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PROFESSOR: Today we're going
to continue our discussion
9
00:00:25,000 --> 00:00:26,780
of parametric curves.
10
00:00:26,780 --> 00:00:29,960
I have to tell you
about arc length.
11
00:00:29,960 --> 00:00:33,590
And let me remind me where
we left off last time.
12
00:00:33,590 --> 00:00:45,820
This is parametric
curves, continued.
13
00:00:45,820 --> 00:00:50,380
Last time, we talked about
the parametric representation
14
00:00:50,380 --> 00:00:51,730
for the circle.
15
00:00:51,730 --> 00:00:55,960
Or one of the parametric
representations for the circle.
16
00:00:55,960 --> 00:00:59,210
Which was this one here.
17
00:00:59,210 --> 00:01:05,590
And first we noted that
this does parameterize,
18
00:01:05,590 --> 00:01:07,770
as we say, the circle.
19
00:01:07,770 --> 00:01:10,390
That satisfies the
equation for the circle.
20
00:01:10,390 --> 00:01:17,990
And it's traced
counterclockwise.
21
00:01:17,990 --> 00:01:20,590
The picture looks like this.
22
00:01:20,590 --> 00:01:22,350
Here's the circle.
23
00:01:22,350 --> 00:01:25,400
And it starts out here
at t = 0 and it gets up
24
00:01:25,400 --> 00:01:31,820
to here at time t = pi / 2.
25
00:01:31,820 --> 00:01:41,330
So now I have to talk
to you about arc length.
26
00:01:41,330 --> 00:01:43,840
In this parametric form.
27
00:01:43,840 --> 00:01:46,170
And the results should
be the same as arc length
28
00:01:46,170 --> 00:01:48,830
around this circle ordinarily.
29
00:01:48,830 --> 00:01:55,840
And we start out with this
basic differential relationship.
30
00:01:55,840 --> 00:02:00,080
ds^2 is dx^2 + dy^2.
31
00:02:00,080 --> 00:02:04,880
And then I'm going to take
the square root, divide by dt,
32
00:02:04,880 --> 00:02:08,540
so the rate of change
with respect to t of s
33
00:02:08,540 --> 00:02:10,780
is going to be the square root.
34
00:02:10,780 --> 00:02:13,220
Well, maybe I'll write
it without dividing.
35
00:02:13,220 --> 00:02:15,130
Just write it as ds.
36
00:02:15,130 --> 00:02:24,640
So this would be
(dx/dt)^2 + (dy/dt)^2, dt.
37
00:02:24,640 --> 00:02:27,050
So this is what you get
formally from this equation.
38
00:02:27,050 --> 00:02:28,920
If you take its
square roots and you
39
00:02:28,920 --> 00:02:32,470
divide by dt squared
in the-- inside
40
00:02:32,470 --> 00:02:35,510
the square root, and you
multiply by dt outside,
41
00:02:35,510 --> 00:02:36,630
so that those cancel.
42
00:02:36,630 --> 00:02:39,220
And this is the formal
connection between the two.
43
00:02:39,220 --> 00:02:41,600
We'll be saying just
a few more words
44
00:02:41,600 --> 00:02:48,430
in a few minutes about how to
make sense of that rigorously.
45
00:02:48,430 --> 00:02:55,020
Alright so that's the set of
formulas for the infinitesimal,
46
00:02:55,020 --> 00:02:57,110
the differential of arc length.
47
00:02:57,110 --> 00:03:00,820
And so to figure it out, I have
to differentiate x with respect
48
00:03:00,820 --> 00:03:02,810
to t.
49
00:03:02,810 --> 00:03:04,225
And remember x is up here.
50
00:03:04,225 --> 00:03:11,330
It's defined by a cos t, so
its derivative is -a sin t.
51
00:03:11,330 --> 00:03:19,850
And similarly, dy/dt = a cos t.
52
00:03:19,850 --> 00:03:22,000
And so I can plug this in.
53
00:03:22,000 --> 00:03:23,620
And I get the arc
length element,
54
00:03:23,620 --> 00:03:36,920
which is the square root of
(-a sin t)^2 + (a cos t)^2, dt.
55
00:03:36,920 --> 00:03:44,670
Which just becomes the square
root of a^2, dt, or a dt.
56
00:03:44,670 --> 00:03:46,370
Now, I was about to divide by t.
57
00:03:46,370 --> 00:03:48,630
Let me do that now.
58
00:03:48,630 --> 00:03:52,050
We can also write the rate
of change of arc length
59
00:03:52,050 --> 00:03:53,430
with respect to t.
60
00:03:53,430 --> 00:03:55,640
And that's a, in this case.
61
00:03:55,640 --> 00:04:01,420
And this gets
interpreted as the speed
62
00:04:01,420 --> 00:04:03,180
of the particle going around.
63
00:04:03,180 --> 00:04:07,270
So not only, let me
trade these two guys,
64
00:04:07,270 --> 00:04:14,020
not only do we have the
direction is counterclockwise,
65
00:04:14,020 --> 00:04:20,010
but we also have that the speed
is, if you like, it's uniform.
66
00:04:20,010 --> 00:04:21,740
It's constant speed.
67
00:04:21,740 --> 00:04:23,650
And the rate is a.
68
00:04:23,650 --> 00:04:26,810
So that's ds/dt.
69
00:04:26,810 --> 00:04:30,840
Travelling around.
70
00:04:30,840 --> 00:04:34,310
And that means that we can
play around with the speed.
71
00:04:34,310 --> 00:04:37,650
And I just want to point out--
So the standard thing, what
72
00:04:37,650 --> 00:04:39,430
you'll have to get
used to, and this
73
00:04:39,430 --> 00:04:42,520
is a standard presentation,
you'll see this everywhere.
74
00:04:42,520 --> 00:04:45,710
In your physics classes and
your other math classes,
75
00:04:45,710 --> 00:04:51,230
if you want to change
the speed, so a new speed
76
00:04:51,230 --> 00:05:01,440
going around this would be, if
I set up the equations this way.
77
00:05:01,440 --> 00:05:05,650
Now I'm tracing around
the same circle.
78
00:05:05,650 --> 00:05:08,120
But the speed is
going to turn out
79
00:05:08,120 --> 00:05:11,470
to be, if you figure
it out, there'll
80
00:05:11,470 --> 00:05:13,200
be an extra factor of k.
81
00:05:13,200 --> 00:05:16,460
So it'll be ak.
82
00:05:16,460 --> 00:05:19,540
That's what we'll work
out to be the speed.
83
00:05:19,540 --> 00:05:22,290
Provided k is positive
and a is positive.
84
00:05:22,290 --> 00:05:30,490
So we're making
these conventions.
85
00:05:30,490 --> 00:05:37,180
The constants that we're
using are positive.
86
00:05:37,180 --> 00:05:40,980
Now, that's the first
and most basic example.
87
00:05:40,980 --> 00:05:42,730
The one that comes
up constantly.
88
00:05:42,730 --> 00:05:46,120
Now, let me just make those
comments about notation
89
00:05:46,120 --> 00:05:47,860
that I wanted to make.
90
00:05:47,860 --> 00:05:51,884
And we've been treating these
squared differentials here
91
00:05:51,884 --> 00:05:53,300
for a little while
and I just want
92
00:05:53,300 --> 00:05:54,810
to pay attention
a little bit more
93
00:05:54,810 --> 00:05:57,280
carefully to these
manipulations.
94
00:05:57,280 --> 00:05:59,660
And what's allowed
and what's not.
95
00:05:59,660 --> 00:06:01,890
And what's justified
and what's not.
96
00:06:01,890 --> 00:06:06,680
So the basis for this was this
approximate calculation that we
97
00:06:06,680 --> 00:06:11,440
had, that (delta s)^2 was
(delta x)^2 + (delta y)^2.
98
00:06:11,440 --> 00:06:16,370
This is how we justified the
arc length formula before.
99
00:06:16,370 --> 00:06:19,530
And let me just show you
that the formula that I
100
00:06:19,530 --> 00:06:22,410
have up here, this
basic formula for arc
101
00:06:22,410 --> 00:06:24,760
length in the
parametric form, follows
102
00:06:24,760 --> 00:06:26,520
just as the other one did.
103
00:06:26,520 --> 00:06:31,370
And now I'm going to do it
slightly more rigorously.
104
00:06:31,370 --> 00:06:34,400
I do the division really
in disguise before I take
105
00:06:34,400 --> 00:06:36,310
the limit of the infinitesimal.
106
00:06:36,310 --> 00:06:40,349
So all I'm really doing
is I'm doing this.
107
00:06:40,349 --> 00:06:42,390
Dividing through by this,
and sorry this is still
108
00:06:42,390 --> 00:06:43,340
approximately equal.
109
00:06:43,340 --> 00:06:46,760
So I'm not dividing by something
that's 0 or infinitesimal.
110
00:06:46,760 --> 00:06:49,320
I'm dividing by
something nonzero.
111
00:06:49,320 --> 00:06:54,120
And here I have ((delta
x)/(delta t))^2 + ((delta
112
00:06:54,120 --> 00:06:58,560
y)/(delta t))^2 And
then in the limit,
113
00:06:58,560 --> 00:07:04,170
I have ds/dt is equal to
the square root of this guy.
114
00:07:04,170 --> 00:07:13,820
Or, if you like, the
square of it, so.
115
00:07:13,820 --> 00:07:17,200
So it's legal to divide by
something that's almost 0
116
00:07:17,200 --> 00:07:20,340
and then take the
limit as we go to 0.
117
00:07:20,340 --> 00:07:22,280
This is really what
derivatives are all about.
118
00:07:22,280 --> 00:07:24,850
That we get a limit here.
119
00:07:24,850 --> 00:07:27,010
As the denominator goes to 0.
120
00:07:27,010 --> 00:07:31,510
Because the numerator's
going to 0 too.
121
00:07:31,510 --> 00:07:32,770
So that's the notation.
122
00:07:32,770 --> 00:07:38,060
And now I want to warn you,
maybe just a little bit,
123
00:07:38,060 --> 00:07:42,420
about misuses, if you
like, of the notation.
124
00:07:42,420 --> 00:07:45,820
We don't do absolutely
everything this way.
125
00:07:45,820 --> 00:07:49,280
This expression that
came up with the squares,
126
00:07:49,280 --> 00:07:55,130
you should never
write it as this.
127
00:07:55,130 --> 00:08:01,600
This, put it on the board
but very quickly, never.
128
00:08:01,600 --> 00:08:02,420
OK.
129
00:08:02,420 --> 00:08:07,130
Don't do that.
130
00:08:07,130 --> 00:08:08,700
We use these square
differentials,
131
00:08:08,700 --> 00:08:12,840
but we don't do it
with these ratios here.
132
00:08:12,840 --> 00:08:15,900
But there was another place
which is slightly confusing.
133
00:08:15,900 --> 00:08:17,680
It looks very
similar, where we did
134
00:08:17,680 --> 00:08:20,247
use the square of the
differential in a denominator.
135
00:08:20,247 --> 00:08:22,580
And I just want to point out
to you that it's different.
136
00:08:22,580 --> 00:08:23,940
It's not the same.
137
00:08:23,940 --> 00:08:25,810
And it is OK.
138
00:08:25,810 --> 00:08:31,660
And that was this one.
139
00:08:31,660 --> 00:08:33,970
This thing here.
140
00:08:33,970 --> 00:08:36,900
This is a second derivative,
it's something else.
141
00:08:36,900 --> 00:08:39,300
And it's got a dt^2
in the denominator.
142
00:08:39,300 --> 00:08:41,230
So it looks rather similar.
143
00:08:41,230 --> 00:08:49,230
But what this represents is
the quantity d/dt squared.
144
00:08:49,230 --> 00:08:51,840
And you can see the
squares came in.
145
00:08:51,840 --> 00:08:53,790
And squared the two expressions.
146
00:08:53,790 --> 00:08:58,970
And then there's
also an x over here.
147
00:08:58,970 --> 00:09:00,600
So that's legal.
148
00:09:00,600 --> 00:09:02,550
Those are notations
that we do use.
149
00:09:02,550 --> 00:09:04,070
And we can even calculate this.
150
00:09:04,070 --> 00:09:05,650
It has a perfectly good meaning.
151
00:09:05,650 --> 00:09:07,540
It's the same as the
derivative with respect
152
00:09:07,540 --> 00:09:10,870
to t of the derivative of x,
which we already know was minus
153
00:09:10,870 --> 00:09:17,530
sine-- sorry, a sin t, I guess.
154
00:09:17,530 --> 00:09:21,120
Not this example,
but the previous one.
155
00:09:21,120 --> 00:09:21,870
Up here.
156
00:09:21,870 --> 00:09:24,340
So the derivative
is this and so I can
157
00:09:24,340 --> 00:09:26,140
differentiate a second time.
158
00:09:26,140 --> 00:09:29,850
And I get -a cos t.
159
00:09:29,850 --> 00:09:31,760
So that's a perfectly
legal operation.
160
00:09:31,760 --> 00:09:33,440
Everything in there makes sense.
161
00:09:33,440 --> 00:09:39,880
Just don't use that.
162
00:09:39,880 --> 00:09:41,770
There's another really
unfortunate thing,
163
00:09:41,770 --> 00:09:45,820
right which is that the 2 creeps
in funny places with sines.
164
00:09:45,820 --> 00:09:48,080
You have sine squared.
165
00:09:48,080 --> 00:09:50,150
It would be out here,
it comes up here
166
00:09:50,150 --> 00:09:51,810
for some strange reason.
167
00:09:51,810 --> 00:09:54,730
This is just because
typographers are lazy
168
00:09:54,730 --> 00:09:56,880
or somebody somewhere
in the history
169
00:09:56,880 --> 00:10:00,620
of mathematical typography
decided to let the 2 migrate.
170
00:10:00,620 --> 00:10:04,650
It would be like
putting the 2 over here.
171
00:10:04,650 --> 00:10:07,940
There's inconsistency
in mathematics, right.
172
00:10:07,940 --> 00:10:11,570
We're not perfect and people
just develop these notations.
173
00:10:11,570 --> 00:10:14,450
So we have to live with them.
174
00:10:14,450 --> 00:10:20,210
The ones that people
accept as conventions.
175
00:10:20,210 --> 00:10:23,230
The next example that
I want to give you
176
00:10:23,230 --> 00:10:24,920
is just slightly different.
177
00:10:24,920 --> 00:10:29,140
It'll be a non-constant
speed parameterization.
178
00:10:29,140 --> 00:10:32,470
Here x = 2 sin t.
179
00:10:32,470 --> 00:10:37,590
And y is, say, cos t.
180
00:10:37,590 --> 00:10:40,450
And let's keep track
of what this one does.
181
00:10:40,450 --> 00:10:43,605
Now, this is a skill
which I'm going
182
00:10:43,605 --> 00:10:45,170
to ask you about quite a bit.
183
00:10:45,170 --> 00:10:46,690
And it's one of several skills.
184
00:10:46,690 --> 00:10:48,970
You'll have to connect
this with some kind
185
00:10:48,970 --> 00:10:50,330
of rectangular equation.
186
00:10:50,330 --> 00:10:51,725
An equation for x and y.
187
00:10:51,725 --> 00:10:54,230
And we'll be doing a certain
amount of this today.
188
00:10:54,230 --> 00:10:56,240
In another context.
189
00:10:56,240 --> 00:11:00,510
Right here, to see the pattern,
we know that the relationship
190
00:11:00,510 --> 00:11:04,160
we're going to want to use
is that sin^2 + cos^2 = 1.
191
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.
192
00:11:11,910 --> 00:11:17,020
And that's going to turn
out to be sin^2 t + cos^2 t.
193
00:11:17,020 --> 00:11:18,310
Which is 1.
194
00:11:18,310 --> 00:11:19,500
So there's the equation.
195
00:11:19,500 --> 00:11:24,690
Here's the rectangular equation
for this parametric curve.
196
00:11:24,690 --> 00:11:32,030
And this describes an ellipse.
197
00:11:32,030 --> 00:11:35,570
That's not the only information
that we can get here.
198
00:11:35,570 --> 00:11:37,180
The other information
that we can get
199
00:11:37,180 --> 00:11:39,570
is this qualitative
information of where
200
00:11:39,570 --> 00:11:42,360
we start, where we're
going, the direction.
201
00:11:42,360 --> 00:11:46,540
It starts out, I
claim, at t = 0.
202
00:11:46,540 --> 00:11:54,630
That's when t = 0, this is
(2 sin 0, cos 0), right?
203
00:11:54,630 --> 00:12:00,330
(2 sin 0, cos 0) is equal
to the point (0, 1).
204
00:12:00,330 --> 00:12:02,360
So it starts up up here.
205
00:12:02,360 --> 00:12:05,140
At (0, 1).
206
00:12:05,140 --> 00:12:08,520
And then the next little
place, so this is one thing
207
00:12:08,520 --> 00:12:11,400
that certainly you
want to do. t = pi/2
208
00:12:11,400 --> 00:12:14,510
is maybe the next
easy point to plot.
209
00:12:14,510 --> 00:12:22,830
And that's going to be
(2 sin(pi/2), cos(pi/2)).
210
00:12:22,830 --> 00:12:27,880
And that's just (2, 0).
211
00:12:27,880 --> 00:12:31,490
And so that's over
here somewhere.
212
00:12:31,490 --> 00:12:34,422
This is (2, 0).
213
00:12:34,422 --> 00:12:36,130
And we know it travels
along the ellipse.
214
00:12:36,130 --> 00:12:40,120
And we know the minor axis is
1, and the major axis is 2,
215
00:12:40,120 --> 00:12:43,000
so it's doing this.
216
00:12:43,000 --> 00:12:45,090
So this is what
happens at t = 0.
217
00:12:45,090 --> 00:12:48,390
This is where we
are at t = pi/2.
218
00:12:48,390 --> 00:12:51,510
And it continues all
the way around, etc.
219
00:12:51,510 --> 00:12:53,370
To the rest of the ellipse.
220
00:12:53,370 --> 00:12:57,750
This is the direction.
221
00:12:57,750 --> 00:13:09,620
So this one happens
to be clockwise.
222
00:13:09,620 --> 00:13:12,640
Alright, now let's keep
track of its speed.
223
00:13:12,640 --> 00:13:25,610
Let's keep track of the speed,
and also the arc length.
224
00:13:25,610 --> 00:13:32,830
So the speed is the square
root of the derivatives here.
225
00:13:32,830 --> 00:13:38,580
That would be (2 cos
t)^2 + (sin t)^2.
226
00:13:42,160 --> 00:13:48,060
And the arc length is what?
227
00:13:48,060 --> 00:13:49,840
Well, if we want to
go all the way around,
228
00:13:49,840 --> 00:13:53,540
we need to know that that
takes a total of 2 pi.
229
00:13:53,540 --> 00:13:55,990
So 0 to 2 pi.
230
00:13:55,990 --> 00:13:59,310
And then we have to integrate
ds, which is this expression,
231
00:13:59,310 --> 00:14:02,620
or ds/dt, dt.
232
00:14:02,620 --> 00:14:11,630
So that's the square root
of 4 cos^2 t + sin^2 t, dt.
233
00:14:20,820 --> 00:14:26,580
The bad news, if you
like, is that this is not
234
00:14:26,580 --> 00:14:38,524
an elementary integral.
235
00:14:38,524 --> 00:14:39,940
In other words,
no matter how long
236
00:14:39,940 --> 00:14:44,240
you try to figure out
how to antidifferentiate
237
00:14:44,240 --> 00:14:47,180
this expression, no matter how
many substitutions you try,
238
00:14:47,180 --> 00:14:50,420
you will fail.
239
00:14:50,420 --> 00:14:52,030
That's the bad news.
240
00:14:52,030 --> 00:14:58,430
The good news is this is
not an elementary integral.
241
00:14:58,430 --> 00:14:59,810
It's not an elementary integral.
242
00:14:59,810 --> 00:15:03,330
Which means that this is
the answer to a question.
243
00:15:03,330 --> 00:15:06,230
Not something that
you have to work on.
244
00:15:06,230 --> 00:15:11,680
So if somebody asks you for
this arc length, you stop here.
245
00:15:11,680 --> 00:15:14,550
That's the answer, so it's
actually better than it looks.
246
00:15:14,550 --> 00:15:17,700
And we'll try to--
I mean, I don't
247
00:15:17,700 --> 00:15:21,230
expect you to know already
what all of the integrals
248
00:15:21,230 --> 00:15:22,680
are that are impossible.
249
00:15:22,680 --> 00:15:24,680
And which ones are hard
and which ones are easy.
250
00:15:24,680 --> 00:15:27,200
So we'll try to coach
you through when
251
00:15:27,200 --> 00:15:28,390
you face these things.
252
00:15:28,390 --> 00:15:31,670
It's not so easy to decide.
253
00:15:31,670 --> 00:15:34,345
I'll give you a few clues, but.
254
00:15:34,345 --> 00:15:34,845
OK.
255
00:15:34,845 --> 00:15:38,310
So this is the arc length.
256
00:15:38,310 --> 00:15:42,270
Now, I want to move on to
the last thing that we did.
257
00:15:42,270 --> 00:15:44,410
Last type of thing
that we did last time.
258
00:15:44,410 --> 00:15:54,140
Which is the surface area.
259
00:15:54,140 --> 00:15:55,230
And yeah, question.
260
00:15:55,230 --> 00:16:03,609
STUDENT: [INAUDIBLE]
261
00:16:03,609 --> 00:16:05,650
PROFESSOR: The question,
this is a good question.
262
00:16:05,650 --> 00:16:08,290
The question is, when
you draw the ellipse,
263
00:16:08,290 --> 00:16:11,650
do you not take into
account what t is.
264
00:16:11,650 --> 00:16:16,460
The answer is that
this is in disguise.
265
00:16:16,460 --> 00:16:20,360
What's going on here
is we have a trouble
266
00:16:20,360 --> 00:16:24,720
with plotting in the plane
what's really happening.
267
00:16:24,720 --> 00:16:29,210
So in other words, it's
kind of in trouble.
268
00:16:29,210 --> 00:16:33,940
So the point is that we have
two functions of t, not one.
269
00:16:33,940 --> 00:16:35,660
x(t) and y(t).
270
00:16:35,660 --> 00:16:38,970
So one thing that I can do if
I plot things in the plane.
271
00:16:38,970 --> 00:16:41,710
In other words, the
main point to make here
272
00:16:41,710 --> 00:16:45,150
is that we're not talking
about the situation
273
00:16:45,150 --> 00:16:46,480
y is a function of x.
274
00:16:46,480 --> 00:16:47,870
We're out of that realm now.
275
00:16:47,870 --> 00:16:49,620
We're somewhere in
a different part
276
00:16:49,620 --> 00:16:51,450
of the universe in our thought.
277
00:16:51,450 --> 00:16:54,760
And you should drop
this point of view.
278
00:16:54,760 --> 00:16:56,990
So this depiction is not
y as a function of x.
279
00:16:56,990 --> 00:17:00,940
Well, that's obvious because
there are two values here,
280
00:17:00,940 --> 00:17:01,710
as opposed to one.
281
00:17:01,710 --> 00:17:02,960
So we're in trouble with that.
282
00:17:02,960 --> 00:17:04,750
And we have that
background parameter,
283
00:17:04,750 --> 00:17:07,170
and that's exactly
why we're using it.
284
00:17:07,170 --> 00:17:08,230
This parameter t.
285
00:17:08,230 --> 00:17:10,430
So that we can depict
the entire curve.
286
00:17:10,430 --> 00:17:14,200
And deal with it as one thing.
287
00:17:14,200 --> 00:17:17,590
So since I can't really draw
it, and since t is nowhere
288
00:17:17,590 --> 00:17:19,840
on the map, you should
sort of imagine it as time,
289
00:17:19,840 --> 00:17:21,760
and there's some kind of
trajectory which is travelling
290
00:17:21,760 --> 00:17:22,260
around.
291
00:17:22,260 --> 00:17:25,590
And then I just labelled
a couple of the places.
292
00:17:25,590 --> 00:17:28,370
If somebody asked you to
draw a picture of this,
293
00:17:28,370 --> 00:17:31,230
well, I'll tell you exactly
where you need the picture
294
00:17:31,230 --> 00:17:33,250
in just one second, alright.
295
00:17:33,250 --> 00:17:36,690
It's going to come up
right now in surface area.
296
00:17:36,690 --> 00:17:39,450
But otherwise, if
nobody asks you to,
297
00:17:39,450 --> 00:17:44,020
you don't even have to put
down t = 0 and t = pi / 2 here.
298
00:17:44,020 --> 00:17:46,402
Because nobody
demanded it of you.
299
00:17:46,402 --> 00:17:47,110
Another question.
300
00:17:47,110 --> 00:17:51,842
STUDENT: [INAUDIBLE]
301
00:17:51,842 --> 00:17:53,550
PROFESSOR: So, another
very good question
302
00:17:53,550 --> 00:17:55,890
which is exactly
connected to this picture.
303
00:17:55,890 --> 00:17:58,060
So how is it that we're
going to use the picture,
304
00:17:58,060 --> 00:18:02,190
and how is it we're going
to use the notion of the t.
305
00:18:02,190 --> 00:18:07,040
The question was, why is
this from t = 0 to t = 2 pi?
306
00:18:07,040 --> 00:18:11,000
That does use the t
information on this diagram.
307
00:18:11,000 --> 00:18:13,150
the point is, we do
know that t starts here.
308
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.
309
00:18:17,620 --> 00:18:19,380
When you go all the
way around once,
310
00:18:19,380 --> 00:18:21,240
it's going to come
back to itself.
311
00:18:21,240 --> 00:18:23,720
These are periodic
functions of period 2 pi.
312
00:18:23,720 --> 00:18:26,780
And they come back to
themselves exactly at 2 pi.
313
00:18:26,780 --> 00:18:29,030
And so that's why we know
in order to get around once,
314
00:18:29,030 --> 00:18:32,250
we need to go from 0 to 2 pi.
315
00:18:32,250 --> 00:18:34,640
And the same thing is going
to come up with surface area
316
00:18:34,640 --> 00:18:35,370
right now.
317
00:18:35,370 --> 00:18:38,720
That's going to be the
issue, is what range of t
318
00:18:38,720 --> 00:18:45,620
we're going to need when we
compute the surface area.
319
00:18:45,620 --> 00:18:52,144
STUDENT: [INAUDIBLE]
320
00:18:52,144 --> 00:18:54,185
PROFESSOR: In a question,
what you might be asked
321
00:18:54,185 --> 00:18:56,180
is what's the
rectangular equation
322
00:18:56,180 --> 00:18:57,650
for a parametric curve?
323
00:18:57,650 --> 00:19:01,720
So that would be
1/4 x^2 + y^2 = 1.
324
00:19:01,720 --> 00:19:03,430
And then you might
be asked, plot it.
325
00:19:03,430 --> 00:19:06,960
Well, that would be a
picture of the ellipse.
326
00:19:06,960 --> 00:19:10,380
OK, those are types of questions
that are legal questions.
327
00:19:10,380 --> 00:19:27,439
STUDENT: [INAUDIBLE]
328
00:19:27,439 --> 00:19:28,980
PROFESSOR: The
question is, do I need
329
00:19:28,980 --> 00:19:30,600
to know any specific formulas?
330
00:19:30,600 --> 00:19:33,250
Any formulas that you know
and remember will help you.
331
00:19:33,250 --> 00:19:35,400
They may be of limited use.
332
00:19:35,400 --> 00:19:37,640
I'm not going to ask
you to memorize anything
333
00:19:37,640 --> 00:19:40,820
except, I guarantee you that
the circle is going to come up.
334
00:19:40,820 --> 00:19:43,500
Not the ellipse, the circle
will come up everywhere
335
00:19:43,500 --> 00:19:44,320
in your life.
336
00:19:44,320 --> 00:19:47,710
So at least at MIT,
your life at MIT.
337
00:19:47,710 --> 00:19:52,002
We're very round here.
338
00:19:52,002 --> 00:19:52,960
Yeah, another question.
339
00:19:52,960 --> 00:19:56,811
STUDENT: I'm just a tiny bit
confused back to the basics.
340
00:19:56,811 --> 00:19:58,810
This is more a question
from yesterday, I guess.
341
00:19:58,810 --> 00:20:04,390
But when you have your
original ds^2 = dx^2 + dy^2,
342
00:20:04,390 --> 00:20:10,060
and then you integrate that to
get arc length, how are you,
343
00:20:10,060 --> 00:20:14,360
the integral has dx's and dy's.
344
00:20:14,360 --> 00:20:18,590
So how are you just
integrating with respect to dx?
345
00:20:18,590 --> 00:20:22,560
PROFESSOR: OK, the question
is how are we just integrating
346
00:20:22,560 --> 00:20:24,200
with respect to x?
347
00:20:24,200 --> 00:20:26,880
So this is a question which
goes back to last time.
348
00:20:26,880 --> 00:20:29,120
And what is it with arc length.
349
00:20:29,120 --> 00:20:30,370
So.
350
00:20:30,370 --> 00:20:35,442
I'm going to have to answer
that question in connection
351
00:20:35,442 --> 00:20:36,400
with what we did today.
352
00:20:36,400 --> 00:20:38,420
So this is a subtle question.
353
00:20:38,420 --> 00:20:42,450
But I want you to realize
that this is actually
354
00:20:42,450 --> 00:20:44,290
an important
conceptual step here.
355
00:20:44,290 --> 00:20:49,810
So shhh, everybody, listen.
356
00:20:49,810 --> 00:20:53,340
If you're representing
one-dimensional objects,
357
00:20:53,340 --> 00:20:56,050
which are curves,
maybe, in space.
358
00:20:56,050 --> 00:20:58,270
Or in two dimensions.
359
00:20:58,270 --> 00:21:00,759
When you're keeping
track of arc length,
360
00:21:00,759 --> 00:21:02,800
you're going to have to
have an integral which is
361
00:21:02,800 --> 00:21:05,180
with respect to some variable.
362
00:21:05,180 --> 00:21:08,510
But that variable,
you get to pick.
363
00:21:08,510 --> 00:21:12,310
And we're launching now
into this variety of choices
364
00:21:12,310 --> 00:21:13,950
of variables with
respect to which you
365
00:21:13,950 --> 00:21:15,980
can represent something.
366
00:21:15,980 --> 00:21:17,580
Now, there are
some disadvantages
367
00:21:17,580 --> 00:21:19,370
on the circle to
representing things
368
00:21:19,370 --> 00:21:21,480
with respect to the variable x.
369
00:21:21,480 --> 00:21:24,732
Because there are two
points on the circle here.
370
00:21:24,732 --> 00:21:26,190
On the other hand,
you actually can
371
00:21:26,190 --> 00:21:27,480
succeed with half the circle.
372
00:21:27,480 --> 00:21:29,620
So you can figure out
the arc length that way.
373
00:21:29,620 --> 00:21:32,521
And then you can set it
up as an integral dx.
374
00:21:32,521 --> 00:21:34,770
But you can also set it up
as an integral with respect
375
00:21:34,770 --> 00:21:37,240
to any parameter you want.
376
00:21:37,240 --> 00:21:40,170
And the uniform parameter
is perhaps the easiest one.
377
00:21:40,170 --> 00:21:43,210
This one is perhaps
the easiest one.
378
00:21:43,210 --> 00:21:47,970
And so now the thing that's
strange about this perspective
379
00:21:47,970 --> 00:21:51,470
- and I'm going to make this
point later in the lecture
380
00:21:51,470 --> 00:21:55,810
as well - is that the
letters x and y-- As I say,
381
00:21:55,810 --> 00:22:00,630
you should drop this notion
that y is a function of x.
382
00:22:00,630 --> 00:22:03,950
This is what we're throwing
away at this point.
383
00:22:03,950 --> 00:22:05,770
What we're thinking
of is, you can
384
00:22:05,770 --> 00:22:08,110
describe things in terms
of any coordinate you want.
385
00:22:08,110 --> 00:22:11,340
You just have to say what each
one is in terms of the others.
386
00:22:11,340 --> 00:22:15,300
And these x and y
over here are where
387
00:22:15,300 --> 00:22:18,380
we are in the Cartesian
coordinate system.
388
00:22:18,380 --> 00:22:20,490
They're not-- And
in this case they're
389
00:22:20,490 --> 00:22:24,610
functions of some
other variable.
390
00:22:24,610 --> 00:22:25,720
Some other variable.
391
00:22:25,720 --> 00:22:27,150
So they're each functions.
392
00:22:27,150 --> 00:22:29,480
So the letters x and
y just changed on you.
393
00:22:29,480 --> 00:22:33,710
They mean something different.
x is no longer the variable.
394
00:22:33,710 --> 00:22:36,810
It's the function.
395
00:22:36,810 --> 00:22:38,542
Right?
396
00:22:38,542 --> 00:22:40,250
You're going to have
to get used to that.
397
00:22:40,250 --> 00:22:42,380
That's because we
run out of letters.
398
00:22:42,380 --> 00:22:44,870
And we kind of want to use
all of them the way we want.
399
00:22:44,870 --> 00:22:48,290
I'll say some more
about that later.
400
00:22:48,290 --> 00:22:51,220
So now I want to do this
surface area example.
401
00:22:51,220 --> 00:22:59,150
I'm going to just take the
surface area of the ellipsoid.
402
00:22:59,150 --> 00:23:11,340
The surface of the
ellipsoid formed
403
00:23:11,340 --> 00:23:19,910
by revolving this previous
example, which was Example 2.
404
00:23:19,910 --> 00:23:28,020
Around the y-axis.
405
00:23:28,020 --> 00:23:30,410
So we want to set up that
surface area integral here
406
00:23:30,410 --> 00:23:32,490
for you.
407
00:23:32,490 --> 00:23:38,160
Now, I remind you that the
area element looks like this.
408
00:23:38,160 --> 00:23:41,190
If you're revolving
around the y-axis,
409
00:23:41,190 --> 00:23:42,815
that means you're
going around this way
410
00:23:42,815 --> 00:23:43,720
and you have some curve.
411
00:23:43,720 --> 00:23:44,990
In this case it's this
piece of an ellipse.
412
00:23:44,990 --> 00:23:46,475
If you sweep it
around you're going
413
00:23:46,475 --> 00:23:48,770
to get what's
called an ellipsoid.
414
00:23:48,770 --> 00:23:53,890
And there's a little chunk here,
that you're wrapping around.
415
00:23:53,890 --> 00:23:58,430
And the important thing you
need besides this ds, this arc
416
00:23:58,430 --> 00:24:04,120
length piece over here, is
the distance to the axis.
417
00:24:04,120 --> 00:24:06,320
So that's this
horizontal distance here.
418
00:24:06,320 --> 00:24:09,850
I'll draw it in another color.
419
00:24:09,850 --> 00:24:15,520
And that horizontal
distance now has a name.
420
00:24:15,520 --> 00:24:18,670
And this is, again, the virtue
of this coordinate system.
421
00:24:18,670 --> 00:24:20,170
The t is something else.
422
00:24:20,170 --> 00:24:21,020
This has a name.
423
00:24:21,020 --> 00:24:22,760
This distance has a name.
424
00:24:22,760 --> 00:24:27,080
This distance is called x.
425
00:24:27,080 --> 00:24:29,570
And it even has a formula.
426
00:24:29,570 --> 00:24:36,090
Its formula is 2 sin t.
427
00:24:36,090 --> 00:24:38,550
In terms of t.
428
00:24:38,550 --> 00:24:41,530
So the full formula
up for the integral
429
00:24:41,530 --> 00:24:46,039
here is, I have to take
the circumference when
430
00:24:46,039 --> 00:24:47,080
I spin this thing around.
431
00:24:47,080 --> 00:24:48,950
And this little
arc length element.
432
00:24:48,950 --> 00:24:53,660
So I have here 2
pi times 2 sin t.
433
00:24:53,660 --> 00:24:55,640
That's the x variable here.
434
00:24:55,640 --> 00:25:00,560
And then I have here ds,
which is kind of a mess.
435
00:25:00,560 --> 00:25:04,170
So unfortunately I don't
quite have room for it.
436
00:25:04,170 --> 00:25:05,650
Plan ahead.
437
00:25:05,650 --> 00:25:15,200
Square root of 4 cos^2 t + sin^2
t, is that what it was, dt.
438
00:25:15,200 --> 00:25:17,740
Alright, I guess I
squeezed it in there.
439
00:25:17,740 --> 00:25:20,090
So that was the arc
length, which I re-copied
440
00:25:20,090 --> 00:25:21,620
from this board above.
441
00:25:21,620 --> 00:25:24,310
That was the ds piece.
442
00:25:24,310 --> 00:25:29,760
It's this whole thing
including the dt.
443
00:25:29,760 --> 00:25:32,360
That's the answer
except for one thing.
444
00:25:32,360 --> 00:25:33,590
What else do we need?
445
00:25:33,590 --> 00:25:35,350
We don't just need
the integrand,
446
00:25:35,350 --> 00:25:37,720
this is half of
setting up an integral.
447
00:25:37,720 --> 00:25:40,990
The other half of setting up
an integral is the limits.
448
00:25:40,990 --> 00:25:42,840
We need specific limits here.
449
00:25:42,840 --> 00:25:46,760
Otherwise we don't have a
number that we can get out.
450
00:25:46,760 --> 00:25:50,370
So we now have to think
about what the limits are.
451
00:25:50,370 --> 00:25:52,550
And maybe somebody can see.
452
00:25:52,550 --> 00:25:54,429
It has something to
do with this diagram
453
00:25:54,429 --> 00:25:55,470
of the ellipse over here.
454
00:25:55,470 --> 00:25:58,520
Can somebody guess what it is?
455
00:25:58,520 --> 00:25:59,480
0 to pi.
456
00:25:59,480 --> 00:26:02,070
Well, that was quick.
457
00:26:02,070 --> 00:26:02,620
That's it.
458
00:26:02,620 --> 00:26:04,709
Because we go from
the top to the bottom,
459
00:26:04,709 --> 00:26:06,250
but we don't want
to continue around.
460
00:26:06,250 --> 00:26:07,640
We don't want to
go from 0 to 2 pi,
461
00:26:07,640 --> 00:26:09,723
because that would be
duplicating what we're going
462
00:26:09,723 --> 00:26:12,020
to get when we spin around.
463
00:26:12,020 --> 00:26:13,730
And we know that we start at 0.
464
00:26:13,730 --> 00:26:15,540
It's interesting
because it descends
465
00:26:15,540 --> 00:26:17,090
when you change
variables to think
466
00:26:17,090 --> 00:26:20,360
of it in terms of the y variable
it's going the opposite way.
467
00:26:20,360 --> 00:26:24,550
But anyway, just one piece
of this is what we want.
468
00:26:24,550 --> 00:26:27,660
So that's this setup.
469
00:26:27,660 --> 00:26:36,230
And now I claim that this is
actually a doable integral.
470
00:26:36,230 --> 00:26:37,850
However, it's long.
471
00:26:37,850 --> 00:26:39,830
I'm going to spare
you, I'll just tell you
472
00:26:39,830 --> 00:26:41,330
how you would get started.
473
00:26:41,330 --> 00:26:45,970
You would use the
substitution u = cos t.
474
00:26:45,970 --> 00:26:53,620
And then the du is
going to be -sin t dt.
475
00:26:53,620 --> 00:26:56,290
But then, unfortunately,
there's a lot more.
476
00:26:56,290 --> 00:26:57,810
There's another
trig substitution
477
00:26:57,810 --> 00:27:01,430
with some other multiple
of the cosine and so forth.
478
00:27:01,430 --> 00:27:02,420
So it goes on and on.
479
00:27:02,420 --> 00:27:06,260
If you want to check
it yourself, you can.
480
00:27:06,260 --> 00:27:08,710
There's an inverse
trig substitution which
481
00:27:08,710 --> 00:27:11,590
isn't compatible with this one.
482
00:27:11,590 --> 00:27:17,090
But it can be done.
483
00:27:17,090 --> 00:27:22,690
Calculated.
484
00:27:22,690 --> 00:27:26,980
In elementary terms.
485
00:27:26,980 --> 00:27:30,547
Yeah, another question.
486
00:27:30,547 --> 00:27:31,380
STUDENT: [INAUDIBLE]
487
00:27:31,380 --> 00:27:33,490
PROFESSOR: So, if you
get this on an exam,
488
00:27:33,490 --> 00:27:35,240
I'm going to have to
coach you through it.
489
00:27:35,240 --> 00:27:37,640
Either I'm going to have to
tell you don't evaluate it
490
00:27:37,640 --> 00:27:40,130
or, you're going to have
to work really hard.
491
00:27:40,130 --> 00:27:42,500
Or here's the first step,
and then the next step
492
00:27:42,500 --> 00:27:44,299
is, keep on going.
493
00:27:44,299 --> 00:27:44,840
Or something.
494
00:27:44,840 --> 00:27:47,890
I'll have to give you some cues.
495
00:27:47,890 --> 00:27:49,260
Because it's quite long.
496
00:27:49,260 --> 00:27:52,860
This is way too long for an
exam, this particular one.
497
00:27:52,860 --> 00:27:53,650
OK.
498
00:27:53,650 --> 00:27:55,359
It's not too long
for a problem set.
499
00:27:55,359 --> 00:27:57,650
This is where I would leave
you off if I were giving it
500
00:27:57,650 --> 00:27:58,320
to you on a problem set.
501
00:27:58,320 --> 00:28:00,220
Just to give you an idea
of the order of magnitude.
502
00:28:00,220 --> 00:28:02,761
Whereas one of the ones that I
did yesterday, I wouldn't even
503
00:28:02,761 --> 00:28:11,120
give you on a problem
set, it was so long.
504
00:28:11,120 --> 00:28:17,630
So now, our next job is to
move on to polar coordinates.
505
00:28:17,630 --> 00:28:20,960
Now, polar coordinates involve
the geometry of circles.
506
00:28:20,960 --> 00:28:23,392
As I said, we really
love circles here.
507
00:28:23,392 --> 00:28:24,100
We're very round.
508
00:28:24,100 --> 00:28:28,210
Just as I love 0, the rest of
the Institute loves circles.
509
00:28:28,210 --> 00:28:47,380
So we're going to
do that right now.
510
00:28:47,380 --> 00:28:58,900
What we're going to talk about
now is polar coordinates.
511
00:28:58,900 --> 00:29:01,010
Which are set up in
the following way.
512
00:29:01,010 --> 00:29:04,640
It's a way of describing
the points in the plane.
513
00:29:04,640 --> 00:29:07,460
Here is a point in
a plane, and here's
514
00:29:07,460 --> 00:29:10,530
what we think of as
the usual x-y axes.
515
00:29:10,530 --> 00:29:12,860
And now this point is
going to be described
516
00:29:12,860 --> 00:29:15,260
by a different pair of
coordinates, different pair
517
00:29:15,260 --> 00:29:16,190
of numbers.
518
00:29:16,190 --> 00:29:26,420
Namely, the distance
to the origin.
519
00:29:26,420 --> 00:29:30,490
And the second parameter
here, second number here,
520
00:29:30,490 --> 00:29:32,550
is this angle theta.
521
00:29:32,550 --> 00:29:41,500
Which is the angle
of ray from origin
522
00:29:41,500 --> 00:29:48,670
with the horizontal axis.
523
00:29:48,670 --> 00:29:50,620
So that's what it
is in language.
524
00:29:50,620 --> 00:29:53,690
And you should put this
in quotation marks,
525
00:29:53,690 --> 00:29:57,320
because it's not
a perfect match.
526
00:29:57,320 --> 00:30:00,800
This is geometrically what
you should always think of,
527
00:30:00,800 --> 00:30:03,720
but the technical
details involve
528
00:30:03,720 --> 00:30:06,530
dealing directly with formulas.
529
00:30:06,530 --> 00:30:09,880
The first formula is
the formula for x.
530
00:30:09,880 --> 00:30:11,590
And this is the
fundamental, these two
531
00:30:11,590 --> 00:30:12,750
are the fundamental ones.
532
00:30:12,750 --> 00:30:16,120
Namely, x = r cos theta.
533
00:30:16,120 --> 00:30:17,860
The second formula
is the formula
534
00:30:17,860 --> 00:30:21,380
for y, which is r sin theta.
535
00:30:21,380 --> 00:30:25,420
So these are the
unambiguous definitions
536
00:30:25,420 --> 00:30:27,100
of polar coordinates.
537
00:30:27,100 --> 00:30:28,790
This is it.
538
00:30:28,790 --> 00:30:32,590
And this is the thing from
which all other almost correct
539
00:30:32,590 --> 00:30:37,180
statements almost follow.
540
00:30:37,180 --> 00:30:39,320
But this is the one you
should trust always.
541
00:30:39,320 --> 00:30:44,980
This is the
unambiguous statement.
542
00:30:44,980 --> 00:30:47,360
So let me give you an
example something that's
543
00:30:47,360 --> 00:30:52,040
close to being a good
formula and is certainly
544
00:30:52,040 --> 00:30:57,530
useful in its way.
545
00:30:57,530 --> 00:31:04,180
Namely, you can think of r as
being the square root of x^2 +
546
00:31:04,180 --> 00:31:05,810
y^2.
547
00:31:05,810 --> 00:31:07,336
That's easy enough
to derive, it's
548
00:31:07,336 --> 00:31:08,460
the distance to the origin.
549
00:31:08,460 --> 00:31:11,320
That's pretty obvious.
550
00:31:11,320 --> 00:31:14,690
And the formula for theta,
which you can also derive,
551
00:31:14,690 --> 00:31:17,480
which is that it's the
inverse tangent of y y/x.
552
00:31:21,050 --> 00:31:24,310
However, let me just warn
you that these formulas are
553
00:31:24,310 --> 00:31:26,870
slightly ambiguous.
554
00:31:26,870 --> 00:31:33,357
So somewhat ambiguous.
555
00:31:33,357 --> 00:31:35,440
In other words, you can't
just apply them blindly.
556
00:31:35,440 --> 00:31:37,023
You actually have
to look at a picture
557
00:31:37,023 --> 00:31:38,180
in order to get them right.
558
00:31:38,180 --> 00:31:43,690
In particular, r could
be plus or minus here.
559
00:31:43,690 --> 00:31:47,950
And when you take
the inverse tangent,
560
00:31:47,950 --> 00:31:52,510
there's an ambiguity between,
it's the same as the inverse
561
00:31:52,510 --> 00:31:56,330
tangent of (-y)/(-x).
562
00:31:56,330 --> 00:32:00,550
So these minus signs are a
plague on your existence.
563
00:32:00,550 --> 00:32:05,050
And you're not going to get a
completely unambiguous answer
564
00:32:05,050 --> 00:32:07,760
out of these formulas
without paying attention
565
00:32:07,760 --> 00:32:08,430
to the diagram.
566
00:32:08,430 --> 00:32:10,550
On the other hand, the
formula up in the box
567
00:32:10,550 --> 00:32:14,337
there always works.
568
00:32:14,337 --> 00:32:15,920
So when people mean
polar coordinates,
569
00:32:15,920 --> 00:32:17,370
they always mean that.
570
00:32:17,370 --> 00:32:22,370
And then they have conventions,
which sometimes match things up
571
00:32:22,370 --> 00:32:27,550
with the formulas over
on this next board.
572
00:32:27,550 --> 00:32:32,670
Let me give you various
examples here first.
573
00:32:32,670 --> 00:32:36,260
But maybe first I
should I should draw
574
00:32:36,260 --> 00:32:38,100
the two coordinate systems.
575
00:32:38,100 --> 00:32:40,560
So the coordinate system
that we're used to
576
00:32:40,560 --> 00:32:43,360
is the rectangular
coordinate system.
577
00:32:43,360 --> 00:32:49,190
And maybe I'll draw it
in orange and green here.
578
00:32:49,190 --> 00:32:59,430
So these are the coordinate
lines y = 0, y = 1, y = 2.
579
00:32:59,430 --> 00:33:01,950
That's how the
coordinate system works.
580
00:33:01,950 --> 00:33:08,427
And over here we have the
rest of the coordinate system.
581
00:33:08,427 --> 00:33:10,510
And this is the way we're
thinking of x and y now.
582
00:33:10,510 --> 00:33:12,570
We're no longer thinking of
y as a function of x and x
583
00:33:12,570 --> 00:33:13,986
as a function of
y, we're thinking
584
00:33:13,986 --> 00:33:16,960
of x as a label of
a place in a plane.
585
00:33:16,960 --> 00:33:20,900
And y as a label of
a place in a plane.
586
00:33:20,900 --> 00:33:27,770
So here we have x =
0, x = 1, x = 2, etc.
587
00:33:27,770 --> 00:33:30,740
Here's x = -1.
588
00:33:30,740 --> 00:33:31,900
So forth.
589
00:33:31,900 --> 00:33:37,100
So that's what the rectangular
coordinate system looks like.
590
00:33:37,100 --> 00:33:41,380
And now I should draw the other
coordinate system that we have.
591
00:33:41,380 --> 00:33:47,900
Which is this guy here.
592
00:33:47,900 --> 00:33:49,610
Well, close enough.
593
00:33:49,610 --> 00:33:54,720
And these guys here.
594
00:33:54,720 --> 00:33:57,730
Kind of this bulls-eye
or target operation.
595
00:33:57,730 --> 00:34:01,480
And this one is,
say, theta = pi/2.
596
00:34:01,480 --> 00:34:03,870
This is theta = 0.
597
00:34:03,870 --> 00:34:07,710
This is theta = -pi/4.
598
00:34:07,710 --> 00:34:11,380
For instance, so I've
just labeled for you three
599
00:34:11,380 --> 00:34:17,870
of the rays on this diagram.
600
00:34:17,870 --> 00:34:23,130
It's kind of like
a radar screen.
601
00:34:23,130 --> 00:34:28,840
And then in pink, this is
maybe r = 2, the radius 2.
602
00:34:28,840 --> 00:34:33,980
And inside is r = 1.
603
00:34:33,980 --> 00:34:38,090
So it's a different coordinate
system for the plane.
604
00:34:38,090 --> 00:34:42,120
And again, the letter
r represents measuring
605
00:34:42,120 --> 00:34:44,930
how far we are from the origin.
606
00:34:44,930 --> 00:34:47,060
The theta represents
something about the angle,
607
00:34:47,060 --> 00:34:50,250
which ray we're on.
608
00:34:50,250 --> 00:34:52,260
And they're just two
different variables.
609
00:34:52,260 --> 00:35:10,880
And this is a very different
kind of coordinate system.
610
00:35:10,880 --> 00:35:15,391
OK so, our main job is
just to get used to this.
611
00:35:15,391 --> 00:35:15,890
For now.
612
00:35:15,890 --> 00:35:18,350
You will be using
this a lot in 18.02.
613
00:35:18,350 --> 00:35:20,570
It's very useful in physics.
614
00:35:20,570 --> 00:35:25,680
And our job is just to
get started with it.
615
00:35:25,680 --> 00:35:29,990
And so, let's try a
few examples here.
616
00:35:29,990 --> 00:35:31,220
Tons of examples.
617
00:35:31,220 --> 00:35:34,590
We'll start out very slow.
618
00:35:34,590 --> 00:35:41,860
If you have (x, y) = (1, -1),
that's a point in the plane.
619
00:35:41,860 --> 00:35:44,380
I can draw that point.
620
00:35:44,380 --> 00:35:46,460
It's down here, right?
621
00:35:46,460 --> 00:35:50,630
This is -1 and this is 1,
and here's my point, (1, -1).
622
00:35:50,630 --> 00:35:53,550
I can figure out what
the representative is
623
00:35:53,550 --> 00:35:56,670
of this in polar coordinates.
624
00:35:56,670 --> 00:36:03,040
So in polar coordinates,
there are actually
625
00:36:03,040 --> 00:36:05,130
a bunch of choices here.
626
00:36:05,130 --> 00:36:09,250
First of all, I'll
tell you one choice.
627
00:36:09,250 --> 00:36:10,970
If I start with the
angle horizontally,
628
00:36:10,970 --> 00:36:14,200
I wrap all the way
around, that would
629
00:36:14,200 --> 00:36:19,350
be to this ray here--
Let's do it in green again.
630
00:36:19,350 --> 00:36:21,820
Alright, I labeled
it actually as -pi/4,
631
00:36:21,820 --> 00:36:27,310
but another way of looking at
it is that it's this angle here.
632
00:36:27,310 --> 00:36:31,440
So that would be r
= square root of 2.
633
00:36:31,440 --> 00:36:34,210
Theta = 7pi/4.
634
00:36:38,150 --> 00:36:41,750
So that's one possibility of
the angle and the distance.
635
00:36:41,750 --> 00:36:45,380
I know the distance is a square
root of 2, that's not hard.
636
00:36:45,380 --> 00:36:47,930
Another way of looking
at it is the way
637
00:36:47,930 --> 00:36:49,640
which was suggested
when I labeled this
638
00:36:49,640 --> 00:36:51,230
with a negative angle.
639
00:36:51,230 --> 00:36:56,850
And that would be r = square
root of 2, theta = -pi/4.
640
00:36:56,850 --> 00:36:58,370
And these are both legal.
641
00:36:58,370 --> 00:37:00,736
These are perfectly
legal representatives.
642
00:37:00,736 --> 00:37:02,110
And that's what
I meant by saying
643
00:37:02,110 --> 00:37:06,180
that these representations over
here are somewhat ambiguous.
644
00:37:06,180 --> 00:37:08,900
There's more than one answer
to this question, of what
645
00:37:08,900 --> 00:37:11,860
the polar representation is.
646
00:37:11,860 --> 00:37:17,190
A third possibility, which is
even more dicey but also legal,
647
00:37:17,190 --> 00:37:21,890
is r equals minus
square root of 2.
648
00:37:21,890 --> 00:37:25,360
Theta = 3pi/4.
649
00:37:25,360 --> 00:37:30,080
Now, what that corresponds to
doing is going around to here.
650
00:37:30,080 --> 00:37:33,490
We're pointing out
3/4 pi direction.
651
00:37:33,490 --> 00:37:37,130
But then going negative
square root of 2 distance.
652
00:37:37,130 --> 00:37:39,710
We're going backwards.
653
00:37:39,710 --> 00:37:42,250
So we're landing
in the same place.
654
00:37:42,250 --> 00:37:44,380
So this is also legal.
655
00:37:44,380 --> 00:37:44,880
Yeah.
656
00:37:44,880 --> 00:37:51,324
STUDENT: [INAUDIBLE]
657
00:37:51,324 --> 00:37:53,240
PROFESSOR: The question
is, don't the radiuses
658
00:37:53,240 --> 00:37:54,989
have to be positive
because they represent
659
00:37:54,989 --> 00:37:56,620
a distance to the origin?
660
00:37:56,620 --> 00:38:00,620
The answer is I
lied to you here.
661
00:38:00,620 --> 00:38:04,770
All of these things that I said
are wrong, except for this.
662
00:38:04,770 --> 00:38:09,020
Which is the rule for what
polar coordinates mean.
663
00:38:09,020 --> 00:38:21,170
So it's maybe plus or minus the
distance, is what it is always.
664
00:38:21,170 --> 00:38:29,090
I try not to lie to you
too much, but I do succeed.
665
00:38:29,090 --> 00:38:36,270
Now, let's do a little
bit more practice here.
666
00:38:36,270 --> 00:38:38,330
There are some easy
examples, which
667
00:38:38,330 --> 00:38:40,580
I will run through
very quickly. r = a,
668
00:38:40,580 --> 00:38:44,100
we already know
this is a circle.
669
00:38:44,100 --> 00:38:51,280
And the 3 theta equals
a constant is a ray.
670
00:38:51,280 --> 00:38:54,820
However, this involves an
implicit assumption, which
671
00:38:54,820 --> 00:38:57,360
I want to point out to you.
672
00:38:57,360 --> 00:38:59,040
So this is Example 3.
673
00:38:59,040 --> 00:39:01,060
Theta's equal to a
constant is a ray.
674
00:39:01,060 --> 00:39:14,070
But this implicitly
assumes 0 <= r < infinity.
675
00:39:14,070 --> 00:39:19,400
If you really wanted to allow
minus infinity < r < infinity
676
00:39:19,400 --> 00:39:22,890
in this example, you
would get a line.
677
00:39:22,890 --> 00:39:28,540
Gives the whole line.
678
00:39:28,540 --> 00:39:30,050
It gives everything behind.
679
00:39:30,050 --> 00:39:33,085
So you go out on some ray,
you go backwards on that ray
680
00:39:33,085 --> 00:39:36,460
and you get the whole line
through the origin, both ways.
681
00:39:36,460 --> 00:39:39,740
If you allow r going to
minus infinity as well.
682
00:39:39,740 --> 00:39:42,310
So the typical
conventions, so here
683
00:39:42,310 --> 00:39:49,680
are the typical conventions.
684
00:39:49,680 --> 00:39:53,140
And you will see people assume
this without even telling you.
685
00:39:53,140 --> 00:39:55,340
So you need to watch out for it.
686
00:39:55,340 --> 00:39:57,450
The typical conventions
are certainly this one,
687
00:39:57,450 --> 00:40:00,270
which is a nice thing to do.
688
00:40:00,270 --> 00:40:04,240
Pretty much all the time,
although not all the time.
689
00:40:04,240 --> 00:40:05,360
Most of the time.
690
00:40:05,360 --> 00:40:11,950
And then you might have
theta ranging from minus pi
691
00:40:11,950 --> 00:40:15,730
to pi, so in other words
symmetric around 0.
692
00:40:15,730 --> 00:40:21,630
Or, another very popular
choice is this one.
693
00:40:21,630 --> 00:40:25,890
Theta's >= 0 and
strictly less than 2pi.
694
00:40:25,890 --> 00:40:29,660
So these are the
two typical ranges
695
00:40:29,660 --> 00:40:33,930
in which all of these
variables are chosen.
696
00:40:33,930 --> 00:40:34,900
But not always.
697
00:40:34,900 --> 00:40:43,210
You'll find that
it's not consistent.
698
00:40:43,210 --> 00:40:46,010
As I said, our job is
to get used to this.
699
00:40:46,010 --> 00:40:49,600
And I need to work up
to some slightly more
700
00:40:49,600 --> 00:40:51,420
complicated examples.
701
00:40:51,420 --> 00:40:57,840
Some of which I'll give
you on next Tuesday.
702
00:40:57,840 --> 00:41:05,780
But let's do a few more.
703
00:41:05,780 --> 00:41:10,820
So, I guess this is Example 4.
704
00:41:10,820 --> 00:41:14,980
Example 4, I'm
going to take y = 1.
705
00:41:14,980 --> 00:41:20,650
That's awfully simple in
rectangular coordinates.
706
00:41:20,650 --> 00:41:23,960
But interestingly,
you might conceivably
707
00:41:23,960 --> 00:41:26,050
want to deal with it
in polar coordinates.
708
00:41:26,050 --> 00:41:29,580
If you do, so here's how
you make the translation.
709
00:41:29,580 --> 00:41:32,850
But this translation
is not so terrible.
710
00:41:32,850 --> 00:41:39,080
What you do is, you plug
in y = r sin(theta).
711
00:41:39,080 --> 00:41:40,710
That's all you have to do.
712
00:41:40,710 --> 00:41:42,760
And so that's going
to be equal to 1.
713
00:41:42,760 --> 00:41:46,240
And that's going to give
us our polar equation.
714
00:41:46,240 --> 00:41:50,330
The polar equation is
r = 1 / sin(theta).
715
00:41:50,330 --> 00:41:54,360
There it is.
716
00:41:54,360 --> 00:41:58,120
And let's draw a picture of it.
717
00:41:58,120 --> 00:42:03,480
So here's a picture
of the line y = 1.
718
00:42:03,480 --> 00:42:11,950
And now we see that if we take
our rays going out from here,
719
00:42:11,950 --> 00:42:17,240
they collide with the
line at various lengths.
720
00:42:17,240 --> 00:42:19,760
So if you take an angle,
theta, here there'll
721
00:42:19,760 --> 00:42:21,364
be a distance r
corresponding to that
722
00:42:21,364 --> 00:42:23,030
and you'll hit this
in exactly one spot.
723
00:42:23,030 --> 00:42:26,600
For each theta you'll
have a different radius.
724
00:42:26,600 --> 00:42:27,810
And it's a variable radius.
725
00:42:27,810 --> 00:42:30,740
It's given by this formula here.
726
00:42:30,740 --> 00:42:33,210
And so to trace this
line out, you actually
727
00:42:33,210 --> 00:42:36,120
have to realize that there's
one more thing involved.
728
00:42:36,120 --> 00:42:40,160
Which is the possible
range of theta.
729
00:42:40,160 --> 00:42:41,730
Again, when you're
doing integrations
730
00:42:41,730 --> 00:42:44,104
you're going to need to know
those limits of integration.
731
00:42:44,104 --> 00:42:46,360
So you're going to
need to know this.
732
00:42:46,360 --> 00:42:48,990
The range here goes
from theta = 0,
733
00:42:48,990 --> 00:42:51,230
that's sort of when
it's out at infinity.
734
00:42:51,230 --> 00:42:53,140
That's when the
denominator is 0 here.
735
00:42:53,140 --> 00:42:55,800
And it goes all the way to pi.
736
00:42:55,800 --> 00:42:57,940
Swing around just one half-turn.
737
00:42:57,940 --> 00:43:03,610
So the range here
is 0 < theta < pi.
738
00:43:03,610 --> 00:43:04,620
Yeah, question.
739
00:43:04,620 --> 00:43:09,676
STUDENT: [INAUDIBLE]
740
00:43:09,676 --> 00:43:11,050
PROFESSOR: The
question is, is it
741
00:43:11,050 --> 00:43:13,940
typical to express r
as a function of theta,
742
00:43:13,940 --> 00:43:16,550
or vice versa, or
does it matter?
743
00:43:16,550 --> 00:43:19,790
The answer is that for the
purposes of this course,
744
00:43:19,790 --> 00:43:24,420
we're almost always going to
be writing things in this form.
745
00:43:24,420 --> 00:43:27,070
r as a function of theta.
746
00:43:27,070 --> 00:43:30,050
And you can do
whatever you want.
747
00:43:30,050 --> 00:43:33,920
This turns out to be what
we'll be doing in this course,
748
00:43:33,920 --> 00:43:37,040
exclusively.
749
00:43:37,040 --> 00:43:40,570
As you'll see when we
get to other examples,
750
00:43:40,570 --> 00:43:42,160
it's the traditional
sort of thing
751
00:43:42,160 --> 00:43:45,060
to do when you're thinking
about observing a planet
752
00:43:45,060 --> 00:43:48,650
or something like that.
753
00:43:48,650 --> 00:43:52,930
You see the angle, and then
you guess far away it is.
754
00:43:52,930 --> 00:43:55,600
But it's not necessary.
755
00:43:55,600 --> 00:43:58,940
The formulas are
often easier this way.
756
00:43:58,940 --> 00:44:00,370
For the examples that we have.
757
00:44:00,370 --> 00:44:02,610
Because it's usually a
trig function of theta.
758
00:44:02,610 --> 00:44:05,110
Whereas the other way, it would
be an inverse trig function.
759
00:44:05,110 --> 00:44:08,930
So it's an uglier expression.
760
00:44:08,930 --> 00:44:10,540
As you can see.
761
00:44:10,540 --> 00:44:12,860
The real reason is that we
choose this thing that's
762
00:44:12,860 --> 00:44:19,410
easier to deal with.
763
00:44:19,410 --> 00:44:22,200
So now let me give you a
slightly more complicated
764
00:44:22,200 --> 00:44:24,410
example of the same type.
765
00:44:24,410 --> 00:44:28,930
Where we use a shortcut.
766
00:44:28,930 --> 00:44:31,680
This is a standard example.
767
00:44:31,680 --> 00:44:33,960
And it comes up a lot.
768
00:44:33,960 --> 00:44:40,730
And so this is an
off-center circle.
769
00:44:40,730 --> 00:44:44,000
A circle is really easy
to describe, but not
770
00:44:44,000 --> 00:44:54,170
necessarily if the center
is on the rim of the circle.
771
00:44:54,170 --> 00:44:56,550
So that's a different problem.
772
00:44:56,550 --> 00:44:59,990
And let's do this with
a circle of radius a.
773
00:44:59,990 --> 00:45:06,120
So this is the point (a,
0) and this is (2a, 0).
774
00:45:06,120 --> 00:45:08,550
And actually, if you
know these two numbers,
775
00:45:08,550 --> 00:45:11,080
you'll be able to remember the
result of this calculation.
776
00:45:11,080 --> 00:45:13,780
Which you'll do about five
or six times and then finally
777
00:45:13,780 --> 00:45:17,310
you'll memorize it during 18.02
when you will need it a lot.
778
00:45:17,310 --> 00:45:21,220
So this is a standard
calculation here.
779
00:45:21,220 --> 00:45:24,350
So the starting place is
the rectangular equation.
780
00:45:24,350 --> 00:45:27,170
And we're going to pass to
the polar representation.
781
00:45:27,170 --> 00:45:33,550
The rectangular representation
is (x-a)^2 + y^2 = a^2.
782
00:45:33,550 --> 00:45:40,290
So this is a circle centered
at (a, 0) of radius a.
783
00:45:40,290 --> 00:45:44,110
And now, if you like, the
slow way of doing this
784
00:45:44,110 --> 00:45:50,145
would be to plug in x = r
cos(theta), y = r sin(theta).
785
00:45:50,145 --> 00:45:51,520
The way I did in
this first step.
786
00:45:51,520 --> 00:45:53,500
And that works perfectly well.
787
00:45:53,500 --> 00:45:56,980
But I'm going to do it
more quickly than that.
788
00:45:56,980 --> 00:46:00,070
Because I can sort of see in
advance how it's going to work.
789
00:46:00,070 --> 00:46:09,810
I'm just going to
expand this out.
790
00:46:09,810 --> 00:46:13,160
And now I see the a^2's cancel.
791
00:46:13,160 --> 00:46:17,120
And not only that,
but x^2 + y^2 = r^2.
792
00:46:17,120 --> 00:46:19,670
So this becomes r^2.
793
00:46:19,670 --> 00:46:28,590
That's x^2 + y^2 - 2ax = 0.
794
00:46:28,590 --> 00:46:32,360
The r came from the fact
that r^2 = x^2 + y^2.
795
00:46:36,100 --> 00:46:37,890
So I'm doing this the rapid way.
796
00:46:37,890 --> 00:46:40,260
You can do it by
plugging in, as I said.
797
00:46:40,260 --> 00:46:43,900
r equals-- So now that
I've simplified it,
798
00:46:43,900 --> 00:46:45,720
I am going to use
that procedure.
799
00:46:45,720 --> 00:46:47,570
I'm going to plug in.
800
00:46:47,570 --> 00:46:57,120
So here I have r^2 -
2ar cos(theta) = 0.
801
00:46:57,120 --> 00:47:00,146
I just plugged in for x.
802
00:47:00,146 --> 00:47:02,270
As I said, I could have
done that at the beginning.
803
00:47:02,270 --> 00:47:06,430
I just simplified first.
804
00:47:06,430 --> 00:47:11,780
And now, this is the same
thing as r^2 = 2ar cos(theta).
805
00:47:11,780 --> 00:47:13,530
And we're almost done.
806
00:47:13,530 --> 00:47:19,230
There's a boring part of this
equation, which is r = 0.
807
00:47:19,230 --> 00:47:21,530
And then there's,
if I divide by r,
808
00:47:21,530 --> 00:47:23,430
there's the interesting
part of the equation.
809
00:47:23,430 --> 00:47:25,830
Which is this.
810
00:47:25,830 --> 00:47:28,810
So this is or r = 0.
811
00:47:28,810 --> 00:47:33,690
Which is already included
in that equation anyway.
812
00:47:33,690 --> 00:47:36,890
So I'm allowed to divide by r
because in the case of r = 0,
813
00:47:36,890 --> 00:47:39,781
this is represented anyway.
814
00:47:39,781 --> 00:47:40,280
Question.
815
00:47:40,280 --> 00:47:44,390
STUDENT: [INAUDIBLE]
816
00:47:44,390 --> 00:47:46,270
PROFESSOR: r = 0
is just one case.
817
00:47:46,270 --> 00:47:48,380
That is, it's the
union of these two.
818
00:47:48,380 --> 00:47:49,550
It's both.
819
00:47:49,550 --> 00:47:50,670
Both are possible.
820
00:47:50,670 --> 00:47:53,270
So r = 0 is one point on it.
821
00:47:53,270 --> 00:47:56,150
And this is all of it.
822
00:47:56,150 --> 00:48:01,230
So we can just ignore this.
823
00:48:01,230 --> 00:48:04,500
So now I want to say one
more important thing.
824
00:48:04,500 --> 00:48:06,600
You need to understand
the range of this.
825
00:48:06,600 --> 00:48:10,840
So wait a second and we're going
to figure out the range here.
826
00:48:10,840 --> 00:48:13,710
The range is very important,
because otherwise you'll
827
00:48:13,710 --> 00:48:18,280
never be able to integrate
using this representation here.
828
00:48:18,280 --> 00:48:19,840
So this is the representation.
829
00:48:19,840 --> 00:48:25,190
But notice when theta =
0, we're out here at 2a.
830
00:48:25,190 --> 00:48:26,780
That's consistent,
and that's actually
831
00:48:26,780 --> 00:48:29,020
how you remember
this factor 2a here.
832
00:48:29,020 --> 00:48:31,570
Because if you remember this
picture and where you land when
833
00:48:31,570 --> 00:48:34,830
theta = 0.
834
00:48:34,830 --> 00:48:36,370
So that's the theta = 0 part.
835
00:48:36,370 --> 00:48:39,440
But now as I tip
up like this, you
836
00:48:39,440 --> 00:48:43,780
see that when we get to
vertical, we're done.
837
00:48:43,780 --> 00:48:44,630
With the circle.
838
00:48:44,630 --> 00:48:46,463
It's gotten shorter and
shorter and shorter,
839
00:48:46,463 --> 00:48:49,020
and at theta = pi/2,
we're down at 0.
840
00:48:49,020 --> 00:48:51,720
Because that's cos(pi/2) = 0.
841
00:48:51,720 --> 00:48:53,770
So it swings up like this.
842
00:48:53,770 --> 00:48:55,400
And it gets up to pi/2.
843
00:48:55,400 --> 00:48:57,110
Similarly, we swing
down like this.
844
00:48:57,110 --> 00:48:59,000
And then we're done.
845
00:48:59,000 --> 00:49:04,510
So the range is
-pi/2 < theta < pi/2.
846
00:49:04,510 --> 00:49:06,650
Or, if you want to
throw in the r = 0 case,
847
00:49:06,650 --> 00:49:08,700
you can throw in this,
this is repeating,
848
00:49:08,700 --> 00:49:11,200
if you like, at the ends.
849
00:49:11,200 --> 00:49:14,100
So this is the range
of this circle.
850
00:49:14,100 --> 00:49:17,150
And let's see.
851
00:49:17,150 --> 00:49:21,300
Next time we'll figure out
area in polar coordinates.