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So, last week we learned how to
do triple integrals in
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00:00:27 --> 00:00:31
rectangular and cylindrical
coordinates.
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00:00:31 --> 00:00:41
And, now we have to learn about
spherical coordinates,
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00:00:41 --> 00:00:49
which you will see are a lot of
fun.
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00:00:49 --> 00:00:53
So, what's the idea of
spherical coordinates?
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00:00:53 --> 00:00:58
Well, you're going to represent
a point in space using the
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00:00:58 --> 00:01:02
distance to the origin and two
angles.
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00:01:02 --> 00:01:06
So, in a way,
you can think of these as a
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space analog of polar
coordinates because you just use
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00:01:09 --> 00:01:12
distance to the origin,
and then you have to use angles
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00:01:12 --> 00:01:15
to determine in which direction
you're going.
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00:01:15 --> 00:01:22
So, somehow they are more polar
than cylindrical coordinates.
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00:01:22 --> 00:01:25
So, how do we do that?
So, let's say that you have a
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00:01:25 --> 00:01:29
point in space at coordinates x,
y, z.
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00:01:29 --> 00:01:32
Then, instead of using x,
y, z, you will use,
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00:01:32 --> 00:01:37
well, one thing you'll use is
the distance from the origin.
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00:01:37 --> 00:01:41
OK, and that is denoted by the
Greek letter which looks like a
24
00:01:41 --> 00:01:43
curly p, but actually it's the
Greek R.
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00:01:43 --> 00:01:57
So -- That's the distance from
the origin.
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00:01:57 --> 00:02:02
And so, that can take values
anywhere between zero and
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00:02:02 --> 00:02:06
infinity.
Then, we have to use two other
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00:02:06 --> 00:02:09
angles.
And, so for that,
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00:02:09 --> 00:02:16
let me actually draw the
vertical half plane that
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00:02:16 --> 00:02:22
contains our point starting from
the z axis.
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00:02:22 --> 00:02:25
OK, so then we have two new
angles.
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00:02:25 --> 00:02:27
Well, one of them is not really
new.
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00:02:27 --> 00:02:30
One is new.
That's phi is the angle
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00:02:30 --> 00:02:34
downwards from the z axis.
And the other one,
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00:02:34 --> 00:02:39
theta, is the angle
counterclockwise from the x
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00:02:39 --> 00:02:51
axis.
OK, so phi, let me do it better.
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00:02:51 --> 00:02:54
So, there's two ways to draw
the letter phi,
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00:02:54 --> 00:02:56
by the way.
And, I recommend this one
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00:02:56 --> 00:02:58
because it doesn't look like a
rho.
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00:02:58 --> 00:03:07
So, that's easier.
That's the angle that you have
41
00:03:07 --> 00:03:13
to go down from the positive z
axis.
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00:03:13 --> 00:03:17
And,
so that angle varies from zero
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00:03:17 --> 00:03:23
when you're on the z axis,
increase to pi over two when
44
00:03:23 --> 00:03:28
you are on the xy plane all the
way to pi or 180� when you are
45
00:03:28 --> 00:03:38
on the negative z axis.
It doesn't go beyond that.
46
00:03:38 --> 00:03:44
OK, so -- Phi is always between
zero and pi.
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00:03:44 --> 00:03:46
And, finally,
the last one,
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00:03:46 --> 00:03:51
theta, is just going to be the
same as before.
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00:03:51 --> 00:03:55
So, it's the angle after you
project to the xy plane.
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00:03:55 --> 00:04:00
That's the angle
counterclockwise from the x
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00:04:00 --> 00:04:02
axis.
OK, so that's a little bit
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00:04:02 --> 00:04:05
overwhelming not just because of
the new letters,
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00:04:05 --> 00:04:07
but also because there is a lot
of angles in there.
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00:04:07 --> 00:04:11
So, let me just try to,
you know, suggest two things
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00:04:11 --> 00:04:13
that might help you a little
bit.
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00:04:13 --> 00:04:17
So, one is, these are called
spherical coordinates because if
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00:04:17 --> 00:04:21
you fix the value of rho,
then you are moving on a sphere
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00:04:21 --> 00:04:27
centered at the origin.
OK, so let's look at what
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00:04:27 --> 00:04:34
happens on a sphere centered at
the origin, so,
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00:04:34 --> 00:04:41
with equation rho equals a.
Well, then phi measures how far
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00:04:41 --> 00:04:44
south you are going,
measures the distance from the
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North Pole.
So,
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00:04:46 --> 00:04:48
if you've learned about
latitude and longitude in
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geography,
well, phi and theta you can
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think of as latitude and
longitude except with slightly
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different conventions.
OK, so, phi is more or less the
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00:04:59 --> 00:05:05
same thing as latitude in the
sense that it measures how far
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00:05:05 --> 00:05:09
north or south you are.
The only difference is in
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geography,
latitude is zero on the equator
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and becomes something north,
something south,
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00:05:16 --> 00:05:18
depending on how far you go
from the equator.
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00:05:18 --> 00:05:21
Here, you measure a latitude
starting from the North Pole
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00:05:21 --> 00:05:24
which is zero,
increasing all the way to the
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00:05:24 --> 00:05:29
South Pole, which is at pi.
And, theta or you can think of
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00:05:29 --> 00:05:33
as longitude,
which measures how far you are
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00:05:33 --> 00:05:36
east or west.
So, the Greenwich Meridian
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00:05:36 --> 00:05:39
would be here,
now, the one on the x axis.
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That's the one you use as the
origin for longitude,
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OK?
Now, if you don't like
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geography, here's another way to
think about it.
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00:05:51 --> 00:05:56
So -- Let's start again from
cylindrical coordinates,
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00:05:56 --> 00:06:02
which hopefully you're kind of
comfortable with now.
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00:06:02 --> 00:06:06
OK, so you know about
cylindrical coordinates where we
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00:06:06 --> 00:06:09
have the z coordinates stay z,
and the xy plane we do R and
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00:06:09 --> 00:06:13
theta polar coordinates.
And now, let's think about what
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00:06:13 --> 00:06:18
happens when you look at just
one of these vertical planes
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00:06:18 --> 00:06:21
containing the z axis.
So, you have the z axis,
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00:06:21 --> 00:06:25
and then you have the direction
away from the z axis,
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00:06:25 --> 00:06:29
which I will call r,
just because that's what r
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00:06:29 --> 00:06:32
measures.
Of course, r goes all around
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00:06:32 --> 00:06:35
the z axis, but I'm just doing a
slice through one of these
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00:06:35 --> 00:06:39
vertical half planes,
fixing the value of theta.
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00:06:39 --> 00:06:43
Then, r of course is a polar
coordinate seen from the point
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00:06:43 --> 00:06:46
of view of the xy plane.
But here, it looks more like
95
00:06:46 --> 00:06:48
you have rectangular coordinates
again.
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00:06:48 --> 00:06:51
So the idea of spherical
coordinate is you're going to
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00:06:51 --> 00:06:54
polar coordinates again in the
rz plane.
98
00:06:54 --> 00:07:01
OK, so if I have a point here,
then rho will be the distance
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00:07:01 --> 00:07:06
from the origin.
And phi will be the angle,
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00:07:06 --> 00:07:10
except it's measured from the
positive z axis,
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00:07:10 --> 00:07:16
not from the horizontal axis.
But, the idea in here,
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00:07:16 --> 00:07:18
see,
let me put that between quotes
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00:07:18 --> 00:07:21
because I'm not sure how correct
that is,
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00:07:21 --> 00:07:28
but in a way,
you can think of this as polar
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00:07:28 --> 00:07:34
coordinates in the rz plane.
So, in particular,
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00:07:34 --> 00:07:38
that's the key to understanding
how to switch between spherical
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00:07:38 --> 00:07:41
coordinates and cylindrical
coordinates,
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00:07:41 --> 00:07:44
and then all the way to x,
y, z if you want,
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00:07:44 --> 00:07:48
right,
because this picture here tells
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00:07:48 --> 00:07:53
us how to express z and r in
terms of rho and phi.
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00:07:53 --> 00:08:03
So, let's see how that works.
If I project here or here,
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00:08:03 --> 00:08:12
so, this line is z.
But, it's also rho times cosine
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00:08:12 --> 00:08:19
phi.
So, I get z equals rho cos phi.
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00:08:19 --> 00:08:21
And, if I look at r,
it's the same thing,
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00:08:21 --> 00:08:31
but on the other side.
So, r will be rho sine phi.
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00:08:31 --> 00:08:34
OK, so you can use this to
switch back and forth between
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00:08:34 --> 00:08:37
spherical and cylindrical.
And of course,
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00:08:37 --> 00:08:43
if you remember what x and y
were in terms of r and theta,
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00:08:43 --> 00:08:49
you can also keep doing this to
figure out, oops.
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00:08:49 --> 00:08:57
So, x is r cos theta.
That becomes rho sine phi cos
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00:08:57 --> 00:09:01
theta.
Y is r sine theta.
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00:09:01 --> 00:09:06
So, that becomes rho sine phi
sine theta.
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00:09:06 --> 00:09:15
And z is rho cos phi.
But, basically you don't really
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00:09:15 --> 00:09:19
need to remember these formulas
as long as you remember how to
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00:09:19 --> 00:09:22
express r in terms of rho sine
phi,
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00:09:22 --> 00:09:29
and x equals r cos theta.
So, now, of course,
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00:09:29 --> 00:09:31
we're going to use spherical
coordinates in situations where
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00:09:31 --> 00:09:33
we have a lot of symmetry,
and in particular,
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00:09:33 --> 00:09:35
where the z axis plays a
special role.
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00:09:35 --> 00:09:38
Actually, that's the same with
cylindrical coordinates.
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00:09:38 --> 00:09:40
Cylindrical and secure
coordinates are set up so that
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00:09:40 --> 00:09:44
the z axis plays a special role.
So, that means whenever you
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00:09:44 --> 00:09:47
have a geometric problem,
and you are not told how to
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00:09:47 --> 00:09:51
choose your coordinates,
it's probably wiser to try to
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00:09:51 --> 00:09:57
center things on the z axis.
That's where these coordinates
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00:09:57 --> 00:10:01
are the best adapted.
And,
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00:10:01 --> 00:10:03
in case you ever need to switch
backwards,
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00:10:03 --> 00:10:07
I just want to point out,
so, rho is the square root of r
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00:10:07 --> 00:10:11
squared plus z squared,
which means it's the square
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00:10:11 --> 00:10:15
root of x squared plus y squared
plus z squared.
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00:10:15 --> 00:10:20
OK, so that's basically all the
formulas about spherical
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00:10:20 --> 00:10:28
coordinates.
OK, any questions about that?
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00:10:28 --> 00:10:31
OK, let's see,
who had seen spherical
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00:10:31 --> 00:10:34
coordinates before just to see?
OK, that's not very many.
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00:10:34 --> 00:10:36
So, I'm sure for,
one of you saw it twice.
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00:10:36 --> 00:10:42
That's great.
Sorry, oops,
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00:10:42 --> 00:10:49
OK, so let's just look quickly
at equations of some of the
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00:10:49 --> 00:10:54
things.
So, as I've said,
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00:10:54 --> 00:11:02
if I set rho equals a,
that will be just a sphere of
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00:11:02 --> 00:11:11
radius a centered at the origin.
More interesting things:
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00:11:11 --> 00:11:14
let's say I give you phi equals
pi over four.
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00:11:14 --> 00:11:18
What do you think that looks
like?
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00:11:18 --> 00:11:29
Actually, let's take a quick
poll on things.
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00:11:29 --> 00:11:31
OK, yeah, everyone seems to be
saying it's a cone,
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00:11:31 --> 00:11:33
and that's indeed the correct
answer.
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00:11:33 --> 00:11:41
So, how do we see that?
Well, remember,
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00:11:41 --> 00:11:44
phi is the angle downward from
the z axis.
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00:11:44 --> 00:11:49
So, let's say that I'm going to
look first at what happens if
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00:11:49 --> 00:11:53
I'm in the right half of a plane
of a blackboard,
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00:11:53 --> 00:11:56
so, in the yz plane.
Then, phi is the angle downward
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00:11:56 --> 00:11:58
from here.
So, if I want to get pi over
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00:11:58 --> 00:12:01
four, that's 45�.
That means I'm going to go
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00:12:01 --> 00:12:03
diagonally like this.
Of course, if I'm in the left
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00:12:03 --> 00:12:06
half of a plane of a blackboard,
it's going to be the same.
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00:12:06 --> 00:12:10
I also take pi over four.
And, I get the other half.
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00:12:10 --> 00:12:13
And, because the equation does
not involve theta,
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00:12:13 --> 00:12:17
it's all the same if I rotate
my vertical plane around the z
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00:12:17 --> 00:12:21
axis.
So, I get the same picture in
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00:12:21 --> 00:12:27
any of these vertical half
planes, actually.
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00:12:27 --> 00:12:32
OK, now, so this is phi equals
pi over four.
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00:12:32 --> 00:12:35
And, just in case,
to point out to you what's
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00:12:35 --> 00:12:39
going on, when phi equals pi
over four, cosine and sine are
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00:12:39 --> 00:12:42
equal to each other.
They are both one over root two.
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00:12:42 --> 00:12:46
So, you can find,
again, the equation of this
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00:12:46 --> 00:12:51
thing in cylindrical
coordinates, which I'll remind
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00:12:51 --> 00:12:54
you was z equals r.
OK, in general,
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00:12:54 --> 00:12:58
phi equals some given number,
or z equals some number times
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00:12:58 --> 00:13:01
r.
That will be a cone centered on
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00:13:01 --> 00:13:04
the z axis.
OK, a special case:
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00:13:04 --> 00:13:07
what if I say phi equals pi
over two?
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00:13:07 --> 00:13:09
Yeah, it's just going to be the
xy plane.
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00:13:09 --> 00:13:13
OK, that's the flattest of all
cones.
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00:13:13 --> 00:13:20
OK, so phi equals pi over two
is going to be just the xy
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00:13:20 --> 00:13:22
plane.
And, in general,
185
00:13:22 --> 00:13:24
if phi is less than pi over
two, then you are in the upper
186
00:13:24 --> 00:13:28
half space.
If phi is more than pi over
187
00:13:28 --> 00:13:32
two, you'll be in the lower half
space.
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00:13:32 --> 00:13:36
OK, so that's pretty much all
we need to know at this point.
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00:13:36 --> 00:13:45
So, what's next?
Well, remember we were trying
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00:13:45 --> 00:13:52
to do triple integrals.
So now we're going to triple
191
00:13:52 --> 00:13:59
integrals in spherical
coordinates.
192
00:13:59 --> 00:14:01
And, for that,
we first need to understand
193
00:14:01 --> 00:14:06
what the volume element is.
What will be dV?
194
00:14:06 --> 00:14:12
OK, so dV will be something,
d rho, d phi,
195
00:14:12 --> 00:14:18
d theta, or in any order that
you want.
196
00:14:18 --> 00:14:23
But, this one is usually the
most convenient.
197
00:14:23 --> 00:14:27
So, to find out what it is,
well, we should look at how we
198
00:14:27 --> 00:14:29
are going to be slicing things
now.
199
00:14:29 --> 00:14:32
OK, so if you integrate d rho,
d phi, d theta,
200
00:14:32 --> 00:14:37
it means that you are actually
slicing your solid into little
201
00:14:37 --> 00:14:40
pieces that live,
somehow,
202
00:14:40 --> 00:14:45
if you set an interval of rows,
OK,
203
00:14:45 --> 00:14:48
sorry, maybe I should,
so, if you first integrate over
204
00:14:48 --> 00:14:51
rho,
it means that you will actually
205
00:14:51 --> 00:14:57
choose first the direction from
the origin even by phi and
206
00:14:57 --> 00:15:00
theta.
And, in that direction,
207
00:15:00 --> 00:15:04
you will try to figure out,
how far does your region
208
00:15:04 --> 00:15:07
extend?
And, of course,
209
00:15:07 --> 00:15:11
how far that goes might depend
on phi and theta.
210
00:15:11 --> 00:15:16
Then, you will vary phi.
So, you have to know,
211
00:15:16 --> 00:15:21
for a given value of theta,
how far down does your solid
212
00:15:21 --> 00:15:22
extend?
And, finally,
213
00:15:22 --> 00:15:25
the value of theta will
correspond to,
214
00:15:25 --> 00:15:28
in which directions around the
z axis do we go?
215
00:15:28 --> 00:15:31
So, we're going to see that in
examples.
216
00:15:31 --> 00:15:34
But before we can do that,
we need to get the volume
217
00:15:34 --> 00:15:36
element.
So, what I would like to
218
00:15:36 --> 00:15:40
suggest is that we need to
figure out,
219
00:15:40 --> 00:15:46
what is the volume of a small
piece of solid which corresponds
220
00:15:46 --> 00:15:49
to a certain change,
delta rho,
221
00:15:49 --> 00:15:52
delta phi,
and delta theta?
222
00:15:52 --> 00:15:56
So, delta rho means that you
have two concentric spheres,
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00:15:56 --> 00:16:01
and you are looking at a very
thin shell in between them.
224
00:16:01 --> 00:16:05
And then, you would be looking
at a piece of that spherical
225
00:16:05 --> 00:16:08
shell corresponding to small
values of phi and theta.
226
00:16:08 --> 00:16:14
So, because I am stretching the
limits of my ability to draw on
227
00:16:14 --> 00:16:18
the board, here's a picture.
I'm going to try to reproduce
228
00:16:18 --> 00:16:21
on the board,
but so let's start by looking
229
00:16:21 --> 00:16:24
just at what happens on the
sphere of radius a,
230
00:16:24 --> 00:16:28
and let's try to figure out the
surface area elements on the
231
00:16:28 --> 00:16:30
sphere in terms of phi and
theta.
232
00:16:30 --> 00:16:39
And then, we'll add the rho
direction.
233
00:16:39 --> 00:16:49
OK, so -- So,
let me say, let's start by
234
00:16:49 --> 00:17:02
understanding surface area on a
sphere of radius a.
235
00:17:02 --> 00:17:12
So, that means we'll be looking
at a little piece of the sphere
236
00:17:12 --> 00:17:21
corresponding to angles delta
phi and in that direction here
237
00:17:21 --> 00:17:26
delta theta.
OK, so when you draw a map of
238
00:17:26 --> 00:17:29
the world on a globe,
that's exactly what the grid
239
00:17:29 --> 00:17:33
lines form for you.
So, what's the area of this guy?
240
00:17:33 --> 00:17:35
Well, of course,
all the sides are curvy.
241
00:17:35 --> 00:17:37
They are all on the sphere.
None of them are straight.
242
00:17:37 --> 00:17:41
But still, if it's small enough
and it looks like a rectangle,
243
00:17:41 --> 00:17:46
so let's just try to figure
out, what are the sides of your
244
00:17:46 --> 00:17:49
rectangle?
OK, so, let's see,
245
00:17:49 --> 00:17:55
well, I think I need to draw a
bigger picture of this guy.
246
00:17:55 --> 00:17:59
OK, so this guy,
so that's a piece of what's
247
00:17:59 --> 00:18:05
called a parallel in geography.
That's a circle that goes
248
00:18:05 --> 00:18:07
east-west.
So now,
249
00:18:07 --> 00:18:10
this parallel as a circle of
radius,
250
00:18:10 --> 00:18:14
well, the radius is less than a
because if your vertical is to
251
00:18:14 --> 00:18:17
the North Pole,
it will be actually much
252
00:18:17 --> 00:18:19
smaller.
So, that's why when you say
253
00:18:19 --> 00:18:22
you're going around the world it
depends on whether you do it at
254
00:18:22 --> 00:18:28
the equator or the North Pole.
It's much easier at the North
255
00:18:28 --> 00:18:33
Pole.
So, anyway, this is a piece of
256
00:18:33 --> 00:18:40
a circle of radius,
well, the radius is what I
257
00:18:40 --> 00:18:49
would call r because that's the
distance from the z axis.
258
00:18:49 --> 00:18:51
OK, that's actually pretty hard
to see now.
259
00:18:51 --> 00:18:58
So if you can see it better on
this one, then so this guy here,
260
00:18:58 --> 00:19:03
this length is r.
And, r is just rho,
261
00:19:03 --> 00:19:07
well, what was a times sine
phi.
262
00:19:07 --> 00:19:09
Remember, we have this angle
phi in here.
263
00:19:09 --> 00:19:14
I should use some color.
It's getting very cluttered.
264
00:19:14 --> 00:19:19
So, we have this phi,
and so r is going to be rho
265
00:19:19 --> 00:19:21
sine phi.
That rho is a.
266
00:19:21 --> 00:19:29
So, let me just put a sine phi.
OK, and the corresponding angle
267
00:19:29 --> 00:19:32
is going to be measured by
theta.
268
00:19:32 --> 00:19:48
So, the length of this is going
to be a sine phi delta theta.
269
00:19:48 --> 00:19:54
That's for this side.
Now, what about that side,
270
00:19:54 --> 00:19:56
the north-south side?
Well, if you're moving
271
00:19:56 --> 00:19:58
north-south, it's not like
east-west.
272
00:19:58 --> 00:20:01
You always have to go all the
way from the North Pole to the
273
00:20:01 --> 00:20:04
South Pole.
So, that's actually a great
274
00:20:04 --> 00:20:08
circle meridian of length,
well, I mean,
275
00:20:08 --> 00:20:13
well, the radius is the radius
of the sphere.
276
00:20:13 --> 00:20:22
Total length is 2pi a.
So, this is a piece of a circle
277
00:20:22 --> 00:20:27
of radius a.
And so, now,
278
00:20:27 --> 00:20:34
the length of this one is going
to be a delta phi.
279
00:20:34 --> 00:20:41
OK, so, just to recap,
this is a sine phi delta theta.
280
00:20:41 --> 00:20:46
And, this guy here is a delta
phi.
281
00:20:46 --> 00:20:59
So, you can't read it because
it's -- And so,
282
00:20:59 --> 00:21:02
that tells us if I take a small
piece of the sphere,
283
00:21:02 --> 00:21:06
then its surface area,
delta s,
284
00:21:06 --> 00:21:15
is going to be approximately a
sine phi delta theta times a
285
00:21:15 --> 00:21:22
delta phi,
which I'm going to rewrite as a
286
00:21:22 --> 00:21:27
squared sine phi delta phi delta
theta.
287
00:21:27 --> 00:21:31
So, what that means is,
say that I want to integrate
288
00:21:31 --> 00:21:34
something just on the surface of
a sphere.
289
00:21:34 --> 00:21:37
Well, I would use phi and theta
as my coordinates.
290
00:21:37 --> 00:21:46
And then, to know how big a
piece of a sphere is,
291
00:21:46 --> 00:21:55
I would just take a squared
sine phi d phi d theta.
292
00:21:55 --> 00:21:59
OK, so that's the surface
element in a sphere.
293
00:21:59 --> 00:22:03
And now, what about going back
into the third dimension,
294
00:22:03 --> 00:22:05
so, adding some depth to these
things?
295
00:22:05 --> 00:22:10
Well, I'm not going to try to
draw a picture because you've
296
00:22:10 --> 00:22:17
seen that's slightly tricky.
Well, let me try anyway just
297
00:22:17 --> 00:22:24
you can have fun with my
completely unreadable diagrams.
298
00:22:24 --> 00:22:28
So anyway, if you look at,
now, something that's a bit
299
00:22:28 --> 00:22:33
like that piece of sphere,
but with some thickness to it.
300
00:22:33 --> 00:22:38
The thickness will be delta
rho, and so the volume will be
301
00:22:38 --> 00:22:44
roughly the area of the thing on
the sphere times the thickness.
302
00:22:44 --> 00:22:48
So, I claim that we will get
basically the volume element
303
00:22:48 --> 00:22:51
just by multiplying things by d
rho.
304
00:22:51 --> 00:23:10
So, let's see that.
So now, if I have a sphere of
305
00:23:10 --> 00:23:19
radius rho, and another one
that's slightly bigger of radius
306
00:23:19 --> 00:23:27
rho plus delta rho,
and then I have a little box in
307
00:23:27 --> 00:23:29
here.
Then,
308
00:23:29 --> 00:23:34
I know that the volume of this
thing will be essentially,
309
00:23:34 --> 00:23:38
well, its thickness,
the thickness is going to be
310
00:23:38 --> 00:23:42
delta rho times the area of its
base,
311
00:23:42 --> 00:23:44
although it doesn't really
matter,
312
00:23:44 --> 00:23:48
which is what we've called
delta s.
313
00:23:48 --> 00:23:55
OK, so we will get,
sorry, a becomes rho now.
314
00:23:55 --> 00:23:57
Square sine phi,
delta rho,
315
00:23:57 --> 00:24:00
delta phi,
delta theta,
316
00:24:00 --> 00:24:04
and so out of that we get the
volume element and spherical
317
00:24:04 --> 00:24:08
coordinates,
which is rho squared sine phi d
318
00:24:08 --> 00:24:09
rho,
d phi,
319
00:24:09 --> 00:24:14
d theta.
And, that's a formula that you
320
00:24:14 --> 00:24:17
should remember.
OK, so whenever we integrate a
321
00:24:17 --> 00:24:20
function,
and we decide to switch to
322
00:24:20 --> 00:24:25
spherical coordinates,
then dx dy dz or r dr d theta
323
00:24:25 --> 00:24:33
dz will become rho squared sine
phi d rho d phi d theta.
324
00:24:33 --> 00:24:40
OK, any questions on that?
No?
325
00:24:40 --> 00:24:58
OK, so let's -- Let's see how
that works.
326
00:24:58 --> 00:25:04
So, as an example,
remember at the end of the last
327
00:25:04 --> 00:25:11
lecture, I tried to set up an
example where we were looking at
328
00:25:11 --> 00:25:16
a sphere sliced by a slanted
plane.
329
00:25:16 --> 00:25:20
And now, we're going to try to
find the volume of that
330
00:25:20 --> 00:25:23
spherical cap again,
but using spherical coordinates
331
00:25:23 --> 00:25:26
instead.
So, I'm going to just be
332
00:25:26 --> 00:25:29
smarter than last time.
So, last time,
333
00:25:29 --> 00:25:33
we had set up these things with
a slanted plane that was cutting
334
00:25:33 --> 00:25:35
things diagonally.
And,
335
00:25:35 --> 00:25:37
if I just want to find the
volume of this cap,
336
00:25:37 --> 00:25:41
then maybe it makes more sense
to rotate things so that my
337
00:25:41 --> 00:25:45
plane is actually horizontal,
and things are going to be
338
00:25:45 --> 00:25:49
centered on the z axis.
So, in case you see that it's
339
00:25:49 --> 00:25:52
the same, then that's great.
If not, then it doesn't really
340
00:25:52 --> 00:25:55
matter.
You can just think of this as a
341
00:25:55 --> 00:26:01
new example.
So, I'm going to try to find
342
00:26:01 --> 00:26:10
the volume of a portion of the
unit sphere -- -- that lies
343
00:26:10 --> 00:26:20
above the horizontal plane,
z equals one over root two.
344
00:26:20 --> 00:26:22
OK, one over root two was the
distance from the origin to our
345
00:26:22 --> 00:26:24
slanted plane.
So, after you rotate,
346
00:26:24 --> 00:26:28
that say you get this value.
Anyway, it's not very important.
347
00:26:28 --> 00:26:31
You can just treat that as a
good example if you want.
348
00:26:31 --> 00:26:36
OK, so we can compute this in
actually pretty much any
349
00:26:36 --> 00:26:39
coordinate system.
And also, of course,
350
00:26:39 --> 00:26:42
we can set up not only the
volume, but we can try to find
351
00:26:42 --> 00:26:44
the moment of inertia about the
central axis,
352
00:26:44 --> 00:26:47
or all sorts of things.
But, we are just doing the
353
00:26:47 --> 00:26:49
volume for simplicity.
So, actually,
354
00:26:49 --> 00:26:52
this would go pretty well in
cylindrical coordinates.
355
00:26:52 --> 00:26:55
But let's do it in spherical
coordinates because that's the
356
00:26:55 --> 00:26:57
topic of today.
A good exercise:
357
00:26:57 --> 00:27:01
do it in cylindrical and see if
you get the same thing.
358
00:27:01 --> 00:27:08
So, how do we do that?
Well, we have to figure out how
359
00:27:08 --> 00:27:14
to set up our triple integral in
spherical coordinates.
360
00:27:14 --> 00:27:18
So, remember we'll be
integrating one dV.
361
00:27:18 --> 00:27:28
So, dV will become rho squared
sign phi d rho d phi d theta.
362
00:27:28 --> 00:27:32
And, now as we start,
we're already facing some
363
00:27:32 --> 00:27:35
serious problem.
We want to set up the bounds
364
00:27:35 --> 00:27:37
for rho for a given,
phi and theta.
365
00:27:37 --> 00:27:39
So, that means we choose
latitude/longitude.
366
00:27:39 --> 00:27:42
We choose which direction we
want to aim for,
367
00:27:42 --> 00:27:45
you know, which point of the
sphere we want to aim at.
368
00:27:45 --> 00:27:50
And, we are going to shoot a
ray from the origin towards this
369
00:27:50 --> 00:27:55
point, and we want to know what
portion of the ray is in our
370
00:27:55 --> 00:28:03
solid.
So -- We are going to choose a
371
00:28:03 --> 00:28:11
value of phi and theta.
And, we are going to try to
372
00:28:11 --> 00:28:16
figure out what part of our ray
is inside this side.
373
00:28:16 --> 00:28:20
So, what should be clear is at
which point we leave the solid,
374
00:28:20 --> 00:28:23
right?
What's the value of rho here?
375
00:28:23 --> 00:28:25
It's just one.
The sphere is rho equals one.
376
00:28:25 --> 00:28:29
That's pretty good.
The question is,
377
00:28:29 --> 00:28:33
where do we enter the region?
So, we enter the region when we
378
00:28:33 --> 00:28:38
go through this plane.
And, the plane is z equals one
379
00:28:38 --> 00:28:41
over root two.
So, what does that tell us
380
00:28:41 --> 00:28:44
about rho?
Well, it tells us,
381
00:28:44 --> 00:28:50
so remember,
z is rho cosine phi.
382
00:28:50 --> 00:28:55
So, the plane is z equals one
over root two.
383
00:28:55 --> 00:29:00
That means rho cosine phi is
one over root two.
384
00:29:00 --> 00:29:05
That means rho equals one over
root two cosine phi or,
385
00:29:05 --> 00:29:11
as some of you know it,
one over root two times second
386
00:29:11 --> 00:29:17
phi.
OK, so if we want to set up the
387
00:29:17 --> 00:29:27
bounds, then we'll start with
one over root two second phi all
388
00:29:27 --> 00:29:32
the way to one.
Now, what's next?
389
00:29:32 --> 00:29:35
Well, so we've done,
I think that's basically the
390
00:29:35 --> 00:29:38
hardest part of the job.
Next, we have to figure out,
391
00:29:38 --> 00:29:41
what's the range for phi?
So, the range for phi,
392
00:29:41 --> 00:29:44
well, we have to figure out how
far to the north and to the
393
00:29:44 --> 00:29:48
south our region goes.
Well, the lower bound for phi
394
00:29:48 --> 00:29:51
is pretty easy,
right, because we go all the
395
00:29:51 --> 00:29:56
way to the North Pole direction.
So, phi starts at zero.
396
00:29:56 --> 00:29:59
The question is,
where does it stop?
397
00:29:59 --> 00:30:02
To find out where it stops,
we have to figure out,
398
00:30:02 --> 00:30:06
what is the value of phi when
we hit the edge of the region?
399
00:30:06 --> 00:30:10
OK, so maybe you see it.
Maybe you don't.
400
00:30:10 --> 00:30:15
One way to do it geometrically
is to just, it's always great to
401
00:30:15 --> 00:30:19
draw a slice of your region.
So, if you slice these things
402
00:30:19 --> 00:30:22
by a vertical plane,
or actually even better,
403
00:30:22 --> 00:30:25
a vertical half plane,
something to delete one half of
404
00:30:25 --> 00:30:28
the picture.
So, I'm going to draw these r
405
00:30:28 --> 00:30:33
and z directions as before.
So, my sphere is here.
406
00:30:33 --> 00:30:38
My plane is here at one over
root two.
407
00:30:38 --> 00:30:43
And, my solid is here.
So now, the question is what is
408
00:30:43 --> 00:30:49
the value of phi when I'm going
to stop hitting the region?
409
00:30:49 --> 00:30:54
And, if you try to figure out
first what is this direction
410
00:30:54 --> 00:30:57
here, that's also one over root
two.
411
00:30:57 --> 00:31:03
And so, this is actually 45�,
also known as pi over four.
412
00:31:03 --> 00:31:09
The other way to think about it
is at this point,
413
00:31:09 --> 00:31:16
well, rho is equal to one
because you are on the sphere.
414
00:31:16 --> 00:31:22
But, you are also on the plane.
So, rho cos phi is one over
415
00:31:22 --> 00:31:26
root two.
So, if you plug rho equals one
416
00:31:26 --> 00:31:31
into here, you get cos phi
equals one over root two which
417
00:31:31 --> 00:31:34
gives you phi equals pi over
four.
418
00:31:34 --> 00:31:37
That's the other way to do it.
You can do it either by
419
00:31:37 --> 00:31:39
calculation or by looking at the
picture.
420
00:31:39 --> 00:31:43
OK, so either way,
we've decided that phi goes
421
00:31:43 --> 00:31:48
from zero to pi over four.
So, this is pi over four.
422
00:31:48 --> 00:31:54
Finally, what about theta?
Well, because we go all around
423
00:31:54 --> 00:32:00
the z axis we are going to go
just zero to 2pi.
424
00:32:00 --> 00:32:06
OK, any questions about these
bounds?
425
00:32:06 --> 00:32:10
OK, so note how the equation of
this horizontal plane in
426
00:32:10 --> 00:32:13
spherical coordinates has become
a little bit weird.
427
00:32:13 --> 00:32:16
But,
if you remember how we do
428
00:32:16 --> 00:32:19
things,
say that you have a line in
429
00:32:19 --> 00:32:21
polar coordinates,
and that line does not pass
430
00:32:21 --> 00:32:23
through the origin,
then you also end up with
431
00:32:23 --> 00:32:26
something like that.
You get something like r equals
432
00:32:26 --> 00:32:31
a second theta or a cos second
theta for horizontal or vertical
433
00:32:31 --> 00:32:33
lines.
And so, it's not surprising you
434
00:32:33 --> 00:32:38
should get this.
That's in line with the idea
435
00:32:38 --> 00:32:44
that we are just doing again,
polar coordinates in the rz
436
00:32:44 --> 00:32:46
directions.
So of course,
437
00:32:46 --> 00:32:48
in general, things can be very
messy.
438
00:32:48 --> 00:32:51
But, generally speaking,
the kinds of regions that we
439
00:32:51 --> 00:32:55
will be setting up things for
are no more complicated or no
440
00:32:55 --> 00:32:59
less complicated than what we
would do in the plane in polar
441
00:32:59 --> 00:33:00
coordinates.
OK, so there's,
442
00:33:00 --> 00:33:03
you know, a small list of
things that you should know how
443
00:33:03 --> 00:33:07
to set up.
But, you won't have some
444
00:33:07 --> 00:33:18
really, really strange thing.
Yes?
445
00:33:18 --> 00:33:20
D rho?
Oh, you mean the bounds for rho?
446
00:33:20 --> 00:33:23
Yes.
So, in the inner integral,
447
00:33:23 --> 00:33:26
we are going to fix values of
phi and theta.
448
00:33:26 --> 00:33:29
So, that means we fix in
advance the direction in which
449
00:33:29 --> 00:33:31
we are going to shoot a ray from
the origin.
450
00:33:31 --> 00:33:35
So now, as we shoot this ray,
we are going to hit our region
451
00:33:35 --> 00:33:37
somewhere.
And, we are going to exit,
452
00:33:37 --> 00:33:40
again, somewhere else.
OK, so first of all we have to
453
00:33:40 --> 00:33:43
figure out where we enter,
where we leave.
454
00:33:43 --> 00:33:46
Well, we enter when the ray
hits the flat face,
455
00:33:46 --> 00:33:50
when we hit the plane.
And, we would leave when we hit
456
00:33:50 --> 00:33:52
the sphere.
So, the lower bound will be
457
00:33:52 --> 00:33:56
given by the plane.
The upper bound will be given
458
00:33:56 --> 00:33:58
by the sphere.
So now, you have to get
459
00:33:58 --> 00:34:01
spherical coordinate equations
for both the plane and the
460
00:34:01 --> 00:34:02
sphere.
For the sphere, that's easy.
461
00:34:02 --> 00:34:05
That's rho equals one.
For the plane,
462
00:34:05 --> 00:34:08
you start with z equals one
over root two.
463
00:34:08 --> 00:34:11
And, you switch it into
spherical coordinates.
464
00:34:11 --> 00:34:14
And then, you solve for rho.
And, that's how you get these
465
00:34:14 --> 00:34:19
bounds.
Is that OK?
466
00:34:19 --> 00:34:26
All right, so that's the setup
part.
467
00:34:26 --> 00:34:29
And, of course,
the evaluation part goes as
468
00:34:29 --> 00:34:30
usual.
469
00:34:30 --> 00:34:42
470
00:34:42 --> 00:34:46
And, since I'm running short of
time, I'm not going to actually
471
00:34:46 --> 00:34:52
do the evaluation.
I'm going to let you figure out
472
00:34:52 --> 00:34:58
how it goes.
Let me just say in case you
473
00:34:58 --> 00:35:07
want to check your answers,
so, at the end you get 2pi over
474
00:35:07 --> 00:35:13
three minus 5pi over six root
two.
475
00:35:13 --> 00:35:17
Yes, it looks quite complicated.
That's basically because you
476
00:35:17 --> 00:35:20
get one over,
well, you get a second square
477
00:35:20 --> 00:35:23
when you integrate C.
When you integrate rho squared,
478
00:35:23 --> 00:35:24
you will get rho cubed over
three.
479
00:35:24 --> 00:35:27
But that rho cubed will give
you a second cube for the lower
480
00:35:27 --> 00:35:29
bound.
And, when you integrate sine
481
00:35:29 --> 00:35:31
phi second cubed phi,
you do a substitution.
482
00:35:31 --> 00:35:37
You see that integrates to one
over second squared with a
483
00:35:37 --> 00:35:42
factor in front.
So, in the second square,
484
00:35:42 --> 00:35:49
when you plug in,
no, that's not quite all of it.
485
00:35:49 --> 00:35:51
Yeah, well, the second square
is one thing,
486
00:35:51 --> 00:35:53
and also the other bound you
get sine phi which integrates to
487
00:35:53 --> 00:35:56
cosine phi.
So, anyways,
488
00:35:56 --> 00:36:04
you get lots of things.
OK, enough about it.
489
00:36:04 --> 00:36:07
So, next, I have to tell you
about applications.
490
00:36:07 --> 00:36:13
And, of course,
well, there's the same
491
00:36:13 --> 00:36:14
applications that we've seen
that last time,
492
00:36:14 --> 00:36:16
finding volumes,
finding masses,
493
00:36:16 --> 00:36:19
finding average values of
functions.
494
00:36:19 --> 00:36:22
In particular,
now, we could say to find the
495
00:36:22 --> 00:36:26
average distance of a point in
this solid to the origin.
496
00:36:26 --> 00:36:28
Well,
spherical coordinates become
497
00:36:28 --> 00:36:32
appealing because the function
you are averaging is just rho
498
00:36:32 --> 00:36:35
while in other coordinate
systems it's a more complicated
499
00:36:35 --> 00:36:37
function.
So, if you are asked to find
500
00:36:37 --> 00:36:41
the average distance from the
origin, spherical coordinates
501
00:36:41 --> 00:36:43
can be interesting.
Also,
502
00:36:43 --> 00:36:47
well, there's moments of
inertia,
503
00:36:47 --> 00:36:50
preferably the one about the z
axis because if you have to
504
00:36:50 --> 00:36:52
integrate something that
involves x or y,
505
00:36:52 --> 00:36:55
then your integrand will
contain that awful rho sine phi
506
00:36:55 --> 00:36:57
sine theta or rho sine phi
cosine theta,
507
00:36:57 --> 00:37:00
and then it won't be much fun
to evaluate.
508
00:37:00 --> 00:37:05
So, that anyway,
there's the usual ones.
509
00:37:05 --> 00:37:08
And then there's a new one.
So, in physics,
510
00:37:08 --> 00:37:16
you've probably seen things
about gravitational attraction.
511
00:37:16 --> 00:37:19
If not, well,
it's what causes apples to fall
512
00:37:19 --> 00:37:22
and other things like that as
well.
513
00:37:22 --> 00:37:26
So, anyway, physics tells you
that if you have two masses,
514
00:37:26 --> 00:37:30
then they attract each other
with a force that's directed
515
00:37:30 --> 00:37:33
towards each other.
And in intensity,
516
00:37:33 --> 00:37:37
it's proportional to the two
masses, and inversely
517
00:37:37 --> 00:37:41
proportional to the square of
the distance between them.
518
00:37:41 --> 00:37:45
So,
if you have a given solid with
519
00:37:45 --> 00:37:50
a certain mass distribution,
and you want to know how it
520
00:37:50 --> 00:37:53
attracts something else that you
will put nearby,
521
00:37:53 --> 00:37:58
then you actually have to,
the first approximation will be
522
00:37:58 --> 00:37:59
to say,
well, let's just put a point
523
00:37:59 --> 00:38:02
mass at its center of mass.
But, if you're solid is
524
00:38:02 --> 00:38:04
actually not homogenous,
or has a weird shape,
525
00:38:04 --> 00:38:07
then that's not actually the
exact answer.
526
00:38:07 --> 00:38:09
So, in general,
you would have to just take
527
00:38:09 --> 00:38:12
every single piece of your
object and figure out how it
528
00:38:12 --> 00:38:14
attracts you,
and then compute the sum of
529
00:38:14 --> 00:38:15
these.
So, for example,
530
00:38:15 --> 00:38:18
if you want to understand why
anything that you drop in this
531
00:38:18 --> 00:38:21
room will fall down,
you have to understand that
532
00:38:21 --> 00:38:24
Boston is actually attracting it
towards Boston.
533
00:38:24 --> 00:38:26
And, Somerville's attracting it
towards Somerville,
534
00:38:26 --> 00:38:29
and lots of things like that.
And, China, which is much
535
00:38:29 --> 00:38:33
further on the other side is
going to attract towards China.
536
00:38:33 --> 00:38:35
But, there's a lot of stuff on
the other side of the Earth.
537
00:38:35 --> 00:38:37
And so, overall,
it's supposed to end up just
538
00:38:37 --> 00:38:41
going down.
OK, so now, how to find this
539
00:38:41 --> 00:38:47
out, well, you have to just
integrate over the entire Earth.
540
00:38:47 --> 00:38:52
OK, so let's try to see how
that goes.
541
00:38:52 --> 00:38:56
So, the setup that's going to
be easiest for us to do
542
00:38:56 --> 00:39:01
computations is going to be that
we are going to be the test mass
543
00:39:01 --> 00:39:04
that's going to be falling.
And, we are going to put
544
00:39:04 --> 00:39:07
ourselves at the origin.
And, the solid that's going to
545
00:39:07 --> 00:39:10
attract us is going to be
wherever we want in space.
546
00:39:10 --> 00:39:13
You'll see, putting yourself at
the origin is going to be
547
00:39:13 --> 00:39:15
better.
Well, you have to put something
548
00:39:15 --> 00:39:17
at the origin.
And, the one that will stay a
549
00:39:17 --> 00:39:21
point mass, I mean,
in my case not really a point,
550
00:39:21 --> 00:39:24
but anyway, let's say that I'm
a point.
551
00:39:24 --> 00:39:27
And then, I have a solid
attracting me.
552
00:39:27 --> 00:39:32
Well,
so then if I take a small piece
553
00:39:32 --> 00:39:37
of it with the mass delta M,
then that portion of the solid
554
00:39:37 --> 00:39:42
exerts a force on me,
which is going to be directed
555
00:39:42 --> 00:39:47
towards it,
and we'll have intensity.
556
00:39:47 --> 00:39:59
So, the gravitational force --
-- exerted by the mass delta M
557
00:39:59 --> 00:40:09
at the point of x,
y, z in space on a mass at the
558
00:40:09 --> 00:40:13
origin.
Well, we know how to express
559
00:40:13 --> 00:40:16
that.
Physics tells us that the
560
00:40:16 --> 00:40:21
magnitude of this force is going
to be, well, G is just a
561
00:40:21 --> 00:40:23
constant.
It's the gravitational
562
00:40:23 --> 00:40:27
constant, and its value depends
on which unit system you use.
563
00:40:27 --> 00:40:33
Usually it's pretty small,
times the mass delta M,
564
00:40:33 --> 00:40:39
times the test mass little m,
divided by the square of the
565
00:40:39 --> 00:40:43
distance.
And, the distance from U to
566
00:40:43 --> 00:40:48
that thing is conveniently
called rho since we've been
567
00:40:48 --> 00:40:51
introducing spherical
coordinates.
568
00:40:51 --> 00:40:54
So, that's the size,
that's the magnitude of the
569
00:40:54 --> 00:40:56
force.
We also need to know the
570
00:40:56 --> 00:41:01
direction of the force.
And, the direction is going to
571
00:41:01 --> 00:41:07
be towards that point.
So, the direction of the force
572
00:41:07 --> 00:41:11
is going to be that of x,
y, z.
573
00:41:11 --> 00:41:13
But if I want a unit vector,
then I should scale this down
574
00:41:13 --> 00:41:22
to length one.
So, let me divide this by rho
575
00:41:22 --> 00:41:32
to get a unit vector.
So, that means that the force
576
00:41:32 --> 00:41:40
I'm getting from this guy is
actually going to be G delta M m
577
00:41:40 --> 00:41:44
over rho cubed times x,
y, z.
578
00:41:44 --> 00:41:50
I'm just multiplying the
magnitude by the unit vector in
579
00:41:50 --> 00:41:54
the correct direction.
OK, so now if I have not just
580
00:41:54 --> 00:41:56
that little p is delta M,
but an entire solid,
581
00:41:56 --> 00:41:59
then I have to sum all these
guys together.
582
00:41:59 --> 00:42:04
And, I will get the vector that
gives me the total force
583
00:42:04 --> 00:42:06
exerted, OK?
So, of course,
584
00:42:06 --> 00:42:09
there's actually three
different calculations in one
585
00:42:09 --> 00:42:12
because you have to sum the x
components to get the x
586
00:42:12 --> 00:42:16
components of a total force.
Same with the y,
587
00:42:16 --> 00:42:28
and same with the z.
So, let me first write down the
588
00:42:28 --> 00:42:36
actual formula.
So, if you integrate over the
589
00:42:36 --> 00:42:39
entire solid,
oh, and I have to remind you,
590
00:42:39 --> 00:42:42
well, what's the mass,
delta M of a small piece of
591
00:42:42 --> 00:42:45
volume delta V?
Well, it's the density times
592
00:42:45 --> 00:42:48
the volume.
So, the mass is going to be,
593
00:42:48 --> 00:42:54
sorry, density is delta.
There is a lot of Greek letters
594
00:42:54 --> 00:43:04
there, times the volume element.
So, you will get that the force
595
00:43:04 --> 00:43:12
is the triple integral over your
solid of G m x,
596
00:43:12 --> 00:43:18
y, z over rho cubed,
delta dV.
597
00:43:18 --> 00:43:21
Now, two observations about
that.
598
00:43:21 --> 00:43:23
So, the first one,
well, of course,
599
00:43:23 --> 00:43:29
these are just constants.
So, they can go out.
600
00:43:29 --> 00:43:31
The second observation,
so here, we are integrating a
601
00:43:31 --> 00:43:33
vector quantity.
So, what does that mean?
602
00:43:33 --> 00:43:38
I just mean the x component of
a force is given by integrating
603
00:43:38 --> 00:43:41
G m x over rho cubed delta dV.
The y components,
604
00:43:41 --> 00:43:43
same thing with y.
The z components,
605
00:43:43 --> 00:43:46
same thing with z.
OK, there's no,
606
00:43:46 --> 00:43:51
like, you know,
just integrate component by
607
00:43:51 --> 00:43:56
component to get each component
of the force.
608
00:43:56 --> 00:44:01
So, now we could very well to
this in rectangular coordinates
609
00:44:01 --> 00:44:04
if we want.
But the annoying thing is this
610
00:44:04 --> 00:44:06
rho cubed.
Rho cubed is going to be x
611
00:44:06 --> 00:44:10
squared plus y squared plus z
squared to the three halves.
612
00:44:10 --> 00:44:13
That's not going to be a very
pleasant thing to integrate.
613
00:44:13 --> 00:44:24
So, it's much better to set up
these integrals in spherical
614
00:44:24 --> 00:44:29
coordinates.
And, if we're going to do it in
615
00:44:29 --> 00:44:32
spherical coordinates,
then probably we don't want to
616
00:44:32 --> 00:44:34
bother too much with x and y
components because those would
617
00:44:34 --> 00:44:38
be unpleasant.
It would give us rho sine phi
618
00:44:38 --> 00:44:47
cos theta or sine theta.
So, the actual way we will set
619
00:44:47 --> 00:44:57
up things, set things up,
is to place the solid so that
620
00:44:57 --> 00:45:04
the z axis is an axis of
symmetry.
621
00:45:04 --> 00:45:07
And, of course,
that only works if the solid
622
00:45:07 --> 00:45:10
has some axis of symmetry.
Like, if you're trying to find
623
00:45:10 --> 00:45:13
the gravitational attraction of
the Pyramid of Giza,
624
00:45:13 --> 00:45:16
then you won't be able to set
up so that it has rotation
625
00:45:16 --> 00:45:18
symmetry.
Well, that's a tough fact of
626
00:45:18 --> 00:45:21
life, and you have to actually
do it in x, y,
627
00:45:21 --> 00:45:24
z coordinates.
But, if at all possible,
628
00:45:24 --> 00:45:27
then you're going to place
things.
629
00:45:27 --> 00:45:30
Well, I guess even then,
you could center it on the z
630
00:45:30 --> 00:45:32
axis.
But anyway, so you're going to
631
00:45:32 --> 00:45:37
mostly place things so that your
solid is actually centered on
632
00:45:37 --> 00:45:41
the z-axis.
And, what you gain by that is
633
00:45:41 --> 00:45:45
that by symmetry,
the gravitational force will be
634
00:45:45 --> 00:45:52
directed along the z axis.
So, you will just have to
635
00:45:52 --> 00:45:58
figure out the z component.
So, then the force will be
636
00:45:58 --> 00:46:03
actually, you know in advance
that it will be given by zero,
637
00:46:03 --> 00:46:11
zero, and some z component.
And then, you just need to
638
00:46:11 --> 00:46:19
compute that component.
And, that component will be
639
00:46:19 --> 00:46:27
just G times m times triple
integral of z over rho cubed
640
00:46:27 --> 00:46:30
delta dV.
OK, so that's the first
641
00:46:30 --> 00:46:35
simplification we can try to do.
The second thing is,
642
00:46:35 --> 00:46:38
well, we have to choose our
favorite coordinate system to do
643
00:46:38 --> 00:46:45
this.
But, I claim that actually
644
00:46:45 --> 00:46:57
spherical coordinates are the
best -- -- because let's see
645
00:46:57 --> 00:47:04
what happens.
So, G times mass times triple
646
00:47:04 --> 00:47:09
integral, well,
a z in spherical coordinates
647
00:47:09 --> 00:47:14
becomes rho cosine phi over rho
cubed.
648
00:47:14 --> 00:47:17
Density, well,
we can't do anything about
649
00:47:17 --> 00:47:21
density.
And then, dV becomes rho
650
00:47:21 --> 00:47:28
squared sine phi d rho d phi d
theta.
651
00:47:28 --> 00:47:34
Well, so, what happens with
that?
652
00:47:34 --> 00:47:37
Well, you see that you have a
rho, a rho squared,
653
00:47:37 --> 00:47:39
and a rho cubed that cancel
each other.
654
00:47:39 --> 00:47:42
So, in fact,
it simplifies quite a bit if
655
00:47:42 --> 00:47:44
you do it in spherical
coordinates.
656
00:47:44 --> 00:48:08
657
00:48:08 --> 00:48:12
OK, so the z component of the
force, sorry,
658
00:48:12 --> 00:48:18
I'm putting a z here to remind
you it's the z component.
659
00:48:18 --> 00:48:19
That is not a partial
derivative, OK?
660
00:48:19 --> 00:48:27
Don't get things mixed up,
just the z component of the
661
00:48:27 --> 00:48:35
force becomes Gm triple integral
of delta cos phi sine phi d rho
662
00:48:35 --> 00:48:40
d phi d theta.
And, so this thing is not dV,
663
00:48:40 --> 00:48:42
of course.
dV is much bigger,
664
00:48:42 --> 00:48:45
but we've somehow canceled out
most of dV with stuff that was
665
00:48:45 --> 00:48:49
in the integrand.
And see, that's actually
666
00:48:49 --> 00:48:55
suddenly much less scary.
OK, so just to give you an
667
00:48:55 --> 00:49:01
example of what you can prove it
this way, you can prove Newton's
668
00:49:01 --> 00:49:06
theorem, which says the
following thing.
669
00:49:06 --> 00:49:23
It says the gravitational
attraction -- -- of a spherical
670
00:49:23 --> 00:49:29
planet,
I should say with uniform
671
00:49:29 --> 00:49:32
density,
or actually it's enough for the
672
00:49:32 --> 00:49:34
density to depend just on
distance to the center.
673
00:49:34 --> 00:49:50
But we just simplify the
statement is equal to that of a
674
00:49:50 --> 00:50:05
point mass -- -- with the same
total mass at its center.
675
00:50:05 --> 00:50:11
OK, so what that means is that,
so the way we would set it up
676
00:50:11 --> 00:50:18
is u would be sitting here and
your planet would be over here.
677
00:50:18 --> 00:50:21
Or, if you're at the surface of
it, then of course you just put
678
00:50:21 --> 00:50:25
it tangent to the xy plane here.
And, you would compute that
679
00:50:25 --> 00:50:27
quantity.
Computation is a little bit
680
00:50:27 --> 00:50:30
annoying if a sphere is sitting
up there because,
681
00:50:30 --> 00:50:31
of course, you have to find
bounds,
682
00:50:31 --> 00:50:33
and that's not going to be very
pleasant.
683
00:50:33 --> 00:50:37
The case that we actually know
how to do fairly well is if you
684
00:50:37 --> 00:50:39
are just at the surface of the
planet.
685
00:50:39 --> 00:50:41
But then,
what the theorem says is that
686
00:50:41 --> 00:50:44
the force that you're going to
feel is exactly the same as if
687
00:50:44 --> 00:50:48
you removed all of the planet
and you just put an equivalent
688
00:50:48 --> 00:50:50
point mass here.
So, if the earth collapsed to a
689
00:50:50 --> 00:50:53
black hole at the center of the
earth with the same mass,
690
00:50:53 --> 00:50:55
well, you wouldn't notice the
difference immediately,
691
00:50:55 --> 00:51:00
or, rather, you would,
but at least not in terms of
692
00:51:00 --> 00:51:04
your weight.
OK, that's the end for today.
693
00:51:04 --> 00:51:04