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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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So, in a way,
you can think of these as a
00:01:06.000 --> 00:01:09.000
space analog of polar
coordinates because you just use
00:01:09.000 --> 00:01:12.000
distance to the origin,
and then you have to use angles
00:01:12.000 --> 00:01:15.000
to determine in which direction
you're going.
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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
00:01:25.000 --> 00:01:29.000
point in space at coordinates x,
y, z.
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Then, instead of using x,
y, z, you will use,
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well, one thing you'll use is
the distance from the origin.
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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So -- That's the distance from
the origin.
00:01:57.000 --> 00:02:02.000
And so, that can take values
anywhere between zero and
00:02:02.000 --> 00:02:06.000
infinity.
Then, we have to use two other
00:02:06.000 --> 00:02:09.000
angles.
And, so for that,
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let me actually draw the
vertical half plane that
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contains our point starting from
the z axis.
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OK, so then we have two new
angles.
00:02:25.000 --> 00:02:27.000
Well, one of them is not really
new.
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One is new.
That's phi is the angle
00:02:30.000 --> 00:02:34.000
downwards from the z axis.
And the other one,
00:02:34.000 --> 00:02:39.000
theta, is the angle
counterclockwise from the x
00:02:39.000 --> 00:02:51.000
axis.
OK, so phi, let me do it better.
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So, there's two ways to draw
the letter phi,
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by the way.
And, I recommend this one
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because it doesn't look like a
rho.
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So, that's easier.
That's the angle that you have
00:03:07.000 --> 00:03:13.000
to go down from the positive z
axis.
00:03:13.000 --> 00:03:17.000
And,
so that angle varies from zero
00:03:17.000 --> 00:03:23.000
when you're on the z axis,
increase to pi over two when
00:03:23.000 --> 00:03:28.000
you are on the xy plane all the
way to pi or 180� when you are
00:03:28.000 --> 00:03:38.000
on the negative z axis.
It doesn't go beyond that.
00:03:38.000 --> 00:03:44.000
OK, so -- Phi is always between
zero and pi.
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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So, it's the angle after you
project to the xy plane.
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
00:04:02.000 --> 00:04:05.000
overwhelming not just because of
the new letters,
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but also because there is a lot
of angles in there.
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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
00:04:21.000 --> 00:04:27.000
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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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
00:04:54.000 --> 00:04:59.000
different conventions.
OK, so, phi is more or less the
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same thing as latitude in the
sense that it measures how far
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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
00:05:21.000 --> 00:05:24.000
which is zero,
increasing all the way to the
00:05:24.000 --> 00:05:29.000
South Pole, which is at pi.
And, theta or you can think of
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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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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.
00:05:51.000 --> 00:05:56.000
So -- Let's start again from
cylindrical coordinates,
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which hopefully you're kind of
comfortable with now.
00:06:02.000 --> 00:06:06.000
OK, so you know about
cylindrical coordinates where we
00:06:06.000 --> 00:06:09.000
have the z coordinates stay z,
and the xy plane we do R and
00:06:09.000 --> 00:06:13.000
theta polar coordinates.
And now, let's think about what
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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and then you have the direction
away from the z axis,
00:06:25.000 --> 00:06:29.000
which I will call r,
just because that's what r
00:06:29.000 --> 00:06:32.000
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
00:06:46.000 --> 00:06:48.000
you have rectangular coordinates
again.
00:06:48.000 --> 00:06:51.000
So the idea of spherical
coordinate is you're going to
00:06:51.000 --> 00:06:54.000
polar coordinates again in the
rz plane.
00:06:54.000 --> 00:07:01.000
OK, so if I have a point here,
then rho will be the distance
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from the origin.
And phi will be the angle,
00:07:06.000 --> 00:07:10.000
except it's measured from the
positive z axis,
00:07:10.000 --> 00:07:16.000
not from the horizontal axis.
But, the idea in here,
00:07:16.000 --> 00:07:18.000
see,
let me put that between quotes
00:07:18.000 --> 00:07:21.000
because I'm not sure how correct
that is,
00:07:21.000 --> 00:07:28.000
but in a way,
you can think of this as polar
00:07:28.000 --> 00:07:34.000
coordinates in the rz plane.
So, in particular,
00:07:34.000 --> 00:07:38.000
that's the key to understanding
how to switch between spherical
00:07:38.000 --> 00:07:41.000
coordinates and cylindrical
coordinates,
00:07:41.000 --> 00:07:44.000
and then all the way to x,
y, z if you want,
00:07:44.000 --> 00:07:48.000
right,
because this picture here tells
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us how to express z and r in
terms of rho and phi.
00:07:53.000 --> 00:08:03.000
So, let's see how that works.
If I project here or here,
00:08:03.000 --> 00:08:12.000
so, this line is z.
But, it's also rho times cosine
00:08:12.000 --> 00:08:19.000
phi.
So, I get z equals rho cos phi.
00:08:19.000 --> 00:08:21.000
And, if I look at r,
it's the same thing,
00:08:21.000 --> 00:08:31.000
but on the other side.
So, r will be rho sine phi.
00:08:31.000 --> 00:08:34.000
OK, so you can use this to
switch back and forth between
00:08:34.000 --> 00:08:37.000
spherical and cylindrical.
And of course,
00:08:37.000 --> 00:08:43.000
if you remember what x and y
were in terms of r and theta,
00:08:43.000 --> 00:08:49.000
you can also keep doing this to
figure out, oops.
00:08:49.000 --> 00:08:57.000
So, x is r cos theta.
That becomes rho sine phi cos
00:08:57.000 --> 00:09:01.000
theta.
Y is r sine theta.
00:09:01.000 --> 00:09:06.000
So, that becomes rho sine phi
sine theta.
00:09:06.000 --> 00:09:15.000
And z is rho cos phi.
But, basically you don't really
00:09:15.000 --> 00:09:19.000
need to remember these formulas
as long as you remember how to
00:09:19.000 --> 00:09:22.000
express r in terms of rho sine
phi,
00:09:22.000 --> 00:09:29.000
and x equals r cos theta.
So, now, of course,
00:09:29.000 --> 00:09:31.000
we're going to use spherical
coordinates in situations where
00:09:31.000 --> 00:09:33.000
we have a lot of symmetry,
and in particular,
00:09:33.000 --> 00:09:35.000
where the z axis plays a
special role.
00:09:35.000 --> 00:09:38.000
Actually, that's the same with
cylindrical coordinates.
00:09:38.000 --> 00:09:40.000
Cylindrical and secure
coordinates are set up so that
00:09:40.000 --> 00:09:44.000
the z axis plays a special role.
So, that means whenever you
00:09:44.000 --> 00:09:47.000
have a geometric problem,
and you are not told how to
00:09:47.000 --> 00:09:51.000
choose your coordinates,
it's probably wiser to try to
00:09:51.000 --> 00:09:57.000
center things on the z axis.
That's where these coordinates
00:09:57.000 --> 00:10:01.000
are the best adapted.
And,
00:10:01.000 --> 00:10:03.000
in case you ever need to switch
backwards,
00:10:03.000 --> 00:10:07.000
I just want to point out,
so, rho is the square root of r
00:10:07.000 --> 00:10:11.000
squared plus z squared,
which means it's the square
00:10:11.000 --> 00:10:15.000
root of x squared plus y squared
plus z squared.
00:10:15.000 --> 00:10:20.000
OK, so that's basically all the
formulas about spherical
00:10:20.000 --> 00:10:28.000
coordinates.
OK, any questions about that?
00:10:28.000 --> 00:10:31.000
OK, let's see,
who had seen spherical
00:10:31.000 --> 00:10:34.000
coordinates before just to see?
OK, that's not very many.
00:10:34.000 --> 00:10:36.000
So, I'm sure for,
one of you saw it twice.
00:10:36.000 --> 00:10:42.000
That's great.
Sorry, oops,
00:10:42.000 --> 00:10:49.000
OK, so let's just look quickly
at equations of some of the
00:10:49.000 --> 00:10:54.000
things.
So, as I've said,
00:10:54.000 --> 00:11:02.000
if I set rho equals a,
that will be just a sphere of
00:11:02.000 --> 00:11:11.000
radius a centered at the origin.
More interesting things:
00:11:11.000 --> 00:11:14.000
let's say I give you phi equals
pi over four.
00:11:14.000 --> 00:11:18.000
What do you think that looks
like?
00:11:18.000 --> 00:11:29.000
Actually, let's take a quick
poll on things.
00:11:29.000 --> 00:11:31.000
OK, yeah, everyone seems to be
saying it's a cone,
00:11:31.000 --> 00:11:33.000
and that's indeed the correct
answer.
00:11:33.000 --> 00:11:41.000
So, how do we see that?
Well, remember,
00:11:41.000 --> 00:11:44.000
phi is the angle downward from
the z axis.
00:11:44.000 --> 00:11:49.000
So, let's say that I'm going to
look first at what happens if
00:11:49.000 --> 00:11:53.000
I'm in the right half of a plane
of a blackboard,
00:11:53.000 --> 00:11:56.000
so, in the yz plane.
Then, phi is the angle downward
00:11:56.000 --> 00:11:58.000
from here.
So, if I want to get pi over
00:11:58.000 --> 00:12:01.000
four, that's 45�.
That means I'm going to go
00:12:01.000 --> 00:12:03.000
diagonally like this.
Of course, if I'm in the left
00:12:03.000 --> 00:12:06.000
half of a plane of a blackboard,
it's going to be the same.
00:12:06.000 --> 00:12:10.000
I also take pi over four.
And, I get the other half.
00:12:10.000 --> 00:12:13.000
And, because the equation does
not involve theta,
00:12:13.000 --> 00:12:17.000
it's all the same if I rotate
my vertical plane around the z
00:12:17.000 --> 00:12:21.000
axis.
So, I get the same picture in
00:12:21.000 --> 00:12:27.000
any of these vertical half
planes, actually.
00:12:27.000 --> 00:12:32.000
OK, now, so this is phi equals
pi over four.
00:12:32.000 --> 00:12:35.000
And, just in case,
to point out to you what's
00:12:35.000 --> 00:12:39.000
going on, when phi equals pi
over four, cosine and sine are
00:12:39.000 --> 00:12:42.000
equal to each other.
They are both one over root two.
00:12:42.000 --> 00:12:46.000
So, you can find,
again, the equation of this
00:12:46.000 --> 00:12:51.000
thing in cylindrical
coordinates, which I'll remind
00:12:51.000 --> 00:12:54.000
you was z equals r.
OK, in general,
00:12:54.000 --> 00:12:58.000
phi equals some given number,
or z equals some number times
00:12:58.000 --> 00:13:01.000
r.
That will be a cone centered on
00:13:01.000 --> 00:13:04.000
the z axis.
OK, a special case:
00:13:04.000 --> 00:13:07.000
what if I say phi equals pi
over two?
00:13:07.000 --> 00:13:09.000
Yeah, it's just going to be the
xy plane.
00:13:09.000 --> 00:13:13.000
OK, that's the flattest of all
cones.
00:13:13.000 --> 00:13:20.000
OK, so phi equals pi over two
is going to be just the xy
00:13:20.000 --> 00:13:22.000
plane.
And, in general,
00:13:22.000 --> 00:13:24.000
if phi is less than pi over
two, then you are in the upper
00:13:24.000 --> 00:13:28.000
half space.
If phi is more than pi over
00:13:28.000 --> 00:13:32.000
two, you'll be in the lower half
space.
00:13:32.000 --> 00:13:36.000
OK, so that's pretty much all
we need to know at this point.
00:13:36.000 --> 00:13:45.000
So, what's next?
Well, remember we were trying
00:13:45.000 --> 00:13:52.000
to do triple integrals.
So now we're going to triple
00:13:52.000 --> 00:13:59.000
integrals in spherical
coordinates.
00:13:59.000 --> 00:14:01.000
And, for that,
we first need to understand
00:14:01.000 --> 00:14:06.000
what the volume element is.
What will be dV?
00:14:06.000 --> 00:14:12.000
OK, so dV will be something,
d rho, d phi,
00:14:12.000 --> 00:14:18.000
d theta, or in any order that
you want.
00:14:18.000 --> 00:14:23.000
But, this one is usually the
most convenient.
00:14:23.000 --> 00:14:27.000
So, to find out what it is,
well, we should look at how we
00:14:27.000 --> 00:14:29.000
are going to be slicing things
now.
00:14:29.000 --> 00:14:32.000
OK, so if you integrate d rho,
d phi, d theta,
00:14:32.000 --> 00:14:37.000
it means that you are actually
slicing your solid into little
00:14:37.000 --> 00:14:40.000
pieces that live,
somehow,
00:14:40.000 --> 00:14:45.000
if you set an interval of rows,
OK,
00:14:45.000 --> 00:14:48.000
sorry, maybe I should,
so, if you first integrate over
00:14:48.000 --> 00:14:51.000
rho,
it means that you will actually
00:14:51.000 --> 00:14:57.000
choose first the direction from
the origin even by phi and
00:14:57.000 --> 00:15:00.000
theta.
And, in that direction,
00:15:00.000 --> 00:15:04.000
you will try to figure out,
how far does your region
00:15:04.000 --> 00:15:07.000
extend?
And, of course,
00:15:07.000 --> 00:15:11.000
how far that goes might depend
on phi and theta.
00:15:11.000 --> 00:15:16.000
Then, you will vary phi.
So, you have to know,
00:15:16.000 --> 00:15:21.000
for a given value of theta,
how far down does your solid
00:15:21.000 --> 00:15:22.000
extend?
And, finally,
00:15:22.000 --> 00:15:25.000
the value of theta will
correspond to,
00:15:25.000 --> 00:15:28.000
in which directions around the
z axis do we go?
00:15:28.000 --> 00:15:31.000
So, we're going to see that in
examples.
00:15:31.000 --> 00:15:34.000
But before we can do that,
we need to get the volume
00:15:34.000 --> 00:15:36.000
element.
So, what I would like to
00:15:36.000 --> 00:15:40.000
suggest is that we need to
figure out,
00:15:40.000 --> 00:15:46.000
what is the volume of a small
piece of solid which corresponds
00:15:46.000 --> 00:15:49.000
to a certain change,
delta rho,
00:15:49.000 --> 00:15:52.000
delta phi,
and delta theta?
00:15:52.000 --> 00:15:56.000
So, delta rho means that you
have two concentric spheres,
00:15:56.000 --> 00:16:01.000
and you are looking at a very
thin shell in between them.
00:16:01.000 --> 00:16:05.000
And then, you would be looking
at a piece of that spherical
00:16:05.000 --> 00:16:08.000
shell corresponding to small
values of phi and theta.
00:16:08.000 --> 00:16:14.000
So, because I am stretching the
limits of my ability to draw on
00:16:14.000 --> 00:16:18.000
the board, here's a picture.
I'm going to try to reproduce
00:16:18.000 --> 00:16:21.000
on the board,
but so let's start by looking
00:16:21.000 --> 00:16:24.000
just at what happens on the
sphere of radius a,
00:16:24.000 --> 00:16:28.000
and let's try to figure out the
surface area elements on the
00:16:28.000 --> 00:16:30.000
sphere in terms of phi and
theta.
00:16:30.000 --> 00:16:39.000
And then, we'll add the rho
direction.
00:16:39.000 --> 00:16:49.000
OK, so -- So,
let me say, let's start by
00:16:49.000 --> 00:17:02.000
understanding surface area on a
sphere of radius a.
00:17:02.000 --> 00:17:12.000
So, that means we'll be looking
at a little piece of the sphere
00:17:12.000 --> 00:17:21.000
corresponding to angles delta
phi and in that direction here
00:17:21.000 --> 00:17:26.000
delta theta.
OK, so when you draw a map of
00:17:26.000 --> 00:17:29.000
the world on a globe,
that's exactly what the grid
00:17:29.000 --> 00:17:33.000
lines form for you.
So, what's the area of this guy?
00:17:33.000 --> 00:17:35.000
Well, of course,
all the sides are curvy.
00:17:35.000 --> 00:17:37.000
They are all on the sphere.
None of them are straight.
00:17:37.000 --> 00:17:41.000
But still, if it's small enough
and it looks like a rectangle,
00:17:41.000 --> 00:17:46.000
so let's just try to figure
out, what are the sides of your
00:17:46.000 --> 00:17:49.000
rectangle?
OK, so, let's see,
00:17:49.000 --> 00:17:55.000
well, I think I need to draw a
bigger picture of this guy.
00:17:55.000 --> 00:17:59.000
OK, so this guy,
so that's a piece of what's
00:17:59.000 --> 00:18:05.000
called a parallel in geography.
That's a circle that goes
00:18:05.000 --> 00:18:07.000
east-west.
So now,
00:18:07.000 --> 00:18:10.000
this parallel as a circle of
radius,
00:18:10.000 --> 00:18:14.000
well, the radius is less than a
because if your vertical is to
00:18:14.000 --> 00:18:17.000
the North Pole,
it will be actually much
00:18:17.000 --> 00:18:19.000
smaller.
So, that's why when you say
00:18:19.000 --> 00:18:22.000
you're going around the world it
depends on whether you do it at
00:18:22.000 --> 00:18:28.000
the equator or the North Pole.
It's much easier at the North
00:18:28.000 --> 00:18:33.000
Pole.
So, anyway, this is a piece of
00:18:33.000 --> 00:18:40.000
a circle of radius,
well, the radius is what I
00:18:40.000 --> 00:18:49.000
would call r because that's the
distance from the z axis.
00:18:49.000 --> 00:18:51.000
OK, that's actually pretty hard
to see now.
00:18:51.000 --> 00:18:58.000
So if you can see it better on
this one, then so this guy here,
00:18:58.000 --> 00:19:03.000
this length is r.
And, r is just rho,
00:19:03.000 --> 00:19:07.000
well, what was a times sine
phi.
00:19:07.000 --> 00:19:09.000
Remember, we have this angle
phi in here.
00:19:09.000 --> 00:19:14.000
I should use some color.
It's getting very cluttered.
00:19:14.000 --> 00:19:19.000
So, we have this phi,
and so r is going to be rho
00:19:19.000 --> 00:19:21.000
sine phi.
That rho is a.
00:19:21.000 --> 00:19:29.000
So, let me just put a sine phi.
OK, and the corresponding angle
00:19:29.000 --> 00:19:32.000
is going to be measured by
theta.
00:19:32.000 --> 00:19:48.000
So, the length of this is going
to be a sine phi delta theta.
00:19:48.000 --> 00:19:54.000
That's for this side.
Now, what about that side,
00:19:54.000 --> 00:19:56.000
the north-south side?
Well, if you're moving
00:19:56.000 --> 00:19:58.000
north-south, it's not like
east-west.
00:19:58.000 --> 00:20:01.000
You always have to go all the
way from the North Pole to the
00:20:01.000 --> 00:20:04.000
South Pole.
So, that's actually a great
00:20:04.000 --> 00:20:08.000
circle meridian of length,
well, I mean,
00:20:08.000 --> 00:20:13.000
well, the radius is the radius
of the sphere.
00:20:13.000 --> 00:20:22.000
Total length is 2pi a.
So, this is a piece of a circle
00:20:22.000 --> 00:20:27.000
of radius a.
And so, now,
00:20:27.000 --> 00:20:34.000
the length of this one is going
to be a delta phi.
00:20:34.000 --> 00:20:41.000
OK, so, just to recap,
this is a sine phi delta theta.
00:20:41.000 --> 00:20:46.000
And, this guy here is a delta
phi.
00:20:46.000 --> 00:20:59.000
So, you can't read it because
it's -- And so,
00:20:59.000 --> 00:21:02.000
that tells us if I take a small
piece of the sphere,
00:21:02.000 --> 00:21:06.000
then its surface area,
delta s,
00:21:06.000 --> 00:21:15.000
is going to be approximately a
sine phi delta theta times a
00:21:15.000 --> 00:21:22.000
delta phi,
which I'm going to rewrite as a
00:21:22.000 --> 00:21:27.000
squared sine phi delta phi delta
theta.
00:21:27.000 --> 00:21:31.000
So, what that means is,
say that I want to integrate
00:21:31.000 --> 00:21:34.000
something just on the surface of
a sphere.
00:21:34.000 --> 00:21:37.000
Well, I would use phi and theta
as my coordinates.
00:21:37.000 --> 00:21:46.000
And then, to know how big a
piece of a sphere is,
00:21:46.000 --> 00:21:55.000
I would just take a squared
sine phi d phi d theta.
00:21:55.000 --> 00:21:59.000
OK, so that's the surface
element in a sphere.
00:21:59.000 --> 00:22:03.000
And now, what about going back
into the third dimension,
00:22:03.000 --> 00:22:05.000
so, adding some depth to these
things?
00:22:05.000 --> 00:22:10.000
Well, I'm not going to try to
draw a picture because you've
00:22:10.000 --> 00:22:17.000
seen that's slightly tricky.
Well, let me try anyway just
00:22:17.000 --> 00:22:24.000
you can have fun with my
completely unreadable diagrams.
00:22:24.000 --> 00:22:28.000
So anyway, if you look at,
now, something that's a bit
00:22:28.000 --> 00:22:33.000
like that piece of sphere,
but with some thickness to it.
00:22:33.000 --> 00:22:38.000
The thickness will be delta
rho, and so the volume will be
00:22:38.000 --> 00:22:44.000
roughly the area of the thing on
the sphere times the thickness.
00:22:44.000 --> 00:22:48.000
So, I claim that we will get
basically the volume element
00:22:48.000 --> 00:22:51.000
just by multiplying things by d
rho.
00:22:51.000 --> 00:23:10.000
So, let's see that.
So now, if I have a sphere of
00:23:10.000 --> 00:23:19.000
radius rho, and another one
that's slightly bigger of radius
00:23:19.000 --> 00:23:27.000
rho plus delta rho,
and then I have a little box in
00:23:27.000 --> 00:23:29.000
here.
Then,
00:23:29.000 --> 00:23:34.000
I know that the volume of this
thing will be essentially,
00:23:34.000 --> 00:23:38.000
well, its thickness,
the thickness is going to be
00:23:38.000 --> 00:23:42.000
delta rho times the area of its
base,
00:23:42.000 --> 00:23:44.000
although it doesn't really
matter,
00:23:44.000 --> 00:23:48.000
which is what we've called
delta s.
00:23:48.000 --> 00:23:55.000
OK, so we will get,
sorry, a becomes rho now.
00:23:55.000 --> 00:23:57.000
Square sine phi,
delta rho,
00:23:57.000 --> 00:24:00.000
delta phi,
delta theta,
00:24:00.000 --> 00:24:04.000
and so out of that we get the
volume element and spherical
00:24:04.000 --> 00:24:08.000
coordinates,
which is rho squared sine phi d
00:24:08.000 --> 00:24:09.000
rho,
d phi,
00:24:09.000 --> 00:24:14.000
d theta.
And, that's a formula that you
00:24:14.000 --> 00:24:17.000
should remember.
OK, so whenever we integrate a
00:24:17.000 --> 00:24:20.000
function,
and we decide to switch to
00:24:20.000 --> 00:24:25.000
spherical coordinates,
then dx dy dz or r dr d theta
00:24:25.000 --> 00:24:33.000
dz will become rho squared sine
phi d rho d phi d theta.
00:24:33.000 --> 00:24:40.000
OK, any questions on that?
No?
00:24:40.000 --> 00:24:58.000
OK, so let's -- Let's see how
that works.
00:24:58.000 --> 00:25:04.000
So, as an example,
remember at the end of the last
00:25:04.000 --> 00:25:11.000
lecture, I tried to set up an
example where we were looking at
00:25:11.000 --> 00:25:16.000
a sphere sliced by a slanted
plane.
00:25:16.000 --> 00:25:20.000
And now, we're going to try to
find the volume of that
00:25:20.000 --> 00:25:23.000
spherical cap again,
but using spherical coordinates
00:25:23.000 --> 00:25:26.000
instead.
So, I'm going to just be
00:25:26.000 --> 00:25:29.000
smarter than last time.
So, last time,
00:25:29.000 --> 00:25:33.000
we had set up these things with
a slanted plane that was cutting
00:25:33.000 --> 00:25:35.000
things diagonally.
And,
00:25:35.000 --> 00:25:37.000
if I just want to find the
volume of this cap,
00:25:37.000 --> 00:25:41.000
then maybe it makes more sense
to rotate things so that my
00:25:41.000 --> 00:25:45.000
plane is actually horizontal,
and things are going to be
00:25:45.000 --> 00:25:49.000
centered on the z axis.
So, in case you see that it's
00:25:49.000 --> 00:25:52.000
the same, then that's great.
If not, then it doesn't really
00:25:52.000 --> 00:25:55.000
matter.
You can just think of this as a
00:25:55.000 --> 00:26:01.000
new example.
So, I'm going to try to find
00:26:01.000 --> 00:26:10.000
the volume of a portion of the
unit sphere -- -- that lies
00:26:10.000 --> 00:26:20.000
above the horizontal plane,
z equals one over root two.
00:26:20.000 --> 00:26:22.000
OK, one over root two was the
distance from the origin to our
00:26:22.000 --> 00:26:24.000
slanted plane.
So, after you rotate,
00:26:24.000 --> 00:26:28.000
that say you get this value.
Anyway, it's not very important.
00:26:28.000 --> 00:26:31.000
You can just treat that as a
good example if you want.
00:26:31.000 --> 00:26:36.000
OK, so we can compute this in
actually pretty much any
00:26:36.000 --> 00:26:39.000
coordinate system.
And also, of course,
00:26:39.000 --> 00:26:42.000
we can set up not only the
volume, but we can try to find
00:26:42.000 --> 00:26:44.000
the moment of inertia about the
central axis,
00:26:44.000 --> 00:26:47.000
or all sorts of things.
But, we are just doing the
00:26:47.000 --> 00:26:49.000
volume for simplicity.
So, actually,
00:26:49.000 --> 00:26:52.000
this would go pretty well in
cylindrical coordinates.
00:26:52.000 --> 00:26:55.000
But let's do it in spherical
coordinates because that's the
00:26:55.000 --> 00:26:57.000
topic of today.
A good exercise:
00:26:57.000 --> 00:27:01.000
do it in cylindrical and see if
you get the same thing.
00:27:01.000 --> 00:27:08.000
So, how do we do that?
Well, we have to figure out how
00:27:08.000 --> 00:27:14.000
to set up our triple integral in
spherical coordinates.
00:27:14.000 --> 00:27:18.000
So, remember we'll be
integrating one dV.
00:27:18.000 --> 00:27:28.000
So, dV will become rho squared
sign phi d rho d phi d theta.
00:27:28.000 --> 00:27:32.000
And, now as we start,
we're already facing some
00:27:32.000 --> 00:27:35.000
serious problem.
We want to set up the bounds
00:27:35.000 --> 00:27:37.000
for rho for a given,
phi and theta.
00:27:37.000 --> 00:27:39.000
So, that means we choose
latitude/longitude.
00:27:39.000 --> 00:27:42.000
We choose which direction we
want to aim for,
00:27:42.000 --> 00:27:45.000
you know, which point of the
sphere we want to aim at.
00:27:45.000 --> 00:27:50.000
And, we are going to shoot a
ray from the origin towards this
00:27:50.000 --> 00:27:55.000
point, and we want to know what
portion of the ray is in our
00:27:55.000 --> 00:28:03.000
solid.
So -- We are going to choose a
00:28:03.000 --> 00:28:11.000
value of phi and theta.
And, we are going to try to
00:28:11.000 --> 00:28:16.000
figure out what part of our ray
is inside this side.
00:28:16.000 --> 00:28:20.000
So, what should be clear is at
which point we leave the solid,
00:28:20.000 --> 00:28:23.000
right?
What's the value of rho here?
00:28:23.000 --> 00:28:25.000
It's just one.
The sphere is rho equals one.
00:28:25.000 --> 00:28:29.000
That's pretty good.
The question is,
00:28:29.000 --> 00:28:33.000
where do we enter the region?
So, we enter the region when we
00:28:33.000 --> 00:28:38.000
go through this plane.
And, the plane is z equals one
00:28:38.000 --> 00:28:41.000
over root two.
So, what does that tell us
00:28:41.000 --> 00:28:44.000
about rho?
Well, it tells us,
00:28:44.000 --> 00:28:50.000
so remember,
z is rho cosine phi.
00:28:50.000 --> 00:28:55.000
So, the plane is z equals one
over root two.
00:28:55.000 --> 00:29:00.000
That means rho cosine phi is
one over root two.
00:29:00.000 --> 00:29:05.000
That means rho equals one over
root two cosine phi or,
00:29:05.000 --> 00:29:11.000
as some of you know it,
one over root two times second
00:29:11.000 --> 00:29:17.000
phi.
OK, so if we want to set up the
00:29:17.000 --> 00:29:27.000
bounds, then we'll start with
one over root two second phi all
00:29:27.000 --> 00:29:32.000
the way to one.
Now, what's next?
00:29:32.000 --> 00:29:35.000
Well, so we've done,
I think that's basically the
00:29:35.000 --> 00:29:38.000
hardest part of the job.
Next, we have to figure out,
00:29:38.000 --> 00:29:41.000
what's the range for phi?
So, the range for phi,
00:29:41.000 --> 00:29:44.000
well, we have to figure out how
far to the north and to the
00:29:44.000 --> 00:29:48.000
south our region goes.
Well, the lower bound for phi
00:29:48.000 --> 00:29:51.000
is pretty easy,
right, because we go all the
00:29:51.000 --> 00:29:56.000
way to the North Pole direction.
So, phi starts at zero.
00:29:56.000 --> 00:29:59.000
The question is,
where does it stop?
00:29:59.000 --> 00:30:02.000
To find out where it stops,
we have to figure out,
00:30:02.000 --> 00:30:06.000
what is the value of phi when
we hit the edge of the region?
00:30:06.000 --> 00:30:10.000
OK, so maybe you see it.
Maybe you don't.
00:30:10.000 --> 00:30:15.000
One way to do it geometrically
is to just, it's always great to
00:30:15.000 --> 00:30:19.000
draw a slice of your region.
So, if you slice these things
00:30:19.000 --> 00:30:22.000
by a vertical plane,
or actually even better,
00:30:22.000 --> 00:30:25.000
a vertical half plane,
something to delete one half of
00:30:25.000 --> 00:30:28.000
the picture.
So, I'm going to draw these r
00:30:28.000 --> 00:30:33.000
and z directions as before.
So, my sphere is here.
00:30:33.000 --> 00:30:38.000
My plane is here at one over
root two.
00:30:38.000 --> 00:30:43.000
And, my solid is here.
So now, the question is what is
00:30:43.000 --> 00:30:49.000
the value of phi when I'm going
to stop hitting the region?
00:30:49.000 --> 00:30:54.000
And, if you try to figure out
first what is this direction
00:30:54.000 --> 00:30:57.000
here, that's also one over root
two.
00:30:57.000 --> 00:31:03.000
And so, this is actually 45�,
also known as pi over four.
00:31:03.000 --> 00:31:09.000
The other way to think about it
is at this point,
00:31:09.000 --> 00:31:16.000
well, rho is equal to one
because you are on the sphere.
00:31:16.000 --> 00:31:22.000
But, you are also on the plane.
So, rho cos phi is one over
00:31:22.000 --> 00:31:26.000
root two.
So, if you plug rho equals one
00:31:26.000 --> 00:31:31.000
into here, you get cos phi
equals one over root two which
00:31:31.000 --> 00:31:34.000
gives you phi equals pi over
four.
00:31:34.000 --> 00:31:37.000
That's the other way to do it.
You can do it either by
00:31:37.000 --> 00:31:39.000
calculation or by looking at the
picture.
00:31:39.000 --> 00:31:43.000
OK, so either way,
we've decided that phi goes
00:31:43.000 --> 00:31:48.000
from zero to pi over four.
So, this is pi over four.
00:31:48.000 --> 00:31:54.000
Finally, what about theta?
Well, because we go all around
00:31:54.000 --> 00:32:00.000
the z axis we are going to go
just zero to 2pi.
00:32:00.000 --> 00:32:06.000
OK, any questions about these
bounds?
00:32:06.000 --> 00:32:10.000
OK, so note how the equation of
this horizontal plane in
00:32:10.000 --> 00:32:13.000
spherical coordinates has become
a little bit weird.
00:32:13.000 --> 00:32:16.000
But,
if you remember how we do
00:32:16.000 --> 00:32:19.000
things,
say that you have a line in
00:32:19.000 --> 00:32:21.000
polar coordinates,
and that line does not pass
00:32:21.000 --> 00:32:23.000
through the origin,
then you also end up with
00:32:23.000 --> 00:32:26.000
something like that.
You get something like r equals
00:32:26.000 --> 00:32:31.000
a second theta or a cos second
theta for horizontal or vertical
00:32:31.000 --> 00:32:33.000
lines.
And so, it's not surprising you
00:32:33.000 --> 00:32:38.000
should get this.
That's in line with the idea
00:32:38.000 --> 00:32:44.000
that we are just doing again,
polar coordinates in the rz
00:32:44.000 --> 00:32:46.000
directions.
So of course,
00:32:46.000 --> 00:32:48.000
in general, things can be very
messy.
00:32:48.000 --> 00:32:51.000
But, generally speaking,
the kinds of regions that we
00:32:51.000 --> 00:32:55.000
will be setting up things for
are no more complicated or no
00:32:55.000 --> 00:32:59.000
less complicated than what we
would do in the plane in polar
00:32:59.000 --> 00:33:00.000
coordinates.
OK, so there's,
00:33:00.000 --> 00:33:03.000
you know, a small list of
things that you should know how
00:33:03.000 --> 00:33:07.000
to set up.
But, you won't have some
00:33:07.000 --> 00:33:18.000
really, really strange thing.
Yes?
00:33:18.000 --> 00:33:20.000
D rho?
Oh, you mean the bounds for rho?
00:33:20.000 --> 00:33:23.000
Yes.
So, in the inner integral,
00:33:23.000 --> 00:33:26.000
we are going to fix values of
phi and theta.
00:33:26.000 --> 00:33:29.000
So, that means we fix in
advance the direction in which
00:33:29.000 --> 00:33:31.000
we are going to shoot a ray from
the origin.
00:33:31.000 --> 00:33:35.000
So now, as we shoot this ray,
we are going to hit our region
00:33:35.000 --> 00:33:37.000
somewhere.
And, we are going to exit,
00:33:37.000 --> 00:33:40.000
again, somewhere else.
OK, so first of all we have to
00:33:40.000 --> 00:33:43.000
figure out where we enter,
where we leave.
00:33:43.000 --> 00:33:46.000
Well, we enter when the ray
hits the flat face,
00:33:46.000 --> 00:33:50.000
when we hit the plane.
And, we would leave when we hit
00:33:50.000 --> 00:33:52.000
the sphere.
So, the lower bound will be
00:33:52.000 --> 00:33:56.000
given by the plane.
The upper bound will be given
00:33:56.000 --> 00:33:58.000
by the sphere.
So now, you have to get
00:33:58.000 --> 00:34:01.000
spherical coordinate equations
for both the plane and the
00:34:01.000 --> 00:34:02.000
sphere.
For the sphere, that's easy.
00:34:02.000 --> 00:34:05.000
That's rho equals one.
For the plane,
00:34:05.000 --> 00:34:08.000
you start with z equals one
over root two.
00:34:08.000 --> 00:34:11.000
And, you switch it into
spherical coordinates.
00:34:11.000 --> 00:34:14.000
And then, you solve for rho.
And, that's how you get these
00:34:14.000 --> 00:34:19.000
bounds.
Is that OK?
00:34:19.000 --> 00:34:26.000
All right, so that's the setup
part.
00:34:26.000 --> 00:34:29.000
And, of course,
the evaluation part goes as
00:34:29.000 --> 00:34:30.000
usual.
00:34:42.000 --> 00:34:46.000
And, since I'm running short of
time, I'm not going to actually
00:34:46.000 --> 00:34:52.000
do the evaluation.
I'm going to let you figure out
00:34:52.000 --> 00:34:58.000
how it goes.
Let me just say in case you
00:34:58.000 --> 00:35:07.000
want to check your answers,
so, at the end you get 2pi over
00:35:07.000 --> 00:35:13.000
three minus 5pi over six root
two.
00:35:13.000 --> 00:35:17.000
Yes, it looks quite complicated.
That's basically because you
00:35:17.000 --> 00:35:20.000
get one over,
well, you get a second square
00:35:20.000 --> 00:35:23.000
when you integrate C.
When you integrate rho squared,
00:35:23.000 --> 00:35:24.000
you will get rho cubed over
three.
00:35:24.000 --> 00:35:27.000
But that rho cubed will give
you a second cube for the lower
00:35:27.000 --> 00:35:29.000
bound.
And, when you integrate sine
00:35:29.000 --> 00:35:31.000
phi second cubed phi,
you do a substitution.
00:35:31.000 --> 00:35:37.000
You see that integrates to one
over second squared with a
00:35:37.000 --> 00:35:42.000
factor in front.
So, in the second square,
00:35:42.000 --> 00:35:49.000
when you plug in,
no, that's not quite all of it.
00:35:49.000 --> 00:35:51.000
Yeah, well, the second square
is one thing,
00:35:51.000 --> 00:35:53.000
and also the other bound you
get sine phi which integrates to
00:35:53.000 --> 00:35:56.000
cosine phi.
So, anyways,
00:35:56.000 --> 00:36:04.000
you get lots of things.
OK, enough about it.
00:36:04.000 --> 00:36:07.000
So, next, I have to tell you
about applications.
00:36:07.000 --> 00:36:13.000
And, of course,
well, there's the same
00:36:13.000 --> 00:36:14.000
applications that we've seen
that last time,
00:36:14.000 --> 00:36:16.000
finding volumes,
finding masses,
00:36:16.000 --> 00:36:19.000
finding average values of
functions.
00:36:19.000 --> 00:36:22.000
In particular,
now, we could say to find the
00:36:22.000 --> 00:36:26.000
average distance of a point in
this solid to the origin.
00:36:26.000 --> 00:36:28.000
Well,
spherical coordinates become
00:36:28.000 --> 00:36:32.000
appealing because the function
you are averaging is just rho
00:36:32.000 --> 00:36:35.000
while in other coordinate
systems it's a more complicated
00:36:35.000 --> 00:36:37.000
function.
So, if you are asked to find
00:36:37.000 --> 00:36:41.000
the average distance from the
origin, spherical coordinates
00:36:41.000 --> 00:36:43.000
can be interesting.
Also,
00:36:43.000 --> 00:36:47.000
well, there's moments of
inertia,
00:36:47.000 --> 00:36:50.000
preferably the one about the z
axis because if you have to
00:36:50.000 --> 00:36:52.000
integrate something that
involves x or y,
00:36:52.000 --> 00:36:55.000
then your integrand will
contain that awful rho sine phi
00:36:55.000 --> 00:36:57.000
sine theta or rho sine phi
cosine theta,
00:36:57.000 --> 00:37:00.000
and then it won't be much fun
to evaluate.
00:37:00.000 --> 00:37:05.000
So, that anyway,
there's the usual ones.
00:37:05.000 --> 00:37:08.000
And then there's a new one.
So, in physics,
00:37:08.000 --> 00:37:16.000
you've probably seen things
about gravitational attraction.
00:37:16.000 --> 00:37:19.000
If not, well,
it's what causes apples to fall
00:37:19.000 --> 00:37:22.000
and other things like that as
well.
00:37:22.000 --> 00:37:26.000
So, anyway, physics tells you
that if you have two masses,
00:37:26.000 --> 00:37:30.000
then they attract each other
with a force that's directed
00:37:30.000 --> 00:37:33.000
towards each other.
And in intensity,
00:37:33.000 --> 00:37:37.000
it's proportional to the two
masses, and inversely
00:37:37.000 --> 00:37:41.000
proportional to the square of
the distance between them.
00:37:41.000 --> 00:37:45.000
So,
if you have a given solid with
00:37:45.000 --> 00:37:50.000
a certain mass distribution,
and you want to know how it
00:37:50.000 --> 00:37:53.000
attracts something else that you
will put nearby,
00:37:53.000 --> 00:37:58.000
then you actually have to,
the first approximation will be
00:37:58.000 --> 00:37:59.000
to say,
well, let's just put a point
00:37:59.000 --> 00:38:02.000
mass at its center of mass.
But, if you're solid is
00:38:02.000 --> 00:38:04.000
actually not homogenous,
or has a weird shape,
00:38:04.000 --> 00:38:07.000
then that's not actually the
exact answer.
00:38:07.000 --> 00:38:09.000
So, in general,
you would have to just take
00:38:09.000 --> 00:38:12.000
every single piece of your
object and figure out how it
00:38:12.000 --> 00:38:14.000
attracts you,
and then compute the sum of
00:38:14.000 --> 00:38:15.000
these.
So, for example,
00:38:15.000 --> 00:38:18.000
if you want to understand why
anything that you drop in this
00:38:18.000 --> 00:38:21.000
room will fall down,
you have to understand that
00:38:21.000 --> 00:38:24.000
Boston is actually attracting it
towards Boston.
00:38:24.000 --> 00:38:26.000
And, Somerville's attracting it
towards Somerville,
00:38:26.000 --> 00:38:29.000
and lots of things like that.
And, China, which is much
00:38:29.000 --> 00:38:33.000
further on the other side is
going to attract towards China.
00:38:33.000 --> 00:38:35.000
But, there's a lot of stuff on
the other side of the Earth.
00:38:35.000 --> 00:38:37.000
And so, overall,
it's supposed to end up just
00:38:37.000 --> 00:38:41.000
going down.
OK, so now, how to find this
00:38:41.000 --> 00:38:47.000
out, well, you have to just
integrate over the entire Earth.
00:38:47.000 --> 00:38:52.000
OK, so let's try to see how
that goes.
00:38:52.000 --> 00:38:56.000
So, the setup that's going to
be easiest for us to do
00:38:56.000 --> 00:39:01.000
computations is going to be that
we are going to be the test mass
00:39:01.000 --> 00:39:04.000
that's going to be falling.
And, we are going to put
00:39:04.000 --> 00:39:07.000
ourselves at the origin.
And, the solid that's going to
00:39:07.000 --> 00:39:10.000
attract us is going to be
wherever we want in space.
00:39:10.000 --> 00:39:13.000
You'll see, putting yourself at
the origin is going to be
00:39:13.000 --> 00:39:15.000
better.
Well, you have to put something
00:39:15.000 --> 00:39:17.000
at the origin.
And, the one that will stay a
00:39:17.000 --> 00:39:21.000
point mass, I mean,
in my case not really a point,
00:39:21.000 --> 00:39:24.000
but anyway, let's say that I'm
a point.
00:39:24.000 --> 00:39:27.000
And then, I have a solid
attracting me.
00:39:27.000 --> 00:39:32.000
Well,
so then if I take a small piece
00:39:32.000 --> 00:39:37.000
of it with the mass delta M,
then that portion of the solid
00:39:37.000 --> 00:39:42.000
exerts a force on me,
which is going to be directed
00:39:42.000 --> 00:39:47.000
towards it,
and we'll have intensity.
00:39:47.000 --> 00:39:59.000
So, the gravitational force --
-- exerted by the mass delta M
00:39:59.000 --> 00:40:09.000
at the point of x,
y, z in space on a mass at the
00:40:09.000 --> 00:40:13.000
origin.
Well, we know how to express
00:40:13.000 --> 00:40:16.000
that.
Physics tells us that the
00:40:16.000 --> 00:40:21.000
magnitude of this force is going
to be, well, G is just a
00:40:21.000 --> 00:40:23.000
constant.
It's the gravitational
00:40:23.000 --> 00:40:27.000
constant, and its value depends
on which unit system you use.
00:40:27.000 --> 00:40:33.000
Usually it's pretty small,
times the mass delta M,
00:40:33.000 --> 00:40:39.000
times the test mass little m,
divided by the square of the
00:40:39.000 --> 00:40:43.000
distance.
And, the distance from U to
00:40:43.000 --> 00:40:48.000
that thing is conveniently
called rho since we've been
00:40:48.000 --> 00:40:51.000
introducing spherical
coordinates.
00:40:51.000 --> 00:40:54.000
So, that's the size,
that's the magnitude of the
00:40:54.000 --> 00:40:56.000
force.
We also need to know the
00:40:56.000 --> 00:41:01.000
direction of the force.
And, the direction is going to
00:41:01.000 --> 00:41:07.000
be towards that point.
So, the direction of the force
00:41:07.000 --> 00:41:11.000
is going to be that of x,
y, z.
00:41:11.000 --> 00:41:13.000
But if I want a unit vector,
then I should scale this down
00:41:13.000 --> 00:41:22.000
to length one.
So, let me divide this by rho
00:41:22.000 --> 00:41:32.000
to get a unit vector.
So, that means that the force
00:41:32.000 --> 00:41:40.000
I'm getting from this guy is
actually going to be G delta M m
00:41:40.000 --> 00:41:44.000
over rho cubed times x,
y, z.
00:41:44.000 --> 00:41:50.000
I'm just multiplying the
magnitude by the unit vector in
00:41:50.000 --> 00:41:54.000
the correct direction.
OK, so now if I have not just
00:41:54.000 --> 00:41:56.000
that little p is delta M,
but an entire solid,
00:41:56.000 --> 00:41:59.000
then I have to sum all these
guys together.
00:41:59.000 --> 00:42:04.000
And, I will get the vector that
gives me the total force
00:42:04.000 --> 00:42:06.000
exerted, OK?
So, of course,
00:42:06.000 --> 00:42:09.000
there's actually three
different calculations in one
00:42:09.000 --> 00:42:12.000
because you have to sum the x
components to get the x
00:42:12.000 --> 00:42:16.000
components of a total force.
Same with the y,
00:42:16.000 --> 00:42:28.000
and same with the z.
So, let me first write down the
00:42:28.000 --> 00:42:36.000
actual formula.
So, if you integrate over the
00:42:36.000 --> 00:42:39.000
entire solid,
oh, and I have to remind you,
00:42:39.000 --> 00:42:42.000
well, what's the mass,
delta M of a small piece of
00:42:42.000 --> 00:42:45.000
volume delta V?
Well, it's the density times
00:42:45.000 --> 00:42:48.000
the volume.
So, the mass is going to be,
00:42:48.000 --> 00:42:54.000
sorry, density is delta.
There is a lot of Greek letters
00:42:54.000 --> 00:43:04.000
there, times the volume element.
So, you will get that the force
00:43:04.000 --> 00:43:12.000
is the triple integral over your
solid of G m x,
00:43:12.000 --> 00:43:18.000
y, z over rho cubed,
delta dV.
00:43:18.000 --> 00:43:21.000
Now, two observations about
that.
00:43:21.000 --> 00:43:23.000
So, the first one,
well, of course,
00:43:23.000 --> 00:43:29.000
these are just constants.
So, they can go out.
00:43:29.000 --> 00:43:31.000
The second observation,
so here, we are integrating a
00:43:31.000 --> 00:43:33.000
vector quantity.
So, what does that mean?
00:43:33.000 --> 00:43:38.000
I just mean the x component of
a force is given by integrating
00:43:38.000 --> 00:43:41.000
G m x over rho cubed delta dV.
The y components,
00:43:41.000 --> 00:43:43.000
same thing with y.
The z components,
00:43:43.000 --> 00:43:46.000
same thing with z.
OK, there's no,
00:43:46.000 --> 00:43:51.000
like, you know,
just integrate component by
00:43:51.000 --> 00:43:56.000
component to get each component
of the force.
00:43:56.000 --> 00:44:01.000
So, now we could very well to
this in rectangular coordinates
00:44:01.000 --> 00:44:04.000
if we want.
But the annoying thing is this
00:44:04.000 --> 00:44:06.000
rho cubed.
Rho cubed is going to be x
00:44:06.000 --> 00:44:10.000
squared plus y squared plus z
squared to the three halves.
00:44:10.000 --> 00:44:13.000
That's not going to be a very
pleasant thing to integrate.
00:44:13.000 --> 00:44:24.000
So, it's much better to set up
these integrals in spherical
00:44:24.000 --> 00:44:29.000
coordinates.
And, if we're going to do it in
00:44:29.000 --> 00:44:32.000
spherical coordinates,
then probably we don't want to
00:44:32.000 --> 00:44:34.000
bother too much with x and y
components because those would
00:44:34.000 --> 00:44:38.000
be unpleasant.
It would give us rho sine phi
00:44:38.000 --> 00:44:47.000
cos theta or sine theta.
So, the actual way we will set
00:44:47.000 --> 00:44:57.000
up things, set things up,
is to place the solid so that
00:44:57.000 --> 00:45:04.000
the z axis is an axis of
symmetry.
00:45:04.000 --> 00:45:07.000
And, of course,
that only works if the solid
00:45:07.000 --> 00:45:10.000
has some axis of symmetry.
Like, if you're trying to find
00:45:10.000 --> 00:45:13.000
the gravitational attraction of
the Pyramid of Giza,
00:45:13.000 --> 00:45:16.000
then you won't be able to set
up so that it has rotation
00:45:16.000 --> 00:45:18.000
symmetry.
Well, that's a tough fact of
00:45:18.000 --> 00:45:21.000
life, and you have to actually
do it in x, y,
00:45:21.000 --> 00:45:24.000
z coordinates.
But, if at all possible,
00:45:24.000 --> 00:45:27.000
then you're going to place
things.
00:45:27.000 --> 00:45:30.000
Well, I guess even then,
you could center it on the z
00:45:30.000 --> 00:45:32.000
axis.
But anyway, so you're going to
00:45:32.000 --> 00:45:37.000
mostly place things so that your
solid is actually centered on
00:45:37.000 --> 00:45:41.000
the z-axis.
And, what you gain by that is
00:45:41.000 --> 00:45:45.000
that by symmetry,
the gravitational force will be
00:45:45.000 --> 00:45:52.000
directed along the z axis.
So, you will just have to
00:45:52.000 --> 00:45:58.000
figure out the z component.
So, then the force will be
00:45:58.000 --> 00:46:03.000
actually, you know in advance
that it will be given by zero,
00:46:03.000 --> 00:46:11.000
zero, and some z component.
And then, you just need to
00:46:11.000 --> 00:46:19.000
compute that component.
And, that component will be
00:46:19.000 --> 00:46:27.000
just G times m times triple
integral of z over rho cubed
00:46:27.000 --> 00:46:30.000
delta dV.
OK, so that's the first
00:46:30.000 --> 00:46:35.000
simplification we can try to do.
The second thing is,
00:46:35.000 --> 00:46:38.000
well, we have to choose our
favorite coordinate system to do
00:46:38.000 --> 00:46:45.000
this.
But, I claim that actually
00:46:45.000 --> 00:46:57.000
spherical coordinates are the
best -- -- because let's see
00:46:57.000 --> 00:47:04.000
what happens.
So, G times mass times triple
00:47:04.000 --> 00:47:09.000
integral, well,
a z in spherical coordinates
00:47:09.000 --> 00:47:14.000
becomes rho cosine phi over rho
cubed.
00:47:14.000 --> 00:47:17.000
Density, well,
we can't do anything about
00:47:17.000 --> 00:47:21.000
density.
And then, dV becomes rho
00:47:21.000 --> 00:47:28.000
squared sine phi d rho d phi d
theta.
00:47:28.000 --> 00:47:34.000
Well, so, what happens with
that?
00:47:34.000 --> 00:47:37.000
Well, you see that you have a
rho, a rho squared,
00:47:37.000 --> 00:47:39.000
and a rho cubed that cancel
each other.
00:47:39.000 --> 00:47:42.000
So, in fact,
it simplifies quite a bit if
00:47:42.000 --> 00:47:44.000
you do it in spherical
coordinates.
00:48:08.000 --> 00:48:12.000
OK, so the z component of the
force, sorry,
00:48:12.000 --> 00:48:18.000
I'm putting a z here to remind
you it's the z component.
00:48:18.000 --> 00:48:19.000
That is not a partial
derivative, OK?
00:48:19.000 --> 00:48:27.000
Don't get things mixed up,
just the z component of the
00:48:27.000 --> 00:48:35.000
force becomes Gm triple integral
of delta cos phi sine phi d rho
00:48:35.000 --> 00:48:40.000
d phi d theta.
And, so this thing is not dV,
00:48:40.000 --> 00:48:42.000
of course.
dV is much bigger,
00:48:42.000 --> 00:48:45.000
but we've somehow canceled out
most of dV with stuff that was
00:48:45.000 --> 00:48:49.000
in the integrand.
And see, that's actually
00:48:49.000 --> 00:48:55.000
suddenly much less scary.
OK, so just to give you an
00:48:55.000 --> 00:49:01.000
example of what you can prove it
this way, you can prove Newton's
00:49:01.000 --> 00:49:06.000
theorem, which says the
following thing.
00:49:06.000 --> 00:49:23.000
It says the gravitational
attraction -- -- of a spherical
00:49:23.000 --> 00:49:29.000
planet,
I should say with uniform
00:49:29.000 --> 00:49:32.000
density,
or actually it's enough for the
00:49:32.000 --> 00:49:34.000
density to depend just on
distance to the center.
00:49:34.000 --> 00:49:50.000
But we just simplify the
statement is equal to that of a
00:49:50.000 --> 00:50:05.000
point mass -- -- with the same
total mass at its center.
00:50:05.000 --> 00:50:11.000
OK, so what that means is that,
so the way we would set it up
00:50:11.000 --> 00:50:18.000
is u would be sitting here and
your planet would be over here.
00:50:18.000 --> 00:50:21.000
Or, if you're at the surface of
it, then of course you just put
00:50:21.000 --> 00:50:25.000
it tangent to the xy plane here.
And, you would compute that
00:50:25.000 --> 00:50:27.000
quantity.
Computation is a little bit
00:50:27.000 --> 00:50:30.000
annoying if a sphere is sitting
up there because,
00:50:30.000 --> 00:50:31.000
of course, you have to find
bounds,
00:50:31.000 --> 00:50:33.000
and that's not going to be very
pleasant.
00:50:33.000 --> 00:50:37.000
The case that we actually know
how to do fairly well is if you
00:50:37.000 --> 00:50:39.000
are just at the surface of the
planet.
00:50:39.000 --> 00:50:41.000
But then,
what the theorem says is that
00:50:41.000 --> 00:50:44.000
the force that you're going to
feel is exactly the same as if
00:50:44.000 --> 00:50:48.000
you removed all of the planet
and you just put an equivalent
00:50:48.000 --> 00:50:50.000
point mass here.
So, if the earth collapsed to a
00:50:50.000 --> 00:50:53.000
black hole at the center of the
earth with the same mass,
00:50:53.000 --> 00:50:55.000
well, you wouldn't notice the
difference immediately,
00:50:55.000 --> 00:51:00.000
or, rather, you would,
but at least not in terms of
00:51:00.000 --> 00:51:04.000
your weight.
OK, that's the end for today.