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So, last week we learned how to
do triple integrals in

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rectangular and cylindrical
coordinates.

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And, now we have to learn about
spherical coordinates,

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which you will see are a lot of
fun.

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So, what's the idea of
spherical coordinates?

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Well, you're going to represent
a point in space using the

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distance to the origin and two
angles.

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00:01:02,000 --> 00:01:06,000
So, in a way,
you can think of these as a

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00:01:06,000 --> 00:01:09,000
space analog of polar
coordinates because you just use

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00:01:09,000 --> 00:01:12,000
distance to the origin,
and then you have to use angles

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to determine in which direction
you're going.

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00:01:15,000 --> 00:01:22,000
So, somehow they are more polar
than cylindrical coordinates.

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So, how do we do that?
So, let's say that you have a

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point in space at coordinates x,
y, z.

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00:01:29,000 --> 00:01:32,000
Then, instead of using x,
y, z, you will use,

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00:01:32,000 --> 00:01:37,000
well, one thing you'll use is
the distance from the origin.

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00:01:37,000 --> 00:01:41,000
OK, and that is denoted by the
Greek letter which looks like a

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curly p, but actually it's the
Greek R.

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00:01:43,000 --> 00:01:57,000
So -- That's the distance from
the origin.

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00:01:57,000 --> 00:02:02,000
And so, that can take values
anywhere between zero and

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infinity.
Then, we have to use two other

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angles.
And, so for that,

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00:02:09,000 --> 00:02:16,000
let me actually draw the
vertical half plane that

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00:02:16,000 --> 00:02:22,000
contains our point starting from
the z axis.

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00:02:22,000 --> 00:02:25,000
OK, so then we have two new
angles.

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00:02:25,000 --> 00:02:27,000
Well, one of them is not really
new.

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00:02:27,000 --> 00:02:30,000
One is new.
That's phi is the angle

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00:02:30,000 --> 00:02:34,000
downwards from the z axis.
And the other one,

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00:02:34,000 --> 00:02:39,000
theta, is the angle
counterclockwise from the x

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00:02:39,000 --> 00:02:51,000
axis.
OK, so phi, let me do it better.

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00:02:51,000 --> 00:02:54,000
So, there's two ways to draw
the letter phi,

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00:02:54,000 --> 00:02:56,000
by the way.
And, I recommend this one

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00:02:56,000 --> 00:02:58,000
because it doesn't look like a
rho.

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00:02:58,000 --> 00:03:07,000
So, that's easier.
That's the angle that you have

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00:03:07,000 --> 00:03:13,000
to go down from the positive z
axis.

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00:03:13,000 --> 00:03:17,000
And,
so that angle varies from zero

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00:03:17,000 --> 00:03:23,000
when you're on the z axis,
increase to pi over two when

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you are on the xy plane all the
way to pi or 180� when you are

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00:03:28,000 --> 00:03:38,000
on the negative z axis.
It doesn't go beyond that.

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00:03:38,000 --> 00:03:44,000
OK, so -- Phi is always between
zero and pi.

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00:03:44,000 --> 00:03:46,000
And, finally,
the last one,

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theta, is just going to be the
same as before.

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00:03:51,000 --> 00:03:55,000
So, it's the angle after you
project to the xy plane.

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00:03:55,000 --> 00:04:00,000
That's the angle
counterclockwise from the x

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axis.
OK, so that's a little bit

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00:04:02,000 --> 00:04:05,000
overwhelming not just because of
the new letters,

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00:04:05,000 --> 00:04:07,000
but also because there is a lot
of angles in there.

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00:04:07,000 --> 00:04:11,000
So, let me just try to,
you know, suggest two things

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that might help you a little
bit.

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So, one is, these are called
spherical coordinates because if

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you fix the value of rho,
then you are moving on a sphere

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centered at the origin.
OK, so let's look at what

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happens on a sphere centered at
the origin, so,

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00:04:34,000 --> 00:04:41,000
with equation rho equals a.
Well, then phi measures how far

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south you are going,
measures the distance from the

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North Pole.
So,

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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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00:04:59,000 --> 00:05:05,000
same thing as latitude in the
sense that it measures how far

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00:05:05,000 --> 00:05:09,000
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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depending on how far you go
from the equator.

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Here, you measure a latitude
starting from the North Pole

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which is zero,
increasing all the way to the

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South Pole, which is at pi.
And, theta or you can think of

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00:05:29,000 --> 00:05:33,000
as longitude,
which measures how far you are

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east or west.
So, the Greenwich Meridian

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would be here,
now, the one on the x axis.

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00:05:39,000 --> 00:05:46,000
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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So -- Let's start again from
cylindrical coordinates,

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00:05:56,000 --> 00:06:02,000
which hopefully you're kind of
comfortable with now.

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OK, so you know about
cylindrical coordinates where we

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00:06:06,000 --> 00:06:09,000
have the z coordinates stay z,
and the xy plane we do R and

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00:06:09,000 --> 00:06:13,000
theta polar coordinates.
And now, let's think about what

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00:06:13,000 --> 00:06:18,000
happens when you look at just
one of these vertical planes

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containing the z axis.
So, you have the z axis,

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00:06:21,000 --> 00:06:25,000
and then you have the direction
away from the z axis,

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00:06:25,000 --> 00:06:29,000
which I will call r,
just because that's what r

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measures.
Of course, r goes all around

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the z axis, but I'm just doing a
slice through one of these

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vertical half planes,
fixing the value of theta.

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Then, r of course is a polar
coordinate seen from the point

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of view of the xy plane.
But here, it looks more like

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you have rectangular coordinates
again.

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00:06:48,000 --> 00:06:51,000
So the idea of spherical
coordinate is you're going to

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polar coordinates again in the
rz plane.

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OK, so if I have a point here,
then rho will be the distance

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00:07:01,000 --> 00:07:06,000
from the origin.
And phi will be the angle,

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00:07:06,000 --> 00:07:10,000
except it's measured from the
positive z axis,

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not from the horizontal axis.
But, the idea in here,

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see,
let me put that between quotes

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because I'm not sure how correct
that is,

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but in a way,
you can think of this as polar

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coordinates in the rz plane.
So, in particular,

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that's the key to understanding
how to switch between spherical

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coordinates and cylindrical
coordinates,

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and then all the way to x,
y, z if you want,

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right,
because this picture here tells

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us how to express z and r in
terms of rho and phi.

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So, let's see how that works.
If I project here or here,

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00:08:03,000 --> 00:08:12,000
so, this line is z.
But, it's also rho times cosine

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phi.
So, I get z equals rho cos phi.

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00:08:19,000 --> 00:08:21,000
And, if I look at r,
it's the same thing,

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but on the other side.
So, r will be rho sine phi.

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00:08:31,000 --> 00:08:34,000
OK, so you can use this to
switch back and forth between

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spherical and cylindrical.
And of course,

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if you remember what x and y
were in terms of r and theta,

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00:08:43,000 --> 00:08:49,000
you can also keep doing this to
figure out, oops.

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00:08:49,000 --> 00:08:57,000
So, x is r cos theta.
That becomes rho sine phi cos

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00:08:57,000 --> 00:09:01,000
theta.
Y is r sine theta.

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00:09:01,000 --> 00:09:06,000
So, that becomes rho sine phi
sine theta.

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00:09:06,000 --> 00:09:15,000
And z is rho cos phi.
But, basically you don't really

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need to remember these formulas
as long as you remember how to

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express r in terms of rho sine
phi,

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00:09:22,000 --> 00:09:29,000
and x equals r cos theta.
So, now, of course,

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00:09:29,000 --> 00:09:31,000
we're going to use spherical
coordinates in situations where

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we have a lot of symmetry,
and in particular,

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00:09:33,000 --> 00:09:35,000
where the z axis plays a
special role.

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00:09:35,000 --> 00:09:38,000
Actually, that's the same with
cylindrical coordinates.

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Cylindrical and secure
coordinates are set up so that

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the z axis plays a special role.
So, that means whenever you

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00:09:44,000 --> 00:09:47,000
have a geometric problem,
and you are not told how to

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choose your coordinates,
it's probably wiser to try to

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00:09:51,000 --> 00:09:57,000
center things on the z axis.
That's where these coordinates

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are the best adapted.
And,

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00:10:01,000 --> 00:10:03,000
in case you ever need to switch
backwards,

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I just want to point out,
so, rho is the square root of r

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squared plus z squared,
which means it's the square

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00:10:11,000 --> 00:10:15,000
root of x squared plus y squared
plus z squared.

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00:10:15,000 --> 00:10:20,000
OK, so that's basically all the
formulas about spherical

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coordinates.
OK, any questions about that?

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OK, let's see,
who had seen spherical

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coordinates before just to see?
OK, that's not very many.

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So, I'm sure for,
one of you saw it twice.

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That's great.
Sorry, oops,

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OK, so let's just look quickly
at equations of some of the

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things.
So, as I've said,

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if I set rho equals a,
that will be just a sphere of

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00:11:02,000 --> 00:11:11,000
radius a centered at the origin.
More interesting things:

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00:11:11,000 --> 00:11:14,000
let's say I give you phi equals
pi over four.

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What do you think that looks
like?

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00:11:18,000 --> 00:11:29,000
Actually, let's take a quick
poll on things.

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OK, yeah, everyone seems to be
saying it's a cone,

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and that's indeed the correct
answer.

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So, how do we see that?
Well, remember,

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phi is the angle downward from
the z axis.

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00:11:44,000 --> 00:11:49,000
So, let's say that I'm going to
look first at what happens if

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I'm in the right half of a plane
of a blackboard,

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so, in the yz plane.
Then, phi is the angle downward

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00:11:56,000 --> 00:11:58,000
from here.
So, if I want to get pi over

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00:11:58,000 --> 00:12:01,000
four, that's 45�.
That means I'm going to go

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00:12:01,000 --> 00:12:03,000
diagonally like this.
Of course, if I'm in the left

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00:12:03,000 --> 00:12:06,000
half of a plane of a blackboard,
it's going to be the same.

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00:12:06,000 --> 00:12:10,000
I also take pi over four.
And, I get the other half.

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00:12:10,000 --> 00:12:13,000
And, because the equation does
not involve theta,

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it's all the same if I rotate
my vertical plane around the z

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axis.
So, I get the same picture in

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any of these vertical half
planes, actually.

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OK, now, so this is phi equals
pi over four.

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00:12:32,000 --> 00:12:35,000
And, just in case,
to point out to you what's

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going on, when phi equals pi
over four, cosine and sine are

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00:12:39,000 --> 00:12:42,000
equal to each other.
They are both one over root two.

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00:12:42,000 --> 00:12:46,000
So, you can find,
again, the equation of this

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00:12:46,000 --> 00:12:51,000
thing in cylindrical
coordinates, which I'll remind

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00:12:51,000 --> 00:12:54,000
you was z equals r.
OK, in general,

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phi equals some given number,
or z equals some number times

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00:12:58,000 --> 00:13:01,000
r.
That will be a cone centered on

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00:13:01,000 --> 00:13:04,000
the z axis.
OK, a special case:

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what if I say phi equals pi
over two?

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00:13:07,000 --> 00:13:09,000
Yeah, it's just going to be the
xy plane.

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OK, that's the flattest of all
cones.

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00:13:13,000 --> 00:13:20,000
OK, so phi equals pi over two
is going to be just the xy

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plane.
And, in general,

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00:13:22,000 --> 00:13:24,000
if phi is less than pi over
two, then you are in the upper

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00:13:24,000 --> 00:13:28,000
half space.
If phi is more than pi over

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00:13:28,000 --> 00:13:32,000
two, you'll be in the lower half
space.

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OK, so that's pretty much all
we need to know at this point.

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00:13:36,000 --> 00:13:45,000
So, what's next?
Well, remember we were trying

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to do triple integrals.
So now we're going to triple

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integrals in spherical
coordinates.

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00:13:59,000 --> 00:14:01,000
And, for that,
we first need to understand

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00:14:01,000 --> 00:14:06,000
what the volume element is.
What will be dV?

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00:14:06,000 --> 00:14:12,000
OK, so dV will be something,
d rho, d phi,

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00:14:12,000 --> 00:14:18,000
d theta, or in any order that
you want.

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00:14:18,000 --> 00:14:23,000
But, this one is usually the
most convenient.

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00:14:23,000 --> 00:14:27,000
So, to find out what it is,
well, we should look at how we

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are going to be slicing things
now.

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00:14:29,000 --> 00:14:32,000
OK, so if you integrate d rho,
d phi, d theta,

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00:14:32,000 --> 00:14:37,000
it means that you are actually
slicing your solid into little

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00:14:37,000 --> 00:14:40,000
pieces that live,
somehow,

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00:14:40,000 --> 00:14:45,000
if you set an interval of rows,
OK,

203
00:14:45,000 --> 00:14:48,000
sorry, maybe I should,
so, if you first integrate over

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00:14:48,000 --> 00:14:51,000
rho,
it means that you will actually

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00:14:51,000 --> 00:14:57,000
choose first the direction from
the origin even by phi and

206
00:14:57,000 --> 00:15:00,000
theta.
And, in that direction,

207
00:15:00,000 --> 00:15:04,000
you will try to figure out,
how far does your region

208
00:15:04,000 --> 00:15:07,000
extend?
And, of course,

209
00:15:07,000 --> 00:15:11,000
how far that goes might depend
on phi and theta.

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00:15:11,000 --> 00:15:16,000
Then, you will vary phi.
So, you have to know,

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00:15:16,000 --> 00:15:21,000
for a given value of theta,
how far down does your solid

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00:15:21,000 --> 00:15:22,000
extend?
And, finally,

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00:15:22,000 --> 00:15:25,000
the value of theta will
correspond to,

214
00:15:25,000 --> 00:15:28,000
in which directions around the
z axis do we go?

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00:15:28,000 --> 00:15:31,000
So, we're going to see that in
examples.

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00:15:31,000 --> 00:15:34,000
But before we can do that,
we need to get the volume

217
00:15:34,000 --> 00:15:36,000
element.
So, what I would like to

218
00:15:36,000 --> 00:15:40,000
suggest is that we need to
figure out,

219
00:15:40,000 --> 00:15:46,000
what is the volume of a small
piece of solid which corresponds

220
00:15:46,000 --> 00:15:49,000
to a certain change,
delta rho,

221
00:15:49,000 --> 00:15:52,000
delta phi,
and delta theta?

222
00:15:52,000 --> 00:15:56,000
So, delta rho means that you
have two concentric spheres,

223
00:15:56,000 --> 00:16:01,000
and you are looking at a very
thin shell in between them.

224
00:16:01,000 --> 00:16:05,000
And then, you would be looking
at a piece of that spherical

225
00:16:05,000 --> 00:16:08,000
shell corresponding to small
values of phi and theta.

226
00:16:08,000 --> 00:16:14,000
So, because I am stretching the
limits of my ability to draw on

227
00:16:14,000 --> 00:16:18,000
the board, here's a picture.
I'm going to try to reproduce

228
00:16:18,000 --> 00:16:21,000
on the board,
but so let's start by looking

229
00:16:21,000 --> 00:16:24,000
just at what happens on the
sphere of radius a,

230
00:16:24,000 --> 00:16:28,000
and let's try to figure out the
surface area elements on the

231
00:16:28,000 --> 00:16:30,000
sphere in terms of phi and
theta.

232
00:16:30,000 --> 00:16:39,000
And then, we'll add the rho
direction.

233
00:16:39,000 --> 00:16:49,000
OK, so -- So,
let me say, let's start by

234
00:16:49,000 --> 00:17:02,000
understanding surface area on a
sphere of radius a.

235
00:17:02,000 --> 00:17:12,000
So, that means we'll be looking
at a little piece of the sphere

236
00:17:12,000 --> 00:17:21,000
corresponding to angles delta
phi and in that direction here

237
00:17:21,000 --> 00:17:26,000
delta theta.
OK, so when you draw a map of

238
00:17:26,000 --> 00:17:29,000
the world on a globe,
that's exactly what the grid

239
00:17:29,000 --> 00:17:33,000
lines form for you.
So, what's the area of this guy?

240
00:17:33,000 --> 00:17:35,000
Well, of course,
all the sides are curvy.

241
00:17:35,000 --> 00:17:37,000
They are all on the sphere.
None of them are straight.

242
00:17:37,000 --> 00:17:41,000
But still, if it's small enough
and it looks like a rectangle,

243
00:17:41,000 --> 00:17:46,000
so let's just try to figure
out, what are the sides of your

244
00:17:46,000 --> 00:17:49,000
rectangle?
OK, so, let's see,

245
00:17:49,000 --> 00:17:55,000
well, I think I need to draw a
bigger picture of this guy.

246
00:17:55,000 --> 00:17:59,000
OK, so this guy,
so that's a piece of what's

247
00:17:59,000 --> 00:18:05,000
called a parallel in geography.
That's a circle that goes

248
00:18:05,000 --> 00:18:07,000
east-west.
So now,

249
00:18:07,000 --> 00:18:10,000
this parallel as a circle of
radius,

250
00:18:10,000 --> 00:18:14,000
well, the radius is less than a
because if your vertical is to

251
00:18:14,000 --> 00:18:17,000
the North Pole,
it will be actually much

252
00:18:17,000 --> 00:18:19,000
smaller.
So, that's why when you say

253
00:18:19,000 --> 00:18:22,000
you're going around the world it
depends on whether you do it at

254
00:18:22,000 --> 00:18:28,000
the equator or the North Pole.
It's much easier at the North

255
00:18:28,000 --> 00:18:33,000
Pole.
So, anyway, this is a piece of

256
00:18:33,000 --> 00:18:40,000
a circle of radius,
well, the radius is what I

257
00:18:40,000 --> 00:18:49,000
would call r because that's the
distance from the z axis.

258
00:18:49,000 --> 00:18:51,000
OK, that's actually pretty hard
to see now.

259
00:18:51,000 --> 00:18:58,000
So if you can see it better on
this one, then so this guy here,

260
00:18:58,000 --> 00:19:03,000
this length is r.
And, r is just rho,

261
00:19:03,000 --> 00:19:07,000
well, what was a times sine
phi.

262
00:19:07,000 --> 00:19:09,000
Remember, we have this angle
phi in here.

263
00:19:09,000 --> 00:19:14,000
I should use some color.
It's getting very cluttered.

264
00:19:14,000 --> 00:19:19,000
So, we have this phi,
and so r is going to be rho

265
00:19:19,000 --> 00:19:21,000
sine phi.
That rho is a.

266
00:19:21,000 --> 00:19:29,000
So, let me just put a sine phi.
OK, and the corresponding angle

267
00:19:29,000 --> 00:19:32,000
is going to be measured by
theta.

268
00:19:32,000 --> 00:19:48,000
So, the length of this is going
to be a sine phi delta theta.

269
00:19:48,000 --> 00:19:54,000
That's for this side.
Now, what about that side,

270
00:19:54,000 --> 00:19:56,000
the north-south side?
Well, if you're moving

271
00:19:56,000 --> 00:19:58,000
north-south, it's not like
east-west.

272
00:19:58,000 --> 00:20:01,000
You always have to go all the
way from the North Pole to the

273
00:20:01,000 --> 00:20:04,000
South Pole.
So, that's actually a great

274
00:20:04,000 --> 00:20:08,000
circle meridian of length,
well, I mean,

275
00:20:08,000 --> 00:20:13,000
well, the radius is the radius
of the sphere.

276
00:20:13,000 --> 00:20:22,000
Total length is 2pi a.
So, this is a piece of a circle

277
00:20:22,000 --> 00:20:27,000
of radius a.
And so, now,

278
00:20:27,000 --> 00:20:34,000
the length of this one is going
to be a delta phi.

279
00:20:34,000 --> 00:20:41,000
OK, so, just to recap,
this is a sine phi delta theta.

280
00:20:41,000 --> 00:20:46,000
And, this guy here is a delta
phi.

281
00:20:46,000 --> 00:20:59,000
So, you can't read it because
it's -- And so,

282
00:20:59,000 --> 00:21:02,000
that tells us if I take a small
piece of the sphere,

283
00:21:02,000 --> 00:21:06,000
then its surface area,
delta s,

284
00:21:06,000 --> 00:21:15,000
is going to be approximately a
sine phi delta theta times a

285
00:21:15,000 --> 00:21:22,000
delta phi,
which I'm going to rewrite as a

286
00:21:22,000 --> 00:21:27,000
squared sine phi delta phi delta
theta.

287
00:21:27,000 --> 00:21:31,000
So, what that means is,
say that I want to integrate

288
00:21:31,000 --> 00:21:34,000
something just on the surface of
a sphere.

289
00:21:34,000 --> 00:21:37,000
Well, I would use phi and theta
as my coordinates.

290
00:21:37,000 --> 00:21:46,000
And then, to know how big a
piece of a sphere is,

291
00:21:46,000 --> 00:21:55,000
I would just take a squared
sine phi d phi d theta.

292
00:21:55,000 --> 00:21:59,000
OK, so that's the surface
element in a sphere.

293
00:21:59,000 --> 00:22:03,000
And now, what about going back
into the third dimension,

294
00:22:03,000 --> 00:22:05,000
so, adding some depth to these
things?

295
00:22:05,000 --> 00:22:10,000
Well, I'm not going to try to
draw a picture because you've

296
00:22:10,000 --> 00:22:17,000
seen that's slightly tricky.
Well, let me try anyway just

297
00:22:17,000 --> 00:22:24,000
you can have fun with my
completely unreadable diagrams.

298
00:22:24,000 --> 00:22:28,000
So anyway, if you look at,
now, something that's a bit

299
00:22:28,000 --> 00:22:33,000
like that piece of sphere,
but with some thickness to it.

300
00:22:33,000 --> 00:22:38,000
The thickness will be delta
rho, and so the volume will be

301
00:22:38,000 --> 00:22:44,000
roughly the area of the thing on
the sphere times the thickness.

302
00:22:44,000 --> 00:22:48,000
So, I claim that we will get
basically the volume element

303
00:22:48,000 --> 00:22:51,000
just by multiplying things by d
rho.

304
00:22:51,000 --> 00:23:10,000
So, let's see that.
So now, if I have a sphere of

305
00:23:10,000 --> 00:23:19,000
radius rho, and another one
that's slightly bigger of radius

306
00:23:19,000 --> 00:23:27,000
rho plus delta rho,
and then I have a little box in

307
00:23:27,000 --> 00:23:29,000
here.
Then,

308
00:23:29,000 --> 00:23:34,000
I know that the volume of this
thing will be essentially,

309
00:23:34,000 --> 00:23:38,000
well, its thickness,
the thickness is going to be

310
00:23:38,000 --> 00:23:42,000
delta rho times the area of its
base,

311
00:23:42,000 --> 00:23:44,000
although it doesn't really
matter,

312
00:23:44,000 --> 00:23:48,000
which is what we've called
delta s.

313
00:23:48,000 --> 00:23:55,000
OK, so we will get,
sorry, a becomes rho now.

314
00:23:55,000 --> 00:23:57,000
Square sine phi,
delta rho,

315
00:23:57,000 --> 00:24:00,000
delta phi,
delta theta,

316
00:24:00,000 --> 00:24:04,000
and so out of that we get the
volume element and spherical

317
00:24:04,000 --> 00:24:08,000
coordinates,
which is rho squared sine phi d

318
00:24:08,000 --> 00:24:09,000
rho,
d phi,

319
00:24:09,000 --> 00:24:14,000
d theta.
And, that's a formula that you

320
00:24:14,000 --> 00:24:17,000
should remember.
OK, so whenever we integrate a

321
00:24:17,000 --> 00:24:20,000
function,
and we decide to switch to

322
00:24:20,000 --> 00:24:25,000
spherical coordinates,
then dx dy dz or r dr d theta

323
00:24:25,000 --> 00:24:33,000
dz will become rho squared sine
phi d rho d phi d theta.

324
00:24:33,000 --> 00:24:40,000
OK, any questions on that?
No?

325
00:24:40,000 --> 00:24:58,000
OK, so let's -- Let's see how
that works.

326
00:24:58,000 --> 00:25:04,000
So, as an example,
remember at the end of the last

327
00:25:04,000 --> 00:25:11,000
lecture, I tried to set up an
example where we were looking at

328
00:25:11,000 --> 00:25:16,000
a sphere sliced by a slanted
plane.

329
00:25:16,000 --> 00:25:20,000
And now, we're going to try to
find the volume of that

330
00:25:20,000 --> 00:25:23,000
spherical cap again,
but using spherical coordinates

331
00:25:23,000 --> 00:25:26,000
instead.
So, I'm going to just be

332
00:25:26,000 --> 00:25:29,000
smarter than last time.
So, last time,

333
00:25:29,000 --> 00:25:33,000
we had set up these things with
a slanted plane that was cutting

334
00:25:33,000 --> 00:25:35,000
things diagonally.
And,

335
00:25:35,000 --> 00:25:37,000
if I just want to find the
volume of this cap,

336
00:25:37,000 --> 00:25:41,000
then maybe it makes more sense
to rotate things so that my

337
00:25:41,000 --> 00:25:45,000
plane is actually horizontal,
and things are going to be

338
00:25:45,000 --> 00:25:49,000
centered on the z axis.
So, in case you see that it's

339
00:25:49,000 --> 00:25:52,000
the same, then that's great.
If not, then it doesn't really

340
00:25:52,000 --> 00:25:55,000
matter.
You can just think of this as a

341
00:25:55,000 --> 00:26:01,000
new example.
So, I'm going to try to find

342
00:26:01,000 --> 00:26:10,000
the volume of a portion of the
unit sphere -- -- that lies

343
00:26:10,000 --> 00:26:20,000
above the horizontal plane,
z equals one over root two.

344
00:26:20,000 --> 00:26:22,000
OK, one over root two was the
distance from the origin to our

345
00:26:22,000 --> 00:26:24,000
slanted plane.
So, after you rotate,

346
00:26:24,000 --> 00:26:28,000
that say you get this value.
Anyway, it's not very important.

347
00:26:28,000 --> 00:26:31,000
You can just treat that as a
good example if you want.

348
00:26:31,000 --> 00:26:36,000
OK, so we can compute this in
actually pretty much any

349
00:26:36,000 --> 00:26:39,000
coordinate system.
And also, of course,

350
00:26:39,000 --> 00:26:42,000
we can set up not only the
volume, but we can try to find

351
00:26:42,000 --> 00:26:44,000
the moment of inertia about the
central axis,

352
00:26:44,000 --> 00:26:47,000
or all sorts of things.
But, we are just doing the

353
00:26:47,000 --> 00:26:49,000
volume for simplicity.
So, actually,

354
00:26:49,000 --> 00:26:52,000
this would go pretty well in
cylindrical coordinates.

355
00:26:52,000 --> 00:26:55,000
But let's do it in spherical
coordinates because that's the

356
00:26:55,000 --> 00:26:57,000
topic of today.
A good exercise:

357
00:26:57,000 --> 00:27:01,000
do it in cylindrical and see if
you get the same thing.

358
00:27:01,000 --> 00:27:08,000
So, how do we do that?
Well, we have to figure out how

359
00:27:08,000 --> 00:27:14,000
to set up our triple integral in
spherical coordinates.

360
00:27:14,000 --> 00:27:18,000
So, remember we'll be
integrating one dV.

361
00:27:18,000 --> 00:27:28,000
So, dV will become rho squared
sign phi d rho d phi d theta.

362
00:27:28,000 --> 00:27:32,000
And, now as we start,
we're already facing some

363
00:27:32,000 --> 00:27:35,000
serious problem.
We want to set up the bounds

364
00:27:35,000 --> 00:27:37,000
for rho for a given,
phi and theta.

365
00:27:37,000 --> 00:27:39,000
So, that means we choose
latitude/longitude.

366
00:27:39,000 --> 00:27:42,000
We choose which direction we
want to aim for,

367
00:27:42,000 --> 00:27:45,000
you know, which point of the
sphere we want to aim at.

368
00:27:45,000 --> 00:27:50,000
And, we are going to shoot a
ray from the origin towards this

369
00:27:50,000 --> 00:27:55,000
point, and we want to know what
portion of the ray is in our

370
00:27:55,000 --> 00:28:03,000
solid.
So -- We are going to choose a

371
00:28:03,000 --> 00:28:11,000
value of phi and theta.
And, we are going to try to

372
00:28:11,000 --> 00:28:16,000
figure out what part of our ray
is inside this side.

373
00:28:16,000 --> 00:28:20,000
So, what should be clear is at
which point we leave the solid,

374
00:28:20,000 --> 00:28:23,000
right?
What's the value of rho here?

375
00:28:23,000 --> 00:28:25,000
It's just one.
The sphere is rho equals one.

376
00:28:25,000 --> 00:28:29,000
That's pretty good.
The question is,

377
00:28:29,000 --> 00:28:33,000
where do we enter the region?
So, we enter the region when we

378
00:28:33,000 --> 00:28:38,000
go through this plane.
And, the plane is z equals one

379
00:28:38,000 --> 00:28:41,000
over root two.
So, what does that tell us

380
00:28:41,000 --> 00:28:44,000
about rho?
Well, it tells us,

381
00:28:44,000 --> 00:28:50,000
so remember,
z is rho cosine phi.

382
00:28:50,000 --> 00:28:55,000
So, the plane is z equals one
over root two.

383
00:28:55,000 --> 00:29:00,000
That means rho cosine phi is
one over root two.

384
00:29:00,000 --> 00:29:05,000
That means rho equals one over
root two cosine phi or,

385
00:29:05,000 --> 00:29:11,000
as some of you know it,
one over root two times second

386
00:29:11,000 --> 00:29:17,000
phi.
OK, so if we want to set up the

387
00:29:17,000 --> 00:29:27,000
bounds, then we'll start with
one over root two second phi all

388
00:29:27,000 --> 00:29:32,000
the way to one.
Now, what's next?

389
00:29:32,000 --> 00:29:35,000
Well, so we've done,
I think that's basically the

390
00:29:35,000 --> 00:29:38,000
hardest part of the job.
Next, we have to figure out,

391
00:29:38,000 --> 00:29:41,000
what's the range for phi?
So, the range for phi,

392
00:29:41,000 --> 00:29:44,000
well, we have to figure out how
far to the north and to the

393
00:29:44,000 --> 00:29:48,000
south our region goes.
Well, the lower bound for phi

394
00:29:48,000 --> 00:29:51,000
is pretty easy,
right, because we go all the

395
00:29:51,000 --> 00:29:56,000
way to the North Pole direction.
So, phi starts at zero.

396
00:29:56,000 --> 00:29:59,000
The question is,
where does it stop?

397
00:29:59,000 --> 00:30:02,000
To find out where it stops,
we have to figure out,

398
00:30:02,000 --> 00:30:06,000
what is the value of phi when
we hit the edge of the region?

399
00:30:06,000 --> 00:30:10,000
OK, so maybe you see it.
Maybe you don't.

400
00:30:10,000 --> 00:30:15,000
One way to do it geometrically
is to just, it's always great to

401
00:30:15,000 --> 00:30:19,000
draw a slice of your region.
So, if you slice these things

402
00:30:19,000 --> 00:30:22,000
by a vertical plane,
or actually even better,

403
00:30:22,000 --> 00:30:25,000
a vertical half plane,
something to delete one half of

404
00:30:25,000 --> 00:30:28,000
the picture.
So, I'm going to draw these r

405
00:30:28,000 --> 00:30:33,000
and z directions as before.
So, my sphere is here.

406
00:30:33,000 --> 00:30:38,000
My plane is here at one over
root two.

407
00:30:38,000 --> 00:30:43,000
And, my solid is here.
So now, the question is what is

408
00:30:43,000 --> 00:30:49,000
the value of phi when I'm going
to stop hitting the region?

409
00:30:49,000 --> 00:30:54,000
And, if you try to figure out
first what is this direction

410
00:30:54,000 --> 00:30:57,000
here, that's also one over root
two.

411
00:30:57,000 --> 00:31:03,000
And so, this is actually 45�,
also known as pi over four.

412
00:31:03,000 --> 00:31:09,000
The other way to think about it
is at this point,

413
00:31:09,000 --> 00:31:16,000
well, rho is equal to one
because you are on the sphere.

414
00:31:16,000 --> 00:31:22,000
But, you are also on the plane.
So, rho cos phi is one over

415
00:31:22,000 --> 00:31:26,000
root two.
So, if you plug rho equals one

416
00:31:26,000 --> 00:31:31,000
into here, you get cos phi
equals one over root two which

417
00:31:31,000 --> 00:31:34,000
gives you phi equals pi over
four.

418
00:31:34,000 --> 00:31:37,000
That's the other way to do it.
You can do it either by

419
00:31:37,000 --> 00:31:39,000
calculation or by looking at the
picture.

420
00:31:39,000 --> 00:31:43,000
OK, so either way,
we've decided that phi goes

421
00:31:43,000 --> 00:31:48,000
from zero to pi over four.
So, this is pi over four.

422
00:31:48,000 --> 00:31:54,000
Finally, what about theta?
Well, because we go all around

423
00:31:54,000 --> 00:32:00,000
the z axis we are going to go
just zero to 2pi.

424
00:32:00,000 --> 00:32:06,000
OK, any questions about these
bounds?

425
00:32:06,000 --> 00:32:10,000
OK, so note how the equation of
this horizontal plane in

426
00:32:10,000 --> 00:32:13,000
spherical coordinates has become
a little bit weird.

427
00:32:13,000 --> 00:32:16,000
But,
if you remember how we do

428
00:32:16,000 --> 00:32:19,000
things,
say that you have a line in

429
00:32:19,000 --> 00:32:21,000
polar coordinates,
and that line does not pass

430
00:32:21,000 --> 00:32:23,000
through the origin,
then you also end up with

431
00:32:23,000 --> 00:32:26,000
something like that.
You get something like r equals

432
00:32:26,000 --> 00:32:31,000
a second theta or a cos second
theta for horizontal or vertical

433
00:32:31,000 --> 00:32:33,000
lines.
And so, it's not surprising you

434
00:32:33,000 --> 00:32:38,000
should get this.
That's in line with the idea

435
00:32:38,000 --> 00:32:44,000
that we are just doing again,
polar coordinates in the rz

436
00:32:44,000 --> 00:32:46,000
directions.
So of course,

437
00:32:46,000 --> 00:32:48,000
in general, things can be very
messy.

438
00:32:48,000 --> 00:32:51,000
But, generally speaking,
the kinds of regions that we

439
00:32:51,000 --> 00:32:55,000
will be setting up things for
are no more complicated or no

440
00:32:55,000 --> 00:32:59,000
less complicated than what we
would do in the plane in polar

441
00:32:59,000 --> 00:33:00,000
coordinates.
OK, so there's,

442
00:33:00,000 --> 00:33:03,000
you know, a small list of
things that you should know how

443
00:33:03,000 --> 00:33:07,000
to set up.
But, you won't have some

444
00:33:07,000 --> 00:33:18,000
really, really strange thing.
Yes?

445
00:33:18,000 --> 00:33:20,000
D rho?
Oh, you mean the bounds for rho?

446
00:33:20,000 --> 00:33:23,000
Yes.
So, in the inner integral,

447
00:33:23,000 --> 00:33:26,000
we are going to fix values of
phi and theta.

448
00:33:26,000 --> 00:33:29,000
So, that means we fix in
advance the direction in which

449
00:33:29,000 --> 00:33:31,000
we are going to shoot a ray from
the origin.

450
00:33:31,000 --> 00:33:35,000
So now, as we shoot this ray,
we are going to hit our region

451
00:33:35,000 --> 00:33:37,000
somewhere.
And, we are going to exit,

452
00:33:37,000 --> 00:33:40,000
again, somewhere else.
OK, so first of all we have to

453
00:33:40,000 --> 00:33:43,000
figure out where we enter,
where we leave.

454
00:33:43,000 --> 00:33:46,000
Well, we enter when the ray
hits the flat face,

455
00:33:46,000 --> 00:33:50,000
when we hit the plane.
And, we would leave when we hit

456
00:33:50,000 --> 00:33:52,000
the sphere.
So, the lower bound will be

457
00:33:52,000 --> 00:33:56,000
given by the plane.
The upper bound will be given

458
00:33:56,000 --> 00:33:58,000
by the sphere.
So now, you have to get

459
00:33:58,000 --> 00:34:01,000
spherical coordinate equations
for both the plane and the

460
00:34:01,000 --> 00:34:02,000
sphere.
For the sphere, that's easy.

461
00:34:02,000 --> 00:34:05,000
That's rho equals one.
For the plane,

462
00:34:05,000 --> 00:34:08,000
you start with z equals one
over root two.

463
00:34:08,000 --> 00:34:11,000
And, you switch it into
spherical coordinates.

464
00:34:11,000 --> 00:34:14,000
And then, you solve for rho.
And, that's how you get these

465
00:34:14,000 --> 00:34:19,000
bounds.
Is that OK?

466
00:34:19,000 --> 00:34:26,000
All right, so that's the setup
part.

467
00:34:26,000 --> 00:34:29,000
And, of course,
the evaluation part goes as

468
00:34:29,000 --> 00:34:30,000
usual.

469
00:34:42,000 --> 00:34:46,000
And, since I'm running short of
time, I'm not going to actually

470
00:34:46,000 --> 00:34:52,000
do the evaluation.
I'm going to let you figure out

471
00:34:52,000 --> 00:34:58,000
how it goes.
Let me just say in case you

472
00:34:58,000 --> 00:35:07,000
want to check your answers,
so, at the end you get 2pi over

473
00:35:07,000 --> 00:35:13,000
three minus 5pi over six root
two.

474
00:35:13,000 --> 00:35:17,000
Yes, it looks quite complicated.
That's basically because you

475
00:35:17,000 --> 00:35:20,000
get one over,
well, you get a second square

476
00:35:20,000 --> 00:35:23,000
when you integrate C.
When you integrate rho squared,

477
00:35:23,000 --> 00:35:24,000
you will get rho cubed over
three.

478
00:35:24,000 --> 00:35:27,000
But that rho cubed will give
you a second cube for the lower

479
00:35:27,000 --> 00:35:29,000
bound.
And, when you integrate sine

480
00:35:29,000 --> 00:35:31,000
phi second cubed phi,
you do a substitution.

481
00:35:31,000 --> 00:35:37,000
You see that integrates to one
over second squared with a

482
00:35:37,000 --> 00:35:42,000
factor in front.
So, in the second square,

483
00:35:42,000 --> 00:35:49,000
when you plug in,
no, that's not quite all of it.

484
00:35:49,000 --> 00:35:51,000
Yeah, well, the second square
is one thing,

485
00:35:51,000 --> 00:35:53,000
and also the other bound you
get sine phi which integrates to

486
00:35:53,000 --> 00:35:56,000
cosine phi.
So, anyways,

487
00:35:56,000 --> 00:36:04,000
you get lots of things.
OK, enough about it.

488
00:36:04,000 --> 00:36:07,000
So, next, I have to tell you
about applications.

489
00:36:07,000 --> 00:36:13,000
And, of course,
well, there's the same

490
00:36:13,000 --> 00:36:14,000
applications that we've seen
that last time,

491
00:36:14,000 --> 00:36:16,000
finding volumes,
finding masses,

492
00:36:16,000 --> 00:36:19,000
finding average values of
functions.

493
00:36:19,000 --> 00:36:22,000
In particular,
now, we could say to find the

494
00:36:22,000 --> 00:36:26,000
average distance of a point in
this solid to the origin.

495
00:36:26,000 --> 00:36:28,000
Well,
spherical coordinates become

496
00:36:28,000 --> 00:36:32,000
appealing because the function
you are averaging is just rho

497
00:36:32,000 --> 00:36:35,000
while in other coordinate
systems it's a more complicated

498
00:36:35,000 --> 00:36:37,000
function.
So, if you are asked to find

499
00:36:37,000 --> 00:36:41,000
the average distance from the
origin, spherical coordinates

500
00:36:41,000 --> 00:36:43,000
can be interesting.
Also,

501
00:36:43,000 --> 00:36:47,000
well, there's moments of
inertia,

502
00:36:47,000 --> 00:36:50,000
preferably the one about the z
axis because if you have to

503
00:36:50,000 --> 00:36:52,000
integrate something that
involves x or y,

504
00:36:52,000 --> 00:36:55,000
then your integrand will
contain that awful rho sine phi

505
00:36:55,000 --> 00:36:57,000
sine theta or rho sine phi
cosine theta,

506
00:36:57,000 --> 00:37:00,000
and then it won't be much fun
to evaluate.

507
00:37:00,000 --> 00:37:05,000
So, that anyway,
there's the usual ones.

508
00:37:05,000 --> 00:37:08,000
And then there's a new one.
So, in physics,

509
00:37:08,000 --> 00:37:16,000
you've probably seen things
about gravitational attraction.

510
00:37:16,000 --> 00:37:19,000
If not, well,
it's what causes apples to fall

511
00:37:19,000 --> 00:37:22,000
and other things like that as
well.

512
00:37:22,000 --> 00:37:26,000
So, anyway, physics tells you
that if you have two masses,

513
00:37:26,000 --> 00:37:30,000
then they attract each other
with a force that's directed

514
00:37:30,000 --> 00:37:33,000
towards each other.
And in intensity,

515
00:37:33,000 --> 00:37:37,000
it's proportional to the two
masses, and inversely

516
00:37:37,000 --> 00:37:41,000
proportional to the square of
the distance between them.

517
00:37:41,000 --> 00:37:45,000
So,
if you have a given solid with

518
00:37:45,000 --> 00:37:50,000
a certain mass distribution,
and you want to know how it

519
00:37:50,000 --> 00:37:53,000
attracts something else that you
will put nearby,

520
00:37:53,000 --> 00:37:58,000
then you actually have to,
the first approximation will be

521
00:37:58,000 --> 00:37:59,000
to say,
well, let's just put a point

522
00:37:59,000 --> 00:38:02,000
mass at its center of mass.
But, if you're solid is

523
00:38:02,000 --> 00:38:04,000
actually not homogenous,
or has a weird shape,

524
00:38:04,000 --> 00:38:07,000
then that's not actually the
exact answer.

525
00:38:07,000 --> 00:38:09,000
So, in general,
you would have to just take

526
00:38:09,000 --> 00:38:12,000
every single piece of your
object and figure out how it

527
00:38:12,000 --> 00:38:14,000
attracts you,
and then compute the sum of

528
00:38:14,000 --> 00:38:15,000
these.
So, for example,

529
00:38:15,000 --> 00:38:18,000
if you want to understand why
anything that you drop in this

530
00:38:18,000 --> 00:38:21,000
room will fall down,
you have to understand that

531
00:38:21,000 --> 00:38:24,000
Boston is actually attracting it
towards Boston.

532
00:38:24,000 --> 00:38:26,000
And, Somerville's attracting it
towards Somerville,

533
00:38:26,000 --> 00:38:29,000
and lots of things like that.
And, China, which is much

534
00:38:29,000 --> 00:38:33,000
further on the other side is
going to attract towards China.

535
00:38:33,000 --> 00:38:35,000
But, there's a lot of stuff on
the other side of the Earth.

536
00:38:35,000 --> 00:38:37,000
And so, overall,
it's supposed to end up just

537
00:38:37,000 --> 00:38:41,000
going down.
OK, so now, how to find this

538
00:38:41,000 --> 00:38:47,000
out, well, you have to just
integrate over the entire Earth.

539
00:38:47,000 --> 00:38:52,000
OK, so let's try to see how
that goes.

540
00:38:52,000 --> 00:38:56,000
So, the setup that's going to
be easiest for us to do

541
00:38:56,000 --> 00:39:01,000
computations is going to be that
we are going to be the test mass

542
00:39:01,000 --> 00:39:04,000
that's going to be falling.
And, we are going to put

543
00:39:04,000 --> 00:39:07,000
ourselves at the origin.
And, the solid that's going to

544
00:39:07,000 --> 00:39:10,000
attract us is going to be
wherever we want in space.

545
00:39:10,000 --> 00:39:13,000
You'll see, putting yourself at
the origin is going to be

546
00:39:13,000 --> 00:39:15,000
better.
Well, you have to put something

547
00:39:15,000 --> 00:39:17,000
at the origin.
And, the one that will stay a

548
00:39:17,000 --> 00:39:21,000
point mass, I mean,
in my case not really a point,

549
00:39:21,000 --> 00:39:24,000
but anyway, let's say that I'm
a point.

550
00:39:24,000 --> 00:39:27,000
And then, I have a solid
attracting me.

551
00:39:27,000 --> 00:39:32,000
Well,
so then if I take a small piece

552
00:39:32,000 --> 00:39:37,000
of it with the mass delta M,
then that portion of the solid

553
00:39:37,000 --> 00:39:42,000
exerts a force on me,
which is going to be directed

554
00:39:42,000 --> 00:39:47,000
towards it,
and we'll have intensity.

555
00:39:47,000 --> 00:39:59,000
So, the gravitational force --
-- exerted by the mass delta M

556
00:39:59,000 --> 00:40:09,000
at the point of x,
y, z in space on a mass at the

557
00:40:09,000 --> 00:40:13,000
origin.
Well, we know how to express

558
00:40:13,000 --> 00:40:16,000
that.
Physics tells us that the

559
00:40:16,000 --> 00:40:21,000
magnitude of this force is going
to be, well, G is just a

560
00:40:21,000 --> 00:40:23,000
constant.
It's the gravitational

561
00:40:23,000 --> 00:40:27,000
constant, and its value depends
on which unit system you use.

562
00:40:27,000 --> 00:40:33,000
Usually it's pretty small,
times the mass delta M,

563
00:40:33,000 --> 00:40:39,000
times the test mass little m,
divided by the square of the

564
00:40:39,000 --> 00:40:43,000
distance.
And, the distance from U to

565
00:40:43,000 --> 00:40:48,000
that thing is conveniently
called rho since we've been

566
00:40:48,000 --> 00:40:51,000
introducing spherical
coordinates.

567
00:40:51,000 --> 00:40:54,000
So, that's the size,
that's the magnitude of the

568
00:40:54,000 --> 00:40:56,000
force.
We also need to know the

569
00:40:56,000 --> 00:41:01,000
direction of the force.
And, the direction is going to

570
00:41:01,000 --> 00:41:07,000
be towards that point.
So, the direction of the force

571
00:41:07,000 --> 00:41:11,000
is going to be that of x,
y, z.

572
00:41:11,000 --> 00:41:13,000
But if I want a unit vector,
then I should scale this down

573
00:41:13,000 --> 00:41:22,000
to length one.
So, let me divide this by rho

574
00:41:22,000 --> 00:41:32,000
to get a unit vector.
So, that means that the force

575
00:41:32,000 --> 00:41:40,000
I'm getting from this guy is
actually going to be G delta M m

576
00:41:40,000 --> 00:41:44,000
over rho cubed times x,
y, z.

577
00:41:44,000 --> 00:41:50,000
I'm just multiplying the
magnitude by the unit vector in

578
00:41:50,000 --> 00:41:54,000
the correct direction.
OK, so now if I have not just

579
00:41:54,000 --> 00:41:56,000
that little p is delta M,
but an entire solid,

580
00:41:56,000 --> 00:41:59,000
then I have to sum all these
guys together.

581
00:41:59,000 --> 00:42:04,000
And, I will get the vector that
gives me the total force

582
00:42:04,000 --> 00:42:06,000
exerted, OK?
So, of course,

583
00:42:06,000 --> 00:42:09,000
there's actually three
different calculations in one

584
00:42:09,000 --> 00:42:12,000
because you have to sum the x
components to get the x

585
00:42:12,000 --> 00:42:16,000
components of a total force.
Same with the y,

586
00:42:16,000 --> 00:42:28,000
and same with the z.
So, let me first write down the

587
00:42:28,000 --> 00:42:36,000
actual formula.
So, if you integrate over the

588
00:42:36,000 --> 00:42:39,000
entire solid,
oh, and I have to remind you,

589
00:42:39,000 --> 00:42:42,000
well, what's the mass,
delta M of a small piece of

590
00:42:42,000 --> 00:42:45,000
volume delta V?
Well, it's the density times

591
00:42:45,000 --> 00:42:48,000
the volume.
So, the mass is going to be,

592
00:42:48,000 --> 00:42:54,000
sorry, density is delta.
There is a lot of Greek letters

593
00:42:54,000 --> 00:43:04,000
there, times the volume element.
So, you will get that the force

594
00:43:04,000 --> 00:43:12,000
is the triple integral over your
solid of G m x,

595
00:43:12,000 --> 00:43:18,000
y, z over rho cubed,
delta dV.

596
00:43:18,000 --> 00:43:21,000
Now, two observations about
that.

597
00:43:21,000 --> 00:43:23,000
So, the first one,
well, of course,

598
00:43:23,000 --> 00:43:29,000
these are just constants.
So, they can go out.

599
00:43:29,000 --> 00:43:31,000
The second observation,
so here, we are integrating a

600
00:43:31,000 --> 00:43:33,000
vector quantity.
So, what does that mean?

601
00:43:33,000 --> 00:43:38,000
I just mean the x component of
a force is given by integrating

602
00:43:38,000 --> 00:43:41,000
G m x over rho cubed delta dV.
The y components,

603
00:43:41,000 --> 00:43:43,000
same thing with y.
The z components,

604
00:43:43,000 --> 00:43:46,000
same thing with z.
OK, there's no,

605
00:43:46,000 --> 00:43:51,000
like, you know,
just integrate component by

606
00:43:51,000 --> 00:43:56,000
component to get each component
of the force.

607
00:43:56,000 --> 00:44:01,000
So, now we could very well to
this in rectangular coordinates

608
00:44:01,000 --> 00:44:04,000
if we want.
But the annoying thing is this

609
00:44:04,000 --> 00:44:06,000
rho cubed.
Rho cubed is going to be x

610
00:44:06,000 --> 00:44:10,000
squared plus y squared plus z
squared to the three halves.

611
00:44:10,000 --> 00:44:13,000
That's not going to be a very
pleasant thing to integrate.

612
00:44:13,000 --> 00:44:24,000
So, it's much better to set up
these integrals in spherical

613
00:44:24,000 --> 00:44:29,000
coordinates.
And, if we're going to do it in

614
00:44:29,000 --> 00:44:32,000
spherical coordinates,
then probably we don't want to

615
00:44:32,000 --> 00:44:34,000
bother too much with x and y
components because those would

616
00:44:34,000 --> 00:44:38,000
be unpleasant.
It would give us rho sine phi

617
00:44:38,000 --> 00:44:47,000
cos theta or sine theta.
So, the actual way we will set

618
00:44:47,000 --> 00:44:57,000
up things, set things up,
is to place the solid so that

619
00:44:57,000 --> 00:45:04,000
the z axis is an axis of
symmetry.

620
00:45:04,000 --> 00:45:07,000
And, of course,
that only works if the solid

621
00:45:07,000 --> 00:45:10,000
has some axis of symmetry.
Like, if you're trying to find

622
00:45:10,000 --> 00:45:13,000
the gravitational attraction of
the Pyramid of Giza,

623
00:45:13,000 --> 00:45:16,000
then you won't be able to set
up so that it has rotation

624
00:45:16,000 --> 00:45:18,000
symmetry.
Well, that's a tough fact of

625
00:45:18,000 --> 00:45:21,000
life, and you have to actually
do it in x, y,

626
00:45:21,000 --> 00:45:24,000
z coordinates.
But, if at all possible,

627
00:45:24,000 --> 00:45:27,000
then you're going to place
things.

628
00:45:27,000 --> 00:45:30,000
Well, I guess even then,
you could center it on the z

629
00:45:30,000 --> 00:45:32,000
axis.
But anyway, so you're going to

630
00:45:32,000 --> 00:45:37,000
mostly place things so that your
solid is actually centered on

631
00:45:37,000 --> 00:45:41,000
the z-axis.
And, what you gain by that is

632
00:45:41,000 --> 00:45:45,000
that by symmetry,
the gravitational force will be

633
00:45:45,000 --> 00:45:52,000
directed along the z axis.
So, you will just have to

634
00:45:52,000 --> 00:45:58,000
figure out the z component.
So, then the force will be

635
00:45:58,000 --> 00:46:03,000
actually, you know in advance
that it will be given by zero,

636
00:46:03,000 --> 00:46:11,000
zero, and some z component.
And then, you just need to

637
00:46:11,000 --> 00:46:19,000
compute that component.
And, that component will be

638
00:46:19,000 --> 00:46:27,000
just G times m times triple
integral of z over rho cubed

639
00:46:27,000 --> 00:46:30,000
delta dV.
OK, so that's the first

640
00:46:30,000 --> 00:46:35,000
simplification we can try to do.
The second thing is,

641
00:46:35,000 --> 00:46:38,000
well, we have to choose our
favorite coordinate system to do

642
00:46:38,000 --> 00:46:45,000
this.
But, I claim that actually

643
00:46:45,000 --> 00:46:57,000
spherical coordinates are the
best -- -- because let's see

644
00:46:57,000 --> 00:47:04,000
what happens.
So, G times mass times triple

645
00:47:04,000 --> 00:47:09,000
integral, well,
a z in spherical coordinates

646
00:47:09,000 --> 00:47:14,000
becomes rho cosine phi over rho
cubed.

647
00:47:14,000 --> 00:47:17,000
Density, well,
we can't do anything about

648
00:47:17,000 --> 00:47:21,000
density.
And then, dV becomes rho

649
00:47:21,000 --> 00:47:28,000
squared sine phi d rho d phi d
theta.

650
00:47:28,000 --> 00:47:34,000
Well, so, what happens with
that?

651
00:47:34,000 --> 00:47:37,000
Well, you see that you have a
rho, a rho squared,

652
00:47:37,000 --> 00:47:39,000
and a rho cubed that cancel
each other.

653
00:47:39,000 --> 00:47:42,000
So, in fact,
it simplifies quite a bit if

654
00:47:42,000 --> 00:47:44,000
you do it in spherical
coordinates.

655
00:48:08,000 --> 00:48:12,000
OK, so the z component of the
force, sorry,

656
00:48:12,000 --> 00:48:18,000
I'm putting a z here to remind
you it's the z component.

657
00:48:18,000 --> 00:48:19,000
That is not a partial
derivative, OK?

658
00:48:19,000 --> 00:48:27,000
Don't get things mixed up,
just the z component of the

659
00:48:27,000 --> 00:48:35,000
force becomes Gm triple integral
of delta cos phi sine phi d rho

660
00:48:35,000 --> 00:48:40,000
d phi d theta.
And, so this thing is not dV,

661
00:48:40,000 --> 00:48:42,000
of course.
dV is much bigger,

662
00:48:42,000 --> 00:48:45,000
but we've somehow canceled out
most of dV with stuff that was

663
00:48:45,000 --> 00:48:49,000
in the integrand.
And see, that's actually

664
00:48:49,000 --> 00:48:55,000
suddenly much less scary.
OK, so just to give you an

665
00:48:55,000 --> 00:49:01,000
example of what you can prove it
this way, you can prove Newton's

666
00:49:01,000 --> 00:49:06,000
theorem, which says the
following thing.

667
00:49:06,000 --> 00:49:23,000
It says the gravitational
attraction -- -- of a spherical

668
00:49:23,000 --> 00:49:29,000
planet,
I should say with uniform

669
00:49:29,000 --> 00:49:32,000
density,
or actually it's enough for the

670
00:49:32,000 --> 00:49:34,000
density to depend just on
distance to the center.

671
00:49:34,000 --> 00:49:50,000
But we just simplify the
statement is equal to that of a

672
00:49:50,000 --> 00:50:05,000
point mass -- -- with the same
total mass at its center.

673
00:50:05,000 --> 00:50:11,000
OK, so what that means is that,
so the way we would set it up

674
00:50:11,000 --> 00:50:18,000
is u would be sitting here and
your planet would be over here.

675
00:50:18,000 --> 00:50:21,000
Or, if you're at the surface of
it, then of course you just put

676
00:50:21,000 --> 00:50:25,000
it tangent to the xy plane here.
And, you would compute that

677
00:50:25,000 --> 00:50:27,000
quantity.
Computation is a little bit

678
00:50:27,000 --> 00:50:30,000
annoying if a sphere is sitting
up there because,

679
00:50:30,000 --> 00:50:31,000
of course, you have to find
bounds,

680
00:50:31,000 --> 00:50:33,000
and that's not going to be very
pleasant.

681
00:50:33,000 --> 00:50:37,000
The case that we actually know
how to do fairly well is if you

682
00:50:37,000 --> 00:50:39,000
are just at the surface of the
planet.

683
00:50:39,000 --> 00:50:41,000
But then,
what the theorem says is that

684
00:50:41,000 --> 00:50:44,000
the force that you're going to
feel is exactly the same as if

685
00:50:44,000 --> 00:50:48,000
you removed all of the planet
and you just put an equivalent

686
00:50:48,000 --> 00:50:50,000
point mass here.
So, if the earth collapsed to a

687
00:50:50,000 --> 00:50:53,000
black hole at the center of the
earth with the same mass,

688
00:50:53,000 --> 00:50:55,000
well, you wouldn't notice the
difference immediately,

689
00:50:55,000 --> 00:51:00,000
or, rather, you would,
but at least not in terms of

690
00:51:00,000 --> 00:51:04,000
your weight.
OK, that's the end for today.