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PROFESSOR: Hi.
00:00:35.860 --> 00:00:39.090
Recall that in our
previous two lectures,
00:00:39.090 --> 00:00:43.320
we have been discussing
the concepts of velocity
00:00:43.320 --> 00:00:47.290
and acceleration of particles
moving along a curve.
00:00:47.290 --> 00:00:49.120
And we had pointed
out originally
00:00:49.120 --> 00:00:53.356
that the most natural definition
was in terms of say, i and j,
00:00:53.356 --> 00:00:55.260
or i, j, and k components--
in other words,
00:00:55.260 --> 00:00:57.020
Cartesian coordinates.
00:00:57.020 --> 00:00:59.750
And then we showed
in the last lecture
00:00:59.750 --> 00:01:03.800
that tangential and
normal components
00:01:03.800 --> 00:01:08.540
form a rather important
system of vectors.
00:01:08.540 --> 00:01:10.160
And now, what we
would like to do
00:01:10.160 --> 00:01:15.740
is to motivate the use of polar
coordinate types of vectors.
00:01:15.740 --> 00:01:18.140
And again, it's
interesting point out
00:01:18.140 --> 00:01:20.330
that we could have
introduced polar coordinates
00:01:20.330 --> 00:01:23.460
into our course at any
one of a number of times.
00:01:23.460 --> 00:01:25.690
In fact, one of the problems
with the traditional
00:01:25.690 --> 00:01:28.560
mathematics curriculum was
that, frequently, topics
00:01:28.560 --> 00:01:32.230
were put in just for the
sake of filling a chapter.
00:01:32.230 --> 00:01:34.180
They fit into the space.
00:01:34.180 --> 00:01:36.530
They didn't need any
additional prerequisites,
00:01:36.530 --> 00:01:37.990
so we put them in there.
00:01:37.990 --> 00:01:41.160
And many a typical traditional
high school algebra
00:01:41.160 --> 00:01:43.050
book could have had
the chapters shuffled
00:01:43.050 --> 00:01:45.170
with no loss of continuity.
00:01:45.170 --> 00:01:48.070
What we're trying to do
in this particular course
00:01:48.070 --> 00:01:51.460
is to motivate why
particular concepts would
00:01:51.460 --> 00:01:53.960
have been invented if
they had not already
00:01:53.960 --> 00:01:55.920
have been invented previously.
00:01:55.920 --> 00:02:00.530
In particular, in dealing with
particles moving in a plane
00:02:00.530 --> 00:02:03.240
or in space, there
comes a time when
00:02:03.240 --> 00:02:06.660
one considers the very important
physical application known
00:02:06.660 --> 00:02:09.032
as a central force field.
00:02:09.032 --> 00:02:10.990
Now, in a central force
field-- and by the way,
00:02:10.990 --> 00:02:13.540
this is where the word polar
coordinates comes up very
00:02:13.540 --> 00:02:15.010
naturally, and why
we call today's
00:02:15.010 --> 00:02:16.450
lecture "Polar Coordinates."
00:02:16.450 --> 00:02:18.250
See, in a central force
field, what we mean
00:02:18.250 --> 00:02:20.950
is that the only
force acting happens
00:02:20.950 --> 00:02:24.230
to be located at
a fixed position.
00:02:24.230 --> 00:02:25.870
And so that the
force of attraction,
00:02:25.870 --> 00:02:29.780
no matter where the particle
is-- wherever that particle is,
00:02:29.780 --> 00:02:34.010
the force is always
directed along the line,
00:02:34.010 --> 00:02:36.350
either towards or away,
depending upon whether it's
00:02:36.350 --> 00:02:38.950
attraction or repulsion.
00:02:41.740 --> 00:02:42.940
It's always acting this way.
00:02:42.940 --> 00:02:45.580
If we're using
Newtonian mechanics,
00:02:45.580 --> 00:02:48.350
since the force is proportional
to the acceleration,
00:02:48.350 --> 00:02:49.270
it says what?
00:02:49.270 --> 00:02:52.290
That a very natural
direction for acceleration
00:02:52.290 --> 00:02:55.550
would be along
the direction that
00:02:55.550 --> 00:02:58.320
connects the origin--
meaning the location
00:02:58.320 --> 00:03:01.710
of the central force--
to the particle.
00:03:01.710 --> 00:03:03.650
Well, you see,
once this is done,
00:03:03.650 --> 00:03:06.320
it becomes a very
natural thing to say, OK,
00:03:06.320 --> 00:03:08.970
if you're going to be talking
about central force fields,
00:03:08.970 --> 00:03:11.720
why not introduce new variables?
00:03:11.720 --> 00:03:13.790
After all, it seems that
the important thing now,
00:03:13.790 --> 00:03:15.510
in measuring the
central force, is
00:03:15.510 --> 00:03:18.780
to know how far the
particle is from the force
00:03:18.780 --> 00:03:22.130
and also, I suppose, to locate
the position of the particle.
00:03:22.130 --> 00:03:23.890
Once you know how
far away it is,
00:03:23.890 --> 00:03:28.480
you would like to know what
angle the radius vector makes
00:03:28.480 --> 00:03:32.070
with the positive x-axis.
00:03:32.070 --> 00:03:34.530
And in this sense,
that would be all
00:03:34.530 --> 00:03:37.740
that we would need to
motivate polar coordinates.
00:03:37.740 --> 00:03:40.370
Now, you see, once the
motivation is there,
00:03:40.370 --> 00:03:43.327
we can always study the
subject independently of
00:03:43.327 --> 00:03:45.410
whether the motivation had
ever been given or not.
00:03:45.410 --> 00:03:46.826
In other words,
now, we say, look,
00:03:46.826 --> 00:03:49.420
it's important to study
polar coordinates.
00:03:49.420 --> 00:03:50.260
Why is it important?
00:03:50.260 --> 00:03:52.860
Because you want to study
central force fields.
00:03:52.860 --> 00:03:54.520
But the structure
of polar coordinates
00:03:54.520 --> 00:03:56.950
is going to be the same,
whether we study central force
00:03:56.950 --> 00:03:58.100
fields or not.
00:03:58.100 --> 00:04:03.810
Then, you see, once we finish
our study of polar coordinates,
00:04:03.810 --> 00:04:06.210
then we say, OK,
now, let's go back,
00:04:06.210 --> 00:04:09.385
as a particular application,
to central force fields.
00:04:09.385 --> 00:04:11.500
At any rate,
without further ado,
00:04:11.500 --> 00:04:14.690
let's tackle the subject
of polar coordinates.
00:04:14.690 --> 00:04:18.839
Again, many of you may
be very, very familiar
00:04:18.839 --> 00:04:21.470
with polar coordinates,
some of you a bit rusty.
00:04:21.470 --> 00:04:26.630
To play it safe, I have divided
the study guide in such a way
00:04:26.630 --> 00:04:30.340
that there will be two units
devoted to polar coordinates,
00:04:30.340 --> 00:04:33.940
but that this one lecture will
cover both of those two units,
00:04:33.940 --> 00:04:36.737
that this lecture, as
usual, will be the overview.
00:04:36.737 --> 00:04:39.070
And I guess this is the best
you can do with mathematics
00:04:39.070 --> 00:04:40.140
beyond a point.
00:04:40.140 --> 00:04:42.120
Beyond a certain
point, mathematics
00:04:42.120 --> 00:04:44.830
ceases to be a spectator
sport, and you just
00:04:44.830 --> 00:04:47.110
have to dirty your hands
with the computations,
00:04:47.110 --> 00:04:50.520
and that the basic theory,
as simple as it may seem,
00:04:50.520 --> 00:04:51.950
is straightforward.
00:04:51.950 --> 00:04:54.870
It's just a case of picking
up the computational know-how
00:04:54.870 --> 00:04:57.390
together with a
familiarity so that you
00:04:57.390 --> 00:04:59.150
feel at ease with the concepts.
00:04:59.150 --> 00:05:01.460
So let's just quickly go
through the highlights
00:05:01.460 --> 00:05:02.830
of polar coordinates.
00:05:02.830 --> 00:05:06.270
They start off in a very
deceptively simple way.
00:05:06.270 --> 00:05:08.580
Namely, given a
point P in the plane,
00:05:08.580 --> 00:05:11.210
if we elect to use
Cartesian coordinates,
00:05:11.210 --> 00:05:15.220
the point P can be
labeled as x comma y.
00:05:15.220 --> 00:05:19.080
On the other hand, using
polar coordinates-- referring
00:05:19.080 --> 00:05:21.760
to this diagram, at
least-- one would
00:05:21.760 --> 00:05:25.440
tend, as we mentioned before, to
label this point r comma theta,
00:05:25.440 --> 00:05:28.490
where r is the distance of
the point from the origin,
00:05:28.490 --> 00:05:31.900
and theta the angle that
we're talking about over here.
00:05:31.900 --> 00:05:34.740
Notice, by the way, that
there is a very simple set
00:05:34.740 --> 00:05:37.230
of formulae by
which we can switch
00:05:37.230 --> 00:05:40.230
from one pair of
representations to the other.
00:05:40.230 --> 00:05:44.640
For example, to express x and
y in terms of r and theta,
00:05:44.640 --> 00:05:47.450
one simply has that x
equals r cosine theta,
00:05:47.450 --> 00:05:50.080
y equals r sine theta.
00:05:50.080 --> 00:05:50.770
OK.
00:05:50.770 --> 00:05:54.940
And conversely, to express r
and theta in terms of x and y,
00:05:54.940 --> 00:05:57.820
one has that r squared
equals x squared
00:05:57.820 --> 00:06:01.504
plus y squared and the
tan theta is y over x.
00:06:01.504 --> 00:06:02.920
By the way, you
might wonder why I
00:06:02.920 --> 00:06:05.160
didn't write r equals
the square root of x
00:06:05.160 --> 00:06:06.930
squared plus y squared.
00:06:06.930 --> 00:06:08.580
The reason I wrote
this is that we're
00:06:08.580 --> 00:06:11.760
going to get into some
complications very shortly,
00:06:11.760 --> 00:06:14.440
complications which
I shall skim over.
00:06:14.440 --> 00:06:15.940
And if you see how
long-winded I am,
00:06:15.940 --> 00:06:18.745
skimming over is not going to be
that short, but short relative
00:06:18.745 --> 00:06:20.120
to the treatment
that we're going
00:06:20.120 --> 00:06:22.060
to give it in the exercises.
00:06:22.060 --> 00:06:23.950
There are certain
complications that set in,
00:06:23.950 --> 00:06:26.356
because r can be negative.
00:06:26.356 --> 00:06:27.730
By the same token,
somebody says,
00:06:27.730 --> 00:06:29.550
instead of writing
tan theta equals y
00:06:29.550 --> 00:06:32.050
over x, why couldn't
you say that theta
00:06:32.050 --> 00:06:36.330
was the inverse tangent y
over x, use the inverse trig
00:06:36.330 --> 00:06:37.260
functions?
00:06:37.260 --> 00:06:40.400
Remember that the inverse
trig functions require
00:06:40.400 --> 00:06:42.640
a principal value domain.
00:06:42.640 --> 00:06:44.020
For the tangent that's what?
00:06:44.020 --> 00:06:47.260
Between minus pi
over 2 and pi over 2.
00:06:47.260 --> 00:06:49.640
And notice that the angle
theta certainly is not
00:06:49.640 --> 00:06:50.734
restricted to that range.
00:06:50.734 --> 00:06:53.150
Theta could be in the second
quadrant, the third quadrant.
00:06:53.150 --> 00:06:54.610
It can wind around.
00:06:54.610 --> 00:06:57.060
In other words,
one of the luxuries
00:06:57.060 --> 00:07:01.130
about Cartesian coordinates
was that they were terribly
00:07:01.130 --> 00:07:03.440
formally stilted.
00:07:03.440 --> 00:07:06.710
There were no two ways to
represent the same point.
00:07:06.710 --> 00:07:08.840
In the study of
Cartesian coordinates,
00:07:08.840 --> 00:07:13.030
one can say, to a point,
position is everything in life,
00:07:13.030 --> 00:07:17.790
that once you've
changed the coordinates,
00:07:17.790 --> 00:07:19.180
you've changed the point.
00:07:19.180 --> 00:07:21.760
For example, there's
no possible way
00:07:21.760 --> 00:07:26.150
for the point x comma y to
equal the point, say, 2 comma 3,
00:07:26.150 --> 00:07:28.810
unless x equals 2 or y equals 3.
00:07:28.810 --> 00:07:31.330
In other words, for
Cartesian coordinates,
00:07:31.330 --> 00:07:34.260
the only way to points could
be equal would be if they were
00:07:34.260 --> 00:07:36.660
equal coordinate by coordinate.
00:07:36.660 --> 00:07:40.010
The complication that sets
in for polar coordinates
00:07:40.010 --> 00:07:43.840
is that, unfortunately, this
simple result no longer remains
00:07:43.840 --> 00:07:44.470
true.
00:07:44.470 --> 00:07:48.530
For example, if I tell you
that the point r_1 comma
00:07:48.530 --> 00:07:51.990
theta_1 is equal to the
point r_2 comma theta_2,
00:07:51.990 --> 00:07:55.840
it does not imply-- that's a
slash through the arrow here.
00:07:55.840 --> 00:07:57.100
This means it's false.
00:07:57.100 --> 00:08:00.770
It does not imply that I can
conclude that r_1 equals r:2
00:08:00.770 --> 00:08:02.590
and that theta_1 equals theta_2.
00:08:02.590 --> 00:08:05.900
To be sure, it's possible
that r_1 equals r_2
00:08:05.900 --> 00:08:08.510
and it's possible that
theta 1 equals theta 2,
00:08:08.510 --> 00:08:10.660
but it doesn't have to happen.
00:08:10.660 --> 00:08:12.380
Well, the nice thing
about mathematics
00:08:12.380 --> 00:08:14.190
is that we can
always give examples
00:08:14.190 --> 00:08:15.680
to back up what we're saying.
00:08:15.680 --> 00:08:18.230
Let's look at a few examples.
00:08:18.230 --> 00:08:20.970
Let's look, for example,
at what happens,
00:08:20.970 --> 00:08:24.510
if we're using radian
measure, if we replace
00:08:24.510 --> 00:08:29.790
the angle by some integral
multiple of 2 pi added
00:08:29.790 --> 00:08:30.930
on to that angle.
00:08:30.930 --> 00:08:33.940
In other words, let's take
the point r_1 comma theta_1
00:08:33.940 --> 00:08:38.900
and compare that point with the
point whose coordinates are r_1
00:08:38.900 --> 00:08:42.770
comma theta_1 plus 2k*pi.
00:08:42.770 --> 00:08:46.010
Look, the r values
are the same here.
00:08:46.010 --> 00:08:50.080
But look at theta_1
and theta_1 plus 2k*pi.
00:08:50.080 --> 00:08:56.700
Notice that, unless k is 0,
these are different angles.
00:08:56.700 --> 00:08:58.200
It's rather interesting.
00:08:58.200 --> 00:09:00.400
I know in high school
this happens a lot.
00:09:00.400 --> 00:09:01.960
Youngsters will
tend to say things
00:09:01.960 --> 00:09:05.940
like 30 degrees is the
same as 390 degrees.
00:09:05.940 --> 00:09:08.730
And in a way, they're right,
except that they say it in such
00:09:08.730 --> 00:09:10.230
a way that it's dangerous.
00:09:10.230 --> 00:09:12.640
Certainly, one should
not confuse 30 degrees
00:09:12.640 --> 00:09:14.050
with 390 degrees.
00:09:14.050 --> 00:09:17.780
What one usually means is that
any trigonometric function
00:09:17.780 --> 00:09:21.390
of 30 degrees is equal to that
same trigonometric function
00:09:21.390 --> 00:09:23.240
of 390 degrees.
00:09:23.240 --> 00:09:25.440
In other words, the
technical way of saying it
00:09:25.440 --> 00:09:27.330
is that the
trigonometric functions
00:09:27.330 --> 00:09:32.190
are periodic with period
2pi in radian measure, 360
00:09:32.190 --> 00:09:34.350
degrees in degree measure.
00:09:34.350 --> 00:09:37.220
In other words, for example,
using radian measure,
00:09:37.220 --> 00:09:41.030
notice that the sine of theta_1
is equal to the sine of theta_1
00:09:41.030 --> 00:09:45.140
plus 2*pi*k, where k
is some integer here.
00:09:45.140 --> 00:09:51.200
But that theta_1 is unequal
to theta_1 plus 2*pi*k if k is
00:09:51.200 --> 00:09:51.900
unequal to 0.
00:09:51.900 --> 00:09:53.440
If k is equal to
0, of course, this
00:09:53.440 --> 00:09:55.100
happens to be a special case.
00:09:55.100 --> 00:09:58.340
You see, the whole
point is that all it
00:09:58.340 --> 00:10:00.010
means in terms of
a graph, if you
00:10:00.010 --> 00:10:03.950
were to plot the
curve y equals sine x,
00:10:03.950 --> 00:10:05.700
it is possible for what?
00:10:05.700 --> 00:10:09.160
Many different values of
theta, many different values
00:10:09.160 --> 00:10:12.850
of x, to give the
same value of sine x.
00:10:12.850 --> 00:10:16.000
What this means, by the way,
in terms of polar coordinates,
00:10:16.000 --> 00:10:18.820
is that it certainly makes
a difference whether we talk
00:10:18.820 --> 00:10:25.550
about a line making
an angle of 30
00:10:25.550 --> 00:10:29.300
degrees with the positive
x-axis or making 390 degrees,
00:10:29.300 --> 00:10:31.220
because, you see,
that 390 degrees
00:10:31.220 --> 00:10:34.720
seems to indicate that you've
made one full circuit and then
00:10:34.720 --> 00:10:36.770
an additional 30 degrees.
00:10:36.770 --> 00:10:39.820
As far as position is concerned,
you're in the same place.
00:10:39.820 --> 00:10:42.490
But for example, in terms
of fuel consumption,
00:10:42.490 --> 00:10:45.770
it uses more fuel, say, to
go through the 390 degrees
00:10:45.770 --> 00:10:47.332
than to go through
the 30 degrees.
00:10:47.332 --> 00:10:49.790
But at any rate, that's not
the point we want to make here.
00:10:49.790 --> 00:10:52.140
The point is, notice in
this particular example
00:10:52.140 --> 00:10:56.020
that this is two different
names for the same point.
00:10:56.020 --> 00:10:59.600
But we cannot conclude that,
coordinate by coordinate,
00:10:59.600 --> 00:11:02.240
the coordinates are equal.
00:11:02.240 --> 00:11:04.040
That's not the worst of it.
00:11:04.040 --> 00:11:07.810
The worst of it is that
r and/or theta don't even
00:11:07.810 --> 00:11:08.860
have to be positive.
00:11:08.860 --> 00:11:11.210
Another way of saying that
is they may be negative.
00:11:11.210 --> 00:11:14.760
For example, to capitalize on
the idea of vector notation,
00:11:14.760 --> 00:11:17.160
what one frequently
does is says,
00:11:17.160 --> 00:11:19.850
look, let's talk about
a negative distance.
00:11:19.850 --> 00:11:22.959
And by a negative distance,
we'll identify that with sense.
00:11:22.959 --> 00:11:25.000
In other words, let's call
this angle here theta.
00:11:27.899 --> 00:11:29.690
See, notice, what you're
really saying here
00:11:29.690 --> 00:11:31.680
is you're assuming that
the radius vector, when
00:11:31.680 --> 00:11:34.880
you're calling this angle theta,
has this particular sense.
00:11:34.880 --> 00:11:37.400
Suppose you wanted to
talk about this vector.
00:11:37.400 --> 00:11:42.420
Notice that this would be the
right sense if the angle here
00:11:42.420 --> 00:11:43.480
happened to be what?
00:11:43.480 --> 00:11:49.680
Theta plus pi-- 180
degrees, pi radians.
00:11:49.680 --> 00:11:51.390
Now, what you say
is, if you're looking
00:11:51.390 --> 00:11:54.310
at this particular point,
notice that with respect
00:11:54.310 --> 00:11:57.440
to this vector,
you must go what?
00:11:57.440 --> 00:11:59.260
Negative r units--
in other words,
00:11:59.260 --> 00:12:03.050
you must move in the opposite
sense of this particular vector
00:12:03.050 --> 00:12:04.780
to get to this point.
00:12:04.780 --> 00:12:08.240
And again, this is a
touchy enough concept
00:12:08.240 --> 00:12:10.550
that we're going to do
this in great detail
00:12:10.550 --> 00:12:12.610
in the learning exercises.
00:12:12.610 --> 00:12:16.560
But the point is, notice
that r comma theta is
00:12:16.560 --> 00:12:20.410
a different name for the same
point as that which is named
00:12:20.410 --> 00:12:23.354
by minus r comma theta plus pi.
00:12:23.354 --> 00:12:24.020
Do you see that?
00:12:24.020 --> 00:12:26.170
Let's look at that one more
time just to make sure.
00:12:26.170 --> 00:12:30.560
See, on the one hand, reading
this angle and this distance,
00:12:30.560 --> 00:12:33.170
this is r comma theta.
00:12:33.170 --> 00:12:36.590
On the other hand, reading
it from this angle,
00:12:36.590 --> 00:12:41.324
the value is minus r and
the angle is theta plus pi.
00:12:41.324 --> 00:12:43.490
And I'll get a different
piece of chalk in a minute,
00:12:43.490 --> 00:12:45.531
because this doesn't seem
to be writing too well.
00:12:45.531 --> 00:12:47.730
But let's not worry about
that for the time being.
00:12:47.730 --> 00:12:49.010
Let me give you an example.
00:12:49.010 --> 00:12:51.450
Look at the curve
whose polar equation is
00:12:51.450 --> 00:12:53.780
r equals sine squared theta.
00:12:53.780 --> 00:12:58.080
Notice that if theta equals pi
over 6, the sine of pi over 6
00:12:58.080 --> 00:12:59.060
is 1/2.
00:12:59.060 --> 00:13:01.380
1/2 squared is 1/4.
00:13:01.380 --> 00:13:05.560
So notice that r equals
1/4, theta equals pi over 6
00:13:05.560 --> 00:13:08.410
satisfies this
particular equation.
00:13:08.410 --> 00:13:10.750
Now, notice that another
name for the same point,
00:13:10.750 --> 00:13:16.970
using this recipe here, is
minus 1/4 comma 7*pi over 6.
00:13:16.970 --> 00:13:18.790
But without even
looking, you should
00:13:18.790 --> 00:13:22.325
be able to see that it's
impossible for this value of r
00:13:22.325 --> 00:13:25.610
and this value of theta
to satisfy this equation.
00:13:25.610 --> 00:13:28.070
In particular, notice
that sine squared
00:13:28.070 --> 00:13:29.480
theta can't be negative.
00:13:29.480 --> 00:13:32.720
Just by reading this equation,
r can never be negative,
00:13:32.720 --> 00:13:35.030
therefore, how can
r equals minus 1/4
00:13:35.030 --> 00:13:37.290
possibly satisfy this equation?
00:13:37.290 --> 00:13:39.030
And this is where
the big complication
00:13:39.030 --> 00:13:41.620
comes in in terms of
simultaneous equations
00:13:41.620 --> 00:13:42.620
and what have you.
00:13:42.620 --> 00:13:43.900
Here, we have what?
00:13:43.900 --> 00:13:46.830
Two different names
for the same point.
00:13:46.830 --> 00:13:51.410
Yet, by one of its names, the
point satisfies the equation,
00:13:51.410 --> 00:13:56.010
and by another name, it
doesn't satisfy the equation.
00:13:56.010 --> 00:13:57.730
And this is a very,
very touchy thing,
00:13:57.730 --> 00:13:59.210
because you'd like
to believe what?
00:13:59.210 --> 00:14:02.410
That a point belongs to a
curve, not the name of a point.
00:14:02.410 --> 00:14:05.330
And that one of the difficulties
with polar coordinates
00:14:05.330 --> 00:14:07.790
is that, to check whether
a point belongs to a curve
00:14:07.790 --> 00:14:12.430
or not, if the point,
as named by one way,
00:14:12.430 --> 00:14:15.070
doesn't satisfy the
equation, you still
00:14:15.070 --> 00:14:16.950
have to check other
possible names
00:14:16.950 --> 00:14:19.700
to see whether they satisfy
the equation or not.
00:14:19.700 --> 00:14:21.770
And this is a very
complicated topic.
00:14:21.770 --> 00:14:24.960
And this is why some of our
learning exercises in this unit
00:14:24.960 --> 00:14:26.100
take so many pages.
00:14:26.100 --> 00:14:29.680
It's more or less devoted to
giving you insight into this
00:14:29.680 --> 00:14:31.530
if you don't already
have this insight.
00:14:31.530 --> 00:14:37.910
But at any rate, let's continue
on in general terms here.
00:14:37.910 --> 00:14:40.530
The next most important
thing to talk about
00:14:40.530 --> 00:14:43.060
is to make sure that
we understand what
00:14:43.060 --> 00:14:45.150
polar coordinates really mean.
00:14:45.150 --> 00:14:47.590
Namely, when someone
says, I am thinking
00:14:47.590 --> 00:14:50.250
of the curve C
whose polar equation
00:14:50.250 --> 00:14:57.660
is r equals sine theta, he does
not mean plot r versus theta
00:14:57.660 --> 00:15:01.200
this way and call that
the curve C. You see,
00:15:01.200 --> 00:15:04.370
notice that even though you're
calling this r and theta,
00:15:04.370 --> 00:15:07.520
using the axes to be at
right angles this way,
00:15:07.520 --> 00:15:10.050
no matter how you
slice it, you're still
00:15:10.050 --> 00:15:12.970
using Cartesian
coordinates here,
00:15:12.970 --> 00:15:16.390
only what theta replacing
the name of x and r
00:15:16.390 --> 00:15:18.030
replacing the name y.
00:15:18.030 --> 00:15:19.606
See, this is not what's meant.
00:15:19.606 --> 00:15:20.980
Remember that when
you're talking
00:15:20.980 --> 00:15:24.810
about polar coordinates, r
specifically measures what?
00:15:24.810 --> 00:15:29.880
The distance of the particular
point on C from the origin,
00:15:29.880 --> 00:15:32.740
and theta measures the angle
that that radius vector
00:15:32.740 --> 00:15:34.780
makes with the x-axis.
00:15:34.780 --> 00:15:37.820
For example, what we do
we mean by the curve whose
00:15:37.820 --> 00:15:41.090
polar equation is
r equals sine theta
00:15:41.090 --> 00:15:51.260
is the circle of radius
1/2 centered on the y-axis,
00:15:51.260 --> 00:15:52.500
you see, at the point what?
00:15:52.500 --> 00:15:55.890
In Cartesian coordinates,
x is 0, y is 1/2.
00:15:55.890 --> 00:15:58.370
And I claim that that's
what we mean by the curve r
00:15:58.370 --> 00:15:59.530
equals sine theta.
00:15:59.530 --> 00:16:01.110
And how can I prove that to you?
00:16:01.110 --> 00:16:03.430
Well, the easiest way
to prove that to you
00:16:03.430 --> 00:16:05.510
is by some elementary geometry.
00:16:05.510 --> 00:16:07.900
Namely, I call this
point r comma theta.
00:16:07.900 --> 00:16:10.180
I now draw in these guidelines.
00:16:10.180 --> 00:16:13.300
Notice that, by definition,
this length is r.
00:16:13.300 --> 00:16:16.710
Since this is an inscribed
angle on a diameter,
00:16:16.710 --> 00:16:18.580
it's a 90-degree angle.
00:16:18.580 --> 00:16:22.360
Therefore, since this angle is
theta and both of these angles
00:16:22.360 --> 00:16:24.420
are complements of
a 90-degree angle,
00:16:24.420 --> 00:16:26.470
this angle must also be theta.
00:16:26.470 --> 00:16:29.360
And now, if I just look at
this particular diagram,
00:16:29.360 --> 00:16:32.920
I read this right triangle,
and immediately, I see what?
00:16:32.920 --> 00:16:36.520
That from this right
triangle, r equals sine theta.
00:16:36.520 --> 00:16:39.200
In other words,
the curve C, whose
00:16:39.200 --> 00:16:42.050
polar equation is r
equals sine theta,
00:16:42.050 --> 00:16:43.810
is this particular curve.
00:16:43.810 --> 00:16:48.400
By the way, we often
get indoctrinated
00:16:48.400 --> 00:16:51.040
into seeing things in a
very natural way in terms
00:16:51.040 --> 00:16:52.610
of a native tongue.
00:16:52.610 --> 00:16:55.390
My father-in-law was
brought up in Russia.
00:16:55.390 --> 00:16:57.200
He had a little grocery store.
00:16:57.200 --> 00:16:59.120
He spoke fluent
English, but when
00:16:59.120 --> 00:17:02.360
he added up customers' bills,
he always added in Russian.
00:17:02.360 --> 00:17:05.420
He was more at home
with Russia numerals.
00:17:05.420 --> 00:17:08.050
And this always impressed
me, until I realized
00:17:08.050 --> 00:17:10.569
that I was the same
way, only I use
00:17:10.569 --> 00:17:13.859
base 10 numerals instead of
base 7 or something like this.
00:17:13.859 --> 00:17:15.960
And the same thing
happens with the polar
00:17:15.960 --> 00:17:17.720
versus Cartesian coordinates.
00:17:17.720 --> 00:17:19.270
Even though polar
coordinates are
00:17:19.270 --> 00:17:21.660
independent of
Cartesian coordinates,
00:17:21.660 --> 00:17:23.530
the fact remains that
we're probably more
00:17:23.530 --> 00:17:25.640
at home with
Cartesian coordinates
00:17:25.640 --> 00:17:27.339
than we are with
polar coordinates.
00:17:27.339 --> 00:17:29.370
Consequently, one
very common trick,
00:17:29.370 --> 00:17:32.560
when we can get away with it,
is to take a polar equation
00:17:32.560 --> 00:17:35.810
and translate it into an
equivalent Cartesian equation.
00:17:35.810 --> 00:17:38.030
Namely, given r
equals sine theta,
00:17:38.030 --> 00:17:41.430
and remembering that r squared
is x squared plus y squared
00:17:41.430 --> 00:17:44.330
and that r sine theta
is y, we multiply
00:17:44.330 --> 00:17:47.410
both sides of this equation
by r to get r squared
00:17:47.410 --> 00:17:49.040
equals r sin(theta).
00:17:49.040 --> 00:17:52.100
This leads to x squared
plus y squared equals y.
00:17:52.100 --> 00:17:54.390
If we then complete
the square, et cetera,
00:17:54.390 --> 00:17:56.770
we find that, in
Cartesian coordinates,
00:17:56.770 --> 00:18:00.210
this is the equation of the
circle centered at the point 0
00:18:00.210 --> 00:18:03.590
comma 1/2 with
radius equal to 1/2.
00:18:03.590 --> 00:18:06.270
The thing that's very,
very important to note here
00:18:06.270 --> 00:18:10.070
is that, in terms of x
and y, the relationship
00:18:10.070 --> 00:18:14.160
x squared plus y squared
equals y is not structurally
00:18:14.160 --> 00:18:17.960
the same as the relationship
r equals sine theta.
00:18:17.960 --> 00:18:20.600
In other words, the
curve C is the same
00:18:20.600 --> 00:18:22.490
whether you use
its Cartesian form
00:18:22.490 --> 00:18:24.570
or whether you use
its polar form.
00:18:24.570 --> 00:18:26.710
But what's very
important to note
00:18:26.710 --> 00:18:29.950
is that the relationship
between r and theta
00:18:29.950 --> 00:18:34.500
is not algebraically the same
as the relationship between x
00:18:34.500 --> 00:18:36.270
and y.
00:18:36.270 --> 00:18:40.250
But the important point
is that, since the curve
00:18:40.250 --> 00:18:44.580
C does not measure r versus
theta as being at right angles
00:18:44.580 --> 00:18:48.070
to each other, that if you
were given r equals sine theta
00:18:48.070 --> 00:18:50.250
and you compute dr / d
theta, which, in this case,
00:18:50.250 --> 00:18:51.390
simply would be what?
00:18:51.390 --> 00:18:55.142
cosine theta-- that does
not have the slope of C.
00:18:55.142 --> 00:18:57.100
You see, this is what I
want you to understand.
00:18:57.100 --> 00:18:59.410
If I compute dr
/ d theta and one
00:18:59.410 --> 00:19:02.850
would interpret that as a slope,
it's not a slope of the curve
00:19:02.850 --> 00:19:06.400
C. It's a slope of this
particular curve which wasn't
00:19:06.400 --> 00:19:09.700
C. In other words,
this is the curve
00:19:09.700 --> 00:19:15.590
that measures r versus theta in
terms of Cartesian coordinates.
00:19:15.590 --> 00:19:17.560
That dr / d theta
just shows you how
00:19:17.560 --> 00:19:19.860
r is changing with
respect to theta,
00:19:19.860 --> 00:19:24.790
it does not tell you how the
curve is rising at that point.
00:19:24.790 --> 00:19:27.600
And again, more
information is given
00:19:27.600 --> 00:19:29.490
on this in terms of exercises.
00:19:29.490 --> 00:19:31.430
But that's what I do
want you to understand.
00:19:31.430 --> 00:19:34.655
It does not mean that you can't
compute the slope of this curve
00:19:34.655 --> 00:19:35.570
at any point.
00:19:35.570 --> 00:19:38.590
It does mean that if
you want the slope
00:19:38.590 --> 00:19:42.110
and you compute it by
letting it equal dr / d theta
00:19:42.110 --> 00:19:44.740
you will get an answer this
way, but it won't be the slope
00:19:44.740 --> 00:19:45.900
that you're looking for.
00:19:45.900 --> 00:19:48.700
In fact, what I hope that
you can see rather simply
00:19:48.700 --> 00:19:50.995
here is the following.
00:19:54.880 --> 00:19:56.680
Say I want the slope
of the tangent line
00:19:56.680 --> 00:19:58.530
to the curve at this
particular point,
00:19:58.530 --> 00:20:00.450
it doesn't really
make much difference
00:20:00.450 --> 00:20:05.560
whether I use the angle phi--
which the tangent line makes
00:20:05.560 --> 00:20:09.372
with the positive x-axis-- or
whether I use phi or "phi,"
00:20:09.372 --> 00:20:14.870
or, I don't know-- This angle
is called either "phi" or "phi".
00:20:14.870 --> 00:20:17.720
And this angle is
called "psi" or "psi."
00:20:17.720 --> 00:20:19.470
I get all mixed up
with the Greek letters.
00:20:19.470 --> 00:20:20.770
You know, in fact,
coming into work today,
00:20:20.770 --> 00:20:22.145
I was listening
to a disc jockey,
00:20:22.145 --> 00:20:24.662
and he was making fun of
how in drugstores now, you
00:20:24.662 --> 00:20:26.370
can get a full course
meal, but you can't
00:20:26.370 --> 00:20:28.306
get any medicines anymore.
00:20:28.306 --> 00:20:29.930
And he was talking
about his friend who
00:20:29.930 --> 00:20:33.284
got all A's in pharmacy school,
and they flunked him out
00:20:33.284 --> 00:20:35.200
because he didn't know
how to make a sandwich.
00:20:35.200 --> 00:20:37.432
I almost flunked out of
math because I have trouble
00:20:37.432 --> 00:20:38.640
with the Greek alphabet here.
00:20:38.640 --> 00:20:40.830
But look, forget about that.
00:20:40.830 --> 00:20:42.600
The thing I want
to see is I want
00:20:42.600 --> 00:20:44.430
to know the direction
of this line.
00:20:44.430 --> 00:20:48.000
If it's convenient to
use phi, I'll use phi.
00:20:48.000 --> 00:20:51.600
If it's convenient to
use psi, I'll use psi.
00:20:51.600 --> 00:20:52.780
What is phi?
00:20:52.780 --> 00:20:55.410
Phi is the angle
that the tangent line
00:20:55.410 --> 00:20:57.110
makes with the positive x-axis.
00:20:57.110 --> 00:20:58.516
What is psi?
00:20:58.516 --> 00:21:01.090
Psi is the angle
that the tangent line
00:21:01.090 --> 00:21:03.180
makes with the radius vector.
00:21:03.180 --> 00:21:06.450
Now, as I'll show you in
some of our exercises,
00:21:06.450 --> 00:21:10.760
to use the definition
that the slope is
00:21:10.760 --> 00:21:15.500
tan phi and translating that
from Cartesian coordinates
00:21:15.500 --> 00:21:19.260
into polar coordinates is
a very, very messy job.
00:21:19.260 --> 00:21:21.940
What turns out to be
very, very interesting,
00:21:21.940 --> 00:21:25.210
at least from my point of view,
is that not only is the angle
00:21:25.210 --> 00:21:29.050
psi-- "psi"-- more natural
to use than phi when
00:21:29.050 --> 00:21:30.800
you're dealing with
polar coordinates,
00:21:30.800 --> 00:21:34.870
but it almost turns out that
because it was more natural,
00:21:34.870 --> 00:21:37.180
there's a simpler
formula for it.
00:21:37.180 --> 00:21:41.290
In other words, whereas
the formula for tan phi
00:21:41.290 --> 00:21:44.990
is very, very complicated
in polar coordinates,
00:21:44.990 --> 00:21:48.120
the formula for tan
psi is very simply
00:21:48.120 --> 00:21:51.920
given by r divided
by dr / d theta.
00:21:51.920 --> 00:21:54.560
In other words, if
I take r, divide
00:21:54.560 --> 00:21:58.860
that by dr / d theta, what I get
is the tangent of this angle.
00:21:58.860 --> 00:21:59.450
Now, look.
00:21:59.450 --> 00:22:01.590
Once I have the
tangent of this angle,
00:22:01.590 --> 00:22:03.466
it's very simple to
construct a tangent line.
00:22:03.466 --> 00:22:04.715
That's what I want you to see.
00:22:04.715 --> 00:22:05.280
Look.
00:22:05.280 --> 00:22:08.570
Suppose this is the point
P, and this is my pole O,
00:22:08.570 --> 00:22:09.870
for polar coordinates.
00:22:09.870 --> 00:22:13.420
And given that r was
some function of theta,
00:22:13.420 --> 00:22:16.230
suppose now I've
computed dr / d theta,
00:22:16.230 --> 00:22:20.780
I've divided that into r, and
I've now found what tan psi is.
00:22:20.780 --> 00:22:23.840
What I do is I take
the line of action that
00:22:23.840 --> 00:22:30.680
joins the origin to P, I
construct the angle psi,
00:22:30.680 --> 00:22:33.620
draw this line, and
whatever that line is,
00:22:33.620 --> 00:22:36.780
that is the line which is
tangent to my particular curve
00:22:36.780 --> 00:22:38.610
at the point P.
00:22:38.610 --> 00:22:40.650
But the thing I want to
see is that we do not
00:22:40.650 --> 00:22:43.550
need Cartesian coordinates
at all in order
00:22:43.550 --> 00:22:47.530
to be able to tackle calculus
properties when an equation is
00:22:47.530 --> 00:22:48.990
written in polar coordinates.
00:22:48.990 --> 00:22:51.070
But what there is a
tendency to do is what?
00:22:51.070 --> 00:22:53.766
That when we're dealing
with polar coordinates,
00:22:53.766 --> 00:22:55.390
since we're more at
home with Cartesian
00:22:55.390 --> 00:22:58.360
coordinates, we will often be
tempted to switch everything
00:22:58.360 --> 00:23:01.750
into Cartesian coordinates,
which is perfectly fair game.
00:23:01.750 --> 00:23:03.530
But the important
thing is to remember
00:23:03.530 --> 00:23:07.560
that all of our calculus results
can be derived independently
00:23:07.560 --> 00:23:10.540
of whether Cartesian
coordinates were ever invented.
00:23:10.540 --> 00:23:12.120
For example, the
final concept I want
00:23:12.120 --> 00:23:16.500
to talk about in this lecture is
how one would have studied area
00:23:16.500 --> 00:23:19.450
if one had only polar
coordinates and had never
00:23:19.450 --> 00:23:22.020
had Cartesian coordinates.
00:23:22.020 --> 00:23:26.100
By the way, notice also that,
in terms of polar coordinates,
00:23:26.100 --> 00:23:29.380
one is more
interested in sectors
00:23:29.380 --> 00:23:30.810
than in rectangular grids.
00:23:30.810 --> 00:23:34.200
In other words, if you look at
something ranging from theta_1
00:23:34.200 --> 00:23:36.560
to theta_2, you
think of something
00:23:36.560 --> 00:23:40.040
caught between two rays here.
00:23:40.040 --> 00:23:42.920
And let's suppose I wanted the
area of this particular region,
00:23:42.920 --> 00:23:45.480
where this particular curve
happened to have the form,
00:23:45.480 --> 00:23:49.140
say, r equals g of theta--
I don't care what it is.
00:23:49.140 --> 00:23:51.390
Let's just call it
r equals g of theta.
00:23:51.390 --> 00:23:53.777
Now, the interesting
point, again-- and I
00:23:53.777 --> 00:23:55.610
keep saying "interesting
point" because they
00:23:55.610 --> 00:23:57.609
are interesting points--
that if you have really
00:23:57.609 --> 00:23:59.860
taken Part 1 of this
course seriously,
00:23:59.860 --> 00:24:03.120
you're going to be amazed
to see how much free mileage
00:24:03.120 --> 00:24:06.630
you get out of Part 2 just
by translating things back
00:24:06.630 --> 00:24:09.580
in to our general
theorems of Part 1.
00:24:09.580 --> 00:24:12.570
Let me see how I could find the
area of this particular region.
00:24:12.570 --> 00:24:16.575
The idea is I'll take a
little increment of area
00:24:16.575 --> 00:24:18.390
here between these
two black lines.
00:24:21.100 --> 00:24:23.630
Remember, my basic building
blocks are now r values.
00:24:23.630 --> 00:24:26.160
What I'll do now is
I'll pick the smallest
00:24:26.160 --> 00:24:30.390
value of r that gives me an arc
that lies inside this segment.
00:24:30.390 --> 00:24:32.880
then I'll pick the
biggest value of r
00:24:32.880 --> 00:24:34.360
that encloses this segment.
00:24:34.360 --> 00:24:39.450
Notice, what I do is I'll
let capital R sub capital
00:24:39.450 --> 00:24:40.590
M denote this radius.
00:24:40.590 --> 00:24:43.300
See, that's this distance
from here to here.
00:24:43.300 --> 00:24:47.270
I'll let little r sub little
m represent this distance.
00:24:47.270 --> 00:24:51.890
What I'm saying is, if
I now swing two arcs,
00:24:51.890 --> 00:24:53.920
my area theorems from
the first semester--
00:24:53.920 --> 00:24:56.580
my area axioms from the
first part of course--
00:24:56.580 --> 00:24:57.300
are still valid.
00:24:57.300 --> 00:25:00.730
Namely, the smaller r--
the smaller sector--
00:25:00.730 --> 00:25:04.090
is contained inside the
delta A that I'm looking for.
00:25:04.090 --> 00:25:06.750
And the larger sector
contains the region
00:25:06.750 --> 00:25:08.720
that I'm looking for.
00:25:08.720 --> 00:25:12.020
In other words, the region
that I'm looking for, delta A,
00:25:12.020 --> 00:25:16.580
is caught between the
areas of these two sectors.
00:25:16.580 --> 00:25:18.540
Now, how do you find
the area of a sector?
00:25:18.540 --> 00:25:21.710
What you do is you take the
area of the entire circle
00:25:21.710 --> 00:25:25.650
and divide it by the
fractional part of the circle
00:25:25.650 --> 00:25:26.940
that you're taking.
00:25:26.940 --> 00:25:30.540
For example, the area of the
circle whose radius is capital
00:25:30.540 --> 00:25:33.930
R sub M is pi R sub M squared.
00:25:33.930 --> 00:25:36.710
Now, what portion of
the circle am I taking?
00:25:36.710 --> 00:25:38.410
The angle is delta theta.
00:25:38.410 --> 00:25:42.780
I'm using radian measure, so
the entire swinging angle would
00:25:42.780 --> 00:25:46.490
have been 2*pi, so I'm
taking delta theta over 2*pi
00:25:46.490 --> 00:25:47.860
of the entire circle.
00:25:47.860 --> 00:25:50.780
So that's the area
of the larger sector.
00:25:50.780 --> 00:25:53.440
What is the area of
the smaller sector?
00:25:53.440 --> 00:25:55.400
The area of the
smaller sector is
00:25:55.400 --> 00:25:59.130
the area of the entire circle--
pi r sub little m squared--
00:25:59.130 --> 00:26:00.460
times, again, what?
00:26:00.460 --> 00:26:02.740
Delta theta over 2*pi.
00:26:02.740 --> 00:26:07.800
And because the region is
embedded between these two,
00:26:07.800 --> 00:26:10.640
its area is caught
between these two areas.
00:26:10.640 --> 00:26:12.570
And if I now solve
for the delta A,
00:26:12.570 --> 00:26:16.210
I simplify, I cancel the
pi's, I get that delta A is
00:26:16.210 --> 00:26:17.170
caught between what?
00:26:17.170 --> 00:26:21.990
1/2 capital R sub M squared
delta theta and 1/2 little r
00:26:21.990 --> 00:26:23.920
sub m squared delta theta.
00:26:23.920 --> 00:26:26.540
I now divide through
by delta theta.
00:26:26.540 --> 00:26:29.060
And I make a non-crucial
assumption here
00:26:29.060 --> 00:26:31.630
that delta theta is
greater than 0, remembering
00:26:31.630 --> 00:26:33.790
that delta theta is negative.
00:26:33.790 --> 00:26:36.900
All I have to do is reverse
the signs of the inequalities.
00:26:36.900 --> 00:26:39.250
The only reason I assumed
that delta theta was positive
00:26:39.250 --> 00:26:42.100
is so that I wouldn't change the
direction of the inequalities.
00:26:42.100 --> 00:26:44.470
A similar
demonstration will hold
00:26:44.470 --> 00:26:46.090
when delta theta is negative.
00:26:46.090 --> 00:26:48.900
The important point is I then
divide through by delta theta.
00:26:48.900 --> 00:26:50.980
I get delta A over delta theta.
00:26:50.980 --> 00:26:54.430
It's caught between 1/2
little r sub m squared and 1/2
00:26:54.430 --> 00:26:56.460
capitalized R sub M squared.
00:26:56.460 --> 00:26:59.361
Now, as I let delta theta
approach 0, what does that
00:26:59.361 --> 00:26:59.860
mean?
00:26:59.860 --> 00:27:02.570
I'm going to let delta
theta close in over here--
00:27:02.570 --> 00:27:03.130
approach 0.
00:27:03.130 --> 00:27:04.830
What's happening here?
00:27:04.830 --> 00:27:06.530
This is a fixed value of r.
00:27:06.530 --> 00:27:10.770
Notice that little r sub
m and capital R sub M,
00:27:10.770 --> 00:27:12.830
as delta theta
approaches 0, they're
00:27:12.830 --> 00:27:15.440
both being pushed
closer and closer to r.
00:27:15.440 --> 00:27:18.360
In other words, as delta
theta approaches 0,
00:27:18.360 --> 00:27:22.520
r sub little m and
capital R sub M
00:27:22.520 --> 00:27:29.952
both approach r, provided that
r is a continuous function
00:27:29.952 --> 00:27:31.660
of theta-- in other
words, that there are
00:27:31.660 --> 00:27:33.770
no breaks in the curve here.
00:27:33.770 --> 00:27:35.810
OK?
00:27:35.810 --> 00:27:37.500
The important point,
then, is what?
00:27:37.500 --> 00:27:40.470
We can then take the limits
as delta theta approaches 0.
00:27:40.470 --> 00:27:42.970
And we find that dA /
d theta is the limit,
00:27:42.970 --> 00:27:45.600
as delta theta approaches
0, delta A divided
00:27:45.600 --> 00:27:48.170
by delta theta, which
simply is what now?
00:27:48.170 --> 00:27:49.360
We come back here.
00:27:49.360 --> 00:27:52.830
As delta theta
approaches 0, little r
00:27:52.830 --> 00:27:56.740
sub m and capital R
sub M both approach r,
00:27:56.740 --> 00:27:58.800
and therefore, this
common limit becomes
00:27:58.800 --> 00:28:02.870
1/2 r squared,
therefore integrating
00:28:02.870 --> 00:28:05.330
this between what limits?
00:28:05.330 --> 00:28:09.630
Between theta_1 and theta_2, I
find that the area of my region
00:28:09.630 --> 00:28:13.610
is the integral from theta_1
to theta_2, 1/2 r squared
00:28:13.610 --> 00:28:15.880
d theta, which, of
course, means what?
00:28:15.880 --> 00:28:21.880
This integral 1/2--
r is g of theta,
00:28:21.880 --> 00:28:23.990
so I square that--
times d theta.
00:28:23.990 --> 00:28:27.030
And notice that this is a
function of theta alone.
00:28:27.030 --> 00:28:29.500
And so I can find
this particular area.
00:28:29.500 --> 00:28:32.904
What's crucial to understand
is that the area does not
00:28:32.904 --> 00:28:34.320
change just because
you're dealing
00:28:34.320 --> 00:28:37.210
with polar coordinates rather
than Cartesian coordinates.
00:28:37.210 --> 00:28:39.860
But rather the form of
the equation changes.
00:28:39.860 --> 00:28:41.980
What I mean by that is
something like this.
00:28:41.980 --> 00:28:43.710
Let's suppose I have
a region like this.
00:28:46.930 --> 00:28:48.077
This region is inanimate.
00:28:48.077 --> 00:28:50.660
It doesn't know whether you're
looking at in polar coordinates
00:28:50.660 --> 00:28:52.470
or in Cartesian coordinates.
00:28:52.470 --> 00:28:55.950
In Cartesian coordinates, it's
bounded above by the curve
00:28:55.950 --> 00:28:59.850
y equals f_1 of x, and it's
bounded below by the curve
00:28:59.850 --> 00:29:02.139
y equals f_2 of x.
00:29:02.139 --> 00:29:03.930
From what we studied
in the first semester,
00:29:03.930 --> 00:29:07.410
the area of the region r
in Cartesian coordinates
00:29:07.410 --> 00:29:14.140
is the integral from 0 to A,
f_1 of x minus f_2 of x, dx.
00:29:14.140 --> 00:29:17.960
On the other hand, if I call
this curve r equals g of theta,
00:29:17.960 --> 00:29:21.050
in terms of polar coordinates,
and the initial angle is
00:29:21.050 --> 00:29:24.520
theta_1, and the terminal
angle here is theta_2,
00:29:24.520 --> 00:29:27.380
then the area of
that same region r
00:29:27.380 --> 00:29:32.180
is given to be 1/2 integral from
theta_1 to theta_2, g of theta
00:29:32.180 --> 00:29:34.610
squared d theta.
00:29:34.610 --> 00:29:37.610
Mathematically, this
integral in terms
00:29:37.610 --> 00:29:40.850
of x and this integral
in terms of theta
00:29:40.850 --> 00:29:43.340
look completely different.
00:29:43.340 --> 00:29:45.790
But the crucial point
is they are simply
00:29:45.790 --> 00:29:49.310
different expressions
for the same answer.
00:29:49.310 --> 00:29:51.700
Which of the two is
the better one to use?
00:29:51.700 --> 00:29:54.030
It depends on the
particular problem.
00:29:54.030 --> 00:29:56.980
If, for example,
a problem begs for
00:29:56.980 --> 00:29:59.070
a polar coordinates
interpretation,
00:29:59.070 --> 00:30:00.840
use polar coordinates.
00:30:00.840 --> 00:30:04.310
In fact, polar coordinates
and Cartesian coordinates
00:30:04.310 --> 00:30:06.560
can both be bad
sets of equations.
00:30:06.560 --> 00:30:07.570
Who knows?
00:30:07.570 --> 00:30:08.890
It's not the point.
00:30:08.890 --> 00:30:11.610
The point is that we now
have two coordinate systems.
00:30:11.610 --> 00:30:14.450
There are others that we will
define as the course goes on.
00:30:14.450 --> 00:30:16.710
But the important point
is that we are now
00:30:16.710 --> 00:30:21.870
ready to tackle motion in the
plane for central force fields
00:30:21.870 --> 00:30:23.720
if we so desire.
00:30:23.720 --> 00:30:26.410
At this particular
moment, I do so desire.
00:30:26.410 --> 00:30:28.230
So the chances are
that next time,
00:30:28.230 --> 00:30:32.310
we will be talking about
velocity and acceleration
00:30:32.310 --> 00:30:34.730
vectors in polar coordinates.
00:30:34.730 --> 00:30:39.160
At any rate, until
next time, good bye.
00:30:39.160 --> 00:30:41.540
Funding for the
publication of this video
00:30:41.540 --> 00:30:46.410
was provided by the Gabriella
and Paul Rosenbaum Foundation.
00:30:46.410 --> 00:30:50.590
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00:30:50.590 --> 00:30:58.290
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