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PROFESSOR: Today we're going
to continue our discussion
00:00:25.000 --> 00:00:26.780
of parametric curves.
00:00:26.780 --> 00:00:29.960
I have to tell you
about arc length.
00:00:29.960 --> 00:00:33.590
And let me remind me where
we left off last time.
00:00:33.590 --> 00:00:45.820
This is parametric
curves, continued.
00:00:45.820 --> 00:00:50.380
Last time, we talked about
the parametric representation
00:00:50.380 --> 00:00:51.730
for the circle.
00:00:51.730 --> 00:00:55.960
Or one of the parametric
representations for the circle.
00:00:55.960 --> 00:00:59.210
Which was this one here.
00:00:59.210 --> 00:01:05.590
And first we noted that
this does parameterize,
00:01:05.590 --> 00:01:07.770
as we say, the circle.
00:01:07.770 --> 00:01:10.390
That satisfies the
equation for the circle.
00:01:10.390 --> 00:01:17.990
And it's traced
counterclockwise.
00:01:17.990 --> 00:01:20.590
The picture looks like this.
00:01:20.590 --> 00:01:22.350
Here's the circle.
00:01:22.350 --> 00:01:25.400
And it starts out here
at t = 0 and it gets up
00:01:25.400 --> 00:01:31.820
to here at time t = pi / 2.
00:01:31.820 --> 00:01:41.330
So now I have to talk
to you about arc length.
00:01:41.330 --> 00:01:43.840
In this parametric form.
00:01:43.840 --> 00:01:46.170
And the results should
be the same as arc length
00:01:46.170 --> 00:01:48.830
around this circle ordinarily.
00:01:48.830 --> 00:01:55.840
And we start out with this
basic differential relationship.
00:01:55.840 --> 00:02:00.080
ds^2 is dx^2 + dy^2.
00:02:00.080 --> 00:02:04.880
And then I'm going to take
the square root, divide by dt,
00:02:04.880 --> 00:02:08.540
so the rate of change
with respect to t of s
00:02:08.540 --> 00:02:10.780
is going to be the square root.
00:02:10.780 --> 00:02:13.220
Well, maybe I'll write
it without dividing.
00:02:13.220 --> 00:02:15.130
Just write it as ds.
00:02:15.130 --> 00:02:24.640
So this would be
(dx/dt)^2 + (dy/dt)^2, dt.
00:02:24.640 --> 00:02:27.050
So this is what you get
formally from this equation.
00:02:27.050 --> 00:02:28.920
If you take its
square roots and you
00:02:28.920 --> 00:02:32.470
divide by dt squared
in the-- inside
00:02:32.470 --> 00:02:35.510
the square root, and you
multiply by dt outside,
00:02:35.510 --> 00:02:36.630
so that those cancel.
00:02:36.630 --> 00:02:39.220
And this is the formal
connection between the two.
00:02:39.220 --> 00:02:41.600
We'll be saying just
a few more words
00:02:41.600 --> 00:02:48.430
in a few minutes about how to
make sense of that rigorously.
00:02:48.430 --> 00:02:55.020
Alright so that's the set of
formulas for the infinitesimal,
00:02:55.020 --> 00:02:57.110
the differential of arc length.
00:02:57.110 --> 00:03:00.820
And so to figure it out, I have
to differentiate x with respect
00:03:00.820 --> 00:03:02.810
to t.
00:03:02.810 --> 00:03:04.225
And remember x is up here.
00:03:04.225 --> 00:03:11.330
It's defined by a cos t, so
its derivative is -a sin t.
00:03:11.330 --> 00:03:19.850
And similarly, dy/dt = a cos t.
00:03:19.850 --> 00:03:22.000
And so I can plug this in.
00:03:22.000 --> 00:03:23.620
And I get the arc
length element,
00:03:23.620 --> 00:03:36.920
which is the square root of
(-a sin t)^2 + (a cos t)^2, dt.
00:03:36.920 --> 00:03:44.670
Which just becomes the square
root of a^2, dt, or a dt.
00:03:44.670 --> 00:03:46.370
Now, I was about to divide by t.
00:03:46.370 --> 00:03:48.630
Let me do that now.
00:03:48.630 --> 00:03:52.050
We can also write the rate
of change of arc length
00:03:52.050 --> 00:03:53.430
with respect to t.
00:03:53.430 --> 00:03:55.640
And that's a, in this case.
00:03:55.640 --> 00:04:01.420
And this gets
interpreted as the speed
00:04:01.420 --> 00:04:03.180
of the particle going around.
00:04:03.180 --> 00:04:07.270
So not only, let me
trade these two guys,
00:04:07.270 --> 00:04:14.020
not only do we have the
direction is counterclockwise,
00:04:14.020 --> 00:04:20.010
but we also have that the speed
is, if you like, it's uniform.
00:04:20.010 --> 00:04:21.740
It's constant speed.
00:04:21.740 --> 00:04:23.650
And the rate is a.
00:04:23.650 --> 00:04:26.810
So that's ds/dt.
00:04:26.810 --> 00:04:30.840
Travelling around.
00:04:30.840 --> 00:04:34.310
And that means that we can
play around with the speed.
00:04:34.310 --> 00:04:37.650
And I just want to point out--
So the standard thing, what
00:04:37.650 --> 00:04:39.430
you'll have to get
used to, and this
00:04:39.430 --> 00:04:42.520
is a standard presentation,
you'll see this everywhere.
00:04:42.520 --> 00:04:45.710
In your physics classes and
your other math classes,
00:04:45.710 --> 00:04:51.230
if you want to change
the speed, so a new speed
00:04:51.230 --> 00:05:01.440
going around this would be, if
I set up the equations this way.
00:05:01.440 --> 00:05:05.650
Now I'm tracing around
the same circle.
00:05:05.650 --> 00:05:08.120
But the speed is
going to turn out
00:05:08.120 --> 00:05:11.470
to be, if you figure
it out, there'll
00:05:11.470 --> 00:05:13.200
be an extra factor of k.
00:05:13.200 --> 00:05:16.460
So it'll be ak.
00:05:16.460 --> 00:05:19.540
That's what we'll work
out to be the speed.
00:05:19.540 --> 00:05:22.290
Provided k is positive
and a is positive.
00:05:22.290 --> 00:05:30.490
So we're making
these conventions.
00:05:30.490 --> 00:05:37.180
The constants that we're
using are positive.
00:05:37.180 --> 00:05:40.980
Now, that's the first
and most basic example.
00:05:40.980 --> 00:05:42.730
The one that comes
up constantly.
00:05:42.730 --> 00:05:46.120
Now, let me just make those
comments about notation
00:05:46.120 --> 00:05:47.860
that I wanted to make.
00:05:47.860 --> 00:05:51.884
And we've been treating these
squared differentials here
00:05:51.884 --> 00:05:53.300
for a little while
and I just want
00:05:53.300 --> 00:05:54.810
to pay attention
a little bit more
00:05:54.810 --> 00:05:57.280
carefully to these
manipulations.
00:05:57.280 --> 00:05:59.660
And what's allowed
and what's not.
00:05:59.660 --> 00:06:01.890
And what's justified
and what's not.
00:06:01.890 --> 00:06:06.680
So the basis for this was this
approximate calculation that we
00:06:06.680 --> 00:06:11.440
had, that (delta s)^2 was
(delta x)^2 + (delta y)^2.
00:06:11.440 --> 00:06:16.370
This is how we justified the
arc length formula before.
00:06:16.370 --> 00:06:19.530
And let me just show you
that the formula that I
00:06:19.530 --> 00:06:22.410
have up here, this
basic formula for arc
00:06:22.410 --> 00:06:24.760
length in the
parametric form, follows
00:06:24.760 --> 00:06:26.520
just as the other one did.
00:06:26.520 --> 00:06:31.370
And now I'm going to do it
slightly more rigorously.
00:06:31.370 --> 00:06:34.400
I do the division really
in disguise before I take
00:06:34.400 --> 00:06:36.310
the limit of the infinitesimal.
00:06:36.310 --> 00:06:40.349
So all I'm really doing
is I'm doing this.
00:06:40.349 --> 00:06:42.390
Dividing through by this,
and sorry this is still
00:06:42.390 --> 00:06:43.340
approximately equal.
00:06:43.340 --> 00:06:46.760
So I'm not dividing by something
that's 0 or infinitesimal.
00:06:46.760 --> 00:06:49.320
I'm dividing by
something nonzero.
00:06:49.320 --> 00:06:54.120
And here I have ((delta
x)/(delta t))^2 + ((delta
00:06:54.120 --> 00:06:58.560
y)/(delta t))^2 And
then in the limit,
00:06:58.560 --> 00:07:04.170
I have ds/dt is equal to
the square root of this guy.
00:07:04.170 --> 00:07:13.820
Or, if you like, the
square of it, so.
00:07:13.820 --> 00:07:17.200
So it's legal to divide by
something that's almost 0
00:07:17.200 --> 00:07:20.340
and then take the
limit as we go to 0.
00:07:20.340 --> 00:07:22.280
This is really what
derivatives are all about.
00:07:22.280 --> 00:07:24.850
That we get a limit here.
00:07:24.850 --> 00:07:27.010
As the denominator goes to 0.
00:07:27.010 --> 00:07:31.510
Because the numerator's
going to 0 too.
00:07:31.510 --> 00:07:32.770
So that's the notation.
00:07:32.770 --> 00:07:38.060
And now I want to warn you,
maybe just a little bit,
00:07:38.060 --> 00:07:42.420
about misuses, if you
like, of the notation.
00:07:42.420 --> 00:07:45.820
We don't do absolutely
everything this way.
00:07:45.820 --> 00:07:49.280
This expression that
came up with the squares,
00:07:49.280 --> 00:07:55.130
you should never
write it as this.
00:07:55.130 --> 00:08:01.600
This, put it on the board
but very quickly, never.
00:08:01.600 --> 00:08:02.420
OK.
00:08:02.420 --> 00:08:07.130
Don't do that.
00:08:07.130 --> 00:08:08.700
We use these square
differentials,
00:08:08.700 --> 00:08:12.840
but we don't do it
with these ratios here.
00:08:12.840 --> 00:08:15.900
But there was another place
which is slightly confusing.
00:08:15.900 --> 00:08:17.680
It looks very
similar, where we did
00:08:17.680 --> 00:08:20.247
use the square of the
differential in a denominator.
00:08:20.247 --> 00:08:22.580
And I just want to point out
to you that it's different.
00:08:22.580 --> 00:08:23.940
It's not the same.
00:08:23.940 --> 00:08:25.810
And it is OK.
00:08:25.810 --> 00:08:31.660
And that was this one.
00:08:31.660 --> 00:08:33.970
This thing here.
00:08:33.970 --> 00:08:36.900
This is a second derivative,
it's something else.
00:08:36.900 --> 00:08:39.300
And it's got a dt^2
in the denominator.
00:08:39.300 --> 00:08:41.230
So it looks rather similar.
00:08:41.230 --> 00:08:49.230
But what this represents is
the quantity d/dt squared.
00:08:49.230 --> 00:08:51.840
And you can see the
squares came in.
00:08:51.840 --> 00:08:53.790
And squared the two expressions.
00:08:53.790 --> 00:08:58.970
And then there's
also an x over here.
00:08:58.970 --> 00:09:00.600
So that's legal.
00:09:00.600 --> 00:09:02.550
Those are notations
that we do use.
00:09:02.550 --> 00:09:04.070
And we can even calculate this.
00:09:04.070 --> 00:09:05.650
It has a perfectly good meaning.
00:09:05.650 --> 00:09:07.540
It's the same as the
derivative with respect
00:09:07.540 --> 00:09:10.870
to t of the derivative of x,
which we already know was minus
00:09:10.870 --> 00:09:17.530
sine-- sorry, a sin t, I guess.
00:09:17.530 --> 00:09:21.120
Not this example,
but the previous one.
00:09:21.120 --> 00:09:21.870
Up here.
00:09:21.870 --> 00:09:24.340
So the derivative
is this and so I can
00:09:24.340 --> 00:09:26.140
differentiate a second time.
00:09:26.140 --> 00:09:29.850
And I get -a cos t.
00:09:29.850 --> 00:09:31.760
So that's a perfectly
legal operation.
00:09:31.760 --> 00:09:33.440
Everything in there makes sense.
00:09:33.440 --> 00:09:39.880
Just don't use that.
00:09:39.880 --> 00:09:41.770
There's another really
unfortunate thing,
00:09:41.770 --> 00:09:45.820
right which is that the 2 creeps
in funny places with sines.
00:09:45.820 --> 00:09:48.080
You have sine squared.
00:09:48.080 --> 00:09:50.150
It would be out here,
it comes up here
00:09:50.150 --> 00:09:51.810
for some strange reason.
00:09:51.810 --> 00:09:54.730
This is just because
typographers are lazy
00:09:54.730 --> 00:09:56.880
or somebody somewhere
in the history
00:09:56.880 --> 00:10:00.620
of mathematical typography
decided to let the 2 migrate.
00:10:00.620 --> 00:10:04.650
It would be like
putting the 2 over here.
00:10:04.650 --> 00:10:07.940
There's inconsistency
in mathematics, right.
00:10:07.940 --> 00:10:11.570
We're not perfect and people
just develop these notations.
00:10:11.570 --> 00:10:14.450
So we have to live with them.
00:10:14.450 --> 00:10:20.210
The ones that people
accept as conventions.
00:10:20.210 --> 00:10:23.230
The next example that
I want to give you
00:10:23.230 --> 00:10:24.920
is just slightly different.
00:10:24.920 --> 00:10:29.140
It'll be a non-constant
speed parameterization.
00:10:29.140 --> 00:10:32.470
Here x = 2 sin t.
00:10:32.470 --> 00:10:37.590
And y is, say, cos t.
00:10:37.590 --> 00:10:40.450
And let's keep track
of what this one does.
00:10:40.450 --> 00:10:43.605
Now, this is a skill
which I'm going
00:10:43.605 --> 00:10:45.170
to ask you about quite a bit.
00:10:45.170 --> 00:10:46.690
And it's one of several skills.
00:10:46.690 --> 00:10:48.970
You'll have to connect
this with some kind
00:10:48.970 --> 00:10:50.330
of rectangular equation.
00:10:50.330 --> 00:10:51.725
An equation for x and y.
00:10:51.725 --> 00:10:54.230
And we'll be doing a certain
amount of this today.
00:10:54.230 --> 00:10:56.240
In another context.
00:10:56.240 --> 00:11:00.510
Right here, to see the pattern,
we know that the relationship
00:11:00.510 --> 00:11:04.160
we're going to want to use
is that sin^2 + cos^2 = 1.
00:11:04.160 --> 00:11:11.910
So in fact the right thing to do
here is to take 1/4 x^2 + y^2.
00:11:11.910 --> 00:11:17.020
And that's going to turn
out to be sin^2 t + cos^2 t.
00:11:17.020 --> 00:11:18.310
Which is 1.
00:11:18.310 --> 00:11:19.500
So there's the equation.
00:11:19.500 --> 00:11:24.690
Here's the rectangular equation
for this parametric curve.
00:11:24.690 --> 00:11:32.030
And this describes an ellipse.
00:11:32.030 --> 00:11:35.570
That's not the only information
that we can get here.
00:11:35.570 --> 00:11:37.180
The other information
that we can get
00:11:37.180 --> 00:11:39.570
is this qualitative
information of where
00:11:39.570 --> 00:11:42.360
we start, where we're
going, the direction.
00:11:42.360 --> 00:11:46.540
It starts out, I
claim, at t = 0.
00:11:46.540 --> 00:11:54.630
That's when t = 0, this is
(2 sin 0, cos 0), right?
00:11:54.630 --> 00:12:00.330
(2 sin 0, cos 0) is equal
to the point (0, 1).
00:12:00.330 --> 00:12:02.360
So it starts up up here.
00:12:02.360 --> 00:12:05.140
At (0, 1).
00:12:05.140 --> 00:12:08.520
And then the next little
place, so this is one thing
00:12:08.520 --> 00:12:11.400
that certainly you
want to do. t = pi/2
00:12:11.400 --> 00:12:14.510
is maybe the next
easy point to plot.
00:12:14.510 --> 00:12:22.830
And that's going to be
(2 sin(pi/2), cos(pi/2)).
00:12:22.830 --> 00:12:27.880
And that's just (2, 0).
00:12:27.880 --> 00:12:31.490
And so that's over
here somewhere.
00:12:31.490 --> 00:12:34.422
This is (2, 0).
00:12:34.422 --> 00:12:36.130
And we know it travels
along the ellipse.
00:12:36.130 --> 00:12:40.120
And we know the minor axis is
1, and the major axis is 2,
00:12:40.120 --> 00:12:43.000
so it's doing this.
00:12:43.000 --> 00:12:45.090
So this is what
happens at t = 0.
00:12:45.090 --> 00:12:48.390
This is where we
are at t = pi/2.
00:12:48.390 --> 00:12:51.510
And it continues all
the way around, etc.
00:12:51.510 --> 00:12:53.370
To the rest of the ellipse.
00:12:53.370 --> 00:12:57.750
This is the direction.
00:12:57.750 --> 00:13:09.620
So this one happens
to be clockwise.
00:13:09.620 --> 00:13:12.640
Alright, now let's keep
track of its speed.
00:13:12.640 --> 00:13:25.610
Let's keep track of the speed,
and also the arc length.
00:13:25.610 --> 00:13:32.830
So the speed is the square
root of the derivatives here.
00:13:32.830 --> 00:13:38.580
That would be (2 cos
t)^2 + (sin t)^2.
00:13:42.160 --> 00:13:48.060
And the arc length is what?
00:13:48.060 --> 00:13:49.840
Well, if we want to
go all the way around,
00:13:49.840 --> 00:13:53.540
we need to know that that
takes a total of 2 pi.
00:13:53.540 --> 00:13:55.990
So 0 to 2 pi.
00:13:55.990 --> 00:13:59.310
And then we have to integrate
ds, which is this expression,
00:13:59.310 --> 00:14:02.620
or ds/dt, dt.
00:14:02.620 --> 00:14:11.630
So that's the square root
of 4 cos^2 t + sin^2 t, dt.
00:14:20.820 --> 00:14:26.580
The bad news, if you
like, is that this is not
00:14:26.580 --> 00:14:38.524
an elementary integral.
00:14:38.524 --> 00:14:39.940
In other words,
no matter how long
00:14:39.940 --> 00:14:44.240
you try to figure out
how to antidifferentiate
00:14:44.240 --> 00:14:47.180
this expression, no matter how
many substitutions you try,
00:14:47.180 --> 00:14:50.420
you will fail.
00:14:50.420 --> 00:14:52.030
That's the bad news.
00:14:52.030 --> 00:14:58.430
The good news is this is
not an elementary integral.
00:14:58.430 --> 00:14:59.810
It's not an elementary integral.
00:14:59.810 --> 00:15:03.330
Which means that this is
the answer to a question.
00:15:03.330 --> 00:15:06.230
Not something that
you have to work on.
00:15:06.230 --> 00:15:11.680
So if somebody asks you for
this arc length, you stop here.
00:15:11.680 --> 00:15:14.550
That's the answer, so it's
actually better than it looks.
00:15:14.550 --> 00:15:17.700
And we'll try to--
I mean, I don't
00:15:17.700 --> 00:15:21.230
expect you to know already
what all of the integrals
00:15:21.230 --> 00:15:22.680
are that are impossible.
00:15:22.680 --> 00:15:24.680
And which ones are hard
and which ones are easy.
00:15:24.680 --> 00:15:27.200
So we'll try to coach
you through when
00:15:27.200 --> 00:15:28.390
you face these things.
00:15:28.390 --> 00:15:31.670
It's not so easy to decide.
00:15:31.670 --> 00:15:34.345
I'll give you a few clues, but.
00:15:34.345 --> 00:15:34.845
OK.
00:15:34.845 --> 00:15:38.310
So this is the arc length.
00:15:38.310 --> 00:15:42.270
Now, I want to move on to
the last thing that we did.
00:15:42.270 --> 00:15:44.410
Last type of thing
that we did last time.
00:15:44.410 --> 00:15:54.140
Which is the surface area.
00:15:54.140 --> 00:15:55.230
And yeah, question.
00:15:55.230 --> 00:16:03.609
STUDENT: [INAUDIBLE]
00:16:03.609 --> 00:16:05.650
PROFESSOR: The question,
this is a good question.
00:16:05.650 --> 00:16:08.290
The question is, when
you draw the ellipse,
00:16:08.290 --> 00:16:11.650
do you not take into
account what t is.
00:16:11.650 --> 00:16:16.460
The answer is that
this is in disguise.
00:16:16.460 --> 00:16:20.360
What's going on here
is we have a trouble
00:16:20.360 --> 00:16:24.720
with plotting in the plane
what's really happening.
00:16:24.720 --> 00:16:29.210
So in other words, it's
kind of in trouble.
00:16:29.210 --> 00:16:33.940
So the point is that we have
two functions of t, not one.
00:16:33.940 --> 00:16:35.660
x(t) and y(t).
00:16:35.660 --> 00:16:38.970
So one thing that I can do if
I plot things in the plane.
00:16:38.970 --> 00:16:41.710
In other words, the
main point to make here
00:16:41.710 --> 00:16:45.150
is that we're not talking
about the situation
00:16:45.150 --> 00:16:46.480
y is a function of x.
00:16:46.480 --> 00:16:47.870
We're out of that realm now.
00:16:47.870 --> 00:16:49.620
We're somewhere in
a different part
00:16:49.620 --> 00:16:51.450
of the universe in our thought.
00:16:51.450 --> 00:16:54.760
And you should drop
this point of view.
00:16:54.760 --> 00:16:56.990
So this depiction is not
y as a function of x.
00:16:56.990 --> 00:17:00.940
Well, that's obvious because
there are two values here,
00:17:00.940 --> 00:17:01.710
as opposed to one.
00:17:01.710 --> 00:17:02.960
So we're in trouble with that.
00:17:02.960 --> 00:17:04.750
And we have that
background parameter,
00:17:04.750 --> 00:17:07.170
and that's exactly
why we're using it.
00:17:07.170 --> 00:17:08.230
This parameter t.
00:17:08.230 --> 00:17:10.430
So that we can depict
the entire curve.
00:17:10.430 --> 00:17:14.200
And deal with it as one thing.
00:17:14.200 --> 00:17:17.590
So since I can't really draw
it, and since t is nowhere
00:17:17.590 --> 00:17:19.840
on the map, you should
sort of imagine it as time,
00:17:19.840 --> 00:17:21.760
and there's some kind of
trajectory which is travelling
00:17:21.760 --> 00:17:22.260
around.
00:17:22.260 --> 00:17:25.590
And then I just labelled
a couple of the places.
00:17:25.590 --> 00:17:28.370
If somebody asked you to
draw a picture of this,
00:17:28.370 --> 00:17:31.230
well, I'll tell you exactly
where you need the picture
00:17:31.230 --> 00:17:33.250
in just one second, alright.
00:17:33.250 --> 00:17:36.690
It's going to come up
right now in surface area.
00:17:36.690 --> 00:17:39.450
But otherwise, if
nobody asks you to,
00:17:39.450 --> 00:17:44.020
you don't even have to put
down t = 0 and t = pi / 2 here.
00:17:44.020 --> 00:17:46.402
Because nobody
demanded it of you.
00:17:46.402 --> 00:17:47.110
Another question.
00:17:47.110 --> 00:17:51.842
STUDENT: [INAUDIBLE]
00:17:51.842 --> 00:17:53.550
PROFESSOR: So, another
very good question
00:17:53.550 --> 00:17:55.890
which is exactly
connected to this picture.
00:17:55.890 --> 00:17:58.060
So how is it that we're
going to use the picture,
00:17:58.060 --> 00:18:02.190
and how is it we're going
to use the notion of the t.
00:18:02.190 --> 00:18:07.040
The question was, why is
this from t = 0 to t = 2 pi?
00:18:07.040 --> 00:18:11.000
That does use the t
information on this diagram.
00:18:11.000 --> 00:18:13.150
the point is, we do
know that t starts here.
00:18:13.150 --> 00:18:17.620
This is pi / 2, this is pi, this
is 3 pi / 2, and this is 2 pi.
00:18:17.620 --> 00:18:19.380
When you go all the
way around once,
00:18:19.380 --> 00:18:21.240
it's going to come
back to itself.
00:18:21.240 --> 00:18:23.720
These are periodic
functions of period 2 pi.
00:18:23.720 --> 00:18:26.780
And they come back to
themselves exactly at 2 pi.
00:18:26.780 --> 00:18:29.030
And so that's why we know
in order to get around once,
00:18:29.030 --> 00:18:32.250
we need to go from 0 to 2 pi.
00:18:32.250 --> 00:18:34.640
And the same thing is going
to come up with surface area
00:18:34.640 --> 00:18:35.370
right now.
00:18:35.370 --> 00:18:38.720
That's going to be the
issue, is what range of t
00:18:38.720 --> 00:18:45.620
we're going to need when we
compute the surface area.
00:18:45.620 --> 00:18:52.144
STUDENT: [INAUDIBLE]
00:18:52.144 --> 00:18:54.185
PROFESSOR: In a question,
what you might be asked
00:18:54.185 --> 00:18:56.180
is what's the
rectangular equation
00:18:56.180 --> 00:18:57.650
for a parametric curve?
00:18:57.650 --> 00:19:01.720
So that would be
1/4 x^2 + y^2 = 1.
00:19:01.720 --> 00:19:03.430
And then you might
be asked, plot it.
00:19:03.430 --> 00:19:06.960
Well, that would be a
picture of the ellipse.
00:19:06.960 --> 00:19:10.380
OK, those are types of questions
that are legal questions.
00:19:10.380 --> 00:19:27.439
STUDENT: [INAUDIBLE]
00:19:27.439 --> 00:19:28.980
PROFESSOR: The
question is, do I need
00:19:28.980 --> 00:19:30.600
to know any specific formulas?
00:19:30.600 --> 00:19:33.250
Any formulas that you know
and remember will help you.
00:19:33.250 --> 00:19:35.400
They may be of limited use.
00:19:35.400 --> 00:19:37.640
I'm not going to ask
you to memorize anything
00:19:37.640 --> 00:19:40.820
except, I guarantee you that
the circle is going to come up.
00:19:40.820 --> 00:19:43.500
Not the ellipse, the circle
will come up everywhere
00:19:43.500 --> 00:19:44.320
in your life.
00:19:44.320 --> 00:19:47.710
So at least at MIT,
your life at MIT.
00:19:47.710 --> 00:19:52.002
We're very round here.
00:19:52.002 --> 00:19:52.960
Yeah, another question.
00:19:52.960 --> 00:19:56.811
STUDENT: I'm just a tiny bit
confused back to the basics.
00:19:56.811 --> 00:19:58.810
This is more a question
from yesterday, I guess.
00:19:58.810 --> 00:20:04.390
But when you have your
original ds^2 = dx^2 + dy^2,
00:20:04.390 --> 00:20:10.060
and then you integrate that to
get arc length, how are you,
00:20:10.060 --> 00:20:14.360
the integral has dx's and dy's.
00:20:14.360 --> 00:20:18.590
So how are you just
integrating with respect to dx?
00:20:18.590 --> 00:20:22.560
PROFESSOR: OK, the question
is how are we just integrating
00:20:22.560 --> 00:20:24.200
with respect to x?
00:20:24.200 --> 00:20:26.880
So this is a question which
goes back to last time.
00:20:26.880 --> 00:20:29.120
And what is it with arc length.
00:20:29.120 --> 00:20:30.370
So.
00:20:30.370 --> 00:20:35.442
I'm going to have to answer
that question in connection
00:20:35.442 --> 00:20:36.400
with what we did today.
00:20:36.400 --> 00:20:38.420
So this is a subtle question.
00:20:38.420 --> 00:20:42.450
But I want you to realize
that this is actually
00:20:42.450 --> 00:20:44.290
an important
conceptual step here.
00:20:44.290 --> 00:20:49.810
So shhh, everybody, listen.
00:20:49.810 --> 00:20:53.340
If you're representing
one-dimensional objects,
00:20:53.340 --> 00:20:56.050
which are curves,
maybe, in space.
00:20:56.050 --> 00:20:58.270
Or in two dimensions.
00:20:58.270 --> 00:21:00.759
When you're keeping
track of arc length,
00:21:00.759 --> 00:21:02.800
you're going to have to
have an integral which is
00:21:02.800 --> 00:21:05.180
with respect to some variable.
00:21:05.180 --> 00:21:08.510
But that variable,
you get to pick.
00:21:08.510 --> 00:21:12.310
And we're launching now
into this variety of choices
00:21:12.310 --> 00:21:13.950
of variables with
respect to which you
00:21:13.950 --> 00:21:15.980
can represent something.
00:21:15.980 --> 00:21:17.580
Now, there are
some disadvantages
00:21:17.580 --> 00:21:19.370
on the circle to
representing things
00:21:19.370 --> 00:21:21.480
with respect to the variable x.
00:21:21.480 --> 00:21:24.732
Because there are two
points on the circle here.
00:21:24.732 --> 00:21:26.190
On the other hand,
you actually can
00:21:26.190 --> 00:21:27.480
succeed with half the circle.
00:21:27.480 --> 00:21:29.620
So you can figure out
the arc length that way.
00:21:29.620 --> 00:21:32.521
And then you can set it
up as an integral dx.
00:21:32.521 --> 00:21:34.770
But you can also set it up
as an integral with respect
00:21:34.770 --> 00:21:37.240
to any parameter you want.
00:21:37.240 --> 00:21:40.170
And the uniform parameter
is perhaps the easiest one.
00:21:40.170 --> 00:21:43.210
This one is perhaps
the easiest one.
00:21:43.210 --> 00:21:47.970
And so now the thing that's
strange about this perspective
00:21:47.970 --> 00:21:51.470
- and I'm going to make this
point later in the lecture
00:21:51.470 --> 00:21:55.810
as well - is that the
letters x and y-- As I say,
00:21:55.810 --> 00:22:00.630
you should drop this notion
that y is a function of x.
00:22:00.630 --> 00:22:03.950
This is what we're throwing
away at this point.
00:22:03.950 --> 00:22:05.770
What we're thinking
of is, you can
00:22:05.770 --> 00:22:08.110
describe things in terms
of any coordinate you want.
00:22:08.110 --> 00:22:11.340
You just have to say what each
one is in terms of the others.
00:22:11.340 --> 00:22:15.300
And these x and y
over here are where
00:22:15.300 --> 00:22:18.380
we are in the Cartesian
coordinate system.
00:22:18.380 --> 00:22:20.490
They're not-- And
in this case they're
00:22:20.490 --> 00:22:24.610
functions of some
other variable.
00:22:24.610 --> 00:22:25.720
Some other variable.
00:22:25.720 --> 00:22:27.150
So they're each functions.
00:22:27.150 --> 00:22:29.480
So the letters x and
y just changed on you.
00:22:29.480 --> 00:22:33.710
They mean something different.
x is no longer the variable.
00:22:33.710 --> 00:22:36.810
It's the function.
00:22:36.810 --> 00:22:38.542
Right?
00:22:38.542 --> 00:22:40.250
You're going to have
to get used to that.
00:22:40.250 --> 00:22:42.380
That's because we
run out of letters.
00:22:42.380 --> 00:22:44.870
And we kind of want to use
all of them the way we want.
00:22:44.870 --> 00:22:48.290
I'll say some more
about that later.
00:22:48.290 --> 00:22:51.220
So now I want to do this
surface area example.
00:22:51.220 --> 00:22:59.150
I'm going to just take the
surface area of the ellipsoid.
00:22:59.150 --> 00:23:11.340
The surface of the
ellipsoid formed
00:23:11.340 --> 00:23:19.910
by revolving this previous
example, which was Example 2.
00:23:19.910 --> 00:23:28.020
Around the y-axis.
00:23:28.020 --> 00:23:30.410
So we want to set up that
surface area integral here
00:23:30.410 --> 00:23:32.490
for you.
00:23:32.490 --> 00:23:38.160
Now, I remind you that the
area element looks like this.
00:23:38.160 --> 00:23:41.190
If you're revolving
around the y-axis,
00:23:41.190 --> 00:23:42.815
that means you're
going around this way
00:23:42.815 --> 00:23:43.720
and you have some curve.
00:23:43.720 --> 00:23:44.990
In this case it's this
piece of an ellipse.
00:23:44.990 --> 00:23:46.475
If you sweep it
around you're going
00:23:46.475 --> 00:23:48.770
to get what's
called an ellipsoid.
00:23:48.770 --> 00:23:53.890
And there's a little chunk here,
that you're wrapping around.
00:23:53.890 --> 00:23:58.430
And the important thing you
need besides this ds, this arc
00:23:58.430 --> 00:24:04.120
length piece over here, is
the distance to the axis.
00:24:04.120 --> 00:24:06.320
So that's this
horizontal distance here.
00:24:06.320 --> 00:24:09.850
I'll draw it in another color.
00:24:09.850 --> 00:24:15.520
And that horizontal
distance now has a name.
00:24:15.520 --> 00:24:18.670
And this is, again, the virtue
of this coordinate system.
00:24:18.670 --> 00:24:20.170
The t is something else.
00:24:20.170 --> 00:24:21.020
This has a name.
00:24:21.020 --> 00:24:22.760
This distance has a name.
00:24:22.760 --> 00:24:27.080
This distance is called x.
00:24:27.080 --> 00:24:29.570
And it even has a formula.
00:24:29.570 --> 00:24:36.090
Its formula is 2 sin t.
00:24:36.090 --> 00:24:38.550
In terms of t.
00:24:38.550 --> 00:24:41.530
So the full formula
up for the integral
00:24:41.530 --> 00:24:46.039
here is, I have to take
the circumference when
00:24:46.039 --> 00:24:47.080
I spin this thing around.
00:24:47.080 --> 00:24:48.950
And this little
arc length element.
00:24:48.950 --> 00:24:53.660
So I have here 2
pi times 2 sin t.
00:24:53.660 --> 00:24:55.640
That's the x variable here.
00:24:55.640 --> 00:25:00.560
And then I have here ds,
which is kind of a mess.
00:25:00.560 --> 00:25:04.170
So unfortunately I don't
quite have room for it.
00:25:04.170 --> 00:25:05.650
Plan ahead.
00:25:05.650 --> 00:25:15.200
Square root of 4 cos^2 t + sin^2
t, is that what it was, dt.
00:25:15.200 --> 00:25:17.740
Alright, I guess I
squeezed it in there.
00:25:17.740 --> 00:25:20.090
So that was the arc
length, which I re-copied
00:25:20.090 --> 00:25:21.620
from this board above.
00:25:21.620 --> 00:25:24.310
That was the ds piece.
00:25:24.310 --> 00:25:29.760
It's this whole thing
including the dt.
00:25:29.760 --> 00:25:32.360
That's the answer
except for one thing.
00:25:32.360 --> 00:25:33.590
What else do we need?
00:25:33.590 --> 00:25:35.350
We don't just need
the integrand,
00:25:35.350 --> 00:25:37.720
this is half of
setting up an integral.
00:25:37.720 --> 00:25:40.990
The other half of setting up
an integral is the limits.
00:25:40.990 --> 00:25:42.840
We need specific limits here.
00:25:42.840 --> 00:25:46.760
Otherwise we don't have a
number that we can get out.
00:25:46.760 --> 00:25:50.370
So we now have to think
about what the limits are.
00:25:50.370 --> 00:25:52.550
And maybe somebody can see.
00:25:52.550 --> 00:25:54.429
It has something to
do with this diagram
00:25:54.429 --> 00:25:55.470
of the ellipse over here.
00:25:55.470 --> 00:25:58.520
Can somebody guess what it is?
00:25:58.520 --> 00:25:59.480
0 to pi.
00:25:59.480 --> 00:26:02.070
Well, that was quick.
00:26:02.070 --> 00:26:02.620
That's it.
00:26:02.620 --> 00:26:04.709
Because we go from
the top to the bottom,
00:26:04.709 --> 00:26:06.250
but we don't want
to continue around.
00:26:06.250 --> 00:26:07.640
We don't want to
go from 0 to 2 pi,
00:26:07.640 --> 00:26:09.723
because that would be
duplicating what we're going
00:26:09.723 --> 00:26:12.020
to get when we spin around.
00:26:12.020 --> 00:26:13.730
And we know that we start at 0.
00:26:13.730 --> 00:26:15.540
It's interesting
because it descends
00:26:15.540 --> 00:26:17.090
when you change
variables to think
00:26:17.090 --> 00:26:20.360
of it in terms of the y variable
it's going the opposite way.
00:26:20.360 --> 00:26:24.550
But anyway, just one piece
of this is what we want.
00:26:24.550 --> 00:26:27.660
So that's this setup.
00:26:27.660 --> 00:26:36.230
And now I claim that this is
actually a doable integral.
00:26:36.230 --> 00:26:37.850
However, it's long.
00:26:37.850 --> 00:26:39.830
I'm going to spare
you, I'll just tell you
00:26:39.830 --> 00:26:41.330
how you would get started.
00:26:41.330 --> 00:26:45.970
You would use the
substitution u = cos t.
00:26:45.970 --> 00:26:53.620
And then the du is
going to be -sin t dt.
00:26:53.620 --> 00:26:56.290
But then, unfortunately,
there's a lot more.
00:26:56.290 --> 00:26:57.810
There's another
trig substitution
00:26:57.810 --> 00:27:01.430
with some other multiple
of the cosine and so forth.
00:27:01.430 --> 00:27:02.420
So it goes on and on.
00:27:02.420 --> 00:27:06.260
If you want to check
it yourself, you can.
00:27:06.260 --> 00:27:08.710
There's an inverse
trig substitution which
00:27:08.710 --> 00:27:11.590
isn't compatible with this one.
00:27:11.590 --> 00:27:17.090
But it can be done.
00:27:17.090 --> 00:27:22.690
Calculated.
00:27:22.690 --> 00:27:26.980
In elementary terms.
00:27:26.980 --> 00:27:30.547
Yeah, another question.
00:27:30.547 --> 00:27:31.380
STUDENT: [INAUDIBLE]
00:27:31.380 --> 00:27:33.490
PROFESSOR: So, if you
get this on an exam,
00:27:33.490 --> 00:27:35.240
I'm going to have to
coach you through it.
00:27:35.240 --> 00:27:37.640
Either I'm going to have to
tell you don't evaluate it
00:27:37.640 --> 00:27:40.130
or, you're going to have
to work really hard.
00:27:40.130 --> 00:27:42.500
Or here's the first step,
and then the next step
00:27:42.500 --> 00:27:44.299
is, keep on going.
00:27:44.299 --> 00:27:44.840
Or something.
00:27:44.840 --> 00:27:47.890
I'll have to give you some cues.
00:27:47.890 --> 00:27:49.260
Because it's quite long.
00:27:49.260 --> 00:27:52.860
This is way too long for an
exam, this particular one.
00:27:52.860 --> 00:27:53.650
OK.
00:27:53.650 --> 00:27:55.359
It's not too long
for a problem set.
00:27:55.359 --> 00:27:57.650
This is where I would leave
you off if I were giving it
00:27:57.650 --> 00:27:58.320
to you on a problem set.
00:27:58.320 --> 00:28:00.220
Just to give you an idea
of the order of magnitude.
00:28:00.220 --> 00:28:02.761
Whereas one of the ones that I
did yesterday, I wouldn't even
00:28:02.761 --> 00:28:11.120
give you on a problem
set, it was so long.
00:28:11.120 --> 00:28:17.630
So now, our next job is to
move on to polar coordinates.
00:28:17.630 --> 00:28:20.960
Now, polar coordinates involve
the geometry of circles.
00:28:20.960 --> 00:28:23.392
As I said, we really
love circles here.
00:28:23.392 --> 00:28:24.100
We're very round.
00:28:24.100 --> 00:28:28.210
Just as I love 0, the rest of
the Institute loves circles.
00:28:28.210 --> 00:28:47.380
So we're going to
do that right now.
00:28:47.380 --> 00:28:58.900
What we're going to talk about
now is polar coordinates.
00:28:58.900 --> 00:29:01.010
Which are set up in
the following way.
00:29:01.010 --> 00:29:04.640
It's a way of describing
the points in the plane.
00:29:04.640 --> 00:29:07.460
Here is a point in
a plane, and here's
00:29:07.460 --> 00:29:10.530
what we think of as
the usual x-y axes.
00:29:10.530 --> 00:29:12.860
And now this point is
going to be described
00:29:12.860 --> 00:29:15.260
by a different pair of
coordinates, different pair
00:29:15.260 --> 00:29:16.190
of numbers.
00:29:16.190 --> 00:29:26.420
Namely, the distance
to the origin.
00:29:26.420 --> 00:29:30.490
And the second parameter
here, second number here,
00:29:30.490 --> 00:29:32.550
is this angle theta.
00:29:32.550 --> 00:29:41.500
Which is the angle
of ray from origin
00:29:41.500 --> 00:29:48.670
with the horizontal axis.
00:29:48.670 --> 00:29:50.620
So that's what it
is in language.
00:29:50.620 --> 00:29:53.690
And you should put this
in quotation marks,
00:29:53.690 --> 00:29:57.320
because it's not
a perfect match.
00:29:57.320 --> 00:30:00.800
This is geometrically what
you should always think of,
00:30:00.800 --> 00:30:03.720
but the technical
details involve
00:30:03.720 --> 00:30:06.530
dealing directly with formulas.
00:30:06.530 --> 00:30:09.880
The first formula is
the formula for x.
00:30:09.880 --> 00:30:11.590
And this is the
fundamental, these two
00:30:11.590 --> 00:30:12.750
are the fundamental ones.
00:30:12.750 --> 00:30:16.120
Namely, x = r cos theta.
00:30:16.120 --> 00:30:17.860
The second formula
is the formula
00:30:17.860 --> 00:30:21.380
for y, which is r sin theta.
00:30:21.380 --> 00:30:25.420
So these are the
unambiguous definitions
00:30:25.420 --> 00:30:27.100
of polar coordinates.
00:30:27.100 --> 00:30:28.790
This is it.
00:30:28.790 --> 00:30:32.590
And this is the thing from
which all other almost correct
00:30:32.590 --> 00:30:37.180
statements almost follow.
00:30:37.180 --> 00:30:39.320
But this is the one you
should trust always.
00:30:39.320 --> 00:30:44.980
This is the
unambiguous statement.
00:30:44.980 --> 00:30:47.360
So let me give you an
example something that's
00:30:47.360 --> 00:30:52.040
close to being a good
formula and is certainly
00:30:52.040 --> 00:30:57.530
useful in its way.
00:30:57.530 --> 00:31:04.180
Namely, you can think of r as
being the square root of x^2 +
00:31:04.180 --> 00:31:05.810
y^2.
00:31:05.810 --> 00:31:07.336
That's easy enough
to derive, it's
00:31:07.336 --> 00:31:08.460
the distance to the origin.
00:31:08.460 --> 00:31:11.320
That's pretty obvious.
00:31:11.320 --> 00:31:14.690
And the formula for theta,
which you can also derive,
00:31:14.690 --> 00:31:17.480
which is that it's the
inverse tangent of y y/x.
00:31:21.050 --> 00:31:24.310
However, let me just warn
you that these formulas are
00:31:24.310 --> 00:31:26.870
slightly ambiguous.
00:31:26.870 --> 00:31:33.357
So somewhat ambiguous.
00:31:33.357 --> 00:31:35.440
In other words, you can't
just apply them blindly.
00:31:35.440 --> 00:31:37.023
You actually have
to look at a picture
00:31:37.023 --> 00:31:38.180
in order to get them right.
00:31:38.180 --> 00:31:43.690
In particular, r could
be plus or minus here.
00:31:43.690 --> 00:31:47.950
And when you take
the inverse tangent,
00:31:47.950 --> 00:31:52.510
there's an ambiguity between,
it's the same as the inverse
00:31:52.510 --> 00:31:56.330
tangent of (-y)/(-x).
00:31:56.330 --> 00:32:00.550
So these minus signs are a
plague on your existence.
00:32:00.550 --> 00:32:05.050
And you're not going to get a
completely unambiguous answer
00:32:05.050 --> 00:32:07.760
out of these formulas
without paying attention
00:32:07.760 --> 00:32:08.430
to the diagram.
00:32:08.430 --> 00:32:10.550
On the other hand, the
formula up in the box
00:32:10.550 --> 00:32:14.337
there always works.
00:32:14.337 --> 00:32:15.920
So when people mean
polar coordinates,
00:32:15.920 --> 00:32:17.370
they always mean that.
00:32:17.370 --> 00:32:22.370
And then they have conventions,
which sometimes match things up
00:32:22.370 --> 00:32:27.550
with the formulas over
on this next board.
00:32:27.550 --> 00:32:32.670
Let me give you various
examples here first.
00:32:32.670 --> 00:32:36.260
But maybe first I
should I should draw
00:32:36.260 --> 00:32:38.100
the two coordinate systems.
00:32:38.100 --> 00:32:40.560
So the coordinate system
that we're used to
00:32:40.560 --> 00:32:43.360
is the rectangular
coordinate system.
00:32:43.360 --> 00:32:49.190
And maybe I'll draw it
in orange and green here.
00:32:49.190 --> 00:32:59.430
So these are the coordinate
lines y = 0, y = 1, y = 2.
00:32:59.430 --> 00:33:01.950
That's how the
coordinate system works.
00:33:01.950 --> 00:33:08.427
And over here we have the
rest of the coordinate system.
00:33:08.427 --> 00:33:10.510
And this is the way we're
thinking of x and y now.
00:33:10.510 --> 00:33:12.570
We're no longer thinking of
y as a function of x and x
00:33:12.570 --> 00:33:13.986
as a function of
y, we're thinking
00:33:13.986 --> 00:33:16.960
of x as a label of
a place in a plane.
00:33:16.960 --> 00:33:20.900
And y as a label of
a place in a plane.
00:33:20.900 --> 00:33:27.770
So here we have x =
0, x = 1, x = 2, etc.
00:33:27.770 --> 00:33:30.740
Here's x = -1.
00:33:30.740 --> 00:33:31.900
So forth.
00:33:31.900 --> 00:33:37.100
So that's what the rectangular
coordinate system looks like.
00:33:37.100 --> 00:33:41.380
And now I should draw the other
coordinate system that we have.
00:33:41.380 --> 00:33:47.900
Which is this guy here.
00:33:47.900 --> 00:33:49.610
Well, close enough.
00:33:49.610 --> 00:33:54.720
And these guys here.
00:33:54.720 --> 00:33:57.730
Kind of this bulls-eye
or target operation.
00:33:57.730 --> 00:34:01.480
And this one is,
say, theta = pi/2.
00:34:01.480 --> 00:34:03.870
This is theta = 0.
00:34:03.870 --> 00:34:07.710
This is theta = -pi/4.
00:34:07.710 --> 00:34:11.380
For instance, so I've
just labeled for you three
00:34:11.380 --> 00:34:17.870
of the rays on this diagram.
00:34:17.870 --> 00:34:23.130
It's kind of like
a radar screen.
00:34:23.130 --> 00:34:28.840
And then in pink, this is
maybe r = 2, the radius 2.
00:34:28.840 --> 00:34:33.980
And inside is r = 1.
00:34:33.980 --> 00:34:38.090
So it's a different coordinate
system for the plane.
00:34:38.090 --> 00:34:42.120
And again, the letter
r represents measuring
00:34:42.120 --> 00:34:44.930
how far we are from the origin.
00:34:44.930 --> 00:34:47.060
The theta represents
something about the angle,
00:34:47.060 --> 00:34:50.250
which ray we're on.
00:34:50.250 --> 00:34:52.260
And they're just two
different variables.
00:34:52.260 --> 00:35:10.880
And this is a very different
kind of coordinate system.
00:35:10.880 --> 00:35:15.391
OK so, our main job is
just to get used to this.
00:35:15.391 --> 00:35:15.890
For now.
00:35:15.890 --> 00:35:18.350
You will be using
this a lot in 18.02.
00:35:18.350 --> 00:35:20.570
It's very useful in physics.
00:35:20.570 --> 00:35:25.680
And our job is just to
get started with it.
00:35:25.680 --> 00:35:29.990
And so, let's try a
few examples here.
00:35:29.990 --> 00:35:31.220
Tons of examples.
00:35:31.220 --> 00:35:34.590
We'll start out very slow.
00:35:34.590 --> 00:35:41.860
If you have (x, y) = (1, -1),
that's a point in the plane.
00:35:41.860 --> 00:35:44.380
I can draw that point.
00:35:44.380 --> 00:35:46.460
It's down here, right?
00:35:46.460 --> 00:35:50.630
This is -1 and this is 1,
and here's my point, (1, -1).
00:35:50.630 --> 00:35:53.550
I can figure out what
the representative is
00:35:53.550 --> 00:35:56.670
of this in polar coordinates.
00:35:56.670 --> 00:36:03.040
So in polar coordinates,
there are actually
00:36:03.040 --> 00:36:05.130
a bunch of choices here.
00:36:05.130 --> 00:36:09.250
First of all, I'll
tell you one choice.
00:36:09.250 --> 00:36:10.970
If I start with the
angle horizontally,
00:36:10.970 --> 00:36:14.200
I wrap all the way
around, that would
00:36:14.200 --> 00:36:19.350
be to this ray here--
Let's do it in green again.
00:36:19.350 --> 00:36:21.820
Alright, I labeled
it actually as -pi/4,
00:36:21.820 --> 00:36:27.310
but another way of looking at
it is that it's this angle here.
00:36:27.310 --> 00:36:31.440
So that would be r
= square root of 2.
00:36:31.440 --> 00:36:34.210
Theta = 7pi/4.
00:36:38.150 --> 00:36:41.750
So that's one possibility of
the angle and the distance.
00:36:41.750 --> 00:36:45.380
I know the distance is a square
root of 2, that's not hard.
00:36:45.380 --> 00:36:47.930
Another way of looking
at it is the way
00:36:47.930 --> 00:36:49.640
which was suggested
when I labeled this
00:36:49.640 --> 00:36:51.230
with a negative angle.
00:36:51.230 --> 00:36:56.850
And that would be r = square
root of 2, theta = -pi/4.
00:36:56.850 --> 00:36:58.370
And these are both legal.
00:36:58.370 --> 00:37:00.736
These are perfectly
legal representatives.
00:37:00.736 --> 00:37:02.110
And that's what
I meant by saying
00:37:02.110 --> 00:37:06.180
that these representations over
here are somewhat ambiguous.
00:37:06.180 --> 00:37:08.900
There's more than one answer
to this question, of what
00:37:08.900 --> 00:37:11.860
the polar representation is.
00:37:11.860 --> 00:37:17.190
A third possibility, which is
even more dicey but also legal,
00:37:17.190 --> 00:37:21.890
is r equals minus
square root of 2.
00:37:21.890 --> 00:37:25.360
Theta = 3pi/4.
00:37:25.360 --> 00:37:30.080
Now, what that corresponds to
doing is going around to here.
00:37:30.080 --> 00:37:33.490
We're pointing out
3/4 pi direction.
00:37:33.490 --> 00:37:37.130
But then going negative
square root of 2 distance.
00:37:37.130 --> 00:37:39.710
We're going backwards.
00:37:39.710 --> 00:37:42.250
So we're landing
in the same place.
00:37:42.250 --> 00:37:44.380
So this is also legal.
00:37:44.380 --> 00:37:44.880
Yeah.
00:37:44.880 --> 00:37:51.324
STUDENT: [INAUDIBLE]
00:37:51.324 --> 00:37:53.240
PROFESSOR: The question
is, don't the radiuses
00:37:53.240 --> 00:37:54.989
have to be positive
because they represent
00:37:54.989 --> 00:37:56.620
a distance to the origin?
00:37:56.620 --> 00:38:00.620
The answer is I
lied to you here.
00:38:00.620 --> 00:38:04.770
All of these things that I said
are wrong, except for this.
00:38:04.770 --> 00:38:09.020
Which is the rule for what
polar coordinates mean.
00:38:09.020 --> 00:38:21.170
So it's maybe plus or minus the
distance, is what it is always.
00:38:21.170 --> 00:38:29.090
I try not to lie to you
too much, but I do succeed.
00:38:29.090 --> 00:38:36.270
Now, let's do a little
bit more practice here.
00:38:36.270 --> 00:38:38.330
There are some easy
examples, which
00:38:38.330 --> 00:38:40.580
I will run through
very quickly. r = a,
00:38:40.580 --> 00:38:44.100
we already know
this is a circle.
00:38:44.100 --> 00:38:51.280
And the 3 theta equals
a constant is a ray.
00:38:51.280 --> 00:38:54.820
However, this involves an
implicit assumption, which
00:38:54.820 --> 00:38:57.360
I want to point out to you.
00:38:57.360 --> 00:38:59.040
So this is Example 3.
00:38:59.040 --> 00:39:01.060
Theta's equal to a
constant is a ray.
00:39:01.060 --> 00:39:14.070
But this implicitly
assumes 0 <= r < infinity.
00:39:14.070 --> 00:39:19.400
If you really wanted to allow
minus infinity < r < infinity
00:39:19.400 --> 00:39:22.890
in this example, you
would get a line.
00:39:22.890 --> 00:39:28.540
Gives the whole line.
00:39:28.540 --> 00:39:30.050
It gives everything behind.
00:39:30.050 --> 00:39:33.085
So you go out on some ray,
you go backwards on that ray
00:39:33.085 --> 00:39:36.460
and you get the whole line
through the origin, both ways.
00:39:36.460 --> 00:39:39.740
If you allow r going to
minus infinity as well.
00:39:39.740 --> 00:39:42.310
So the typical
conventions, so here
00:39:42.310 --> 00:39:49.680
are the typical conventions.
00:39:49.680 --> 00:39:53.140
And you will see people assume
this without even telling you.
00:39:53.140 --> 00:39:55.340
So you need to watch out for it.
00:39:55.340 --> 00:39:57.450
The typical conventions
are certainly this one,
00:39:57.450 --> 00:40:00.270
which is a nice thing to do.
00:40:00.270 --> 00:40:04.240
Pretty much all the time,
although not all the time.
00:40:04.240 --> 00:40:05.360
Most of the time.
00:40:05.360 --> 00:40:11.950
And then you might have
theta ranging from minus pi
00:40:11.950 --> 00:40:15.730
to pi, so in other words
symmetric around 0.
00:40:15.730 --> 00:40:21.630
Or, another very popular
choice is this one.
00:40:21.630 --> 00:40:25.890
Theta's >= 0 and
strictly less than 2pi.
00:40:25.890 --> 00:40:29.660
So these are the
two typical ranges
00:40:29.660 --> 00:40:33.930
in which all of these
variables are chosen.
00:40:33.930 --> 00:40:34.900
But not always.
00:40:34.900 --> 00:40:43.210
You'll find that
it's not consistent.
00:40:43.210 --> 00:40:46.010
As I said, our job is
to get used to this.
00:40:46.010 --> 00:40:49.600
And I need to work up
to some slightly more
00:40:49.600 --> 00:40:51.420
complicated examples.
00:40:51.420 --> 00:40:57.840
Some of which I'll give
you on next Tuesday.
00:40:57.840 --> 00:41:05.780
But let's do a few more.
00:41:05.780 --> 00:41:10.820
So, I guess this is Example 4.
00:41:10.820 --> 00:41:14.980
Example 4, I'm
going to take y = 1.
00:41:14.980 --> 00:41:20.650
That's awfully simple in
rectangular coordinates.
00:41:20.650 --> 00:41:23.960
But interestingly,
you might conceivably
00:41:23.960 --> 00:41:26.050
want to deal with it
in polar coordinates.
00:41:26.050 --> 00:41:29.580
If you do, so here's how
you make the translation.
00:41:29.580 --> 00:41:32.850
But this translation
is not so terrible.
00:41:32.850 --> 00:41:39.080
What you do is, you plug
in y = r sin(theta).
00:41:39.080 --> 00:41:40.710
That's all you have to do.
00:41:40.710 --> 00:41:42.760
And so that's going
to be equal to 1.
00:41:42.760 --> 00:41:46.240
And that's going to give
us our polar equation.
00:41:46.240 --> 00:41:50.330
The polar equation is
r = 1 / sin(theta).
00:41:50.330 --> 00:41:54.360
There it is.
00:41:54.360 --> 00:41:58.120
And let's draw a picture of it.
00:41:58.120 --> 00:42:03.480
So here's a picture
of the line y = 1.
00:42:03.480 --> 00:42:11.950
And now we see that if we take
our rays going out from here,
00:42:11.950 --> 00:42:17.240
they collide with the
line at various lengths.
00:42:17.240 --> 00:42:19.760
So if you take an angle,
theta, here there'll
00:42:19.760 --> 00:42:21.364
be a distance r
corresponding to that
00:42:21.364 --> 00:42:23.030
and you'll hit this
in exactly one spot.
00:42:23.030 --> 00:42:26.600
For each theta you'll
have a different radius.
00:42:26.600 --> 00:42:27.810
And it's a variable radius.
00:42:27.810 --> 00:42:30.740
It's given by this formula here.
00:42:30.740 --> 00:42:33.210
And so to trace this
line out, you actually
00:42:33.210 --> 00:42:36.120
have to realize that there's
one more thing involved.
00:42:36.120 --> 00:42:40.160
Which is the possible
range of theta.
00:42:40.160 --> 00:42:41.730
Again, when you're
doing integrations
00:42:41.730 --> 00:42:44.104
you're going to need to know
those limits of integration.
00:42:44.104 --> 00:42:46.360
So you're going to
need to know this.
00:42:46.360 --> 00:42:48.990
The range here goes
from theta = 0,
00:42:48.990 --> 00:42:51.230
that's sort of when
it's out at infinity.
00:42:51.230 --> 00:42:53.140
That's when the
denominator is 0 here.
00:42:53.140 --> 00:42:55.800
And it goes all the way to pi.
00:42:55.800 --> 00:42:57.940
Swing around just one half-turn.
00:42:57.940 --> 00:43:03.610
So the range here
is 0 < theta < pi.
00:43:03.610 --> 00:43:04.620
Yeah, question.
00:43:04.620 --> 00:43:09.676
STUDENT: [INAUDIBLE]
00:43:09.676 --> 00:43:11.050
PROFESSOR: The
question is, is it
00:43:11.050 --> 00:43:13.940
typical to express r
as a function of theta,
00:43:13.940 --> 00:43:16.550
or vice versa, or
does it matter?
00:43:16.550 --> 00:43:19.790
The answer is that for the
purposes of this course,
00:43:19.790 --> 00:43:24.420
we're almost always going to
be writing things in this form.
00:43:24.420 --> 00:43:27.070
r as a function of theta.
00:43:27.070 --> 00:43:30.050
And you can do
whatever you want.
00:43:30.050 --> 00:43:33.920
This turns out to be what
we'll be doing in this course,
00:43:33.920 --> 00:43:37.040
exclusively.
00:43:37.040 --> 00:43:40.570
As you'll see when we
get to other examples,
00:43:40.570 --> 00:43:42.160
it's the traditional
sort of thing
00:43:42.160 --> 00:43:45.060
to do when you're thinking
about observing a planet
00:43:45.060 --> 00:43:48.650
or something like that.
00:43:48.650 --> 00:43:52.930
You see the angle, and then
you guess far away it is.
00:43:52.930 --> 00:43:55.600
But it's not necessary.
00:43:55.600 --> 00:43:58.940
The formulas are
often easier this way.
00:43:58.940 --> 00:44:00.370
For the examples that we have.
00:44:00.370 --> 00:44:02.610
Because it's usually a
trig function of theta.
00:44:02.610 --> 00:44:05.110
Whereas the other way, it would
be an inverse trig function.
00:44:05.110 --> 00:44:08.930
So it's an uglier expression.
00:44:08.930 --> 00:44:10.540
As you can see.
00:44:10.540 --> 00:44:12.860
The real reason is that we
choose this thing that's
00:44:12.860 --> 00:44:19.410
easier to deal with.
00:44:19.410 --> 00:44:22.200
So now let me give you a
slightly more complicated
00:44:22.200 --> 00:44:24.410
example of the same type.
00:44:24.410 --> 00:44:28.930
Where we use a shortcut.
00:44:28.930 --> 00:44:31.680
This is a standard example.
00:44:31.680 --> 00:44:33.960
And it comes up a lot.
00:44:33.960 --> 00:44:40.730
And so this is an
off-center circle.
00:44:40.730 --> 00:44:44.000
A circle is really easy
to describe, but not
00:44:44.000 --> 00:44:54.170
necessarily if the center
is on the rim of the circle.
00:44:54.170 --> 00:44:56.550
So that's a different problem.
00:44:56.550 --> 00:44:59.990
And let's do this with
a circle of radius a.
00:44:59.990 --> 00:45:06.120
So this is the point (a,
0) and this is (2a, 0).
00:45:06.120 --> 00:45:08.550
And actually, if you
know these two numbers,
00:45:08.550 --> 00:45:11.080
you'll be able to remember the
result of this calculation.
00:45:11.080 --> 00:45:13.780
Which you'll do about five
or six times and then finally
00:45:13.780 --> 00:45:17.310
you'll memorize it during 18.02
when you will need it a lot.
00:45:17.310 --> 00:45:21.220
So this is a standard
calculation here.
00:45:21.220 --> 00:45:24.350
So the starting place is
the rectangular equation.
00:45:24.350 --> 00:45:27.170
And we're going to pass to
the polar representation.
00:45:27.170 --> 00:45:33.550
The rectangular representation
is (x-a)^2 + y^2 = a^2.
00:45:33.550 --> 00:45:40.290
So this is a circle centered
at (a, 0) of radius a.
00:45:40.290 --> 00:45:44.110
And now, if you like, the
slow way of doing this
00:45:44.110 --> 00:45:50.145
would be to plug in x = r
cos(theta), y = r sin(theta).
00:45:50.145 --> 00:45:51.520
The way I did in
this first step.
00:45:51.520 --> 00:45:53.500
And that works perfectly well.
00:45:53.500 --> 00:45:56.980
But I'm going to do it
more quickly than that.
00:45:56.980 --> 00:46:00.070
Because I can sort of see in
advance how it's going to work.
00:46:00.070 --> 00:46:09.810
I'm just going to
expand this out.
00:46:09.810 --> 00:46:13.160
And now I see the a^2's cancel.
00:46:13.160 --> 00:46:17.120
And not only that,
but x^2 + y^2 = r^2.
00:46:17.120 --> 00:46:19.670
So this becomes r^2.
00:46:19.670 --> 00:46:28.590
That's x^2 + y^2 - 2ax = 0.
00:46:28.590 --> 00:46:32.360
The r came from the fact
that r^2 = x^2 + y^2.
00:46:36.100 --> 00:46:37.890
So I'm doing this the rapid way.
00:46:37.890 --> 00:46:40.260
You can do it by
plugging in, as I said.
00:46:40.260 --> 00:46:43.900
r equals-- So now that
I've simplified it,
00:46:43.900 --> 00:46:45.720
I am going to use
that procedure.
00:46:45.720 --> 00:46:47.570
I'm going to plug in.
00:46:47.570 --> 00:46:57.120
So here I have r^2 -
2ar cos(theta) = 0.
00:46:57.120 --> 00:47:00.146
I just plugged in for x.
00:47:00.146 --> 00:47:02.270
As I said, I could have
done that at the beginning.
00:47:02.270 --> 00:47:06.430
I just simplified first.
00:47:06.430 --> 00:47:11.780
And now, this is the same
thing as r^2 = 2ar cos(theta).
00:47:11.780 --> 00:47:13.530
And we're almost done.
00:47:13.530 --> 00:47:19.230
There's a boring part of this
equation, which is r = 0.
00:47:19.230 --> 00:47:21.530
And then there's,
if I divide by r,
00:47:21.530 --> 00:47:23.430
there's the interesting
part of the equation.
00:47:23.430 --> 00:47:25.830
Which is this.
00:47:25.830 --> 00:47:28.810
So this is or r = 0.
00:47:28.810 --> 00:47:33.690
Which is already included
in that equation anyway.
00:47:33.690 --> 00:47:36.890
So I'm allowed to divide by r
because in the case of r = 0,
00:47:36.890 --> 00:47:39.781
this is represented anyway.
00:47:39.781 --> 00:47:40.280
Question.
00:47:40.280 --> 00:47:44.390
STUDENT: [INAUDIBLE]
00:47:44.390 --> 00:47:46.270
PROFESSOR: r = 0
is just one case.
00:47:46.270 --> 00:47:48.380
That is, it's the
union of these two.
00:47:48.380 --> 00:47:49.550
It's both.
00:47:49.550 --> 00:47:50.670
Both are possible.
00:47:50.670 --> 00:47:53.270
So r = 0 is one point on it.
00:47:53.270 --> 00:47:56.150
And this is all of it.
00:47:56.150 --> 00:48:01.230
So we can just ignore this.
00:48:01.230 --> 00:48:04.500
So now I want to say one
more important thing.
00:48:04.500 --> 00:48:06.600
You need to understand
the range of this.
00:48:06.600 --> 00:48:10.840
So wait a second and we're going
to figure out the range here.
00:48:10.840 --> 00:48:13.710
The range is very important,
because otherwise you'll
00:48:13.710 --> 00:48:18.280
never be able to integrate
using this representation here.
00:48:18.280 --> 00:48:19.840
So this is the representation.
00:48:19.840 --> 00:48:25.190
But notice when theta =
0, we're out here at 2a.
00:48:25.190 --> 00:48:26.780
That's consistent,
and that's actually
00:48:26.780 --> 00:48:29.020
how you remember
this factor 2a here.
00:48:29.020 --> 00:48:31.570
Because if you remember this
picture and where you land when
00:48:31.570 --> 00:48:34.830
theta = 0.
00:48:34.830 --> 00:48:36.370
So that's the theta = 0 part.
00:48:36.370 --> 00:48:39.440
But now as I tip
up like this, you
00:48:39.440 --> 00:48:43.780
see that when we get to
vertical, we're done.
00:48:43.780 --> 00:48:44.630
With the circle.
00:48:44.630 --> 00:48:46.463
It's gotten shorter and
shorter and shorter,
00:48:46.463 --> 00:48:49.020
and at theta = pi/2,
we're down at 0.
00:48:49.020 --> 00:48:51.720
Because that's cos(pi/2) = 0.
00:48:51.720 --> 00:48:53.770
So it swings up like this.
00:48:53.770 --> 00:48:55.400
And it gets up to pi/2.
00:48:55.400 --> 00:48:57.110
Similarly, we swing
down like this.
00:48:57.110 --> 00:48:59.000
And then we're done.
00:48:59.000 --> 00:49:04.510
So the range is
-pi/2 < theta < pi/2.
00:49:04.510 --> 00:49:06.650
Or, if you want to
throw in the r = 0 case,
00:49:06.650 --> 00:49:08.700
you can throw in this,
this is repeating,
00:49:08.700 --> 00:49:11.200
if you like, at the ends.
00:49:11.200 --> 00:49:14.100
So this is the range
of this circle.
00:49:14.100 --> 00:49:17.150
And let's see.
00:49:17.150 --> 00:49:21.300
Next time we'll figure out
area in polar coordinates.