.

00:05:13.670 --> 00:05:15.650
So there's like
one big-- I guess

00:05:15.650 --> 00:05:18.470
you could imagine a
sideways dumbbell.

00:05:18.470 --> 00:05:20.440
So a sideways
dumbbell-- and I'll

00:05:20.440 --> 00:05:23.570
maybe curve it out a little bit.

00:05:23.570 --> 00:05:27.780
So this is the kind of the top
of the dumbbell-- or an hour

00:05:27.780 --> 00:05:29.780
glass, I guess you could
say, or a dumbbell.

00:05:29.780 --> 00:05:32.650
It would look
something like that.

00:05:32.650 --> 00:05:35.650
So I'm trying my
best to draw it.

00:05:35.650 --> 00:05:37.190
It would look
something like that.

00:05:37.190 --> 00:05:42.670
And the reason why this is
not definable in this way-- it

00:05:42.670 --> 00:05:45.540
becomes obvious if you kind of
look at a cross section of it.

00:05:45.540 --> 00:05:49.280
There's no way to define
only two functions that's

00:05:49.280 --> 00:05:53.490
a lower bound and an
upper bound in terms of z.

00:05:53.490 --> 00:05:57.310
So even if you say, hey, maybe
my domain will be all of the x,

00:05:57.310 --> 00:05:59.470
y values that can
be taken on-- let

00:05:59.470 --> 00:06:01.610
me see how well I can draw this.

00:06:01.610 --> 00:06:03.750
So you say my x,
y values-- so let

00:06:03.750 --> 00:06:06.130
me try to draw this whole
thing a little bit better,

00:06:06.130 --> 00:06:08.240
a better attempt.

00:06:08.240 --> 00:06:11.130
So you might say, OK, for
something like a dumbbell--

00:06:11.130 --> 00:06:15.220
let me clear out
that part as well.

00:06:15.220 --> 00:06:20.866
For something like a dumbbell--
so let me erase that.

00:06:20.866 --> 00:06:22.490
So for something like
a dumbbell, maybe

00:06:22.490 --> 00:06:27.100
my domain is right over here.

00:06:27.100 --> 00:06:30.140
So these are all the x, y
values that you can take on.

00:06:30.140 --> 00:06:34.553
But in order to have a
dumbbell shape, for any one x,

00:06:34.553 --> 00:06:36.980
y, z is going to
take on-- there's not

00:06:36.980 --> 00:06:39.040
just an upper and a lower
bound, and z doesn't

00:06:39.040 --> 00:06:40.430
take on all values in between.

00:06:40.430 --> 00:06:43.110
Well, let me just draw
it a more clearly.

00:06:43.110 --> 00:06:46.010
So our dumbbell-- maybe
it's centered on the z-axis.

00:06:46.010 --> 00:06:47.610
This is the middle
of our dumbbell,

00:06:47.610 --> 00:06:49.800
and then it comes out like that.

00:06:49.800 --> 00:06:54.350
And then up here, the z-axis--
so it looks like that.

00:06:54.350 --> 00:06:56.320
And then it goes
below the x, y plane,

00:06:56.320 --> 00:06:58.360
and it does kind
of a similar thing.

00:06:58.360 --> 00:07:03.220
It goes below the x, y plane
and looks something like that.

00:07:03.220 --> 00:07:05.690
So notice, for any
given x, y, what

00:07:05.690 --> 00:07:08.810
would be-- if you attempted
to make it a type 1 region,

00:07:08.810 --> 00:07:14.410
you would say, well, maybe
this is the top surface.

00:07:14.410 --> 00:07:20.030
And maybe you would say down
here is the bottom surface.

00:07:20.030 --> 00:07:23.590
But notice, z can't take
on every value in between.

00:07:23.590 --> 00:07:25.025
You kind of have
to break this up

00:07:25.025 --> 00:07:27.150
if you wanted to be able
to do something like that.

00:07:27.150 --> 00:07:29.524
You would have to break this
up into two separate regions

00:07:29.524 --> 00:07:32.540
where this would be
the bottom region,

00:07:32.540 --> 00:07:37.410
and then this right over here
would be another top region.

00:07:37.410 --> 00:07:41.100
So this dumbbell shape itself
is not a type 1 region,

00:07:41.100 --> 00:07:42.520
but you could
actually break it up

00:07:42.520 --> 00:07:45.160
into two, separate
type 1 regions.

00:07:45.160 --> 00:07:46.410
So, hopefully, that helps out.

00:07:46.410 --> 00:07:48.118
And actually, another
way to think about,

00:07:48.118 --> 00:07:49.910
this might be an
easier way-- if we

00:07:49.910 --> 00:07:51.819
were to look at it
from this direction,

00:07:51.819 --> 00:07:53.527
and if we were to just
think about the z,

00:07:53.527 --> 00:07:55.990
y, if we were just
thinking about what's

00:07:55.990 --> 00:07:58.970
happening on the z, y
plane-- so that's a z,

00:07:58.970 --> 00:08:03.680
and this is y right over
here-- our dumbbell shape would

00:08:03.680 --> 00:08:09.950
look something like
this, my best attempt

00:08:09.950 --> 00:08:11.920
to draw our dumbbell shape.

00:08:11.920 --> 00:08:15.220
And so if you get a given
x or y, maybe x is even 0,

00:08:15.220 --> 00:08:19.960
and you're sitting right here
on the y-axis, notice z is not,

00:08:19.960 --> 00:08:23.370
even up here, cannot be
a function of just y.

00:08:23.370 --> 00:08:26.696
On this top part, there's
two possible z-values

00:08:26.696 --> 00:08:28.445
that we need to take
on for that given y--

00:08:28.445 --> 00:08:31.990
two possible z-values
for that given y.

00:08:31.990 --> 00:08:34.130
So you can't define
it simply in terms

00:08:34.130 --> 00:08:38.469
of just one lower bound function
and one upper bound function.


Probability and the Monty Hall problem

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.850
Let's now tackle a
classic thought experiment

00:00:02.850 --> 00:00:06.217
in probability, called
the Monte Hall problem.

00:00:06.217 --> 00:00:07.800
And it's called the
Monty Hall problem

00:00:07.800 --> 00:00:11.560
because Monty Hall was the game
show host in Let's Make a Deal,

00:00:11.560 --> 00:00:14.400
where they would set up
a situation very similar

00:00:14.400 --> 00:00:17.260
to the Monte Hall problem
that we're about to say.

00:00:17.260 --> 00:00:19.080
So let's say that
on the show, you're

00:00:19.080 --> 00:00:20.330
presented with three curtains.

00:00:20.330 --> 00:00:22.830
So you're the contestant, this
little chef-looking character

00:00:22.830 --> 00:00:23.610
right over there.

00:00:23.610 --> 00:00:25.526
You're presented with
three curtains-- curtain

00:00:25.526 --> 00:00:29.340
number one, curtain number
two, and curtain number three.

00:00:29.340 --> 00:00:31.950
And you're told that behind
one of these three curtains,

00:00:31.950 --> 00:00:34.730
there's a fabulous prize,
something that you really want,

00:00:34.730 --> 00:00:38.270
a car, or a vacation, or some
type of large amount of cash.

00:00:38.270 --> 00:00:40.600
And then behind the
other two, and we

00:00:40.600 --> 00:00:42.800
don't know which they
are, there is something

00:00:42.800 --> 00:00:43.770
that you do not want.

00:00:43.770 --> 00:00:46.850
A new pet goat or an
ostrich or something

00:00:46.850 --> 00:00:48.410
like that, or a
beach ball, something

00:00:48.410 --> 00:00:50.510
that is not as good
as the cash prize.

00:00:50.510 --> 00:00:53.650
And so your goal is to try
to find the cash prize.

00:00:53.650 --> 00:00:55.670
And they say guess
which one, or which one

00:00:55.670 --> 00:00:57.030
would you like to select?

00:00:57.030 --> 00:00:59.750
And so let's say that you
select door number one,

00:00:59.750 --> 00:01:01.280
or curtain number one.

00:01:01.280 --> 00:01:03.190
Then the Monte
Hall and Let's Make

00:01:03.190 --> 00:01:05.960
a Deal crew will make it a
little bit more interesting.

00:01:05.960 --> 00:01:08.320
They want to show you
whether or not you won.

00:01:08.320 --> 00:01:11.180
They'll then show you one
of the other two doors,

00:01:11.180 --> 00:01:12.770
or one of the
other two curtains.

00:01:12.770 --> 00:01:14.853
And they'll show you one
of the other two curtains

00:01:14.853 --> 00:01:16.156
that does not have the prize.

00:01:16.156 --> 00:01:17.530
And no matter
which one you pick,

00:01:17.530 --> 00:01:19.570
there'll always be at
least one other curtain

00:01:19.570 --> 00:01:20.900
that does not have the prize.

00:01:20.900 --> 00:01:22.400
There might be two,
if you picked right.

00:01:22.400 --> 00:01:24.524
But there will always be
at least one other curtain

00:01:24.524 --> 00:01:26.054
that does not have the prize.

00:01:26.054 --> 00:01:27.470
And then they will
show it to you.

00:01:27.470 --> 00:01:29.844
And so let's say that they
show you curtain number three.

00:01:29.844 --> 00:01:33.320
And so curtain number
three has the goat.

00:01:33.320 --> 00:01:35.820
And then they'll
ask you, do you want

00:01:35.820 --> 00:01:41.910
to switch to curtain number two?

00:01:41.910 --> 00:01:44.860
And the question here is,
does it make a difference?

00:01:44.860 --> 00:01:48.150
Are you better off holding
fast, sticking to your guns,

00:01:48.150 --> 00:01:49.850
staying with the original guess?

00:01:49.850 --> 00:01:53.560
Are you better off switching
to whatever curtain is left?

00:01:53.560 --> 00:01:55.020
Or does it not matter?

00:01:55.020 --> 00:01:56.770
It's just random
probability, and it's not

00:01:56.770 --> 00:01:59.660
going to make a difference
whether you switch or not.

00:01:59.660 --> 00:02:02.520
So that is the brain teaser.

00:02:02.520 --> 00:02:03.850
Pause the video now.

00:02:03.850 --> 00:02:05.550
I encourage you
to think about it.

00:02:05.550 --> 00:02:09.180
In the next video, we will
start to analyze the solution

00:02:09.180 --> 00:02:13.757
a little bit deeper, whether
it makes any difference at all.

00:02:13.757 --> 00:02:15.590
So now I've assumed
that you've unpaused it.

00:02:15.590 --> 00:02:16.881
You've thought deeply about it.

00:02:16.881 --> 00:02:18.590
Perhaps you have
an opinion on it.

00:02:18.590 --> 00:02:20.260
Now let's work it
through a little bit.

00:02:20.260 --> 00:02:21.634
And at any point,
I encourage you

00:02:21.634 --> 00:02:24.090
to pause it and kind of
extrapolate beyond what I've

00:02:24.090 --> 00:02:25.960
already talked about.

00:02:25.960 --> 00:02:27.640
So let's think
about the game show

00:02:27.640 --> 00:02:29.680
from the show's point of view.

00:02:29.680 --> 00:02:31.359
So the show knows
where there's the goat

00:02:31.359 --> 00:02:32.650
and where there isn't the goat.

00:02:32.650 --> 00:02:35.350
So let's door number
one, door number two,

00:02:35.350 --> 00:02:37.810
and door number three.

00:02:37.810 --> 00:02:42.390
So let's say that our
prize is right over here.

00:02:42.390 --> 00:02:45.590
So our prize is the car.

00:02:45.590 --> 00:02:50.105
Our prize is the car, and
that we have a goat over here,

00:02:50.105 --> 00:02:51.646
and over here we
also have-- maybe we

00:02:51.646 --> 00:02:53.477
have two goats in
this situation.

00:02:53.477 --> 00:02:55.310
So what are we going
to do as the game show?

00:02:55.310 --> 00:02:57.060
Remember, the contestants
don't know this.

00:02:57.060 --> 00:02:58.220
We know this.

00:02:58.220 --> 00:03:02.340
So the contestant picks door
number one right over here.

00:03:02.340 --> 00:03:04.266
Then we can't lift
door number two

00:03:04.266 --> 00:03:05.640
because there's
a car back there.

00:03:05.640 --> 00:03:07.440
We're going to lift
door number three,

00:03:07.440 --> 00:03:10.700
and we're going to
expose this goat.

00:03:10.700 --> 00:03:12.360
In which case, it
probably would be

00:03:12.360 --> 00:03:15.330
good for the person to switch.

00:03:15.330 --> 00:03:21.700
If the person picks door number
two, then we as the game show

00:03:21.700 --> 00:03:24.930
can show either door number
one or door number three,

00:03:24.930 --> 00:03:26.600
and then it actually
does not make sense

00:03:26.600 --> 00:03:28.780
for the person to switch.

00:03:28.780 --> 00:03:31.440
If they picked
door number three,

00:03:31.440 --> 00:03:33.690
then we have to
show door number one

00:03:33.690 --> 00:03:35.670
because we can't
pick door number two.

00:03:35.670 --> 00:03:38.380
And in that case, it
actually makes a lot

00:03:38.380 --> 00:03:41.410
of sense for the
person to switch.

00:03:41.410 --> 00:03:43.170
Now, with that out of
the way, let's think

00:03:43.170 --> 00:03:46.310
about the probabilities
given the two strategies.

00:03:46.310 --> 00:03:51.720
So if you don't
switch, or another way

00:03:51.720 --> 00:03:54.310
to think about this strategy is
you always stick to your guns.

00:03:54.310 --> 00:03:56.700
You always stick to
your first guess.

00:03:56.700 --> 00:04:02.380
Well, in that situation, what
is your probability of winning?

00:04:02.380 --> 00:04:03.730
Well, there's three doors.

00:04:03.730 --> 00:04:07.159
The prize is equally likely
to be behind any one of them.

00:04:07.159 --> 00:04:08.450
So there's three possibilities.

00:04:08.450 --> 00:04:10.340
One has the outcome
that you desire.

00:04:10.340 --> 00:04:14.210
The probability of winning will
be 1/3 if you don't switch.

00:04:14.210 --> 00:04:18.696
Likewise, your
probability of losing,

00:04:18.696 --> 00:04:20.529
well, there's two ways
that you can lose out

00:04:20.529 --> 00:04:21.487
of three possibilities.

00:04:21.487 --> 00:04:22.610
It's going to be 2/3.

00:04:22.610 --> 00:04:24.151
And these are the
only possibilities,

00:04:24.151 --> 00:04:25.920
and these add up to
one right over here.

00:04:25.920 --> 00:04:28.410
So don't switch, 1/3
chance of winning.

00:04:28.410 --> 00:04:30.770
Now let's think about
the switching situation.

00:04:30.770 --> 00:04:34.940
So let's say always--
when you always switch.

00:04:34.940 --> 00:04:39.640
Let's think about how
this might play out.

00:04:39.640 --> 00:04:43.761
What is your
probability of winning?

00:04:43.761 --> 00:04:45.260
And before we even
think about that,

00:04:45.260 --> 00:04:48.680
think about how you would
win if you always switch.

00:04:48.680 --> 00:04:52.409
So if you pick wrong
the first time,

00:04:52.409 --> 00:04:53.700
they're going to show you this.

00:04:53.700 --> 00:04:54.910
And so you should always switch.

00:04:54.910 --> 00:04:56.640
So if you pick door
number one, they're

00:04:56.640 --> 00:04:58.190
going to show you
door number three.

00:04:58.190 --> 00:04:59.500
You should switch.

00:04:59.500 --> 00:05:01.200
If you picked wrong
door number three,

00:05:01.200 --> 00:05:02.950
they're going to show
you door number one.

00:05:02.950 --> 00:05:04.440
You should switch.

00:05:04.440 --> 00:05:09.820
So if you picked wrong and
switch, you will always win.

00:05:09.820 --> 00:05:11.251
Let me write this down.

00:05:11.251 --> 00:05:13.750
And this insight actually came
from one of the middle school

00:05:13.750 --> 00:05:16.166
students in the summer camp
that Khan Academy was running.

00:05:16.166 --> 00:05:18.510
It's actually a fabulous
way to think about this.

00:05:18.510 --> 00:05:21.210
So if you pick
wrong, so step one,

00:05:21.210 --> 00:05:28.540
so initial pick is wrong,
so you pick one of the two

00:05:28.540 --> 00:05:36.110
wrong doors, and then step
two, you always switch,

00:05:36.110 --> 00:05:38.110
you will land on the car.

00:05:38.110 --> 00:05:40.050
Because if you picked
one of the wrong doors,

00:05:40.050 --> 00:05:42.434
they're going to have to
show the other wrong door.

00:05:42.434 --> 00:05:43.850
And so if you
switch, you're going

00:05:43.850 --> 00:05:45.562
to end up on the right answer.

00:05:45.562 --> 00:05:48.020
So what is the probability of
winning if you always switch?

00:05:48.020 --> 00:05:50.395
Well, it's going to be the
probability that you initially

00:05:50.395 --> 00:05:51.002
picked wrong.

00:05:51.002 --> 00:05:52.960
Well, what's the probability
that you initially

00:05:52.960 --> 00:05:53.910
picked wrong?

00:05:53.910 --> 00:05:56.090
Well, there's two
out of the three ways

00:05:56.090 --> 00:05:57.660
to initially pick wrong.

00:05:57.660 --> 00:06:00.467
So you actually have a
2/3 chance of winning.

00:06:00.467 --> 00:06:02.800
There's a 2/3 chance you're
going to pick wrong and then

00:06:02.800 --> 00:06:04.900
switch into the right one.

00:06:04.900 --> 00:06:06.450
Likewise, what's
your probability

00:06:06.450 --> 00:06:11.550
of losing, given that you're
always going to switch?

00:06:11.550 --> 00:06:18.310
Well, the way that you would
lose is you pick right,

00:06:18.310 --> 00:06:19.630
you picked correctly.

00:06:19.630 --> 00:06:21.870
In step two, they're
going to show

00:06:21.870 --> 00:06:25.126
one of the two empty
or non-prize doors.

00:06:25.126 --> 00:06:26.500
And then step
three, you're going

00:06:26.500 --> 00:06:33.517
to switch into the
other empty door.

00:06:33.517 --> 00:06:35.600
But either way, you're
definitely going to switch.

00:06:35.600 --> 00:06:38.270
So the only way to lose, if
you're always going to switch,

00:06:38.270 --> 00:06:40.582
is to pick the right
the first time.

00:06:40.582 --> 00:06:42.040
Well, what's the
probability of you

00:06:42.040 --> 00:06:43.700
picking right the first time?

00:06:43.700 --> 00:06:46.110
Well, that is only 1/3.

00:06:46.110 --> 00:06:48.240
So you see here, it's
sometimes counter-intuitive,

00:06:48.240 --> 00:06:50.795
but hopefully this makes
sense as to why it isn't.

00:06:50.795 --> 00:06:54.580
You have a 1/3 chance of winning
if you stick to your guns,

00:06:54.580 --> 00:06:58.490
and a 2/3 chance of winning
if you always switch.

00:06:58.490 --> 00:07:00.420
Another way to
think about it is,

00:07:00.420 --> 00:07:02.780
when you first make
your initial pick,

00:07:02.780 --> 00:07:06.960
there's a 1/3 chance
that it's there,

00:07:06.960 --> 00:07:10.920
and there's a 2/3
chance that it's

00:07:10.920 --> 00:07:12.740
in one of the other two doors.

00:07:12.740 --> 00:07:14.630
And they're going to
empty out one of them.

00:07:14.630 --> 00:07:16.088
So when you switch,
you essentially

00:07:16.088 --> 00:07:18.470
are capturing that
2/3 probability.

00:07:18.470 --> 00:07:20.260
And we see that right there.

00:07:20.260 --> 00:07:23.040
So hopefully you enjoyed that.


Why we got zero flux in divergence theorem example 1

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:02.700
In the last video we used
the divergence theorem

00:00:02.700 --> 00:00:04.530
to show that the
flux across this

00:00:04.530 --> 00:00:08.510
surface right now, which is
equal to the divergence of f

00:00:08.510 --> 00:00:11.660
along or summed up
throughout the entire region,

00:00:11.660 --> 00:00:12.804
it's equal to 0.

00:00:12.804 --> 00:00:15.220
And what I want to do in this
video is think a little bit.

00:00:15.220 --> 00:00:16.860
When you get an
answer like 0, you

00:00:16.860 --> 00:00:18.110
want to think about why is it?

00:00:18.110 --> 00:00:22.170
This is saying that there's
no net flux across the surface

00:00:22.170 --> 00:00:23.000
right over here.

00:00:23.000 --> 00:00:26.260
Or if you sum up all of the
divergences in this volume,

00:00:26.260 --> 00:00:28.050
you are getting 0.

00:00:28.050 --> 00:00:29.270
So why is that?

00:00:29.270 --> 00:00:30.880
Well, the simple way
to think about it

00:00:30.880 --> 00:00:33.410
is, when we took the divergence
of f, this vector field f

00:00:33.410 --> 00:00:35.790
is hard to visualize,
but the divergence of f

00:00:35.790 --> 00:00:37.620
is fairly easy to visualize.

00:00:37.620 --> 00:00:40.140
The divergence is
equal to 2 times x.

00:00:40.140 --> 00:00:41.990
So over here you're
going to get,

00:00:41.990 --> 00:00:44.710
as you go further and
further in this direction,

00:00:44.710 --> 00:00:46.600
as x becomes larger,
your divergence

00:00:46.600 --> 00:00:48.060
becomes more and more positive.

00:00:48.060 --> 00:00:51.950
So you have kind of a
divergence of 2 right over here.

00:00:51.950 --> 00:00:54.620
You have a divergence
of 1 along that line.

00:00:54.620 --> 00:00:56.620
And you have a divergence
of 0 right there.

00:00:56.620 --> 00:00:58.780
And that's also true,
obviously, as you go higher.

00:00:58.780 --> 00:01:00.238
Because you're just
changing the z.

00:01:00.238 --> 00:01:01.400
You're not changing the x.

00:01:01.400 --> 00:01:04.004
So all over here you
have positive divergence.

00:01:04.004 --> 00:01:05.670
Over there you have
positive divergence,

00:01:05.670 --> 00:01:07.060
and not just along that plane.

00:01:07.060 --> 00:01:09.530
But if you go in the
x direction as well,

00:01:09.530 --> 00:01:13.250
this whole region of space,
you have positive divergence,

00:01:13.250 --> 00:01:17.125
I guess you could say, in the
positive x side of our octant.

00:01:17.125 --> 00:01:19.250
But then as you go on that
side, on the other side,

00:01:19.250 --> 00:01:20.950
you have negative divergence.

00:01:20.950 --> 00:01:26.340
And this diagram is symmetric
with respect to the zy plane.

00:01:26.340 --> 00:01:28.850
And so those divergences
cancel them out.

00:01:28.850 --> 00:01:30.790
You would have had
a positive flux

00:01:30.790 --> 00:01:33.940
across the surface or a
positive value right over here.

00:01:33.940 --> 00:01:36.400
Instead of calculating
it for this region,

00:01:36.400 --> 00:01:38.600
we had calculated
for region that

00:01:38.600 --> 00:01:40.407
was just between x is 0 and 1.

00:01:40.407 --> 00:01:41.990
So let's just think
about that region.

00:01:41.990 --> 00:01:46.160
So that region that would have
been cut off right over here.

00:01:46.160 --> 00:01:49.020
That would have been cut
off right over there.

00:01:49.020 --> 00:01:50.770
And so the back--
I guess you could

00:01:50.770 --> 00:01:55.140
say the back wall of that,
the back wall of this,

00:01:55.140 --> 00:01:59.330
would have been the zy plane.

00:01:59.330 --> 00:02:02.770
Now if we care
about this volume--

00:02:02.770 --> 00:02:05.670
so we're essentially
eliminating the rest of it.

00:02:05.670 --> 00:02:08.310
So let me try to eliminate
it as best as I can,

00:02:08.310 --> 00:02:10.039
change the colors.

00:02:10.039 --> 00:02:13.800
So if I eliminate that part
of it over there and all

00:02:13.800 --> 00:02:16.260
maybe even what we
see, all of that-- I

00:02:16.260 --> 00:02:17.880
should have deleted that first.

00:02:17.880 --> 00:02:20.520
So if we eliminate
the back part of it

00:02:20.520 --> 00:02:25.420
and we're just dealing
with it when x is positive,

00:02:25.420 --> 00:02:28.360
then our entire solution
that we did in the last video

00:02:28.360 --> 00:02:30.560
would have been the
exact same, except now x

00:02:30.560 --> 00:02:33.230
is going to vary between--
instead of negative 1 and 1,

00:02:33.230 --> 00:02:35.370
it'll vary between 0 and 1.

00:02:35.370 --> 00:02:37.050
And so our bounds
of integration, x

00:02:37.050 --> 00:02:40.130
is going to go between 0 and 1.

00:02:40.130 --> 00:02:42.990
And then in that situation,
our final answer-- this part,

00:02:42.990 --> 00:02:44.610
this would be between 0 and 1.

00:02:44.610 --> 00:02:46.060
That would all be 0.

00:02:46.060 --> 00:02:49.980
And we would be left
with 3/2 minus 1/2.

00:02:49.980 --> 00:02:56.810
3/2 minus 1/2 is 1 minus 1/6,
which is just going to be 5/6.

00:02:56.810 --> 00:02:59.030
And so when you just think
about this part of it--

00:02:59.030 --> 00:03:01.880
this side of it, this one
that I've just drawn-- you

00:03:01.880 --> 00:03:04.090
had a positive flux of 5/6.

00:03:04.090 --> 00:03:07.560
On the other side you had
a negative flux of 5/6.

00:03:07.560 --> 00:03:08.910
And then they canceled out.

00:03:08.910 --> 00:03:13.330
One way to interpret
that is, if we thought

00:03:13.330 --> 00:03:16.360
about the entire surface, like
what we saw in the last video,

00:03:16.360 --> 00:03:18.896
then you had an aggregate
inward flux in the last video.

00:03:18.896 --> 00:03:20.020
That's why it was negative.

00:03:20.020 --> 00:03:22.790
And then it's completely
offset by an outward flux

00:03:22.790 --> 00:03:24.796
right over here, the positive.

00:03:24.796 --> 00:03:26.920
You had a negative divergence
in that other region,

00:03:26.920 --> 00:03:29.040
and you have a positive
divergence that completely

00:03:29.040 --> 00:03:33.130
offsets it in this
region right over here.


Divergence theorem example 1

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.620
Let's see if we might be able to
make some use of the divergence

00:00:03.620 --> 00:00:04.440
theorem.

00:00:04.440 --> 00:00:08.460
So I have this region, this
simple solid right over here.

00:00:08.460 --> 00:00:10.700
x can go between
negative 1 and 1.

00:00:10.700 --> 00:00:13.309
z, this kind of arch
part right over here,

00:00:13.309 --> 00:00:14.600
is going to be a function of x.

00:00:14.600 --> 00:00:15.766
That's the upper bound on z.

00:00:15.766 --> 00:00:17.880
The lower bound on z is just 0.

00:00:17.880 --> 00:00:19.655
And then y could go
anywhere between 0,

00:00:19.655 --> 00:00:22.920
and then it's bounded
here by this plane, where

00:00:22.920 --> 00:00:26.860
we can express y as a
function of z. y is 2 minus z

00:00:26.860 --> 00:00:29.390
along this plane
right over here.

00:00:29.390 --> 00:00:31.440
And we're given this
crazy vector field.

00:00:31.440 --> 00:00:34.950
It has natural logs
and tangents in it.

00:00:34.950 --> 00:00:37.340
And we're asked to evaluate
the surface integral,

00:00:37.340 --> 00:00:40.080
or actually, I should say
the flux of our vector field

00:00:40.080 --> 00:00:41.880
across the boundary
of this region,

00:00:41.880 --> 00:00:45.390
across the surface of this
region right over here.

00:00:45.390 --> 00:00:48.740
And surface integrals are
messy as is, especially

00:00:48.740 --> 00:00:51.952
when you have a crazy
vector field like this.

00:00:51.952 --> 00:00:53.910
But you could imagine
that there might be a way

00:00:53.910 --> 00:00:57.980
to simplify this, perhaps
using the divergence theorem.

00:00:57.980 --> 00:01:00.700
The divergence theorem
tells us that the flux

00:01:00.700 --> 00:01:05.239
across the boundary of
this simple solid region

00:01:05.239 --> 00:01:09.030
is going to be the same
thing as the triple integral

00:01:09.030 --> 00:01:10.810
over the volume of
it, or I'll just

00:01:10.810 --> 00:01:19.570
call it over the region, of
the divergence of F dv, where

00:01:19.570 --> 00:01:22.760
dv is some combination
of dx, dy, dz.

00:01:22.760 --> 00:01:25.070
The divergence times
each little cubic volume,

00:01:25.070 --> 00:01:30.060
infinitesimal cubic
volume, so times dv.

00:01:30.060 --> 00:01:32.890
So let's see if this
simplifies things a bit.

00:01:32.890 --> 00:01:36.040
So let's calculate the
divergence of F first.

00:01:36.040 --> 00:01:40.120
So the divergence
of F is going to be

00:01:40.120 --> 00:01:42.870
the partial of the x component,
or the partial of the-- you

00:01:42.870 --> 00:01:46.110
could say the i component or the
x component with respect to x.

00:01:46.110 --> 00:01:48.990
Well, the derivative of this
with respect to x is just x.

00:01:48.990 --> 00:01:51.290
The derivative of this
with respect to x, luckily,

00:01:51.290 --> 00:01:51.920
is just 0.

00:01:51.920 --> 00:01:53.812
This is a constant
in terms of x.

00:01:53.812 --> 00:01:55.270
Now let's go over
here, the partial

00:01:55.270 --> 00:01:56.880
of this with respect to y.

00:01:56.880 --> 00:02:00.170
The partial of this with
respect to y is just x.

00:02:00.170 --> 00:02:02.230
And then this is just a
constant in terms of y,

00:02:02.230 --> 00:02:03.688
so it's just going
to be 0 when you

00:02:03.688 --> 00:02:05.566
take the derivative
with respect to y.

00:02:05.566 --> 00:02:07.940
And then, finally, the partial
of this with respect to z,

00:02:07.940 --> 00:02:10.320
well, this is just a
constant in terms of z.

00:02:10.320 --> 00:02:12.050
Doesn't change when z changes.

00:02:12.050 --> 00:02:15.760
So the partial with respect to
z is just going to be 0 here.

00:02:15.760 --> 00:02:17.950
And so taking the divergence
really, really, really

00:02:17.950 --> 00:02:18.940
simplified things.

00:02:18.940 --> 00:02:22.700
The divergence of F
simplified down to 2x.

00:02:22.700 --> 00:02:24.610
And so now we can
restate the flux

00:02:24.610 --> 00:02:28.950
across the surface as a
triple integral of 2x.

00:02:28.950 --> 00:02:30.990
So let me just write 2x here.

00:02:30.990 --> 00:02:33.190
And let's think
about the ordering.

00:02:33.190 --> 00:02:39.420
So y can go between 0 and this
plane that is a function of z.

00:02:39.420 --> 00:02:40.600
So let's write that down.

00:02:40.600 --> 00:02:47.377
So y is bounded below by 0 and
above by this plane 2 minus z.

00:02:53.530 --> 00:02:57.370
z is bounded below
by 0 and above

00:02:57.370 --> 00:03:00.845
by-- you could call them these
parabolas of 1 minus x squared.

00:03:03.670 --> 00:03:07.350
And then x is bounded
below by negative 1

00:03:07.350 --> 00:03:09.370
and bounded above by 1.

00:03:09.370 --> 00:03:12.110
So negative 1 is less than
or equal to x is less than

00:03:12.110 --> 00:03:13.580
or equal to 1.

00:03:13.580 --> 00:03:16.010
And so this is probably a
good order of integration.

00:03:16.010 --> 00:03:18.710
We can integrate with
respect to y first,

00:03:18.710 --> 00:03:20.460
and then we'll get
a function of z.

00:03:20.460 --> 00:03:22.291
Then we can integrate
with respect to z,

00:03:22.291 --> 00:03:23.540
and we'll get a function of x.

00:03:23.540 --> 00:03:26.130
And then we can integrate
with respect to x.

00:03:26.130 --> 00:03:27.930
So let's do it in that order.

00:03:27.930 --> 00:03:31.523
So first we'll integrate with
respect to y, so we have dy.

00:03:34.460 --> 00:03:40.930
y is bounded below at 0 and
above by the plane 2 minus z.

00:03:40.930 --> 00:03:44.100
So this right over here is
a plane y is equal to 0.

00:03:44.100 --> 00:03:47.860
And this up over here is the
plane y is equal to 2 minus z.

00:03:50.500 --> 00:03:53.025
Then we can integrate
with respect to z.

00:03:56.910 --> 00:04:01.070
And z, once again,
is bounded below by 0

00:04:01.070 --> 00:04:05.570
and bounded above by these
parabolas, 1 minus x squared.

00:04:05.570 --> 00:04:08.800
And then, finally, we can
integrate with respect to x.

00:04:08.800 --> 00:04:11.750
And x is bounded
below by negative 1

00:04:11.750 --> 00:04:13.570
and bounded above by 1.

00:04:13.570 --> 00:04:15.880
So let's do some
integration here.

00:04:15.880 --> 00:04:18.440
So the first thing, when
we're integrating with respect

00:04:18.440 --> 00:04:21.540
to x-- sorry, when we're
integrating with respect to y,

00:04:21.540 --> 00:04:23.140
2x is just a constant.

00:04:23.140 --> 00:04:27.220
So this expression
right over here

00:04:27.220 --> 00:04:34.770
is just going to be 2x
times y, and then we're

00:04:34.770 --> 00:04:40.470
going to evaluate it
from 0 to 2 minus z.

00:04:40.470 --> 00:04:44.890
So it's going to be 2x times
2 minus z minus 2x times 0.

00:04:44.890 --> 00:04:47.050
Well, that second part's
just going to be 0.

00:04:47.050 --> 00:04:53.980
So this is going to
simplify as-- I'll

00:04:53.980 --> 00:04:59.180
write it this way--
2x times 2 minus z.

00:05:04.300 --> 00:05:06.230
And actually, I'll just
leave it like that.

00:05:06.230 --> 00:05:08.680
And then we're going to
integrate this with respect

00:05:08.680 --> 00:05:09.300
to z.

00:05:12.730 --> 00:05:16.170
And that's going to go from
0 to 1 minus x squared,

00:05:16.170 --> 00:05:17.450
and then we have our dz there.

00:05:17.450 --> 00:05:19.866
And then after that, we're
going to integrate with respect

00:05:19.866 --> 00:05:25.970
to x, negative 1 to 1 dx.

00:05:25.970 --> 00:05:29.887
So let's take the antiderivative
here with respect to z.

00:05:29.887 --> 00:05:31.720
This you really can
just view as a constant.

00:05:31.720 --> 00:05:33.580
We can actually even
bring it out front,

00:05:33.580 --> 00:05:34.890
but I'll leave it there.

00:05:34.890 --> 00:05:36.410
So this piece right
over here-- I'll

00:05:36.410 --> 00:05:39.310
do it in z's color--
this piece right

00:05:39.310 --> 00:05:43.270
over here, see, we can
leave the 2x out front.

00:05:43.270 --> 00:05:46.120
Actually, I'll leave the 2x
out front of the whole thing.

00:05:46.120 --> 00:05:49.970
It's going to be 2x times--
so the antiderivative of this

00:05:49.970 --> 00:05:52.650
with respect to z
is going to be 2z.

00:05:52.650 --> 00:06:00.740
Antiderivative of this is
negative z squared over 2,

00:06:00.740 --> 00:06:03.850
and we are going to
evaluate this from 0

00:06:03.850 --> 00:06:05.550
to 1 minus x squared.

00:06:10.250 --> 00:06:11.730
When we evaluate
them at 0, we're

00:06:11.730 --> 00:06:13.502
just going to get
0 right over here.

00:06:13.502 --> 00:06:14.960
And so we really
just have to worry

00:06:14.960 --> 00:06:17.737
about when z is equal
to 1 minus x squared.

00:06:17.737 --> 00:06:18.570
Did I do that right?

00:06:18.570 --> 00:06:19.820
Yep.

00:06:19.820 --> 00:06:23.780
2z, and then minus
z squared over 2.

00:06:23.780 --> 00:06:25.980
You take the derivative,
you get negative z.

00:06:25.980 --> 00:06:27.850
Take the derivative
here, you just get 2.

00:06:27.850 --> 00:06:28.790
So that's right.

00:06:28.790 --> 00:06:33.680
So this is going
to be equal to 2x--

00:06:33.680 --> 00:06:41.770
let me do that same color-- it's
going to be equal to 2x times--

00:06:41.770 --> 00:06:46.830
let me get this right, let me go
into that pink color-- 2x times

00:06:46.830 --> 00:06:47.332
2z.

00:06:47.332 --> 00:06:49.040
Well, z is going to
be 1 minus x squared,

00:06:49.040 --> 00:06:52.440
so it's going to be
2 minus 2x squared.

00:06:52.440 --> 00:06:54.010
That was just 2 times that.

00:06:54.010 --> 00:06:59.230
And then minus-- I'll just
write 1/2 times this quantity

00:06:59.230 --> 00:07:00.060
squared.

00:07:00.060 --> 00:07:05.510
So this quantity squared is
going to be 1 minus 2x squared

00:07:05.510 --> 00:07:07.450
plus x to the fourth.

00:07:07.450 --> 00:07:09.457
That's just some basic
algebra right over there.

00:07:09.457 --> 00:07:11.290
And then from that,
you're going to subtract

00:07:11.290 --> 00:07:13.599
this thing evaluated at 0,
which is just going to be 0.

00:07:13.599 --> 00:07:15.640
So [? y, you ?] just won't
even think about that.

00:07:15.640 --> 00:07:18.340
And now we need to
simplify this a little bit.

00:07:18.340 --> 00:07:20.700
And we are going to get,
if we simplify this,

00:07:20.700 --> 00:07:25.980
we get 2 minus 2x
squared minus 1/2,

00:07:25.980 --> 00:07:30.780
and then plus-- so this is
negative 1/2 times negative 2x

00:07:30.780 --> 00:07:31.280
squared.

00:07:31.280 --> 00:07:34.980
So it's going to be
positive x squared

00:07:34.980 --> 00:07:39.380
minus 1/2 x to the fourth.

00:07:39.380 --> 00:07:41.670
Now, let's see, can
we simplify this part?

00:07:41.670 --> 00:07:44.130
Let me just make sure we
know what we're doing here.

00:07:44.130 --> 00:07:46.140
So we have this 2x
right over there.

00:07:46.140 --> 00:07:47.890
I want to make sure I
got the signs right.

00:07:47.890 --> 00:07:49.260
Yep, looks like I did.

00:07:49.260 --> 00:07:50.490
And now let's look at this.

00:07:50.490 --> 00:07:52.240
So let's see, can I
simplify a little bit?

00:07:52.240 --> 00:07:56.230
I have 2 minus
1/2, which is 3/2.

00:07:56.230 --> 00:07:58.170
So I have 3/2.

00:07:58.170 --> 00:08:02.120
That's that term and that
term take into account.

00:08:02.120 --> 00:08:05.220
And then I have negative
2x squared plus x squared.

00:08:05.220 --> 00:08:08.650
So that's just going to
result in negative x squared,

00:08:08.650 --> 00:08:12.450
if I take that
term and that term.

00:08:12.450 --> 00:08:17.050
And then I have negative
1/2 x to the fourth,

00:08:17.050 --> 00:08:19.845
and I'm multiplying
this whole thing by 2x.

00:08:22.400 --> 00:08:30.530
And so that's going to give us--
we have, let's see, 2x times

00:08:30.530 --> 00:08:31.320
3/2.

00:08:31.320 --> 00:08:31.920
And I want to make sure.

00:08:31.920 --> 00:08:33.130
I'm doing this
slowly, so I don't

00:08:33.130 --> 00:08:34.254
make any careless mistakes.

00:08:34.254 --> 00:08:35.558
The 2's cancel out.

00:08:35.558 --> 00:08:39.710
You get 3x, and then
2x times negative

00:08:39.710 --> 00:08:43.250
x squared is negative
2x to the third.

00:08:43.250 --> 00:08:45.040
And then 2x times
this right over here.

00:08:45.040 --> 00:08:46.706
The 2 cancels out
with the negative 1/2,

00:08:46.706 --> 00:08:50.690
you have negative
x to the fifth.

00:08:50.690 --> 00:08:53.050
So all of this simplifies
to this right over here.

00:08:53.050 --> 00:08:56.330
So our whole thing simplifies
to an integral with respect

00:08:56.330 --> 00:08:59.090
to x. x will go
from negative 1 to 1

00:08:59.090 --> 00:09:05.760
of this business of 3x
minus 2x to the third minus

00:09:05.760 --> 00:09:11.740
x to the fifth, and
then we have dx.

00:09:11.740 --> 00:09:14.300
And now we just take the
antiderivative with respect

00:09:14.300 --> 00:09:21.670
to x, which is going to be 3/2
x squared minus-- let's see, x

00:09:21.670 --> 00:09:24.100
to the fourth power--
minus 1/2, because it's

00:09:24.100 --> 00:09:26.600
going to be 2/4,
x to the fourth.

00:09:26.600 --> 00:09:27.330
Is that right?

00:09:27.330 --> 00:09:29.740
Because if you multiply
it, you're going to 2.

00:09:29.740 --> 00:09:34.630
Yep, x to the third, and then
minus x to the sixth over 6.

00:09:34.630 --> 00:09:40.940
And it's going to go from 1 to
negative 1 or negative 1 to 1.

00:09:40.940 --> 00:09:42.714
So when you evaluate
it at 1-- I'll

00:09:42.714 --> 00:09:43.880
just write it out real fast.

00:09:43.880 --> 00:09:45.380
So first, when you
evaluate it at 1,

00:09:45.380 --> 00:09:50.800
you get 3/2 minus 1/2 minus 1/6.

00:09:50.800 --> 00:09:57.340
And then from that, we are
going to subtract 3/2 minus 1/2

00:09:57.340 --> 00:09:59.386
plus 1/6.

00:09:59.386 --> 00:10:00.760
Or actually, no,
they're actually

00:10:00.760 --> 00:10:02.430
all going to cancel out.

00:10:02.430 --> 00:10:03.570
Is that right?

00:10:03.570 --> 00:10:06.510
Are they all going
to cancel out?

00:10:06.510 --> 00:10:08.040
Yep, I think that's right.

00:10:08.040 --> 00:10:09.250
They all cancel out.

00:10:09.250 --> 00:10:14.680
So it's actually going to be
plus, or I should say minus 1/6

00:10:14.680 --> 00:10:16.216
right over here.

00:10:16.216 --> 00:10:17.590
And then all of
these cancel out.

00:10:17.590 --> 00:10:18.660
That cancels with that.

00:10:18.660 --> 00:10:20.730
That cancels with
that because we're

00:10:20.730 --> 00:10:22.280
subtracting the negative 1/2.

00:10:22.280 --> 00:10:23.935
And that cancels with that.

00:10:23.935 --> 00:10:27.670
And so we are
actually left with 0.

00:10:27.670 --> 00:10:30.410
So after doing all of that
work, this whole thing

00:10:30.410 --> 00:10:32.710
evaluates to 0,
which was actually

00:10:32.710 --> 00:10:35.570
kind of a neat simplification.

00:10:35.570 --> 00:10:38.010
So this whole thing
right over here

00:10:38.010 --> 00:10:43.806
evaluated, very conveniently,
evaluated to be equal to 0.