1
00:00:00,000 --> 00:00:07,470
JOEL LEWIS: Hi.

2
00:00:07,470 --> 00:00:08,990
Welcome back to recitation.

3
00:00:08,990 --> 00:00:11,540
In lecture, you've been
learning about line integrals

4
00:00:11,540 --> 00:00:15,080
and computing them around
curves and closed curves

5
00:00:15,080 --> 00:00:17,140
and in various different ways.

6
00:00:17,140 --> 00:00:21,820
So here I have some problems
on line integrals for you.

7
00:00:21,820 --> 00:00:27,070
So in all cases I want C to
be the circle of radius b.

8
00:00:27,070 --> 00:00:30,090
So b is some constant,
some positive constant.

9
00:00:30,090 --> 00:00:32,320
It's the circle of radius
b centered at the origin,

10
00:00:32,320 --> 00:00:34,322
and I want to orient
it counterclockwise.

11
00:00:34,322 --> 00:00:35,780
And then what I'd
like you to do is

12
00:00:35,780 --> 00:00:38,360
for each of the following
vector fields F,

13
00:00:38,360 --> 00:00:43,160
I'd like you to compute the line
integral around C of F dot dr.

14
00:00:43,160 --> 00:00:47,850
So in the first case,
where F is x*i plus y*j.

15
00:00:47,850 --> 00:00:52,200
In the second, where F is g
of x, y times x*i plus y*j.

16
00:00:52,200 --> 00:00:55,680
So here g of x, y is
some scalar function.

17
00:00:55,680 --> 00:00:59,810
But you don't know a
formula for this function.

18
00:00:59,810 --> 00:01:02,445
So your answer might be in
terms of g, for example.

19
00:01:06,550 --> 00:01:08,550
You can assume it's a
continuous, differentiable

20
00:01:08,550 --> 00:01:10,440
nice function.

21
00:01:10,440 --> 00:01:14,630
And then the third one,
F is minus y*i plus x*j.

22
00:01:14,630 --> 00:01:16,690
Now before you start,
I want to give you

23
00:01:16,690 --> 00:01:18,570
a little suggestion,
which is often

24
00:01:18,570 --> 00:01:21,650
when we're given a line
integral like this,

25
00:01:21,650 --> 00:01:26,250
the first thing you want
to do is jump in and do

26
00:01:26,250 --> 00:01:28,200
a parameterization right
away for the curve,

27
00:01:28,200 --> 00:01:32,450
and then you get a normal
single variable integral.

28
00:01:32,450 --> 00:01:34,790
So what I'd like you to
do for these problems

29
00:01:34,790 --> 00:01:37,600
is to think about the
setup and think about

30
00:01:37,600 --> 00:01:40,240
whether you can do this
without ever parameterizing

31
00:01:40,240 --> 00:01:46,415
C, so without ever substituting
in cosine and sine or whatever.

32
00:01:46,415 --> 00:01:48,040
So for all three
parts of this problem.

33
00:01:48,040 --> 00:01:50,349
So if you can use some
sort of geometric reasoning

34
00:01:50,349 --> 00:01:51,890
to save yourself a
little bit of work

35
00:01:51,890 --> 00:01:53,960
without ever going to
the parameterization.

36
00:01:53,960 --> 00:01:56,614
So why don't you pause the
video, spend some time,

37
00:01:56,614 --> 00:01:59,030
work that out, come back, and
we can work it out together.

38
00:02:06,940 --> 00:02:09,190
Hopefully you had some luck
working on these problems.

39
00:02:09,190 --> 00:02:11,540
Let's get started.

40
00:02:11,540 --> 00:02:14,070
So let's do the
first problem first.

41
00:02:14,070 --> 00:02:17,660
Let's think about what this
vector field F looks like.

42
00:02:17,660 --> 00:02:19,570
This first vector field.

43
00:02:19,570 --> 00:02:23,360
So let me just draw a
little picture over here.

44
00:02:23,360 --> 00:02:28,330
So here's our
circle of radius b.

45
00:02:28,330 --> 00:02:32,535
And this vector field F
given by x*i plus y*j.

46
00:02:32,535 --> 00:02:41,250
At every point (x,
y), the vector F

47
00:02:41,250 --> 00:02:44,110
is the same as the position
vector of that point.

48
00:02:44,110 --> 00:02:47,220
So over here the
vector's like that.

49
00:02:47,220 --> 00:02:51,310
Over here, the
vector's like that.

50
00:02:51,310 --> 00:02:54,700
Up here, the vector
is like that.

51
00:02:54,700 --> 00:02:58,589
So these are just a
few little values of F

52
00:02:58,589 --> 00:02:59,630
that I've drawn in there.

53
00:02:59,630 --> 00:03:06,520
And so down here,
say, F is like that.

54
00:03:06,520 --> 00:03:09,820
So in particular, so that's
just sort of-- you know,

55
00:03:09,820 --> 00:03:13,590
if you wanted, you could
draw in some more vectors,

56
00:03:13,590 --> 00:03:17,050
get a full vector field picture.

57
00:03:17,050 --> 00:03:20,600
So the thing to observe
here is that a circle

58
00:03:20,600 --> 00:03:23,040
is a really nice curve.

59
00:03:23,040 --> 00:03:27,010
So the circle has the
property that the position

60
00:03:27,010 --> 00:03:30,900
vector at a point is orthogonal
to the tangent vector

61
00:03:30,900 --> 00:03:31,640
to the circle.

62
00:03:31,640 --> 00:03:34,180
At every point on the
circle, the tangent vector

63
00:03:34,180 --> 00:03:37,440
to the circle is perpendicular
to the position vector.

64
00:03:37,440 --> 00:03:43,190
So that means it's perpendicular
to F, because F is the same,

65
00:03:43,190 --> 00:03:45,960
in fact, but is parallel
to the position vector.

66
00:03:45,960 --> 00:03:53,175
So in Part a, you
have that F dot

67
00:03:53,175 --> 00:03:56,580
the tangent vector
to your curve is

68
00:03:56,580 --> 00:04:00,490
equal to zero at every
point on the entire curve.

69
00:04:00,490 --> 00:04:01,290
All right?

70
00:04:01,290 --> 00:04:06,530
So your field F dot your
tangent vector is always zero.

71
00:04:06,530 --> 00:04:13,030
So that means that the
integral around C of F dot dr,

72
00:04:13,030 --> 00:04:17,610
well, we know that dr is T ds.

73
00:04:17,610 --> 00:04:20,090
So this is F dot T ds.

74
00:04:20,090 --> 00:04:21,420
But that's just zero.

75
00:04:21,420 --> 00:04:24,649
It's just an integral and the
integrand is zero everywhere.

76
00:04:24,649 --> 00:04:27,190
And whenever you take a definite
integral of something that's

77
00:04:27,190 --> 00:04:29,040
zero everywhere, you get zero.

78
00:04:29,040 --> 00:04:30,632
So this is just zero right away.

79
00:04:30,632 --> 00:04:32,840
We didn't have to parameterize
the curve or anything.

80
00:04:32,840 --> 00:04:34,600
We just had to look
at this picture

81
00:04:34,600 --> 00:04:37,110
to sort of understand
that this kind of field,

82
00:04:37,110 --> 00:04:38,870
it's called a
radial vector field,

83
00:04:38,870 --> 00:04:43,830
where the vector F is always
pointed directly outwards.

84
00:04:43,830 --> 00:04:45,600
When you integrate a
radial vector field

85
00:04:45,600 --> 00:04:47,680
around a circle
centered at the origin,

86
00:04:47,680 --> 00:04:50,160
you get zero, because the
contribution at every point

87
00:04:50,160 --> 00:04:51,720
is zero.

88
00:04:51,720 --> 00:04:52,930
So that's Part a.

89
00:04:52,930 --> 00:04:56,050
Part b is actually
exactly the same.

90
00:04:56,050 --> 00:04:59,030
If we look back at our
formula over here in Part b,

91
00:04:59,030 --> 00:05:03,370
we have that F is given by
some function g of x, y times

92
00:05:03,370 --> 00:05:05,040
x i hat plus y j hat.

93
00:05:05,040 --> 00:05:07,230
Well, what is this
g of x, y doing?

94
00:05:07,230 --> 00:05:08,660
It's just rescaling.

95
00:05:08,660 --> 00:05:11,180
It's telling you at
every point you can scale

96
00:05:11,180 --> 00:05:13,530
that vector by some amount.

97
00:05:13,530 --> 00:05:17,130
So if we looked over at this
picture, maybe over here

98
00:05:17,130 --> 00:05:19,350
you would scale some of
these vectors to be longer,

99
00:05:19,350 --> 00:05:20,849
and over here they
might be shorter,

100
00:05:20,849 --> 00:05:22,680
or you might switch
them to be negative,

101
00:05:22,680 --> 00:05:25,930
but you don't change the
direction of any vector

102
00:05:25,930 --> 00:05:27,970
in the field from Part a.

103
00:05:27,970 --> 00:05:29,630
You just change their length.

104
00:05:29,630 --> 00:05:31,640
So you still have a
radial vector field.

105
00:05:31,640 --> 00:05:34,250
And you still have the
property that at every point

106
00:05:34,250 --> 00:05:36,760
on our curve, the tangent
vector to the curve

107
00:05:36,760 --> 00:05:40,150
is orthogonal to the vector
F. So the tangent vector

108
00:05:40,150 --> 00:05:42,510
is orthogonal to F, so
that means you again

109
00:05:42,510 --> 00:05:44,890
have F dot T is equal to zero.

110
00:05:44,890 --> 00:05:49,220
And so F dot dr is
also equal to 0 ds,

111
00:05:49,220 --> 00:05:51,380
and so when you integrate
that, you just get zero.

112
00:05:51,380 --> 00:05:53,560
So that's also what
happens in Part b.

113
00:05:53,560 --> 00:05:58,000
So Part b, I'm just
going to write ditto.

114
00:05:58,000 --> 00:06:01,600
The exact same reasoning applies
in Part b as applied in Part a.

115
00:06:01,600 --> 00:06:05,614
And you also get
zero as your integral

116
00:06:05,614 --> 00:06:07,530
without having to
parameterize, without having

117
00:06:07,530 --> 00:06:09,890
to do any tricky
calculations at all.

118
00:06:09,890 --> 00:06:10,390
All right.

119
00:06:10,390 --> 00:06:12,600
So let's now look at Part c.

120
00:06:12,600 --> 00:06:16,330
I'm going to draw
another little picture.

121
00:06:16,330 --> 00:06:20,620
So in Part c,
there's your curve.

122
00:06:20,620 --> 00:06:25,270
At the point (x,y)-- so I'm
going to draw some choices of F

123
00:06:25,270 --> 00:06:26,370
again.

124
00:06:26,370 --> 00:06:30,500
So in Part c, at
the point (x,y),

125
00:06:30,500 --> 00:06:42,600
your vector field F is
minus y i hat plus x j hat.

126
00:06:42,600 --> 00:06:44,590
Now if you draw
that on the picture

127
00:06:44,590 --> 00:06:49,480
here, over there
that's that vector.

128
00:06:49,480 --> 00:06:52,270
Over here, so at the
point (0,1), say,

129
00:06:52,270 --> 00:06:54,970
that gives you the
vector [-1, 0].

130
00:06:54,970 --> 00:06:57,569
So that's horizontal
to the left.

131
00:06:57,569 --> 00:06:58,360
Here are some more.

132
00:06:58,360 --> 00:07:01,520
There's one there,
there's one there.

133
00:07:01,520 --> 00:07:04,450
There's another one
over here and so on.

134
00:07:04,450 --> 00:07:08,460
In fact, what you'll notice
is that this vector F is just

135
00:07:08,460 --> 00:07:12,280
parallel to the tangent vector
of the circle everywhere.

136
00:07:12,280 --> 00:07:15,360
This field is a
tangential field.

137
00:07:15,360 --> 00:07:18,236
It's always pointing
parallel to the curve.

138
00:07:18,236 --> 00:07:18,735
OK?

139
00:07:22,270 --> 00:07:24,020
It's perpendicular to
the position vector.

140
00:07:24,020 --> 00:07:26,350
It's in the same direction
as the tangent vector

141
00:07:26,350 --> 00:07:27,050
at every point.

142
00:07:27,050 --> 00:07:29,300
So this is something that
you've seen before, I think.

143
00:07:29,300 --> 00:07:31,750
That this vector
field is giving you

144
00:07:31,750 --> 00:07:35,160
a sort of nice rotating motion.

145
00:07:35,160 --> 00:07:39,640
You know, at every point it's
circulating counterclockwise.

146
00:07:39,640 --> 00:07:40,610
So what does that mean?

147
00:07:40,610 --> 00:07:44,490
Well, again, it's not exactly
the same as Part a and b,

148
00:07:44,490 --> 00:07:47,060
but again we'll be able
to compute this integral

149
00:07:47,060 --> 00:07:48,280
without parameterizing.

150
00:07:48,280 --> 00:07:48,960
Why?

151
00:07:48,960 --> 00:07:54,170
Because F dot T in this
case-- well, so, let's see.

152
00:07:54,170 --> 00:07:57,070
What is the norm of F?

153
00:07:57,070 --> 00:07:59,530
The magnitude of F is
just the square root of

154
00:07:59,530 --> 00:08:00,940
(x squared plus y squared).

155
00:08:00,940 --> 00:08:04,850
So on our circle of radius b,
that means the magnitude of F

156
00:08:04,850 --> 00:08:05,940
is b.

157
00:08:05,940 --> 00:08:10,020
And the magnitude of T, the
unit tangent vector, is 1,

158
00:08:10,020 --> 00:08:11,562
and they point in
the same direction.

159
00:08:11,562 --> 00:08:13,186
So when you have two
vectors that point

160
00:08:13,186 --> 00:08:15,090
in the same direction,
their dot product

161
00:08:15,090 --> 00:08:18,190
is just the product
of their magnitudes.

162
00:08:18,190 --> 00:08:21,340
So that means F dot
T is equal to b.

163
00:08:21,340 --> 00:08:23,630
This is a constant.

164
00:08:23,630 --> 00:08:25,050
F dot T is equal to b.

165
00:08:25,050 --> 00:08:32,970
So when you integrate
around the circle, F dot dr,

166
00:08:32,970 --> 00:08:38,150
well, this is equal
to the integral

167
00:08:38,150 --> 00:08:43,760
around a circle of F dot the
tangent vector with respect

168
00:08:43,760 --> 00:08:44,900
to arc length.

169
00:08:44,900 --> 00:08:47,810
But this integrand, F
dot the tangent vector,

170
00:08:47,810 --> 00:08:49,250
is this constant b.

171
00:08:49,250 --> 00:08:54,210
So you're integrating
over the curve b ds.

172
00:08:54,210 --> 00:08:56,210
And when you integrate
a constant ds,

173
00:08:56,210 --> 00:08:58,180
well, that just gives
you the total arc length.

174
00:08:58,180 --> 00:09:01,730
So this is b times
the total arc length.

175
00:09:01,730 --> 00:09:04,310
And this is a
circle of radius b.

176
00:09:04,310 --> 00:09:12,060
So that's b times 2 pi b, which
we could also write as 2 pi

177
00:09:12,060 --> 00:09:13,734
b squared.

178
00:09:13,734 --> 00:09:14,400
So there you go.

179
00:09:14,400 --> 00:09:17,890
So in this third case, you have
a nice tangential vector field.

180
00:09:17,890 --> 00:09:20,700
So that means the
integrand actually

181
00:09:20,700 --> 00:09:23,630
works out to be constant.

182
00:09:23,630 --> 00:09:25,226
Because the integrand
is constant,

183
00:09:25,226 --> 00:09:27,100
we don't ever have to
parameterize the curve.

184
00:09:27,100 --> 00:09:28,766
We can just use the
fact that we already

185
00:09:28,766 --> 00:09:32,170
know its arc length in order
to compute this integral.

186
00:09:32,170 --> 00:09:35,430
Again, we could do
all of these integrals

187
00:09:35,430 --> 00:09:37,660
if we wanted by
parameterizing the circle,

188
00:09:37,660 --> 00:09:42,110
by x equals b cosine
t, y equals b sine t,

189
00:09:42,110 --> 00:09:45,190
and going through and writing
this as an integral from t

190
00:09:45,190 --> 00:09:47,510
equals 0 to 2 pi, and so on.

191
00:09:47,510 --> 00:09:50,687
But these are
examples of problems

192
00:09:50,687 --> 00:09:53,020
where it's helpful to think
about what's going on first,

193
00:09:53,020 --> 00:09:56,320
see if you can understand the
geometry of your situation.

194
00:09:56,320 --> 00:10:00,190
And sometimes you'll have
a problem like this where

195
00:10:00,190 --> 00:10:05,080
you'll-- either in this class
or elsewhere in your life--

196
00:10:05,080 --> 00:10:08,000
where something that might
seem complicated has a simple

197
00:10:08,000 --> 00:10:09,410
geometric explanation.

198
00:10:09,410 --> 00:10:11,010
And so when that
does happen, it's

199
00:10:11,010 --> 00:10:12,640
nice when you can
take advantage of it.

200
00:10:12,640 --> 00:10:14,639
Sometimes that won't
happen and sometimes you'll

201
00:10:14,639 --> 00:10:17,020
have to do the parameterization
and the computation.

202
00:10:17,020 --> 00:10:20,000
But in these cases we have
these nice three examples

203
00:10:20,000 --> 00:10:23,240
where with a radial
vector field,

204
00:10:23,240 --> 00:10:25,550
you get that the
integrand is always zero,

205
00:10:25,550 --> 00:10:29,120
or with a tangential
vector field, you have

206
00:10:29,120 --> 00:10:30,690
that the integrand is constant.

207
00:10:30,690 --> 00:10:31,190
All right.

208
00:10:31,190 --> 00:10:33,134
So, I'll stop there.