1
00:00:00,000 --> 00:00:07,090

2
00:00:07,090 --> 00:00:07,540
JOEL LEWIS: Hi.

3
00:00:07,540 --> 00:00:09,330
Welcome back to recitation.

4
00:00:09,330 --> 00:00:12,510
I've got a nice exercise here
on Stokes' Theorem for you.

5
00:00:12,510 --> 00:00:15,450
Now this problem is a little bit
more sophisticated than a

6
00:00:15,450 --> 00:00:17,630
lot of problems we've been
doing in recitation.

7
00:00:17,630 --> 00:00:20,400
So it requires a little bit
more thought, and it also

8
00:00:20,400 --> 00:00:22,940
involves more mathematical
sophistication.

9
00:00:22,940 --> 00:00:28,230
So we're doing a clever kind
of proof here that I like.

10
00:00:28,230 --> 00:00:31,330
So hopefully you'll
like this one.

11
00:00:31,330 --> 00:00:33,310
It's a little bit in a different
style than some of

12
00:00:33,310 --> 00:00:35,600
the ones we've done.

13
00:00:35,600 --> 00:00:37,860
So I think I need to talk about
it a little bit before

14
00:00:37,860 --> 00:00:38,630
we get started.

15
00:00:38,630 --> 00:00:41,580
So let's let F be the
field x, y, z.

16
00:00:41,580 --> 00:00:45,180
So this is our radial field
that we've seen a lot in

17
00:00:45,180 --> 00:00:45,870
recitation.

18
00:00:45,870 --> 00:00:49,510
So what I'd like you to do is
prove that this field is not

19
00:00:49,510 --> 00:00:52,200
the curl of any field
G. All right.

20
00:00:52,200 --> 00:00:56,450
So I'd like you to show that
there's no field G such that F

21
00:00:56,450 --> 00:00:58,790
is equal to the curl of G.

22
00:00:58,790 --> 00:01:02,750
Now, rather than just saying
that to you and letting you

23
00:01:02,750 --> 00:01:05,760
run off, I have a suggestion
for an interesting way you

24
00:01:05,760 --> 00:01:07,750
could go about this.

25
00:01:07,750 --> 00:01:11,340
And this interesting way is
going to use Stokes' Theorem.

26
00:01:11,340 --> 00:01:16,400
So what I'd like you to do is
a proof by contradiction.

27
00:01:16,400 --> 00:01:20,020
OK, so what you're going to do
is you're going to assume that

28
00:01:20,020 --> 00:01:21,740
F is a curl.

29
00:01:21,740 --> 00:01:22,240
OK?

30
00:01:22,240 --> 00:01:27,020
So you're going to assume that
there is some G such that F is

31
00:01:27,020 --> 00:01:31,750
curl G. And then you're going
to use that to get a

32
00:01:31,750 --> 00:01:33,890
ridiculous conclusion.

33
00:01:33,890 --> 00:01:36,090
So you're going to start with
that premise, and you're going

34
00:01:36,090 --> 00:01:38,960
to end up with a
contradiction.

35
00:01:38,960 --> 00:01:44,540
So these two arrows colliding
into each other is a symbol

36
00:01:44,540 --> 00:01:46,810
that mathematicians use
for a contradiction.

37
00:01:46,810 --> 00:01:49,035
So you're going to start from
this premise, and you're going

38
00:01:49,035 --> 00:01:50,460
to reach a contradiction.

39
00:01:50,460 --> 00:01:52,200
And what that's going
to show is that your

40
00:01:52,200 --> 00:01:54,300
premise couldn't be right.

41
00:01:54,300 --> 00:01:54,570
Yeah?

42
00:01:54,570 --> 00:01:57,670
Because if you start from a true
premise, well then all

43
00:01:57,670 --> 00:02:00,050
your conclusions should
be true as well.

44
00:02:00,050 --> 00:02:03,170
So if you reach a false
conclusion, then you must have

45
00:02:03,170 --> 00:02:05,470
had a false premise.

46
00:02:05,470 --> 00:02:09,170
So what you're going to do is
you're going to take a sphere

47
00:02:09,170 --> 00:02:11,590
of radius b centered
at the origin.

48
00:02:11,590 --> 00:02:13,750
And a curve C on the sphere.

49
00:02:13,750 --> 00:02:16,640
You know, a simple,
closed curve.

50
00:02:16,640 --> 00:02:21,230
So assuming that F is this curl
of G, what I'd like you

51
00:02:21,230 --> 00:02:26,620
to do is use Stokes' Theorem to
interpret the line integral

52
00:02:26,620 --> 00:02:31,170
of G dot dr over C in
two different ways.

53
00:02:31,170 --> 00:02:32,110
OK?

54
00:02:32,110 --> 00:02:35,070
And interpreting this line
integral in two different

55
00:02:35,070 --> 00:02:37,660
ways, you're going to reach a
contradiction, and that will

56
00:02:37,660 --> 00:02:40,130
show that F really
isn't a curl.

57
00:02:40,130 --> 00:02:41,760
So that's what I'd
like you to do.

58
00:02:41,760 --> 00:02:44,320
So why don't you pause the
video, go ahead and see if you

59
00:02:44,320 --> 00:02:46,410
can work that out, come
back, and we'll

60
00:02:46,410 --> 00:02:47,660
talk about it together.

61
00:02:47,660 --> 00:02:55,660

62
00:02:55,660 --> 00:02:57,910
I hope you enjoyed working
on this problem.

63
00:02:57,910 --> 00:02:59,320
Let's get started on it.

64
00:02:59,320 --> 00:03:03,810
So as I was saying before we
started, what we're going to

65
00:03:03,810 --> 00:03:06,280
do is we're looking for a
proof by contradiction.

66
00:03:06,280 --> 00:03:08,205
So as the problem says, we're
going to start with a sphere.

67
00:03:08,205 --> 00:03:11,520

68
00:03:11,520 --> 00:03:14,170
And I'm going to take
this curve C--

69
00:03:14,170 --> 00:03:17,700
some simply connected closed
curve that's going to go

70
00:03:17,700 --> 00:03:21,360
around the back of the sphere,
and it's going to be oriented

71
00:03:21,360 --> 00:03:23,780
one way or the other-- and
it's going to divide this

72
00:03:23,780 --> 00:03:25,580
sphere into two pieces.

73
00:03:25,580 --> 00:03:30,050
So there's the one cap on
one side of it, S1.

74
00:03:30,050 --> 00:03:31,020
And then there's--

75
00:03:31,020 --> 00:03:33,950
whatever the other piece on
the other side of it--

76
00:03:33,950 --> 00:03:35,430
S2.

77
00:03:35,430 --> 00:03:36,030
OK.

78
00:03:36,030 --> 00:03:39,480
And so what we're going to do is
we're going to think about,

79
00:03:39,480 --> 00:03:41,140
what is this line integral?

80
00:03:41,140 --> 00:03:41,420
OK.

81
00:03:41,420 --> 00:03:45,340
So this is our curve C
here on the sphere.

82
00:03:45,340 --> 00:03:55,200
So the integral over C of G dot
dr. So this is what the

83
00:03:55,200 --> 00:03:56,910
problem suggests
we think about.

84
00:03:56,910 --> 00:04:02,510
So this is a line integral of
a field dot dr over the

85
00:04:02,510 --> 00:04:03,640
boundary of two surfaces.

86
00:04:03,640 --> 00:04:04,890
Right?

87
00:04:04,890 --> 00:04:08,440

88
00:04:08,440 --> 00:04:11,490
C is the boundary of S1,
and C-- if we orient

89
00:04:11,490 --> 00:04:12,750
it the other way--

90
00:04:12,750 --> 00:04:18,730
is the boundary of S2, when we
orient them both outwards.

91
00:04:18,730 --> 00:04:21,660
OK, so what is this?

92
00:04:21,660 --> 00:04:23,460
So Stokes' Theorem tells
us something

93
00:04:23,460 --> 00:04:24,470
about this line integral.

94
00:04:24,470 --> 00:04:28,300
So let's first think about
this as the top cap--

95
00:04:28,300 --> 00:04:29,720
that cap S1--

96
00:04:29,720 --> 00:04:32,770
with boundary C oriented
so that they

97
00:04:32,770 --> 00:04:33,720
agree with each other.

98
00:04:33,720 --> 00:04:37,350
So the normal is outwards
on the sphere, and C is

99
00:04:37,350 --> 00:04:38,980
proceeding in the direction
that I've

100
00:04:38,980 --> 00:04:40,390
drawn the arrow here.

101
00:04:40,390 --> 00:04:44,810
Well, in that circumstance, we
have that the integral around

102
00:04:44,810 --> 00:04:50,790
C of G dot dr by Stokes'
Theorem is equal to the

103
00:04:50,790 --> 00:05:03,030
surface integral over S1 of curl
of G dot n with respect

104
00:05:03,030 --> 00:05:04,970
to surface area.

105
00:05:04,970 --> 00:05:05,210
Right?

106
00:05:05,210 --> 00:05:06,290
So this is just Stokes'
Theorem.

107
00:05:06,290 --> 00:05:09,380
Stokes' Theorem says the line
integral of G around the

108
00:05:09,380 --> 00:05:12,945
boundary curve is equal to the
surface integral of the curl

109
00:05:12,945 --> 00:05:15,710
of G over the region,
provided all of our

110
00:05:15,710 --> 00:05:17,460
orientations are correct.

111
00:05:17,460 --> 00:05:17,870
OK.

112
00:05:17,870 --> 00:05:21,470
Well, we know though what
curl of G is, because by

113
00:05:21,470 --> 00:05:27,952
assumption, F is equal to curl
of G. OK, so this is equal to

114
00:05:27,952 --> 00:05:36,050
the surface integral over
S1 of F dot n dS.

115
00:05:36,050 --> 00:05:38,300
So in the first step, we
use Stokes' Theorem.

116
00:05:38,300 --> 00:05:42,300
In the second step, we use our
assumption that curl G is

117
00:05:42,300 --> 00:05:45,410
equal to F. Well, now what.

118
00:05:45,410 --> 00:05:46,550
But we know what F is.

119
00:05:46,550 --> 00:05:46,750
Right?

120
00:05:46,750 --> 00:05:49,430
F is this radial
field x, y, z.

121
00:05:49,430 --> 00:05:52,760
So F and n are pointing
in the same direction.

122
00:05:52,760 --> 00:05:55,580
They're parallel to each other.
n is a unit vector, so

123
00:05:55,580 --> 00:05:58,890
this is just the length of F.
This F dot n is just the

124
00:05:58,890 --> 00:06:02,450
length of F. And since we're
on a sphere of radius b,

125
00:06:02,450 --> 00:06:04,310
this is just b.

126
00:06:04,310 --> 00:06:06,680
OK, so the integrand
is just b.

127
00:06:06,680 --> 00:06:17,940
So this is the integral over S1
of b dS, which is b times

128
00:06:17,940 --> 00:06:23,580
the area of S1.

129
00:06:23,580 --> 00:06:24,930
OK.

130
00:06:24,930 --> 00:06:27,240
One thing I'd like you to notice
is that in particular,

131
00:06:27,240 --> 00:06:29,250
this is a positive number.

132
00:06:29,250 --> 00:06:32,410
b is positive and the
area is positive.

133
00:06:32,410 --> 00:06:32,840
OK.

134
00:06:32,840 --> 00:06:35,150
So that's our first
interpretation.

135
00:06:35,150 --> 00:06:41,830
So we take our field G that
we suppose exists, and we

136
00:06:41,830 --> 00:06:45,550
integrate it around this curve
C, and we apply Stokes'

137
00:06:45,550 --> 00:06:49,720
Theorem, and then the fact that
we know what F is means

138
00:06:49,720 --> 00:06:54,290
that we know what F dot n is,
and so that makes our surface

139
00:06:54,290 --> 00:06:55,860
integral very easy to compute.

140
00:06:55,860 --> 00:06:59,730
And it turns out to be b times
the area of S, which I just

141
00:06:59,730 --> 00:07:03,360
happened to notice is
a positive number.

142
00:07:03,360 --> 00:07:04,140
OK.

143
00:07:04,140 --> 00:07:07,210
Well, now we can do the
same trick on the

144
00:07:07,210 --> 00:07:09,210
other half of the sphere.

145
00:07:09,210 --> 00:07:09,480
Right?

146
00:07:09,480 --> 00:07:11,160
So we just did the
top cap here.

147
00:07:11,160 --> 00:07:12,100
We did S1.

148
00:07:12,100 --> 00:07:14,920
So now we have the bottom
cap, or whatever.

149
00:07:14,920 --> 00:07:18,180
All the rest of the
sphere, S2.

150
00:07:18,180 --> 00:07:18,450
OK.

151
00:07:18,450 --> 00:07:23,050
So we can also get the integral
over C of G dot dr.

152
00:07:23,050 --> 00:07:25,200
We can interpret it in terms
of Stokes' Theorem.

153
00:07:25,200 --> 00:07:28,530
But notice then that in
C we still want to

154
00:07:28,530 --> 00:07:29,390
use the same normal.

155
00:07:29,390 --> 00:07:31,140
We like outwards pointing
normals.

156
00:07:31,140 --> 00:07:34,190
So we're going to have to orient
C the other way in

157
00:07:34,190 --> 00:07:36,480
order to make Stokes'
Theorem make sense.

158
00:07:36,480 --> 00:07:39,730
So let's walk over here
where we have

159
00:07:39,730 --> 00:07:40,760
some empty board space.

160
00:07:40,760 --> 00:07:42,160
So we want to orient
C the other way.

161
00:07:42,160 --> 00:07:45,730
So in other words, we're going
to take the negative of this

162
00:07:45,730 --> 00:07:47,070
line integral.

163
00:07:47,070 --> 00:07:53,170
So it's minus G dot dr. And if
we apply Stokes' Theorem to

164
00:07:53,170 --> 00:07:56,940
this line integral-- so this is
the same line integral, but

165
00:07:56,940 --> 00:07:59,290
with the opposite orientation
on C and so with

166
00:07:59,290 --> 00:08:02,530
the opposite sign--

167
00:08:02,530 --> 00:08:08,720
by Stokes' Theorem, this is
equal to the integral over S2

168
00:08:08,720 --> 00:08:16,520
of the curl of G dot n dS.

169
00:08:16,520 --> 00:08:19,350
And so the way I've set this up,
this is still my outward

170
00:08:19,350 --> 00:08:20,730
pointing normal.

171
00:08:20,730 --> 00:08:21,550
OK.

172
00:08:21,550 --> 00:08:25,580
But again, we can use our
assumption and we can get curl

173
00:08:25,580 --> 00:08:30,170
of G is equal to F, because
we're assuming

174
00:08:30,170 --> 00:08:31,760
that G has this property.

175
00:08:31,760 --> 00:08:43,220
So this is equal to the integral
over S2 of F dot n

176
00:08:43,220 --> 00:08:44,340
with respect to surface area.

177
00:08:44,340 --> 00:08:46,990
And again, n is the outward
pointing normal, and F is

178
00:08:46,990 --> 00:08:50,350
parallel to it.

179
00:08:50,350 --> 00:08:53,430
So this dot product is just the
length of F. The outward

180
00:08:53,430 --> 00:08:54,930
pointing unit normal.

181
00:08:54,930 --> 00:08:57,050
So this is just the length
of F, which is b.

182
00:08:57,050 --> 00:09:04,120
So this is equal to b times
the area of S2,

183
00:09:04,120 --> 00:09:06,880
which is also positive.

184
00:09:06,880 --> 00:09:08,730
So what have we just shown?

185
00:09:08,730 --> 00:09:12,150
Well, we started from the
assumption that there exists a

186
00:09:12,150 --> 00:09:15,800
G such that F is the curl of
G. And starting from that

187
00:09:15,800 --> 00:09:17,480
assumption--

188
00:09:17,480 --> 00:09:18,430
let's look--

189
00:09:18,430 --> 00:09:25,940
we showed that the line integral
around C of G dot dr

190
00:09:25,940 --> 00:09:29,170
is equal to some positive
number.

191
00:09:29,170 --> 00:09:34,580
And we also showed, over here,
that the negative of the line

192
00:09:34,580 --> 00:09:39,300
integral of G around C is equal
to some positive number.

193
00:09:39,300 --> 00:09:40,700
Well, this is clearly absurd.

194
00:09:40,700 --> 00:09:42,420
That can't be true.

195
00:09:42,420 --> 00:09:45,950
So starting from our assumption
that F was the curl

196
00:09:45,950 --> 00:09:48,990
of G-- that there is a G such
the F is the curl of G--

197
00:09:48,990 --> 00:09:50,590
we reached an absurd
conclusion.

198
00:09:50,590 --> 00:09:52,970
We reached a conclusion that
the same number is both

199
00:09:52,970 --> 00:09:55,210
positive and negative.

200
00:09:55,210 --> 00:09:56,280
But that can't happen.

201
00:09:56,280 --> 00:09:58,730
So that means our premise
had to be false.

202
00:09:58,730 --> 00:09:59,360
OK.

203
00:09:59,360 --> 00:10:00,610
So this is a contradiction.

204
00:10:00,610 --> 00:10:22,284

205
00:10:22,284 --> 00:10:24,230
So our assumption is false.

206
00:10:24,230 --> 00:10:26,690
And our assumption that we used
to get this whole thing

207
00:10:26,690 --> 00:10:31,170
started was that F was the curl
of some G. All right.

208
00:10:31,170 --> 00:10:32,310
So what have we shown?

209
00:10:32,310 --> 00:10:35,530
So we used a nice argument here
with Stokes' Theorem in

210
00:10:35,530 --> 00:10:39,080
order to show that certain
fields aren't the curl of

211
00:10:39,080 --> 00:10:39,930
other fields.

212
00:10:39,930 --> 00:10:43,370
So Stokes' Theorem limits the
kind of fields that can be

213
00:10:43,370 --> 00:10:44,630
curls of other fields.

214
00:10:44,630 --> 00:10:48,170
Now perhaps, you may have
thought of other theorems that

215
00:10:48,170 --> 00:10:50,520
you can use that also
limit what sorts of

216
00:10:50,520 --> 00:10:51,930
fields can be curls.

217
00:10:51,930 --> 00:10:55,090
And so there are other ways to
reach this true conclusion

218
00:10:55,090 --> 00:10:56,500
that our field F--

219
00:10:56,500 --> 00:10:58,760
whose components are x,
y, and z-- is not

220
00:10:58,760 --> 00:11:00,310
the curl of any field.

221
00:11:00,310 --> 00:11:02,700
This isn't the only way to
reach that conclusion.

222
00:11:02,700 --> 00:11:05,680
But this is a nice way that
shows that Stokes' Theorem

223
00:11:05,680 --> 00:11:08,820
puts some limitations on what
fields can behave like if

224
00:11:08,820 --> 00:11:10,380
they're going to be curls.

225
00:11:10,380 --> 00:11:12,190
I'll stop there.

226
00:11:12,190 --> 00:11:12,472