Surface integral example part 1: Parameterizing the unit sphere

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:04.050
What we will attempt to start
to do in this video is take

00:00:04.050 --> 00:00:08.330
the surface integral
of the function

00:00:08.330 --> 00:00:10.910
x squared over
our surface, where

00:00:10.910 --> 00:00:14.000
the surface in question,
the surface we're

00:00:14.000 --> 00:00:17.580
going to care about is
going to be the unit sphere.

00:00:17.580 --> 00:00:19.200
So it could be
defined by x squared

00:00:19.200 --> 00:00:22.917
plus y squared plus z
squared is equal to 1.

00:00:22.917 --> 00:00:25.500
And what I'm going to focus on
in this first video, because it

00:00:25.500 --> 00:00:27.670
will take us several
videos to do it,

00:00:27.670 --> 00:00:31.880
is just the parameterization of
this surface right over here.

00:00:31.880 --> 00:00:34.310
And as you'll see, this
is often the hardest part

00:00:34.310 --> 00:00:36.610
because it takes a little
bit of visualization.

00:00:36.610 --> 00:00:38.840
And then after that,
it's kind of mechanical,

00:00:38.840 --> 00:00:40.950
but it can be kind of
hairy at the same time.

00:00:40.950 --> 00:00:42.560
So it's worth going through.

00:00:42.560 --> 00:00:44.974
So first, let's think about
how we can parameterize--

00:00:44.974 --> 00:00:46.640
and I have trouble
even saying the word.

00:00:46.640 --> 00:00:49.210
How we can parameterize
this unit sphere

00:00:49.210 --> 00:00:52.240
as a function of two parameters.

00:00:52.240 --> 00:00:53.830
So let's think about it.

00:00:53.830 --> 00:00:55.570
Let's think about
it a little bit.

00:00:55.570 --> 00:00:59.240
So first, let's just think
about the unit sphere.

00:00:59.240 --> 00:01:02.100
I'm going to take a side
view of the unit sphere.

00:01:02.100 --> 00:01:04.970
So let's take the unit sphere.

00:01:04.970 --> 00:01:07.770
So this right over
here is our z-axis.

00:01:07.770 --> 00:01:09.240
That's our z-axis.

00:01:09.240 --> 00:01:11.390
And then over here,
I'm going to draw--

00:01:11.390 --> 00:01:14.230
this is going to be not
just the x or the y-axis.

00:01:14.230 --> 00:01:18.650
This is going to be entire
xy-plane viewed from the side.

00:01:18.650 --> 00:01:23.970
That is the xy-plane.

00:01:23.970 --> 00:01:28.110
Now, our sphere,
our unit sphere,

00:01:28.110 --> 00:01:31.150
might look something like this.

00:01:31.150 --> 00:01:35.030
The unit sphere itself is
not too hard to visualize.

00:01:35.030 --> 00:01:36.990
It might look
something like that.

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The radius-- let me
make it very clear.

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The radius at any point is 1.

00:01:41.730 --> 00:01:44.080
So this length right
over here is 1.

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That length right
over there is 1.

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And this is a sphere,
not just a circle.

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So I could even shade
it in a little bit,

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just to make it clear
that this thing has

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some dimensionality to it.

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So that's shading it in.

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It kind of makes it look a
little bit more spherical.

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Now, let's attempt
to parameterize this.

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And as a first step,
let's just think.

00:02:02.850 --> 00:02:05.192
If we didn't have to think
above and below the xy-plane,

00:02:05.192 --> 00:02:07.650
if we just thought about where
this unit sphere intersected

00:02:07.650 --> 00:02:10.432
the xy-plane, how we
could parameterize that.

00:02:10.432 --> 00:02:11.640
So let's just think about it.

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So where it intersects
the xy-plane.

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It intersects it there, and
there, and actually everywhere.

00:02:16.440 --> 00:02:18.082
So it intersects it
right over there.

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So let's just draw
the xy-plane and think

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about that intersection,
and then we

00:02:21.210 --> 00:02:22.710
could think about
what happens as we

00:02:22.710 --> 00:02:25.270
go above and below the xy-plane.

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So on the xy-plane, this little
region where we just shaded in.

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So let me draw.

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So now you could view
this as almost a top view.

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The z-axis is now going to be
pointing straight out at you,

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straight out of the screen.

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So that's x.

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So let me draw it.

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So that's x, and then this
right over there is y.

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So this thing that we
were viewing sideways,

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now we're viewing
it from the top.

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And so now our unit
sphere is going

00:02:53.760 --> 00:03:00.052
to look something like
this viewed from above.

00:03:00.052 --> 00:03:02.260
What I just drew, this dotted
circle right over here,

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this is going to be where
our unit sphere intersects--

00:03:05.530 --> 00:03:06.650
I labeled that y.

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That should be x.

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Don't want to
confuse you already.

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Let me clear that.

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So this is our x-axis.

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This is our x-axis.

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So this little
dotted blue circle,

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this is where our unit sphere
intersects the xy-plane.

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And so using this,
we can start to think

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about how to parameterize
at least our x-

00:03:27.220 --> 00:03:30.380
and y-values, our x- and
y-coordinates, as a function

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of a first parameter.

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So the first
parameter, we can think

00:03:33.500 --> 00:03:36.810
of something that is-- so this
is the z-axis popping straight

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out at us.

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So we're essentially,
if we're rotating around

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that z-axis viewed from above,
we could imagine an angle.

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I will call that angle
s, which is essentially

00:03:49.980 --> 00:03:53.560
saying how much we're
rotating from the x-axis

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towards the y-axis.

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You could think about
it in the xy-plane

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or in a plane that is
parallel to the xy-plane.

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Or you could say, going
around the z-axis.

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The z-axis popping
straight up at us.

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And the radius here is always 1.

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It's a unit sphere.

00:04:08.510 --> 00:04:10.430
So given this
parameter s, what would

00:04:10.430 --> 00:04:11.730
be your x- and y-coordinates?

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And now we're thinking
about it right

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if we're sitting
in the xy-plane.

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Well, the x-coordinate--
this goes back

00:04:17.149 --> 00:04:20.089
to the unit circle definition
of our trig functions.

00:04:20.089 --> 00:04:23.050
The x-coordinate is
going to be cosine of s.

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It would be the radius, which
is 1, times the cosine of s.

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And the y-coordinate would
be 1 times the sine of s.

00:04:30.520 --> 00:04:33.390
That's actually where we get our
definitions for cosine and sine

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from.

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So that's pretty
straightforward.

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And in this case, z is
obviously equal to 0.

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So if we wanted to add our
z-coordinate here, z is 0.

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We are sitting in the xy-plane.

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But now, let's think
about what happens

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if we go above and
below the xy-plane.

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Remember, this is
in any plane that

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is parallel to the xy-plane.

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This is saying how we are
rotated around the z-axis.

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Now, let's think about if
we go above and below it.

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And to figure out how
far above or below it,

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I'm going to introduce
another parameter.

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And this new parameter I'm
going to introduce is t.

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t is how much we've rotated
above and below the xy-plane.

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Now, what's
interesting about that

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is if we take any other cross
section that is parallel

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to the xy-plane now, we are
going to have a smaller radius.

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Let me make that clear.

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So if we're right over there,
now where this plane intersects

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our unit sphere, the
radius is smaller.

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The radius is smaller
than it was before.

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Well, what would
be this new radius?

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Well, a little bit
of trigonometry.

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It's the same as this
length right over here,

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which is going to
be cosine of t.

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So the radius is going
to be cosine of t.

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And it still works
over here because if t

00:05:58.270 --> 00:06:01.040
goes all the way to
0, cosine of 0 is 1.

00:06:01.040 --> 00:06:02.550
And then that works
right over there

00:06:02.550 --> 00:06:04.520
when we're in the xy-plane.

00:06:04.520 --> 00:06:08.150
So the radius over here
is going to be-- so

00:06:08.150 --> 00:06:10.920
that right over
there is cosine of 0.

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So this is when t is equal to 0.

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And we haven't rotated
above or below the xy-plane.

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But if we have rotated
above the xy-plane,

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the radius has changed.

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It is now cosine of t.

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And now we can use that to truly
parameterize x and y anywhere.

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So now, let's look at
this cross section.

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So we're not necessarily
in the xy-plane,

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we're in something that's
parallel to the xy-plane.

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And so if we're up here,
now all of a sudden,

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the cross section-- if
we view it from above,

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might look something like this.

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It might look
something like this.

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We're viewing it from
above, this cross section

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right over here.

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Our radius right over
here is cosine of t.

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And so given that--
I guess altitude

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that we're at, what would now
be the parameterization using s

00:07:05.190 --> 00:07:06.440
of x and y?

00:07:06.440 --> 00:07:10.250
Well, it's the exact same
thing, except now our radius

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isn't a fixed 1.

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It is now a function of t.

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So we're now a
little bit higher.

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So now, our
x-coordinate is going

00:07:19.450 --> 00:07:24.550
to be our radius,
which is cosine of t.

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That's just our radius.

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Times cosine of s.

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Times cosine of s, how
much we've angled around.

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And in this case, s has gone
all the way around here.

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S has gone all the
way around there,

00:07:41.190 --> 00:07:44.940
so it's going to be cosine
of t times cosine of s.

00:07:44.940 --> 00:07:46.780
And then, our
y-coordinate is going

00:07:46.780 --> 00:07:50.860
to be our radius, which is
cosine of t times sine of s.

00:07:50.860 --> 00:07:53.740
Same exact logic here, except
now we have a different radius.

00:07:53.740 --> 00:07:55.620
Our radius is no longer 1.

00:07:55.620 --> 00:07:58.570
Times sine of s.

00:07:58.570 --> 00:08:01.560
I'm running out of space,
let me scroll to the right

00:08:01.560 --> 00:08:03.049
a little bit.

00:08:03.049 --> 00:08:04.590
And I know this
looks very confusing,

00:08:04.590 --> 00:08:06.900
but you just have to say,
at any given level we are,

00:08:06.900 --> 00:08:08.440
we're parallel to the x-axis.

00:08:08.440 --> 00:08:10.540
We're kind of tracing
out another circle where

00:08:10.540 --> 00:08:12.960
another plane intersects
our unit sphere.

00:08:12.960 --> 00:08:15.600
We're now then
rotating around with s.

00:08:15.600 --> 00:08:17.080
And so our radius will change.

00:08:17.080 --> 00:08:19.440
It's a function of how
much above and below

00:08:19.440 --> 00:08:22.692
we've rotated-- how much
above or below the xy-plane

00:08:22.692 --> 00:08:23.275
we've rotated.

00:08:23.275 --> 00:08:25.290
So this is just our
radius instead of 1.

00:08:25.290 --> 00:08:28.740
And then, s is how much we've
rotated around the z-axis.

00:08:28.740 --> 00:08:30.404
Same there for the y-coordinate.

00:08:30.404 --> 00:08:32.570
And then the z-coordinate
is pretty straightforward.

00:08:32.570 --> 00:08:34.817
It's going to be
completely a function of t.

00:08:34.817 --> 00:08:37.150
It's not dependent on how
much we've rotated around here

00:08:37.150 --> 00:08:39.480
at any given altitude.

00:08:39.480 --> 00:08:41.570
It is what our
altitude actually is.

00:08:41.570 --> 00:08:44.150
Now, we can go straight to
this diagram right over here.

00:08:44.150 --> 00:08:48.750
Our z-coordinate is just
going to be the sine of t.

00:08:48.750 --> 00:08:53.290
So our z is equal to sine of t.

00:08:53.290 --> 00:08:54.490
So let me write that down.

00:08:54.490 --> 00:08:59.530
So the z is going to
be equal to sine of t.

00:08:59.530 --> 00:09:03.130
So now, every point
on this sphere

00:09:03.130 --> 00:09:07.389
can be described as a
function of t and s.

00:09:07.389 --> 00:09:09.180
Now, we have to think
about over what range

00:09:09.180 --> 00:09:10.570
will they be defined.

00:09:10.570 --> 00:09:13.770
Well, s is going to go-- at any
given level, you could think.

00:09:13.770 --> 00:09:16.304
For any given t, s is going
to go all the way around.

00:09:16.304 --> 00:09:17.470
We see that right over here.

00:09:17.470 --> 00:09:19.011
At any given level
viewed from above,

00:09:19.011 --> 00:09:20.870
s is going to go
all the way around.

00:09:20.870 --> 00:09:22.750
So thinking about
in radians, s is

00:09:22.750 --> 00:09:26.680
going to be between 0 and 2 pi.

00:09:26.680 --> 00:09:30.490
And t is essentially our
altitude in the z-direction.

00:09:30.490 --> 00:09:32.160
So t can go all
the way down here,

00:09:32.160 --> 00:09:34.410
which would be
negative pi over 2.

00:09:34.410 --> 00:09:37.950
So t can be between
negative pi over 2.

00:09:37.950 --> 00:09:40.610
And it can go all the
way up to pi over 2.

00:09:40.610 --> 00:09:42.860
It doesn't need to go all
the way back down again.

00:09:42.860 --> 00:09:47.290
And so it goes all the way
back-- it goes only up to pi

00:09:47.290 --> 00:09:48.040
over 2.

00:09:48.040 --> 00:09:50.502
And then we have our
parameterization.

00:09:50.502 --> 00:09:52.210
Let me write this down
in a form that you

00:09:52.210 --> 00:09:53.660
might recognize even more.

00:09:53.660 --> 00:09:58.250
If we wanted to write our
surface as a position vector

00:09:58.250 --> 00:10:01.110
function, we could
write it like this.

00:10:01.110 --> 00:10:06.250
We could write it r is
a function of s and t,

00:10:06.250 --> 00:10:09.330
and it is equal to
our x-component.

00:10:09.330 --> 00:10:19.250
Our i-component is going to
be cosine of t cosine of s i.

00:10:19.250 --> 00:10:28.820
And then plus our y-component
is cosine of t sine of s

00:10:28.820 --> 00:10:32.190
plus our z-component,
which is the sine-- which

00:10:32.190 --> 00:10:35.850
is just-- oh, I
forgot our j vector.

00:10:35.850 --> 00:10:46.670
j plus the z-component, which
is just sine of t sine k.

00:10:46.670 --> 00:10:47.660
And we're done.

00:10:47.660 --> 00:10:51.040
And these are the ranges that
those parameters will take on.

00:10:51.040 --> 00:10:52.290
So that's just the first step.

00:10:52.290 --> 00:10:54.880
We've parameterized
this surface.

00:10:54.880 --> 00:10:56.630
Now we're going to
have to actually set up

00:10:56.630 --> 00:10:57.480
the surface integral.

00:10:57.480 --> 00:10:58.900
It's going to involve
a little bit of taking

00:10:58.900 --> 00:11:00.800
a cross product,
which can get hairy,

00:11:00.800 --> 00:11:04.007
and then we can actually
evaluate the integral itself.


Conceptual clarification for 2D divergence theorem

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:04.929
Let's revisit the line integral F.n ds

00:00:04.929 --> 00:00:07.792
right over here because I want to make sure we have the proper conception

00:00:07.792 --> 00:00:10.090
and I was little "loosey goosey" with it in the last video

00:00:10.090 --> 00:00:13.775
and in this video I want to get a little bit more exacting and actually use units

00:00:13.775 --> 00:00:16.557
so that we really understand what's going on here

00:00:16.557 --> 00:00:18.654
So I've drawn our path "C"

00:00:18.654 --> 00:00:22.119
and we're traversing it in the positive counterclockwise direction

00:00:22.119 --> 00:00:24.939
and then I've taken a few sample points for F

00:00:24.939 --> 00:00:26.559
at any point in the x-y plane

00:00:26.559 --> 00:00:29.113
that associates a 2-dimensional vector

00:00:29.113 --> 00:00:31.110
maybe at that point the 2-dimensional vector looks like that,

00:00:31.110 --> 00:00:33.822
maybe at that point the 2-dimensional vector looks like that

00:00:33.822 --> 00:00:41.992
and then n is of course the unit normal vector at any point on our curve

00:00:41.992 --> 00:00:47.409
the outward pointing unit vector at any point on our curve

00:00:47.409 --> 00:00:52.119
Now in the last video, I talked about F as being some type of a velocity function

00:00:52.119 --> 00:00:54.574
that at any point it gives you the velocity of the particles there

00:00:54.574 --> 00:00:56.518
and that wasn't exactly right

00:00:56.518 --> 00:00:58.289
in order to really understand what's happening here

00:00:58.289 --> 00:01:02.339
in order to really conceptualize this as kind of flux through the boundary

00:01:02.339 --> 00:01:05.984
the rate of mass exiting this boundary here

00:01:05.984 --> 00:01:08.953
we actually have to introduce a density aspect to F

00:01:08.953 --> 00:01:15.219
So right over here, I've rewritten F, and I've rewritten it as of a product of a scalar function and a vector function

00:01:15.219 --> 00:01:17.986
so the scalar part right over here Rho of x,y

00:01:17.986 --> 00:01:22.736
Rho is a Greek letter often used to represent density of some kind

00:01:22.736 --> 00:01:24.604
in this case its mass density

00:01:24.604 --> 00:01:30.590
so at any given x,y point this tells us what the mass density is

00:01:30.590 --> 00:01:34.680
mass density will be some mass in a 2-dimensional world

00:01:34.680 --> 00:01:36.754
so it's mass per area

00:01:36.754 --> 00:01:39.212
and if we want particular units for our example-

00:01:39.212 --> 00:01:41.718
once again, this isn't the only way that this can be conceived of

00:01:41.718 --> 00:01:45.536
there's other applications, but this is the easiest way for my brain to process it

00:01:45.536 --> 00:01:51.907
we can imagine this as kilogram per square meter

00:01:51.907 --> 00:01:53.857
and this right over here is the velocity vector

00:01:53.857 --> 00:01:57.399
it tells us what is the velocity of the particles of that point

00:01:57.399 --> 00:02:00.726
so this is kind of saying, "How much particles do you have at a kind of a point?

00:02:00.726 --> 00:02:05.615
How dense are they?" and this is "How fast are they going and in what direction?"

00:02:05.615 --> 00:02:09.864
and this whole thing is a vector, it's a velocity vector

00:02:09.864 --> 00:02:11.276
but the components right over here

00:02:11.276 --> 00:02:15.290
M of x,y is just a number and you multiply that times a vector

00:02:15.290 --> 00:02:19.231
so M of x,y right over here is going to be a scalar function

00:02:19.231 --> 00:02:21.074
when you multiply by i it becomes a vector

00:02:21.074 --> 00:02:23.460
that's going to give you a speed

00:02:23.460 --> 00:02:26.879
and then N of x,y is also going to give you a speed

00:02:26.879 --> 00:02:29.330
and it tells you a speed in a j direction

00:02:29.330 --> 00:02:32.628
so it becomes a vector and a speed in the i direction becomes a vector as well

00:02:32.628 --> 00:02:39.646
but these speeds, the units of speed (let me write this over here)

00:02:39.646 --> 00:02:41.887
so now we're talking about in particular M of x,y and N of x,y

00:02:41.887 --> 00:02:49.877
that would be in units of distance per time and so maybe for this example we'll say

00:02:49.877 --> 00:02:52.285
the units are meters per second

00:02:52.285 --> 00:02:55.442
So let's think about the units will be for this function

00:02:55.442 --> 00:03:01.347
if we distribute the Rho, because really at any given x,y, it really is just a number

00:03:01.347 --> 00:03:08.253
so if we do that, we're going to get F- I'm not going to keep writing F of x,y

00:03:08.253 --> 00:03:12.692
we'll just understand that F, Rho, M and N are functions of x,y

00:03:12.692 --> 00:03:22.469
F is going to be equal to Rho times M times the unit vector i

00:03:22.469 --> 00:03:31.267
plus Rho times N times the unit vector j

00:03:31.267 --> 00:03:36.143
now what are the units here? what's Rho times M- what units are we going to get there?

00:03:36.143 --> 00:03:38.803
and we're gonna get the same units when we do Rho times N

00:03:38.803 --> 00:03:42.801
we'll we're gonna have, if we pick these particular units, we're going to have

00:03:42.801 --> 00:03:51.482
kilograms per meter squared times meters per second

00:03:51.482 --> 00:03:53.780
so a little bit of dimensional analysis here

00:03:53.780 --> 00:03:57.634
this meter in the numerator will cancel out with one of the meters in the denominator

00:03:57.634 --> 00:04:00.566
and we are left with something kind of strange

00:04:00.566 --> 00:04:05.163
kilograms per meter second

00:04:05.163 --> 00:04:11.445
which is essentially what the- if you view this vector has a magnitude in some direction

00:04:11.445 --> 00:04:16.266
the magnitude component is going to have these units right over here

00:04:16.266 --> 00:04:19.999
and then we're going to take this and we're dotting it with N

00:04:19.999 --> 00:04:22.211
N just only gives us a direction

00:04:22.211 --> 00:04:26.610
it is a unit-less vector- it's only specifying a direction at any point in the curve

00:04:26.610 --> 00:04:34.978
so when I take a dot product with this, it's going to give us essentially what is the magnitude of F

00:04:34.978 --> 00:04:43.240
going in the direction of N. So this right over here, when you take the dot, it's essentially

00:04:43.240 --> 00:04:47.291
a part of the magnitude of F going in N's direction

00:04:47.291 --> 00:04:50.058
and it's going to have the same exact units as F

00:04:50.058 --> 00:04:56.323
so the units of this part, you're going to have kilograms per meter second

00:04:56.323 --> 00:04:58.648
and let me make this very clear-

00:04:58.648 --> 00:05:01.918
So let's say we're focusing on this point over here

00:05:01.918 --> 00:05:08.803
F looks like that, its magnitude, the length of that vector is going to be in kilograms per meter second

00:05:08.803 --> 00:05:11.914
then we have a normal vector right over there

00:05:11.914 --> 00:05:13.579
and when you take the dot product, you're essentially saying

00:05:13.579 --> 00:05:16.953
"What's the magnitude that's going in the normal direction?"

00:05:16.953 --> 00:05:21.317
so essentially, what's the magnitude of that vector right over there

00:05:21.317 --> 00:05:23.568
it's going to be in kilograms per meter second

00:05:23.568 --> 00:05:25.752
and we're multiplying it times ds

00:05:25.752 --> 00:05:30.717
we're multiplying it times this infinitesimally small segment of the curve

00:05:30.717 --> 00:05:32.584
we're going to multiply that times ds

00:05:32.584 --> 00:05:35.769
well, what are the units of ds? it's going to be unit of length

00:05:35.769 --> 00:05:37.464
we'll just go with meters

00:05:37.464 --> 00:05:39.650
so this right over here is going to be meters

00:05:39.650 --> 00:05:44.830
there's this whole integral, you're going to have kilogram per meter second times meters

00:05:44.830 --> 00:05:54.994
so if you have kilograms per meter second and you were to multiply that times meters, what do you get?

00:05:54.994 --> 00:05:59.191
well this meters is going to cancel out that meters and then you get something that kinda starts to make sense

00:05:59.191 --> 00:06:01.955
you have kilogram per second

00:06:01.955 --> 00:06:05.308
and so this hopefully this makes it clear what's going on here

00:06:05.308 --> 00:06:09.602
this is telling us how much mass is crossing that little ds

00:06:09.602 --> 00:06:12.439
that little section of the curve per second

00:06:12.439 --> 00:06:15.022
and if you were to add up- and that's what integrals are all about,

00:06:15.022 --> 00:06:19.086
adding up an infinite number of these infinitesimally small ds's

00:06:19.086 --> 00:06:21.179
if you add all of that up

00:06:21.179 --> 00:06:24.854
you're going to get- the value of this entire integral is going to be in kilograms

00:06:24.854 --> 00:06:27.945
and kilograms per second, and it's essentially going to say,

00:06:27.945 --> 00:06:34.446
"How much mass is exiting this this curve at any given point? Or at any given time?"

00:06:34.446 --> 00:06:46.128
So this whole integral (let me rewrite it) of F.n ds tells us

00:06:46.128 --> 00:06:57.493
the mass exiting the curve per second

00:06:57.493 --> 00:06:59.402
and this should also be consistent in the last video

00:06:59.402 --> 00:07:04.772
we saw that this is equivalent to- and this is where we kinda view it as a 2-dimensional divergence theorem

00:07:04.772 --> 00:07:11.753
in the last video, we saw that this is equivalent to the double integral over the area of the divergence of f

00:07:11.753 --> 00:07:13.900
which is essentially just- well, I could write it 2 ways

00:07:13.900 --> 00:07:28.521
the divergence of f and this right over here, that's just the partial of the i component with respect to x

00:07:28.521 --> 00:07:32.213
(let me write it over here, I don't want to do this too fast and loose)

00:07:32.213 --> 00:07:40.699
so this right over here is going to be the partial of Rho M (let me write it like this), Rho M

00:07:40.699 --> 00:07:47.360
with respect to x plus the partial of the y component

00:07:47.360 --> 00:07:58.856
Rho N with respect to y times each little chunk of area

00:07:58.856 --> 00:08:02.843
Well, what are the units of this going to be right over here?

00:08:02.843 --> 00:08:08.638
we know what Rho M is- Rho M gives us kilogram per meter second

00:08:08.638 --> 00:08:12.555
but if we take the derivative with respect to meters again,

00:08:12.555 --> 00:08:18.950
the units for either of these characters

00:08:18.950 --> 00:08:25.170
are going to be kilograms per meter second per second

00:08:25.170 --> 00:08:29.356
because we're taking the derivative with respect - sorry per METER, we're taking the derivative

00:08:29.356 --> 00:08:33.692
with respect to another unit of distance so you're going to take per meter

00:08:33.692 --> 00:08:36.856
so you're going to have another meter right in the denominator

00:08:36.856 --> 00:08:38.637
that's going to be the units here

00:08:38.637 --> 00:08:44.649
and then you're multiplying it times an area

00:08:44.649 --> 00:08:49.994
so that would be meters squared, this right over here is square meters

00:08:49.994 --> 00:08:51.941
they cancel it out, and once again

00:08:51.941 --> 00:08:54.156
this whole part here that you're summing up

00:08:54.156 --> 00:08:59.167
gives us kilograms per second, so you're having a bunch of kilograms per second

00:08:59.167 --> 00:09:07.966
and you're just adding them up throughout the entire area right over here

00:09:07.966 --> 00:09:13.395
So hopefully this makes a little more sense, about how kinda how to conceptualize this vector function F

00:09:13.395 --> 00:09:18.830
if it confuses you, try your best to ignore it I guess

00:09:18.830 --> 00:09:26.988
for me at least, this helped me out having a stronger conception of what vector F could kind of represent


2D divergence theorem

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:02.267
- [Instructor] Now that
we know a little bit about

00:00:02.267 --> 00:00:03.694
how to construct a unit normal vector

00:00:03.694 --> 00:00:05.401
at any point in a curve,

00:00:05.401 --> 00:00:07.443
that's what we did in the last video,

00:00:07.443 --> 00:00:10.921
I want to start exploring
an interesting expression.

00:00:10.921 --> 00:00:15.921
So the expression is the line
integral around a closed loop,

00:00:16.598 --> 00:00:18.239
and we're gonna go in the positive,

00:00:18.239 --> 00:00:22.366
counterclockwise orientation,
of a vector field F,

00:00:22.366 --> 00:00:24.081
dotted with the unit normal vector

00:00:24.081 --> 00:00:25.553
at any point on that curve ds,

00:00:25.553 --> 00:00:27.970
I'll write ds in magenta, ds.

00:00:33.335 --> 00:00:36.209
So first let's conceptualize
what this is even saying,

00:00:36.209 --> 00:00:38.127
and then let's manipulate it a little bit

00:00:38.127 --> 00:00:40.647
to see if we can come up with
and interesting conclusion.

00:00:40.647 --> 00:00:41.981
We actually will manipulate it.

00:00:41.981 --> 00:00:43.527
we'll use Green's theorem,

00:00:43.527 --> 00:00:44.467
and we're gonna actually come up

00:00:44.467 --> 00:00:46.917
with a two-dimensional version
of the divergence theorem,

00:00:46.917 --> 00:00:48.435
which all sounds very complicated,

00:00:48.435 --> 00:00:50.879
but hopefully we can get a
little bit of an intuition for it

00:00:50.879 --> 00:00:54.512
as to why it actually a
little bit of common sense.

00:00:54.512 --> 00:00:55.900
So first, lets just think about this.

00:00:55.900 --> 00:00:59.471
So let me draw a coordinate plane here.

00:00:59.471 --> 00:01:01.554
So let me do it in white.

00:01:02.518 --> 00:01:06.113
So this right over
here, that's our y-axis.

00:01:06.113 --> 00:01:08.226
That over there is our x-axis.

00:01:08.226 --> 00:01:11.506
Let me draw ourselves my curve.

00:01:11.506 --> 00:01:14.355
So my curve might look something like...

00:01:14.355 --> 00:01:17.104
I'll do it in the blue color.

00:01:17.104 --> 00:01:20.058
So my curve might look
something like this, my contour,

00:01:20.058 --> 00:01:22.754
and its going in the positive,

00:01:22.754 --> 00:01:25.923
counterclockwise direction just like that.

00:01:25.923 --> 00:01:27.548
And now we have our vector field,

00:01:27.548 --> 00:01:29.200
and, just a reminder, we've
seen this multiple times.

00:01:29.200 --> 00:01:31.839
My vector field will associate a vector

00:01:31.839 --> 00:01:34.919
with any point on the x-y plane,

00:01:34.919 --> 00:01:39.042
and it can be defined as
some function of x and y,

00:01:39.042 --> 00:01:41.029
actually, I'll call that P,

00:01:41.029 --> 00:01:44.520
some function of x and y
times the i unit vector.

00:01:44.520 --> 00:01:47.083
So it says what the i component
of the vector field is

00:01:47.083 --> 00:01:48.879
for any x and y point,

00:01:48.879 --> 00:01:51.126
and then what the j component,

00:01:51.126 --> 00:01:53.626
or what we multiply the j component by

00:01:53.626 --> 00:01:57.319
or the vertical component
by for any x and y point.

00:01:57.319 --> 00:01:59.517
So some function of x and y times i

00:01:59.517 --> 00:02:02.987
plus some other scalar
function of x and y times j.

00:02:02.987 --> 00:02:04.566
And so, if you give me any point,

00:02:04.566 --> 00:02:06.572
there'll be an associated vector with it.

00:02:06.572 --> 00:02:08.808
Any point, there's an associated vector,

00:02:08.808 --> 00:02:10.978
depending on how we define this function.

00:02:10.978 --> 00:02:13.692
But this expression right over here,

00:02:13.692 --> 00:02:15.167
we're taking a line integral.

00:02:15.167 --> 00:02:19.132
We care specifically about
the points along this curve,

00:02:19.132 --> 00:02:21.535
along this contour right over here.

00:02:21.535 --> 00:02:24.069
And so let's think about what
this piece right over here

00:02:24.069 --> 00:02:27.300
is actually telling us before we sum up

00:02:27.300 --> 00:02:30.538
all of these infinitesimally small pieces.

00:02:30.538 --> 00:02:32.425
So if we just take F dot n,

00:02:32.425 --> 00:02:34.151
so let's just think about
a point on this curve.

00:02:34.151 --> 00:02:38.267
A point on this curve, maybe
this point right over here.

00:02:38.267 --> 00:02:40.276
So associated with that
point, there is a vector.

00:02:40.276 --> 00:02:43.165
That's what the vector field does.

00:02:43.165 --> 00:02:47.021
F might look something
like that right over there.

00:02:47.021 --> 00:02:49.437
So that might be F at that point,

00:02:49.437 --> 00:02:50.701
and then we're gonna dot it

00:02:50.701 --> 00:02:53.010
with the unit normal vector at that point.

00:02:53.010 --> 00:02:55.225
So the unit normal vector
might look something like that.

00:02:55.225 --> 00:02:57.357
That would be n hat at that point.

00:02:57.357 --> 00:03:00.538
This is the vector field at that point.

00:03:00.538 --> 00:03:05.089
When you take the dot product,
you get a scaler quantity.

00:03:05.089 --> 00:03:07.649
You essentially just get a
number and you might remember it

00:03:07.649 --> 00:03:09.638
and there's several
videos where we go into

00:03:09.638 --> 00:03:10.974
detail about this but
that tells you how much

00:03:10.974 --> 00:03:13.746
those two vectors go together.

00:03:13.746 --> 00:03:16.424
It's essentially, if they're
completely orthogonal

00:03:16.424 --> 00:03:18.685
to each other you're gonna
get zero and if they go

00:03:18.685 --> 00:03:21.176
completely in the same
direction it's essentially

00:03:21.176 --> 00:03:24.077
just gonna multiply their
magnitudes times each other.

00:03:24.077 --> 00:03:26.477
And since you have a
unit normal vector here,

00:03:26.477 --> 00:03:31.199
what this is essentially going
to give you is the magnitude

00:03:31.199 --> 00:03:35.866
of the vector field F that
goes in the normal direction.

00:03:39.872 --> 00:03:41.233
So, think of it this way.

00:03:41.233 --> 00:03:42.396
So let's think about the component of this

00:03:42.396 --> 00:03:43.610
that goes in the normal direction.

00:03:43.610 --> 00:03:45.721
It might look something
like that, this would be

00:03:45.721 --> 00:03:47.278
the component that goes
the tangential direction.

00:03:47.278 --> 00:03:50.315
So, this expression right
over here is going to give us

00:03:50.315 --> 00:03:53.292
the magnitude of this vector.

00:03:53.292 --> 00:03:56.402
Let me write this down,
this right over here is the

00:03:56.402 --> 00:04:01.069
magnitude of the component
of F, in the normal direction

00:04:10.535 --> 00:04:11.874
or in the same direction
as that unit normal vector.

00:04:11.874 --> 00:04:13.291
Normal direction.

00:04:15.224 --> 00:04:19.070
And then were gonna multiply
that times a very infinitely

00:04:19.070 --> 00:04:21.035
small length of our contour,

00:04:21.035 --> 00:04:22.945
of our curve right around that point.

00:04:22.945 --> 00:04:24.751
So were gonna multiply that thing,

00:04:24.751 --> 00:04:26.377
times this right over here.

00:04:26.377 --> 00:04:29.372
And so you might say, "Well,
you know, okay, I kind of

00:04:29.372 --> 00:04:31.445
"get what that is saying but

00:04:31.445 --> 00:04:32.876
"how could this ever
be physically relevant,

00:04:32.876 --> 00:04:34.422
"or what could be the intuition for what

00:04:34.422 --> 00:04:37.361
"this expression is even measuring?"

00:04:37.361 --> 00:04:38.826
And to think about
that, I always visualize

00:04:38.826 --> 00:04:41.351
this is in two dimensions.

00:04:41.351 --> 00:04:42.764
You will later do things like this

00:04:42.764 --> 00:04:44.237
in three dimensions, so imagine this is

00:04:44.237 --> 00:04:46.794
a two dimensional universe
and we're studying gases.

00:04:46.794 --> 00:04:50.194
And so you have all these gas
particles in a two dimensional

00:04:50.194 --> 00:04:52.884
universe so they only can have
kind of an x and y coordinate

00:04:52.884 --> 00:04:55.527
and this vector field is essentially

00:04:55.527 --> 00:04:58.296
telling you the velocity
at any point there.

00:04:58.296 --> 00:05:00.411
So this is the velocity of
the particles at this point,

00:05:00.411 --> 00:05:02.820
this is the velocity of
particles at that point,

00:05:02.820 --> 00:05:06.101
that is the velocity of
particles at that point.

00:05:06.101 --> 00:05:08.503
And so, when you take F right
on this curve, that's the

00:05:08.503 --> 00:05:10.364
velocity of the particles at that point

00:05:10.364 --> 00:05:12.260
they're going in that direction.

00:05:12.260 --> 00:05:15.353
When you dot it with N it tells you,

00:05:15.353 --> 00:05:17.504
essentially, the speed going straight out

00:05:17.504 --> 00:05:19.796
right at that point, and then when

00:05:19.796 --> 00:05:21.213
you multiply that times ds,
you're essentially saying,

00:05:21.213 --> 00:05:25.130
at any given moment, how
fast, or at that point

00:05:29.115 --> 00:05:30.640
right over there on the curve,
how fast are the particles

00:05:30.640 --> 00:05:32.140
exiting the curve?

00:05:33.847 --> 00:05:35.628
And so if you were to sum up all of them

00:05:35.628 --> 00:05:38.625
which is essentially what this
integral is doing with the

00:05:38.625 --> 00:05:40.382
slight integral, it's essentially saying,

00:05:40.382 --> 00:05:44.439
how fast are the particles
exiting this contour?

00:05:44.439 --> 00:05:46.975
Are they even entering
the contour if you get

00:05:46.975 --> 00:05:48.189
a negative number?

00:05:48.189 --> 00:05:50.100
But since were taking the
unit normal vector that

00:05:50.100 --> 00:05:53.154
is outward-pointing, it's saying,

00:05:53.154 --> 00:05:55.607
"How fast are they exiting this thing?"

00:05:55.607 --> 00:05:57.010
If you've got a negative
number, that means

00:05:57.010 --> 00:05:58.647
there might be some net entrance.

00:05:58.647 --> 00:06:02.525
So, this whole expression
is if you take that analogy,

00:06:02.525 --> 00:06:04.134
it doesn't have to have that
physical representation,

00:06:04.134 --> 00:06:08.884
how fast are particles, are
two dimensional gas particles

00:06:14.110 --> 00:06:15.777
exiting the contour?

00:06:19.552 --> 00:06:21.318
In the future, you can do three dimensions

00:06:21.318 --> 00:06:22.815
where you have a surface and you can say,

00:06:22.815 --> 00:06:25.099
"How fast are things
exiting this surface?"

00:06:25.099 --> 00:06:26.984
And so let's start, now
that we hopefully have

00:06:26.984 --> 00:06:29.767
a decent conceptual understanding of what

00:06:29.767 --> 00:06:32.635
this could represent let's play
around with it a little bit,

00:06:32.635 --> 00:06:35.116
especially because we know
how to define a normal vector.

00:06:35.116 --> 00:06:37.100
So, let's rewrite it using what we know

00:06:37.100 --> 00:06:39.584
about how to construct a normal vector.

00:06:39.584 --> 00:06:41.577
So, if we rewrite it our
integral becomes this.

00:06:41.577 --> 00:06:46.410
We have our vector field F,
dotted with the normal vector.

00:06:49.119 --> 00:06:50.678
The normal vector we can write this way.

00:06:50.678 --> 00:06:55.261
A normal vector we saw was
dy times y minus dx times j.

00:06:57.750 --> 00:06:59.921
And then we had to divide
it by its magnitude.

00:06:59.921 --> 00:07:02.204
In order to make it a unit normal vector,

00:07:02.204 --> 00:07:03.428
the magnitude was this right over here,

00:07:03.428 --> 00:07:05.518
dx squared by dy squared,
which is the same thing

00:07:05.518 --> 00:07:09.699
as ds which is the same thing
as that little mini arc length

00:07:09.699 --> 00:07:13.366
that infinitely small
length of our contour.

00:07:15.710 --> 00:07:17.422
So, we're going to divide it by ds.

00:07:17.422 --> 00:07:20.097
We're going to divide it by ds.

00:07:20.097 --> 00:07:22.480
And then we're going to
multiply it times ds.

00:07:22.480 --> 00:07:23.811
We're going to multiply it by ds.

00:07:23.811 --> 00:07:26.200
And ds is just a scanner quality,

00:07:26.200 --> 00:07:28.145
and so we can actually even multiply

00:07:28.145 --> 00:07:31.875
this thing times ds before
taking the dot product

00:07:31.875 --> 00:07:33.517
or vice versa.

00:07:33.517 --> 00:07:37.622
But these two things
are going to cancel out.

00:07:37.622 --> 00:07:40.492
And so we're essentially
left with F dot this thing

00:07:40.492 --> 00:07:41.559
right over here.

00:07:41.559 --> 00:07:43.679
But we have F defined
right over here so let's

00:07:43.679 --> 00:07:45.174
take the dot products.

00:07:45.174 --> 00:07:47.606
I'll just write the line
integral symbol again.

00:07:47.606 --> 00:07:50.681
We're going in the
counterclockwise direction.

00:07:50.681 --> 00:07:54.060
And when we evaluate and now let's in--

00:07:54.060 --> 00:07:55.600
let me pick a color that I have not used.

00:07:55.600 --> 00:07:59.506
I may have used many colors
so I'll do yellow again.

00:07:59.506 --> 00:08:01.211
So now, let's evaluate
F dot this business.

00:08:01.211 --> 00:08:03.612
So, dot product fairly straightforward.

00:08:03.612 --> 00:08:07.029
You take the product of the x-components,

00:08:09.011 --> 00:08:13.397
or essentially the magnitude
of the x-components.

00:08:13.397 --> 00:08:17.647
So, it's going to be P of
Y, P of xy, times the dy,

00:08:22.990 --> 00:08:27.407
plus the product of the
magnitudes of the y component

00:08:31.408 --> 00:08:32.779
or the j component.

00:08:32.779 --> 00:08:37.091
So it's going to be plus
Q of xy times minus dx,

00:08:37.091 --> 00:08:38.450
times negative dx.

00:08:38.450 --> 00:08:40.756
Well, that's going to give us

00:08:40.756 --> 00:08:44.423
negative Q of xy,
negative Q of xy times dx.

00:08:46.504 --> 00:08:48.330
So, this is kind of interesting statement.

00:08:48.330 --> 00:08:51.588
We've seen something not too
different from this before,

00:08:51.588 --> 00:08:54.416
when we saw even just the
definition of Green's theorem.

00:08:54.416 --> 00:08:58.256
Let me rewrite here just so we remember.

00:08:58.256 --> 00:08:59.869
So, the definition when we
learned about Green's theorem,

00:08:59.869 --> 00:09:02.398
it told us if we're taking a
line integral over this contour

00:09:02.398 --> 00:09:05.387
and there's multiple ways
to write it, but one way

00:09:05.387 --> 00:09:08.512
that we often still see it,
and we've explored it already

00:09:08.512 --> 00:09:12.845
in our videos is if we were
to say M times dx plus N

00:09:14.747 --> 00:09:17.080
times dy, this is equal to--

00:09:18.338 --> 00:09:20.730
and I'm just restating
Green's theorem right here--

00:09:20.730 --> 00:09:23.572
this is equal to the double
integral over the region

00:09:23.572 --> 00:09:26.322
that this contour surrounds of...

00:09:28.764 --> 00:09:31.704
And whatever function
you have here times dy,

00:09:31.704 --> 00:09:35.621
you take the partial of
that with respect to X,

00:09:39.827 --> 00:09:41.437
the partial of N with respect to X,

00:09:41.437 --> 00:09:44.681
and from that we subtract
whatever was on the dx side,

00:09:44.681 --> 00:09:49.024
so the partial of M with
respect to Y, and then

00:09:49.024 --> 00:09:52.286
you could say times df,
dy, or you could say da,

00:09:52.286 --> 00:09:56.577
for the infinitesimally
small little chunk of area.

00:09:56.577 --> 00:09:59.160
So, I'm going to write DA here.

00:10:00.451 --> 00:10:03.534
All right, dx, dy, or it could be dy.

00:10:05.889 --> 00:10:08.521
Or we might as well write DA here,

00:10:08.521 --> 00:10:10.678
since we're speaking in generalizations,

00:10:10.678 --> 00:10:14.212
where DA is an infinitely
small little chunk of area.

00:10:14.212 --> 00:10:15.877
So this here, this is just a
restatement of Green's theorem

00:10:15.877 --> 00:10:17.877
so we already know this.

00:10:20.099 --> 00:10:21.969
This is a restatement of Green's Theorem

00:10:21.969 --> 00:10:23.534
and how can we apply it here?

00:10:23.534 --> 00:10:25.134
Well, it's the same thing.
You have a little bit

00:10:25.134 --> 00:10:28.177
of sine differences but you
can apply Green's theorem here.

00:10:28.177 --> 00:10:31.711
This is going to be equal
to the double integral

00:10:31.711 --> 00:10:35.378
over the region that
this contour surrounds.

00:10:36.515 --> 00:10:38.322
And then, what we want to do is,

00:10:38.322 --> 00:10:41.382
we want to look at whatever
is the function that

00:10:41.382 --> 00:10:43.013
is being multiplied by the dy.

00:10:43.013 --> 00:10:45.150
In this case, this is the function

00:10:45.150 --> 00:10:46.850
that is being multiplied times the dy.

00:10:46.850 --> 00:10:48.368
And we want to take the partial
of that in respect to X.

00:10:48.368 --> 00:10:53.132
So, we're going to take the
partial of P with respect to X.

00:10:53.132 --> 00:10:56.382
So, the partial of P with respect to X.

00:10:57.869 --> 00:11:01.224
And then from that we
are going to subtract.

00:11:01.224 --> 00:11:03.443
We're going to subtract
the other function,

00:11:03.443 --> 00:11:05.131
whatever's being multiplied by dx,

00:11:05.131 --> 00:11:09.453
we're going to take the partial
of that with respect to Y.

00:11:09.453 --> 00:11:11.802
So here, we're going to take the partial

00:11:11.802 --> 00:11:13.212
of this whole thing with respect to Y,

00:11:13.212 --> 00:11:14.493
but we have a negative out here

00:11:14.493 --> 00:11:18.481
so it's going to be minus partial of Q

00:11:18.481 --> 00:11:21.534
with respect to Y and then we have DA,

00:11:21.534 --> 00:11:23.134
and obviously, these two negatives,

00:11:23.134 --> 00:11:25.597
subtracting the negative
just gives you a positive.

00:11:25.597 --> 00:11:28.380
So this is going to be
equal to the double integral

00:11:28.380 --> 00:11:30.146
over the region.

00:11:30.146 --> 00:11:31.819
And maybe you already
see where this is going,

00:11:31.819 --> 00:11:34.180
maybe getting a little bit excited.

00:11:34.180 --> 00:11:36.797
The partial of P with respect to X plus

00:11:36.797 --> 00:11:40.047
the partial of Q with respect to Y, dA.

00:11:41.577 --> 00:11:44.202
Now, okay, I took the partial of...

00:11:44.202 --> 00:11:45.976
"Well, what is this telling me, Sal?"

00:11:45.976 --> 00:11:47.555
Well, look at this thing right over here.

00:11:47.555 --> 00:11:51.076
P was originally the
function that was telling us

00:11:51.076 --> 00:11:53.181
the magnitude in the X direction.

00:11:53.181 --> 00:11:55.229
Q was telling us the
magnitude of the Y direction.

00:11:55.229 --> 00:11:57.421
We're taking the partial
of this with respect to X,

00:11:57.421 --> 00:11:59.755
we're taking the partial
of this with respect

00:11:59.755 --> 00:12:00.966
to Y and we're summing them.

00:12:00.966 --> 00:12:03.966
This is exactly the divergence of F.

00:12:05.519 --> 00:12:06.825
And if that doesn't make any sense,

00:12:06.825 --> 00:12:08.706
go watch the video on divergence.

00:12:08.706 --> 00:12:12.373
This right over here
is the divergence of F.

00:12:14.696 --> 00:12:17.661
This is the divergence
by definition, really,

00:12:17.661 --> 00:12:20.072
this is the divergence
of our vector field F.

00:12:20.072 --> 00:12:22.097
And so, we have a very interesting thing.

00:12:22.097 --> 00:12:24.412
This thing that we saw was
the original expression

00:12:24.412 --> 00:12:26.582
that we started studying
which is essentially saying,

00:12:26.582 --> 00:12:28.072
"What's the speed at which the particles

00:12:28.072 --> 00:12:30.128
"are exiting this surface?"

00:12:30.128 --> 00:12:32.901
We now get it in terms of
this little expression.

00:12:32.901 --> 00:12:35.255
And we'll interpret in an intuitive way

00:12:35.255 --> 00:12:38.663
and a little bit of this in a little bit.

00:12:38.663 --> 00:12:42.885
So this is equal to the
divergent of F times dA

00:12:42.885 --> 00:12:46.529
over the whole region
so the double integral.

00:12:46.529 --> 00:12:50.637
So, we're summing up the
divergence of F times

00:12:50.637 --> 00:12:52.778
the infinite, really small
little chunk over there

00:12:52.778 --> 00:12:54.667
and we're summing them
up over the whole region.

00:12:54.667 --> 00:12:57.083
Now, why does that make intuitive sense?

00:12:57.083 --> 00:13:00.041
And for you to realize why
it makes intuitive sense,

00:13:00.041 --> 00:13:02.387
you just have to remind
yourself what the divergence is.

00:13:02.387 --> 00:13:06.738
Divergence is a measure of
whether things are expanding

00:13:06.738 --> 00:13:09.574
or diverging or kind of contracting.

00:13:09.574 --> 00:13:13.574
If you have a point over
here where around that,

00:13:14.697 --> 00:13:17.207
the particles are kind of
moving away from each other,

00:13:17.207 --> 00:13:19.281
you would have a positive divergence here.

00:13:19.281 --> 00:13:21.976
If you have a point,
sometimes called a sync,

00:13:21.976 --> 00:13:26.096
where all of the particles
are kind of condensing

00:13:26.096 --> 00:13:30.045
or converging you would have
a negative divergence here.

00:13:30.045 --> 00:13:32.016
And now, so this should
make a lot of sense.

00:13:32.016 --> 00:13:35.474
You take any little
infinitely small area in this,

00:13:35.474 --> 00:13:37.729
you take any little infinitely small area,

00:13:37.729 --> 00:13:42.018
and then multiply that
times the divergence there.

00:13:42.018 --> 00:13:43.894
So, the more divergence, you're going

00:13:43.894 --> 00:13:45.476
to get a larger number and then you sum

00:13:45.476 --> 00:13:48.403
it up across the entire region.

00:13:48.403 --> 00:13:50.142
That makes a lot of sense.

00:13:50.142 --> 00:13:53.015
The more diverging that's
going on through your region,

00:13:53.015 --> 00:13:55.325
obviously, the more stuff that's going

00:13:55.325 --> 00:13:59.305
to be exiting your boundary over here.

00:13:59.305 --> 00:14:00.539
So, it actually makes complete sense,

00:14:00.539 --> 00:14:02.674
or hopefully it makes a
little bit of complete sense.

00:14:02.674 --> 00:14:06.526
If you were to see how fast are things

00:14:06.526 --> 00:14:07.873
exiting the surface, it's
really the two dimensional flux,

00:14:07.873 --> 00:14:10.315
how fast are things exiting the surface,

00:14:10.315 --> 00:14:12.215
and you take the sum of all of them,

00:14:12.215 --> 00:14:14.281
that's going to be the
same thing of summing

00:14:14.281 --> 00:14:18.192
all of the divergences over this area

00:14:18.192 --> 00:14:20.802
that the contour is surrounding.

00:14:20.802 --> 00:14:22.835
So, hopefully that makes a
little bit of sense to you.

00:14:22.835 --> 00:14:24.745
And it's actually another way of

00:14:24.745 --> 00:14:26.627
kind of thinking about Green's Theorem.

00:14:26.627 --> 00:14:29.405
And we also have just
explored what we just said,

00:14:29.405 --> 00:14:31.552
this expression that the
divergence summed over

00:14:31.552 --> 00:14:34.045
this region over here is the
same thing as the F dot n

00:14:34.045 --> 00:14:35.482
over the contour.

00:14:35.482 --> 00:14:38.398
That is, essentially, the two dimensional

00:14:38.398 --> 00:14:39.981
divergence theorem.


Constructing a unit normal vector to a curve

WEBVTT
Kind: captions
Language: en

00:00:00.624 --> 00:00:02.920
- [Voiceover] So let's say
that we've got the curve R

00:00:02.920 --> 00:00:04.878
defined, so this is our curve R.

00:00:04.878 --> 00:00:06.461
It's X of T times I

00:00:08.040 --> 00:00:10.966
plus Y T times J, it's a curve

00:00:10.966 --> 00:00:13.941
in two dimensions on the XY plane.

00:00:13.941 --> 00:00:17.914
And let's graph it, just
graph it in kind of a

00:00:17.914 --> 00:00:19.662
a generalized form.

00:00:19.662 --> 00:00:21.093
So that's our Y-axis.

00:00:21.093 --> 00:00:22.607
This is our X-axis.

00:00:22.607 --> 00:00:25.258
Our curve R might look
something like this.

00:00:25.258 --> 00:00:27.431
It might look something,
let me draw a little

00:00:27.431 --> 00:00:28.691
bit more of a.

00:00:28.691 --> 00:00:30.796
Maybe it looks something like this.

00:00:30.796 --> 00:00:32.177
Maybe that's just part of it.

00:00:32.177 --> 00:00:34.880
And as T increases, we're
going in that direction

00:00:34.880 --> 00:00:36.300
right over there.

00:00:36.300 --> 00:00:37.733
What I want to do in this video,

00:00:37.733 --> 00:00:41.430
this is really more vector
algebra than vector calculus.

00:00:41.430 --> 00:00:43.616
Is think about at any given point here

00:00:43.616 --> 00:00:45.679
whether we can figure out a normal vector.

00:00:45.679 --> 00:00:48.016
In particular, a unit normal vector.

00:00:48.016 --> 00:00:49.466
Obviously you can figure
out a normal vector

00:00:49.466 --> 00:00:51.226
you can just divide it by its magnitude

00:00:51.226 --> 00:00:53.380
and you will get the unit normal vector.

00:00:53.380 --> 00:00:55.430
So I want to figure out at any given point

00:00:55.430 --> 00:00:58.803
a vector that's popping
straight out in that direction.

00:00:58.803 --> 00:01:00.358
And has a magnitude one.

00:01:00.358 --> 00:01:03.080
So that would be our unit normal vector.

00:01:03.080 --> 00:01:04.778
And to do that, first we'll think about

00:01:04.778 --> 00:01:06.372
what a tangent vector is.

00:01:06.372 --> 00:01:08.139
And from a tangent
vector we can figure out

00:01:08.139 --> 00:01:09.682
the normal vector.

00:01:09.682 --> 00:01:11.735
And it really goes back
to some of what you might

00:01:11.735 --> 00:01:13.727
have done in algebra one, or algebra two

00:01:13.727 --> 00:01:15.422
of if you have a slope of a line

00:01:15.422 --> 00:01:17.433
the negative reciprocal of
that slope is going to be

00:01:17.433 --> 00:01:19.058
the slope of that negative line.

00:01:19.058 --> 00:01:20.772
And we'll see a very similar thing

00:01:20.772 --> 00:01:23.313
when we do it right over
here with the vector

00:01:23.313 --> 00:01:25.101
with this vector algebra.

00:01:25.101 --> 00:01:26.738
So the first thing I
want to think about is

00:01:26.738 --> 00:01:27.978
how do we construct

00:01:27.978 --> 00:01:31.452
how do we construct a tangent line.

00:01:31.452 --> 00:01:33.951
Well, you could imagine at sum T

00:01:33.951 --> 00:01:35.362
at sum T

00:01:35.362 --> 00:01:38.003
this is what our position
vector is going to look like.

00:01:38.003 --> 00:01:40.026
So call that R one.

00:01:40.026 --> 00:01:41.518
R one right over there.

00:01:41.518 --> 00:01:44.423
And then if we wait, if we
allow T to go up a little bit,

00:01:44.423 --> 00:01:46.426
if T is time, we'll wait a little while

00:01:46.426 --> 00:01:48.994
a few seconds, or however
we were measuring things.

00:01:48.994 --> 00:01:51.270
And then R two might
look something like this.

00:01:51.270 --> 00:01:54.672
This is when T has gotten
a little bit larger.

00:01:54.672 --> 00:01:56.695
We're further down the path.

00:01:56.695 --> 00:01:59.394
And so one way that you can approximate

00:01:59.394 --> 00:02:02.160
the slope of the tangent line

00:02:02.160 --> 00:02:06.662
or the slope between
these two points, for now,

00:02:06.662 --> 00:02:09.201
is essentially the difference
between these two vectors.

00:02:09.201 --> 00:02:11.640
The difference between
these two vectors is

00:02:11.640 --> 00:02:13.030
you could view that

00:02:13.030 --> 00:02:14.778
you could view that as

00:02:14.778 --> 00:02:16.211
delta

00:02:16.211 --> 00:02:17.227
delta R.

00:02:17.227 --> 00:02:19.746
This vector plus that vector
is equal to that vector.

00:02:19.746 --> 00:02:22.347
Or, R two minus R one is going to give you

00:02:22.347 --> 00:02:24.306
this delta R right over here.

00:02:24.306 --> 00:02:26.553
And as R two, as that increment

00:02:26.553 --> 00:02:28.431
between R one and R two gets smaller

00:02:28.431 --> 00:02:29.458
and smaller and smaller.

00:02:29.458 --> 00:02:32.027
As we have a smaller
and smaller T increment

00:02:32.027 --> 00:02:34.782
as we get a smaller and
smaller T increment.

00:02:34.782 --> 00:02:37.045
So we get a smaller
and smaller T increment

00:02:37.045 --> 00:02:40.083
the slope of that delta R is going to more

00:02:40.083 --> 00:02:41.962
and more approximate the slope

00:02:41.962 --> 00:02:43.579
of the tangent line.

00:02:43.579 --> 00:02:45.082
All the way to the point that if you have

00:02:45.082 --> 00:02:47.215
an infinitely small change in T.

00:02:47.215 --> 00:02:48.240
So you have a DT.

00:02:48.240 --> 00:02:49.915
So you go from R then you just

00:02:49.915 --> 00:02:52.487
you change T a very small amount

00:02:52.487 --> 00:02:54.754
that delta R, and we can
kind of conceptualize that,

00:02:54.754 --> 00:02:57.537
as DR, that does approximate

00:02:57.537 --> 00:02:59.996
the A tangent vector.

00:02:59.996 --> 00:03:02.646
So if you have a very small change in T

00:03:02.646 --> 00:03:04.313
then your very small

00:03:05.510 --> 00:03:07.350
del DR I'll call it

00:03:07.350 --> 00:03:09.443
because now we're talking
about a differential.

00:03:09.443 --> 00:03:11.292
Your very small differential.

00:03:11.292 --> 00:03:12.450
Right over here.

00:03:12.450 --> 00:03:14.491
That is a tangent

00:03:14.491 --> 00:03:16.798
that is a tangent vector.

00:03:16.798 --> 00:03:18.330
So DR

00:03:18.330 --> 00:03:19.163
DR is

00:03:20.496 --> 00:03:21.329
a tangent

00:03:22.777 --> 00:03:25.349
tangent vector at any

00:03:25.349 --> 00:03:27.169
at any given point.

00:03:27.169 --> 00:03:29.729
And once again, all of this
is a little bit of review.

00:03:29.729 --> 00:03:31.700
But DR, we can write as

00:03:31.700 --> 00:03:33.117
DR is equal to DX

00:03:34.625 --> 00:03:36.485
times I plus the

00:03:36.485 --> 00:03:38.707
infinite small change in X

00:03:38.707 --> 00:03:40.143
times the I unit vector

00:03:40.143 --> 00:03:42.559
plus the infinite small change in Y

00:03:42.559 --> 00:03:44.145
times the J unit vector.

00:03:44.145 --> 00:03:45.811
And you see that, you see that

00:03:45.811 --> 00:03:47.030
if I were to draw

00:03:47.030 --> 00:03:48.684
if I were to draw a curve.

00:03:48.684 --> 00:03:50.432
Let me just draw another one.

00:03:50.432 --> 00:03:52.354
Actually, I don't even
have to draw the axis.

00:03:52.354 --> 00:03:54.750
If our DR looks like that

00:03:54.750 --> 00:03:56.771
if that is our DR

00:03:56.771 --> 00:03:58.986
then, we can break that down

00:03:58.986 --> 00:04:01.891
into its vertical and
horizontal components.

00:04:01.891 --> 00:04:04.646
This right over here is DY.

00:04:04.646 --> 00:04:06.483
And that right over there

00:04:06.483 --> 00:04:08.799
that right over there is

00:04:08.799 --> 00:04:09.716
that is DX.

00:04:10.790 --> 00:04:13.484
And so we see that DX times I.

00:04:13.484 --> 00:04:15.637
Actually, this is DX times I.

00:04:15.637 --> 00:04:17.272
And this is DY

00:04:17.272 --> 00:04:19.324
this is DY times J.

00:04:19.324 --> 00:04:20.554
DY is the magnitude,

00:04:20.554 --> 00:04:22.322
J gives us the direction.

00:04:22.322 --> 00:04:23.875
DX is the magnitude.

00:04:23.875 --> 00:04:25.410
I tells us that we're moving

00:04:25.410 --> 00:04:26.812
in the horizontal direction.

00:04:26.812 --> 00:04:28.945
Over here, this actually
would be a negative.

00:04:28.945 --> 00:04:30.986
This must be a negative
value right over here

00:04:30.986 --> 00:04:32.530
and this must be a positive value

00:04:32.530 --> 00:04:34.663
based on the way that I drew it.

00:04:34.663 --> 00:04:36.959
So that gives us a tangent vector.

00:04:36.959 --> 00:04:39.233
And now we want to from
that tangent vector

00:04:39.233 --> 00:04:41.217
figure out a normal vector.

00:04:41.217 --> 00:04:43.258
A vector that is essentially perpendicular

00:04:43.258 --> 00:04:44.976
to this vector right over here.

00:04:44.976 --> 00:04:46.244
And there's actually going to be two

00:04:46.244 --> 00:04:47.402
vectors like that.

00:04:47.402 --> 00:04:48.917
There's going to be the vector

00:04:48.917 --> 00:04:51.742
that kind of is perpendicular
in the right direction

00:04:51.742 --> 00:04:53.036
because we care about direction.

00:04:53.036 --> 00:04:55.547
Or the vector that's perpendicular
in the left direction.

00:04:55.547 --> 00:04:56.786
And we can pick either one.

00:04:56.786 --> 00:04:59.012
But for this video, I'm
gonna focus on the one

00:04:59.012 --> 00:05:00.374
that goes in the right direction.

00:05:00.374 --> 00:05:02.383
We're gonna see that
that's gonna be useful

00:05:02.383 --> 00:05:04.457
in the next video when we
start doing a little bit of

00:05:04.457 --> 00:05:06.337
vector calculus.

00:05:06.337 --> 00:05:08.560
And so let's think about
what that might be.

00:05:08.560 --> 00:05:10.836
And what I'll do to make
it a little bit clearer.

00:05:10.836 --> 00:05:12.930
Let me draw a DR again.

00:05:12.930 --> 00:05:15.113
I'll draw a DR like this.

00:05:15.113 --> 00:05:16.363
This is our DR.

00:05:17.295 --> 00:05:18.619
This is DR.

00:05:18.619 --> 00:05:21.057
And then this, right over here.

00:05:21.057 --> 00:05:25.202
This right over there, we
already said this is DY

00:05:25.202 --> 00:05:26.595
times I.

00:05:26.595 --> 00:05:28.247
And then this, sorry,

00:05:28.247 --> 00:05:29.744
this is DY times J.

00:05:29.744 --> 00:05:31.340
We're going in the vertical direction.

00:05:31.340 --> 00:05:33.035
DY times J.

00:05:33.035 --> 00:05:34.783
And then in a different color

00:05:34.783 --> 00:05:37.090
this right now if I
already used that color.

00:05:37.090 --> 00:05:38.768
I haven't used, oh,

00:05:38.768 --> 00:05:41.347
I haven't used or had blue yet.

00:05:41.347 --> 00:05:43.778
So this right over here is DX

00:05:43.778 --> 00:05:44.695
DX times I.

00:05:45.870 --> 00:05:48.134
So we know from our algebra courses

00:05:48.134 --> 00:05:49.699
you take the negative reciprocal

00:05:49.699 --> 00:05:51.336
so there's gonna be
something about swapping

00:05:51.336 --> 00:05:53.266
these two things around,
and then taking the

00:05:53.266 --> 00:05:54.107
negative one.

00:05:54.107 --> 00:05:56.412
But to figure out, we want the
one that goes to the right.

00:05:56.412 --> 00:05:58.273
So which one should we use?

00:05:58.273 --> 00:06:00.284
So let's think about it a little bit.

00:06:00.284 --> 00:06:01.951
If we, if we take DY

00:06:03.650 --> 00:06:04.600
times I.

00:06:04.600 --> 00:06:05.574
So we take this length

00:06:05.574 --> 00:06:07.403
in the I direction,

00:06:07.403 --> 00:06:08.634
we're gonna get

00:06:08.634 --> 00:06:10.504
we're gonna get this.

00:06:10.504 --> 00:06:12.170
We're gonna get that, so this is

00:06:12.170 --> 00:06:13.003
DY

00:06:14.098 --> 00:06:15.595
times I.

00:06:15.595 --> 00:06:17.513
And then if we were to

00:06:17.513 --> 00:06:19.545
if we were to take D,
if we were to just take

00:06:19.545 --> 00:06:20.856
DX

00:06:20.856 --> 00:06:21.844
times

00:06:21.844 --> 00:06:23.693
J, that would take us down.

00:06:23.693 --> 00:06:25.042
'Cause DX it must be negative here

00:06:25.042 --> 00:06:26.758
since it's pointed to the left.

00:06:26.758 --> 00:06:30.975
So we have to swap the
sign of DX to go upwards.

00:06:30.975 --> 00:06:33.474
So we swap the sign of DX

00:06:33.474 --> 00:06:34.795
to go upwards.

00:06:34.795 --> 00:06:36.806
'Cause I was here it was a negative sign.

00:06:36.806 --> 00:06:39.383
It went leftwards, we
want it to go upwards.

00:06:39.383 --> 00:06:40.542
So this is gonna be negative

00:06:40.542 --> 00:06:42.524
negative DX times J.

00:06:42.524 --> 00:06:44.361
We're now moving in
the vertical direction.

00:06:44.361 --> 00:06:46.648
And that, at least visually,
this isn't kind of a

00:06:46.648 --> 00:06:48.395
rigorous proof that I'm giving you.

00:06:48.395 --> 00:06:51.057
But this is hopefully good
a good visual representation

00:06:51.057 --> 00:06:52.631
that that does

00:06:52.631 --> 00:06:54.491
that that does get you.

00:06:54.491 --> 00:06:56.135
I should have drawn it a little bit.

00:06:56.135 --> 00:06:58.546
That does get you pretty

00:06:58.546 --> 00:07:01.326
that gets you pretty close,
just visually inspecting it

00:07:01.326 --> 00:07:03.297
to what looks like the perpendicular line.

00:07:03.297 --> 00:07:05.085
It's consistent with what
you learned in algebra class,

00:07:05.085 --> 00:07:06.142
as well.

00:07:06.142 --> 00:07:08.141
That we're taking the negative reciprocal,

00:07:08.141 --> 00:07:10.803
we're swapping the X's and the Y's.

00:07:10.803 --> 00:07:12.977
Or the change in X and the change in Y.

00:07:12.977 --> 00:07:15.284
And we're taking the
negative of one of them.

00:07:15.284 --> 00:07:17.438
And so we have our normal line

00:07:17.438 --> 00:07:18.840
just like that.

00:07:18.840 --> 00:07:20.372
Our normal vector.

00:07:20.372 --> 00:07:23.237
So a normal vector is going to be

00:07:23.237 --> 00:07:24.070
DYI

00:07:24.962 --> 00:07:25.795
minus DXJ.

00:07:26.892 --> 00:07:28.376
But then if we want a normalize it

00:07:28.376 --> 00:07:30.044
we want to divide by

00:07:30.044 --> 00:07:31.283
by that magnitude.

00:07:31.283 --> 00:07:33.347
So a normal, let me write it this way.

00:07:33.347 --> 00:07:34.911
A normal vector.

00:07:34.911 --> 00:07:36.191
So let me call this

00:07:36.191 --> 00:07:37.999
I'll just call it A.

00:07:37.999 --> 00:07:39.979
A normal vector

00:07:39.979 --> 00:07:42.022
is going to be DY

00:07:42.022 --> 00:07:42.855
times I.

00:07:43.758 --> 00:07:44.753
Is going to be

00:07:44.753 --> 00:07:45.647
DY

00:07:45.647 --> 00:07:46.959
times I.

00:07:46.959 --> 00:07:48.376
Minus DX times J.

00:07:49.313 --> 00:07:51.287
Minus DX times J.

00:07:51.287 --> 00:07:52.526
I'll do that same blue color.

00:07:52.526 --> 00:07:53.359
Minus DX

00:07:54.872 --> 00:07:55.705
times J.

00:07:56.761 --> 00:07:58.813
Now, if we want this to
be a unit normal vector

00:07:58.813 --> 00:08:01.251
we have to divide it
by the magnitude of A.

00:08:01.251 --> 00:08:02.949
But what is the magnitude of A?

00:08:02.949 --> 00:08:05.022
The magnitude of A

00:08:05.022 --> 00:08:07.265
is going to be equal to

00:08:07.265 --> 00:08:09.634
it's going to be equal to the square root

00:08:09.634 --> 00:08:12.173
of, and I'll just start
with the DX squared.

00:08:12.173 --> 00:08:14.153
So it's the negative DX squared.

00:08:14.153 --> 00:08:15.749
Which is just going to be DX squared.

00:08:15.749 --> 00:08:18.176
The same thing as positive DX squared.

00:08:18.176 --> 00:08:20.199
It's going to be DX squared

00:08:20.199 --> 00:08:22.097
plus DY squared.

00:08:22.097 --> 00:08:22.930
Plus DY

00:08:24.169 --> 00:08:25.002
squared.

00:08:25.002 --> 00:08:26.883
I could have put the
negative right in here

00:08:26.883 --> 00:08:29.686
but then when you square it,
that negative would disappear.

00:08:29.686 --> 00:08:31.028
But this thing right over here

00:08:31.028 --> 00:08:34.377
and we saw this when we first
started exploring arc length.

00:08:34.377 --> 00:08:37.454
This thing right over here
is the exact same thing

00:08:37.454 --> 00:08:38.287
as DS,

00:08:39.538 --> 00:08:41.031
and I know there's no DS

00:08:41.031 --> 00:08:43.126
that we've shown right over here.

00:08:43.126 --> 00:08:45.288
But we've seen it multiple times.

00:08:45.288 --> 00:08:47.523
When you're thinking of about if you

00:08:47.523 --> 00:08:50.936
if you think about the length of DR as DS

00:08:50.936 --> 00:08:54.015
that's exactly what
this thing over here is.

00:08:54.015 --> 00:08:56.422
So this can also be written as

00:08:56.422 --> 00:08:57.255
DS.

00:08:57.255 --> 00:08:59.176
So the infinite hasn't really changed

00:08:59.176 --> 00:09:00.770
in the arc length

00:09:00.770 --> 00:09:02.172
but it's a scaler quantity.

00:09:02.172 --> 00:09:03.513
You're not concerned

00:09:03.513 --> 00:09:05.176
you're just concerned with
the absolute distance.

00:09:05.176 --> 00:09:07.689
You're not concerned so
much with the direction.

00:09:07.689 --> 00:09:09.118
Another way to be do it is

00:09:09.118 --> 00:09:10.583
it's the magnitude

00:09:10.583 --> 00:09:13.589
it's the magnitude right over here of DR.

00:09:13.589 --> 00:09:16.020
So now we have everything
we need to construct

00:09:16.020 --> 00:09:18.321
our unit normal vector.

00:09:18.321 --> 00:09:20.600
Our unit normal vector at any point.

00:09:20.600 --> 00:09:21.871
And I'll now write N

00:09:21.871 --> 00:09:23.801
and I'll put a hat on top of it.

00:09:23.801 --> 00:09:25.435
Say that this is a unit normal vector.

00:09:25.435 --> 00:09:27.620
We'll have magnitude one is going to be

00:09:27.620 --> 00:09:30.413
equal to A divided by this.

00:09:30.413 --> 00:09:32.485
Or we could even write it this way.

00:09:32.485 --> 00:09:34.386
So we could write it as

00:09:34.386 --> 00:09:35.573
there are multiple ways we can write it.

00:09:35.573 --> 00:09:37.434
We can write it as

00:09:37.434 --> 00:09:38.917
I'll write it in this color.

00:09:38.917 --> 00:09:39.750
As DY

00:09:40.937 --> 00:09:41.802
times I

00:09:41.802 --> 00:09:43.365
minus DX

00:09:43.365 --> 00:09:44.374
times J.

00:09:44.374 --> 00:09:46.954
And then all of that times

00:09:46.954 --> 00:09:49.625
or maybe not times, divided by,

00:09:49.625 --> 00:09:50.458
DS.

00:09:50.458 --> 00:09:52.510
Divided by the magnitude of this.

00:09:52.510 --> 00:09:54.135
So divided by

00:09:54.135 --> 00:09:54.968
divided by

00:09:56.054 --> 00:09:56.887
DS.

00:09:56.887 --> 00:09:59.052
And obviously I can
distribute it on each of these

00:09:59.052 --> 00:10:00.637
by on each of these terms.

00:10:00.637 --> 00:10:02.473
But this right here, we've
been able to construct

00:10:02.473 --> 00:10:05.086
a unit normal vector at any point

00:10:05.086 --> 00:10:06.253
on this curve.


Three pointer vs free throwing probability

WEBVTT
Kind: captions
Language: en

00:00:03.057 --> 00:00:04.265
LEBRON JAMES: Hey, everybody.

00:00:04.265 --> 00:00:05.763
LeBron here.

00:00:05.763 --> 00:00:08.272
Got another quick
brain teaser for you.

00:00:08.272 --> 00:00:10.940
Do I have a better odds
of making 3 free throws

00:00:10.940 --> 00:00:13.560
in a row or 1 three-pointer?

00:00:13.560 --> 00:00:16.750
Here's my friend,
Sal, with the answer.

00:00:16.750 --> 00:00:18.260
SAL: Excellent question, LeBron.

00:00:18.260 --> 00:00:20.460
But before I answer
it, I want to point out

00:00:20.460 --> 00:00:22.560
an interesting trend
related to your question

00:00:22.560 --> 00:00:23.660
that I just dug up.

00:00:23.660 --> 00:00:26.420
This is from the New York
Times, October, 2009.

00:00:26.420 --> 00:00:28.890
So it's a couple of years old,
but it's really interesting.

00:00:28.890 --> 00:00:31.730
It shows that since
three-pointers were first

00:00:31.730 --> 00:00:34.190
introduced-- they were
first introduced in the 1979

00:00:34.190 --> 00:00:37.040
to 1980 season--
that three-pointers

00:00:37.040 --> 00:00:38.800
have become more
and more frequent.

00:00:38.800 --> 00:00:42.270
So what they are showing here is
the average attempts per team,

00:00:42.270 --> 00:00:43.320
season by season.

00:00:43.320 --> 00:00:45.790
And it looks like there's just
a steady upward trend here

00:00:45.790 --> 00:00:46.790
related to our question.

00:00:46.790 --> 00:00:49.126
There's a couple
of anomalies here.

00:00:49.126 --> 00:00:50.500
And the ones that
really jump out

00:00:50.500 --> 00:00:52.730
are these three seasons
in the late '90s.

00:00:52.730 --> 00:00:54.950
And that's because the
actual three-pointer line

00:00:54.950 --> 00:00:57.379
was pulled in to essentially
get higher scoring games.

00:00:57.379 --> 00:00:58.920
So people attempted
more, but then it

00:00:58.920 --> 00:01:00.854
was put back to where
it was originally.

00:01:00.854 --> 00:01:02.020
This was a shortened season.

00:01:02.020 --> 00:01:03.700
And I'm not really
sure what happened

00:01:03.700 --> 00:01:08.180
in this, what is this, the
2001, 2002 season here.

00:01:08.180 --> 00:01:09.850
But it's something
to think about.

00:01:09.850 --> 00:01:11.990
There is just this trend
that more and more people

00:01:11.990 --> 00:01:13.280
are taking three-pointers.

00:01:13.280 --> 00:01:15.030
Now, with that out of
the way, let's think

00:01:15.030 --> 00:01:16.640
about your actual question.

00:01:16.640 --> 00:01:19.480
And to answer it, I dug up
your stats right over here

00:01:19.480 --> 00:01:20.470
from nba.com.

00:01:20.470 --> 00:01:22.650
And we'll use your career stats.

00:01:22.650 --> 00:01:27.360
So we want to compare 3 free
throws to a three-pointer.

00:01:27.360 --> 00:01:30.190
So right over here, we have
your three-pointer percentage,

00:01:30.190 --> 00:01:31.750
and this is in your career.

00:01:31.750 --> 00:01:33.790
And I'll round it to
the nearest hundredths,

00:01:33.790 --> 00:01:36.180
so it looks like it's about 33%.

00:01:36.180 --> 00:01:41.615
So your three-point percentage
is-- we'll just call it 33%.

00:01:41.615 --> 00:01:44.170
And then your free
throw percentage,

00:01:44.170 --> 00:01:46.696
your career free
throw percentage--

00:01:46.696 --> 00:01:48.070
so this is free
throw percentage,

00:01:48.070 --> 00:01:48.900
and this is in your career.

00:01:48.900 --> 00:01:50.700
We'll round to the
nearest hundredths.

00:01:50.700 --> 00:01:52.890
So we'll round up
right over here.

00:01:52.890 --> 00:01:57.380
That gets us to
right at about 75%.

00:01:57.380 --> 00:01:59.750
So clearly looking at these
numbers right over here,

00:01:59.750 --> 00:02:02.440
you're much more likely
to make a given free throw

00:02:02.440 --> 00:02:04.131
than making a given
three-pointer.

00:02:04.131 --> 00:02:06.380
You're more than twice as
likely to make a free throw.

00:02:06.380 --> 00:02:07.630
But that's not what you asked.

00:02:07.630 --> 00:02:10.909
You asked what about 3
free throws in a row.

00:02:10.909 --> 00:02:12.950
And so what we'll do is
we'll do an analysis very

00:02:12.950 --> 00:02:14.970
similar to the last
time when we asked

00:02:14.970 --> 00:02:16.830
about 10 free throws in a row.

00:02:16.830 --> 00:02:18.630
So let's think about
the first free throw.

00:02:18.630 --> 00:02:21.510
So free throw number one.

00:02:21.510 --> 00:02:23.870
If we imagined a
gazillion-- a billion--

00:02:23.870 --> 00:02:25.210
a gazillion's not a real number.

00:02:25.210 --> 00:02:28.020
If we imagined a billion LeBron
Jameses, identical LeBron

00:02:28.020 --> 00:02:30.860
Jameses all taking
that first free throw,

00:02:30.860 --> 00:02:33.590
we would expect, on
average, that 75%

00:02:33.590 --> 00:02:35.420
would make that
first free throw.

00:02:35.420 --> 00:02:38.250
So 75% is 3/4.

00:02:38.250 --> 00:02:41.840
So 75% would make
that first free throw.

00:02:41.840 --> 00:02:44.657
And 25%, we would
expect, on average--

00:02:44.657 --> 00:02:47.240
wouldn't always be the case, but
this is what we would expect.

00:02:47.240 --> 00:02:50.740
25% would miss that
first free throw.

00:02:50.740 --> 00:02:56.980
Now, let's go to the second free
throw, free throw number two.

00:02:56.980 --> 00:02:59.330
And we only care about
the LeBron Jameses

00:02:59.330 --> 00:03:01.600
that keep making
their free throws.

00:03:01.600 --> 00:03:04.970
So let's think about of the
75% that made that first one.

00:03:04.970 --> 00:03:07.649
Some of the 25% might make
that second one and then maybe

00:03:07.649 --> 00:03:08.440
even the third one.

00:03:08.440 --> 00:03:10.940
But let's just think about the
ones that made the first one.

00:03:10.940 --> 00:03:12.440
Of the ones that
made the first one,

00:03:12.440 --> 00:03:16.080
we would expect 75% of them
to make the second one.

00:03:16.080 --> 00:03:20.330
So 75% of the 75%,
that's 1/2 of the 75%.

00:03:20.330 --> 00:03:24.100
That's about 75%
of the 75% would

00:03:24.100 --> 00:03:28.070
make that second free throw
and the first free throw.

00:03:28.070 --> 00:03:34.190
So now we have-- this is
going to be 75% times 75%.

00:03:34.190 --> 00:03:36.497
And, of course, there's
other combinations

00:03:36.497 --> 00:03:38.830
out here where someone's
missed at least one of the free

00:03:38.830 --> 00:03:39.730
throws.

00:03:39.730 --> 00:03:43.900
Now, let's go to the third free
throw, free throw number three.

00:03:43.900 --> 00:03:47.250
What percentage of these
LeBron Jameses right

00:03:47.250 --> 00:03:49.030
here will make the
third free throw?

00:03:49.030 --> 00:03:51.980
Well 75% of these will
make the third free throw.

00:03:51.980 --> 00:03:54.410
So 75% of this number.

00:03:54.410 --> 00:03:55.770
So let me just draw it visually.

00:03:55.770 --> 00:03:56.350
That's 1/2.

00:03:56.350 --> 00:03:59.500
That's about 75% of that number.

00:03:59.500 --> 00:04:01.530
They will make the
third one, as well.

00:04:01.530 --> 00:04:09.500
And so this is 75% of this
number, which is 75% of 75%.

00:04:09.500 --> 00:04:13.180
This is how many LeBron
Jameses are going to make all 3

00:04:13.180 --> 00:04:14.530
of the free throws.

00:04:14.530 --> 00:04:16.160
And, once again,
we can write this

00:04:16.160 --> 00:04:17.890
as-- we could either
multiply it out

00:04:17.890 --> 00:04:21.740
or we can just write this as
75% to the third power, which

00:04:21.740 --> 00:04:26.700
is the same thing as 75%
literally means 75 per 100.

00:04:26.700 --> 00:04:30.820
Same thing as 75 over 100
to the third power, which

00:04:30.820 --> 00:04:36.280
is the same thing as
0.75 to the third power.

00:04:36.280 --> 00:04:38.010
And so let's calculate it.

00:04:38.010 --> 00:04:39.562
Get the calculator out.

00:04:39.562 --> 00:04:41.770
And actually, let me show
you we get the same result.

00:04:41.770 --> 00:04:46.450
We can write 0.75 times
0.75-- and on this calculator,

00:04:46.450 --> 00:04:48.490
that little snowflake
looking thing,

00:04:48.490 --> 00:04:52.110
it means multiplication--
times 0.75.

00:04:52.110 --> 00:04:54.789
And then we get 0.42.

00:04:54.789 --> 00:04:56.330
I'll round to the
nearest hundredths.

00:04:56.330 --> 00:04:57.746
And that's the
same thing we would

00:04:57.746 --> 00:05:00.610
get if we got 0.75
to the third power.

00:05:00.610 --> 00:05:02.690
Once again, 0.42.

00:05:02.690 --> 00:05:04.070
So let me write that.

00:05:04.070 --> 00:05:09.240
So this gets us to
approximately 0.42,

00:05:09.240 --> 00:05:12.380
which is the same thing as 42%.

00:05:12.380 --> 00:05:15.480
So your probability of
making 3 free throws in a row

00:05:15.480 --> 00:05:22.180
is 42%, which is still higher
than making 1 three-pointer.

00:05:22.180 --> 00:05:24.480
So I'll leave you there,
but I want the people

00:05:24.480 --> 00:05:27.550
who are watching this video to
think about what would happen

00:05:27.550 --> 00:05:28.800
if the numbers were different?

00:05:28.800 --> 00:05:31.930
Or maybe look up some NBA
players, or maybe some college

00:05:31.930 --> 00:05:35.560
players, and figure out
and compare the probability

00:05:35.560 --> 00:05:37.540
of making 3 free throws
to a three-pointer,

00:05:37.540 --> 00:05:39.999
and see if you can
find any players where

00:05:39.999 --> 00:05:41.790
their probability of
making a three-pointer

00:05:41.790 --> 00:05:46.280
is actually higher than
making 3 free throws in a row.


Why Europe is worried about Greece

WEBVTT
Kind: captions
Language: en

00:00:00.311 --> 00:00:02.827
Given all the data that we`ve explored in the last video

00:00:02.827 --> 00:00:06.915
and that we have over here, the very high debt to GDP burden that Greece has and

00:00:06.915 --> 00:00:09.977
the very weak economy, it`s already in a deep recession.

00:00:09.977 --> 00:00:12.665
It`s especially apparent when you look at its unemployment rate.

00:00:12.665 --> 00:00:18.064
This is the period October 2010 to March 2012, unemployment rate was already high

00:00:18.064 --> 00:00:22.678
in 2010 and it`s just been going through the roof, it`s already in the low 20%

00:00:22.678 --> 00:00:28.427
which is a huge number and even that is understating how bad things are on the ground in Greece

00:00:28.427 --> 00:00:31.514
because the unemployment is disproportionately affecting the young

00:00:31.514 --> 00:00:33.633
so if you looked at the unemployment rate for people in their 20`s

00:00:33.633 --> 00:00:39.100
it would be much much much higher than even this already unbelievably high unemployment rate.

00:00:39.100 --> 00:00:43.669
And we've already explored that the bond markets are expressing that, by asing for higher and higher

00:00:43.669 --> 00:00:48.363
interest rates from Greece - these are the long term interest rates in Greece.

00:00:48.363 --> 00:00:53.783
And so the possible outcomes that we're talking about here --

00:00:53.783 --> 00:00:56.324
So this is Greece -

00:00:56.324 --> 00:00:58.427
--the possible outcomes for Greece--

00:00:58.427 --> 00:01:01.479
The only way that they are able to stay in the Eurozone

00:01:01.479 --> 00:01:05.466
is if someone essentially writes them a huge check, that takes care of a good bit of their debt.

00:01:05.466 --> 00:01:15.659
So one option is that they get bailed out. They get bailed out by the rest of Europe.

00:01:15.659 --> 00:01:19.607
And Germany is a major actor here, because they are the largest economy of Europe.

00:01:19.607 --> 00:01:23.810
But with the bailout, people are saying: "Hey, look, if we are going to write you a

00:01:23.810 --> 00:01:28.567
big check, you've got to cut some of your excesses, you've got to cut some of your mismanagement."

00:01:28.567 --> 00:01:35.181
So the bailout packages are all tied to some form of austerity.

00:01:35.181 --> 00:01:41.109
Now, we've already talked about why austerity is definitely not politically popular in Greece

00:01:41.109 --> 00:01:43.768
but why it's also - it might just be a scary thing to do.

00:01:43.768 --> 00:01:47.214
Because you already have an ultra weak economy,

00:01:47.214 --> 00:01:49.833
you already have the unemployment rate going through the roof,

00:01:49.833 --> 00:01:52.844
what happens if you get even more austere from where you are now,

00:01:52.844 --> 00:01:56.826
if you do a drastic cut to government spending from where you are now,

00:01:56.826 --> 00:01:59.728
this could go to who knows where, to a large degree

00:01:59.728 --> 00:02:04.560
this rise of unemployment is due to the last few rounds of bailouts and austerity

00:02:04.560 --> 00:02:08.519
and so that's why the Greek people are getting very suspect of this austerity.

00:02:08.519 --> 00:02:12.168
But if that doesn't happen, and we've talked about this in the last video,

00:02:12.168 --> 00:02:16.442
if that doesn't happen, the only viable option for Greece

00:02:16.442 --> 00:02:19.710
is to go off the Euro.

00:02:19.710 --> 00:02:23.276
So, leave the Eurozone.

00:02:23.276 --> 00:02:30.866
The Eurozone is the subset of European union countries that use the Euro as their currency.

00:02:30.866 --> 00:02:34.624
And we've already talked about this, it would not be a painless process.

00:02:34.624 --> 00:02:41.684
They would leave the Eurozone, the new currency would be the new Greek drachma

00:02:41.684 --> 00:02:45.932
and with this it would be probably hugely devalued relative to the Euro

00:02:45.932 --> 00:02:47.856
so you would have savings wiped out,

00:02:47.856 --> 00:02:52.792
maybe not fully wiped out, but savings devalued for Greek savers,

00:02:52.792 --> 00:02:58.228
and they know that already, that's why, because this outcome is looking more and more likely,

00:02:58.228 --> 00:03:01.915
there is essentially a run on banks in Greece right now.

00:03:01.915 --> 00:03:05.695
People are going, withdrawing their Euros, so they can stuff them in their mattresses,

00:03:05.695 --> 00:03:08.307
so they won't be converted in devalued drachmas.

00:03:08.307 --> 00:03:11.854
But this is having the effect of really weak [..?..]

00:03:11.854 --> 00:03:14.097
potentially breaking down the banking system.

00:03:14.097 --> 00:03:21.057
So you could have bank failures.

00:03:21.057 --> 00:03:25.158
And the whole strategy of leaving the Eurozone,

00:03:25.158 --> 00:03:26.800
leaving the Euro and going to the drachma,

00:03:26.800 --> 00:03:29.366
would be to inflate away your liabilities,

00:03:29.366 --> 00:03:33.599
would be literally to inflate away your debt obligations and your entitlement obligations,

00:03:33.599 --> 00:03:37.726
but that inflation, and this has definitively happened in the past with countries in this situation,

00:03:37.726 --> 00:03:43.255
could very easily turn to hyper inflation.

00:03:43.255 --> 00:03:46.527
And so whether you look at austerity in this reality, where you get bailed out,

00:03:46.527 --> 00:03:49.107
or you look at this reality over here,

00:03:49.107 --> 00:03:54.375
in either situation, in the medium term, the economy can really fall apart,

00:03:54.375 --> 00:03:57.947
it's really not clear how, outside of maybe extra generous bailout,

00:03:57.947 --> 00:04:01.679
how that can be avoided.

00:04:01.679 --> 00:04:03.695
And so in that situation, the economy falls apart.

00:04:03.695 --> 00:04:11.283
And when economy falls apart, in a big way,

00:04:11.283 --> 00:04:16.762
and you have major unemployment, it is a scary thing,

00:04:16.762 --> 00:04:18.330
it can lead to social unrest.

00:04:18.330 --> 00:04:22.531
And I'm not talking about 8%, 9% or 10% of uneployment rate like we have in US,

00:04:22.531 --> 00:04:26.068
we are talking about 20%, 30%, 40% unemployment rate, even higher

00:04:26.068 --> 00:04:29.314
in the folks that are likely to be unrestful socially,

00:04:29.314 --> 00:04:32.797
which is the young people, so this could lead to social unrest,

00:04:32.797 --> 00:04:41.130
and even radicalization, when people start worrying about whether they get

00:04:41.130 --> 00:04:43.408
food on the table and they don't have a job,

00:04:43.408 --> 00:04:48.196
they might start supporting people who have more radical views.

00:04:48.196 --> 00:04:52.059
So just that by itself, is very, very scary proposition.

00:04:52.059 --> 00:04:56.024
The history of Europe tells us that even if relatively small countries in Europe

00:04:56.024 --> 00:05:01.134
fall apart in this way, it could have repercussions in the rest of Europe.

00:05:01.134 --> 00:05:04.241
This is how the World War I and World War II got started.

00:05:04.241 --> 00:05:13.183
Just that reason alone is reason enough for people to think very seriously about bailing Greece out.

00:05:13.183 --> 00:05:15.901
And it doesn't come without moral consequences,

00:05:15.901 --> 00:05:18.977
the counter-argument is - if we bailed them out

00:05:18.977 --> 00:05:29.812
doesn't this reward mismanagement and overspending?

00:05:29.812 --> 00:05:35.530
And even a little bit of shady accounting on their national economic statistics.

00:05:35.530 --> 00:05:37.833
So, there's a very good reason, if you are a German tax payer,

00:05:37.833 --> 00:05:41.167
then you are very suspicious of these bailouts.

00:05:41.167 --> 00:05:42.684
Why we are keep running the cheques to the Greeks

00:05:42.684 --> 00:05:44.701
if they are not willing to take some pain?

00:05:44.701 --> 00:05:46.746
Now the Greek side of it, they say "Look, we've already taken amount of pain,

00:05:46.746 --> 00:05:50.646
we are already kind of on the brink, possibly across the brink,

00:05:50.646 --> 00:05:54.999
if you force even more pain on us,

00:05:54.999 --> 00:05:57.910
then we are going to be in really, our society

00:05:57.910 --> 00:06:00.795
is at the risk of falling apart. So, we are in a desperate situation.

00:06:00.795 --> 00:06:05.965
This is not a time to kind of force a moral point on us."

00:06:05.965 --> 00:06:08.633
So that by itself is a reason why people are worried

00:06:08.633 --> 00:06:10.666
but then there is even a bigger reason

00:06:10.666 --> 00:06:13.199
that Greece might not be the only,

00:06:13.199 --> 00:06:15.369
it's only the first, maybe the worst, of the situations

00:06:15.369 --> 00:06:22.988
but if Greece falls apart, and especially if the Eurozone - or, I should say, the European Union or Europe - is not able to bail out Greece

00:06:22.988 --> 00:06:26.281
that's an implicit signal to the rest of world

00:06:26.281 --> 00:06:30.401
that Europe is not able to essentially bail out its countries.

00:06:30.401 --> 00:06:34.675
So, you see on this chart right over here,

00:06:34.675 --> 00:06:37.200
Greece is not alone, it is definitely the worst,

00:06:37.200 --> 00:06:39.658
but right behind it, you have Portugal.

00:06:39.658 --> 00:06:42.683
And its debt, you have long-term debt with interest rates

00:06:42.683 --> 00:06:45.379
looks like around the 12 or 13% range.

00:06:45.379 --> 00:06:51.477
Portugal has a 93% debt to GDP ratio and already has very high unemployment rate.

00:06:51.477 --> 00:06:54.462
If Greece is allowed to leave Eurozone and does not get bailed out,

00:06:54.462 --> 00:06:57.326
investors are going to start wondering

00:06:57.326 --> 00:07:00.706
"Well, hey, maybe Portugal is not going to get bailed out".

00:07:00.706 --> 00:07:03.431
This going to make people expect more interest from Portugal

00:07:03.431 --> 00:07:05.433
in order to lend them money, and they need money

00:07:05.433 --> 00:07:09.004
in order to continue operating in the way they are operating,

00:07:09.004 --> 00:07:11.763
but every increase in a percentage point, right over here,

00:07:11.763 --> 00:07:16.106
is going to really eat into Portuguese GDP and kind of force them down its debt spiral.

00:07:16.106 --> 00:07:18.197
Their debt as percentage of their GDP is almost 100%

00:07:18.197 --> 00:07:20.753
if they have to pay an extra one percent on that,

00:07:20.753 --> 00:07:24.482
that 1% is going to eat into their GDP

00:07:24.482 --> 00:07:28.165
and it is going to force Portugal, the economy to slow down even more

00:07:28.165 --> 00:07:29.908
and make it even more and more onerous

00:07:29.908 --> 00:07:34.328
and they would just kind of go in the same direction as Greece.

00:07:34.328 --> 00:07:38.645
And once again, Portugal is not alone, Italy has a very high debt to GDP,

00:07:38.645 --> 00:07:40.840
Ireland - very high debt to GDP ratio,

00:07:40.840 --> 00:07:43.801
Spain has a very high unemployment rate.

00:07:43.801 --> 00:07:46.724
So, the fear here - and I guess there are two major fears -

00:07:46.724 --> 00:07:51.316
one is that Greece by itself can become a point of instability in Europe,

00:07:51.316 --> 00:07:53.420
but the second fear, if this thing isn't solved,

00:07:53.420 --> 00:07:57.278
that this could cause a kind of contagion or a chain reaction through Europe,

00:07:57.278 --> 00:08:02.291
that people start getting freaked out, they start wanting to not lend to these countries,

00:08:02.291 --> 00:08:04.196
that becomes a self-fulfilling prophecy,

00:08:04.196 --> 00:08:08.067
they might also have to leave the Eurozone.