1
00:00:00,000 --> 00:00:07,880

2
00:00:07,880 --> 00:00:08,330
JOEL LEWIS: Hi.

3
00:00:08,330 --> 00:00:09,870
Welcome back to recitation.

4
00:00:09,870 --> 00:00:12,040
In lecture, you've been learning
about the divergence

5
00:00:12,040 --> 00:00:15,990
theorem, also known as Gauss'
theorem, and flux, and all

6
00:00:15,990 --> 00:00:16,700
that good stuff.

7
00:00:16,700 --> 00:00:21,300
So I have a nice exercise
on it for you here.

8
00:00:21,300 --> 00:00:24,780
So what I want is I want you to
take F, and I want it to be

9
00:00:24,780 --> 00:00:30,160
the field whose components are
x over rho cubed, y over rho

10
00:00:30,160 --> 00:00:31,970
cubed, and z over rho cubed.

11
00:00:31,970 --> 00:00:36,260
So here, rho is your usual rho
from spherical coordinates.

12
00:00:36,260 --> 00:00:38,420
Rho is equal to the square
root of x squared plus y

13
00:00:38,420 --> 00:00:39,990
squared plus z squared.

14
00:00:39,990 --> 00:00:43,520
And I want S to be the surface
of the box whose vertices are

15
00:00:43,520 --> 00:00:47,640
plus or minus 2, plus or minus
2, plus or minus 2.

16
00:00:47,640 --> 00:00:49,880
So it's a cubicle box.

17
00:00:49,880 --> 00:00:52,140
So what I'd like you to do, is
first in part a, I'd like you

18
00:00:52,140 --> 00:00:56,380
to show that the divergence of
F is 0, wherever the field F

19
00:00:56,380 --> 00:00:57,890
is defined.

20
00:00:57,890 --> 00:01:00,860
In part b, what I'd like you to
think about is whether we

21
00:01:00,860 --> 00:01:05,530
can conclude from that, that the
flux through the surface

22
00:01:05,530 --> 00:01:07,205
of S is equal to 0.

23
00:01:07,205 --> 00:01:08,040
All right.

24
00:01:08,040 --> 00:01:13,730
And in part c, what I'd like
you to do is to use the

25
00:01:13,730 --> 00:01:15,990
extended version of Gauss'
theorem-- or the extended

26
00:01:15,990 --> 00:01:18,050
version of the divergence
theorem--

27
00:01:18,050 --> 00:01:20,960
in order to actually compute
the flux through S by

28
00:01:20,960 --> 00:01:22,240
computing an integral.

29
00:01:22,240 --> 00:01:30,640
So why don't you pause the video
for a couple of minutes,

30
00:01:30,640 --> 00:01:32,350
work out this problem,
come back, and we

31
00:01:32,350 --> 00:01:33,600
can work it out together.

32
00:01:33,600 --> 00:01:41,260

33
00:01:41,260 --> 00:01:42,960
Hopefully you had some luck
with this problem.

34
00:01:42,960 --> 00:01:44,410
Let's get started.

35
00:01:44,410 --> 00:01:49,920
Part a asks you to compute the
divergence of F. So in order

36
00:01:49,920 --> 00:01:51,960
to compute that, we're going
to need to take the partial

37
00:01:51,960 --> 00:01:55,020
derivatives of the components of
F. And in order to do that,

38
00:01:55,020 --> 00:01:56,610
at some point I'm going to
need to take a partial

39
00:01:56,610 --> 00:01:57,540
derivative of rho.

40
00:01:57,540 --> 00:02:00,120
So let me first compute the
partial derivatives of rho,

41
00:02:00,120 --> 00:02:02,180
and that will save me a tiny
bit of work later.

42
00:02:02,180 --> 00:02:07,960
So rho is equal to the square
root of x squared plus y

43
00:02:07,960 --> 00:02:10,940
squared plus z squared.

44
00:02:10,940 --> 00:02:17,200
So partial rho partial x--

45
00:02:17,200 --> 00:02:19,970
well, you just apply your usual
chain rule here-- and I

46
00:02:19,970 --> 00:02:23,140
guess we get a half, but then we
get a 2 that cancels it, so

47
00:02:23,140 --> 00:02:26,390
I think this works out to x
divided by the square root of

48
00:02:26,390 --> 00:02:28,335
x squared plus y squared plus
z squared, so that's

49
00:02:28,335 --> 00:02:29,915
x divided by rho.

50
00:02:29,915 --> 00:02:30,440
All right.

51
00:02:30,440 --> 00:02:33,990
And I'm just going to keep rho
around here, because otherwise

52
00:02:33,990 --> 00:02:35,650
I have to write out the square
root of x squared plus y

53
00:02:35,650 --> 00:02:37,390
squared plus z squared over and
over again, and this is

54
00:02:37,390 --> 00:02:40,620
going to save me some effort
and would save you

55
00:02:40,620 --> 00:02:42,450
some effort as well.

56
00:02:42,450 --> 00:02:42,830
So OK.

57
00:02:42,830 --> 00:02:44,060
So this is rho.

58
00:02:44,060 --> 00:02:45,310
So this is d rho dx.

59
00:02:45,310 --> 00:02:54,840

60
00:02:54,840 --> 00:02:58,220
So we want to take the x partial
of the first component

61
00:02:58,220 --> 00:03:06,320
of F. So that's the x partial
of x over rho cubed.

62
00:03:06,320 --> 00:03:06,990
OK.

63
00:03:06,990 --> 00:03:10,920
And you just apply your
usual quotient rule,

64
00:03:10,920 --> 00:03:11,880
so what do we get?

65
00:03:11,880 --> 00:03:13,970
We get the derivative
of the top.

66
00:03:13,970 --> 00:03:18,540
So that's rho cubed minus--

67
00:03:18,540 --> 00:03:22,030
OK, so the top is x times the
derivative of the bottom,

68
00:03:22,030 --> 00:03:27,160
which is going to be 3 rho
squared times x over rho--

69
00:03:27,160 --> 00:03:28,890
so that's 3--

70
00:03:28,890 --> 00:03:34,430
so we have an x-- so it's 3x
squared rho, divided by the

71
00:03:34,430 --> 00:03:36,850
bottom squared, which
is rho to the sixth.

72
00:03:36,850 --> 00:03:39,500
And I guess there's a common
factor of rho everywhere that

73
00:03:39,500 --> 00:03:40,700
we can cancel out.

74
00:03:40,700 --> 00:03:47,560
So this is equal to rho squared
minus 3x squared

75
00:03:47,560 --> 00:03:50,490
divided by rho to the fifth.

76
00:03:50,490 --> 00:03:50,990
OK.

77
00:03:50,990 --> 00:03:54,500
So that's the x-partial
derivative of the first

78
00:03:54,500 --> 00:03:56,070
component of F.

79
00:03:56,070 --> 00:03:58,540
Now we need the y-partial
derivative of the second

80
00:03:58,540 --> 00:04:00,910
component of F, and the
z-partial derivative of the

81
00:04:00,910 --> 00:04:04,450
third component of F. But if you
go and look back at what

82
00:04:04,450 --> 00:04:06,810
the formula for F was, you see
that this is a very, very

83
00:04:06,810 --> 00:04:08,120
symmetric formula.

84
00:04:08,120 --> 00:04:11,290
So in order to get from the
first component to the second

85
00:04:11,290 --> 00:04:14,870
component, we just change x to
y, and to get from the second

86
00:04:14,870 --> 00:04:16,990
component to the third, we just
change y to z, because of

87
00:04:16,990 --> 00:04:19,820
course rho treats x,
y, and z the same.

88
00:04:19,820 --> 00:04:20,650
So what does that mean?

89
00:04:20,650 --> 00:04:22,560
Well, that means that the
partial derivatives are easy

90
00:04:22,560 --> 00:04:22,910
to compute.

91
00:04:22,910 --> 00:04:28,340
Having computed this x-partial
derivative, we also get that

92
00:04:28,340 --> 00:04:36,610
partial over partial y of the
second component-- which is y

93
00:04:36,610 --> 00:04:38,720
over rho cubed--

94
00:04:38,720 --> 00:04:44,310
is equal to rho squared
minus 3y squared,

95
00:04:44,310 --> 00:04:46,410
over rho to the fifth.

96
00:04:46,410 --> 00:04:55,180
And the last one we get, partial
over partial z of z

97
00:04:55,180 --> 00:05:02,000
over rho cubed is equal
to rho squared minus--

98
00:05:02,000 --> 00:05:03,440
I'm getting a little
cramped here--

99
00:05:03,440 --> 00:05:08,530
3z squared, over rho
to the fifth.

100
00:05:08,530 --> 00:05:17,210
And so adding these up, we get
that div F is equal to the sum

101
00:05:17,210 --> 00:05:18,980
of those three things.

102
00:05:18,980 --> 00:05:20,000
So let's see what we've got.

103
00:05:20,000 --> 00:05:21,620
We've got a 3--

104
00:05:21,620 --> 00:05:24,200
so the denominators are
all rho to the fifth.

105
00:05:24,200 --> 00:05:28,520
And we've got 3 rho squared
minus 3x squared minus 3y

106
00:05:28,520 --> 00:05:30,070
squared minus 3z squared.

107
00:05:30,070 --> 00:05:35,520
So this is equal to 3 rho
squared minus 3x squared minus

108
00:05:35,520 --> 00:05:41,780
3y squared minus 3z squared,
all over rho to the fifth.

109
00:05:41,780 --> 00:05:44,960
But of course, rho squared is x
squared plus y squared plus

110
00:05:44,960 --> 00:05:47,260
z squared, so this numerator
is just 0.

111
00:05:47,260 --> 00:05:49,660
So this is equal to 0.

112
00:05:49,660 --> 00:05:50,090
OK.

113
00:05:50,090 --> 00:05:52,005
Which is what we thought
it should be.

114
00:05:52,005 --> 00:05:52,650
All right.

115
00:05:52,650 --> 00:05:53,170
Good.

116
00:05:53,170 --> 00:05:54,250
So that's part a.

117
00:05:54,250 --> 00:05:56,980
We just computed the partial
derivatives of F, and then

118
00:05:56,980 --> 00:05:59,180
added them together to
get the divergence.

119
00:05:59,180 --> 00:06:01,480
And we found that, in fact,
yes, the divergence

120
00:06:01,480 --> 00:06:02,610
was equal to 0.

121
00:06:02,610 --> 00:06:02,970
Great.

122
00:06:02,970 --> 00:06:04,070
So that's part a.

123
00:06:04,070 --> 00:06:06,580
So let's go look at
what part b was.

124
00:06:06,580 --> 00:06:10,490
Part b asks, can we conclude
that the flux through the

125
00:06:10,490 --> 00:06:12,305
surface S is 0?

126
00:06:12,305 --> 00:06:12,930
All right.

127
00:06:12,930 --> 00:06:15,180
Now remember what the divergence
theorem says.

128
00:06:15,180 --> 00:06:18,860
The divergence theorem says
that the flux through a

129
00:06:18,860 --> 00:06:23,140
surface of a field is equal to
the triple integral of the

130
00:06:23,140 --> 00:06:26,390
divergence of that field over
the interior, provided the

131
00:06:26,390 --> 00:06:28,400
field is defined and
differentiable

132
00:06:28,400 --> 00:06:29,730
and nice, or whatever.

133
00:06:29,730 --> 00:06:31,610
Everywhere inside.

134
00:06:31,610 --> 00:06:32,170
OK?

135
00:06:32,170 --> 00:06:34,730
But this field has a problem.

136
00:06:34,730 --> 00:06:39,240
Almost everywhere, this field
is nicely behaved, but at 0,

137
00:06:39,240 --> 00:06:40,130
we have a real problem.

138
00:06:40,130 --> 00:06:42,110
We're dividing by 0.

139
00:06:42,110 --> 00:06:42,340
Right?

140
00:06:42,340 --> 00:06:45,090
So this field is not
defined at 0.

141
00:06:45,090 --> 00:06:49,230
So there's a single point in the
middle of this cube where

142
00:06:49,230 --> 00:06:51,310
this field behaves badly.

143
00:06:51,310 --> 00:06:53,460
And that means we can't
apply the divergence

144
00:06:53,460 --> 00:06:55,560
theorem inside this cube.

145
00:06:55,560 --> 00:06:58,100
So since we can't apply the
divergence theorem, we aren't

146
00:06:58,100 --> 00:07:02,260
allowed to conclude immediately
that the flux

147
00:07:02,260 --> 00:07:04,230
through this surface is 0.

148
00:07:04,230 --> 00:07:04,960
OK.

149
00:07:04,960 --> 00:07:07,070
So the answer is no.

150
00:07:07,070 --> 00:07:09,750
We can't conclude that the flux
through S is 0, because

151
00:07:09,750 --> 00:07:11,765
one of the hypotheses
of the divergence

152
00:07:11,765 --> 00:07:13,300
theorem isn't satisfied.

153
00:07:13,300 --> 00:07:14,700
Namely, the field isn't defined

154
00:07:14,700 --> 00:07:19,650
everywhere inside the surface.

155
00:07:19,650 --> 00:07:20,590
OK.

156
00:07:20,590 --> 00:07:25,930
So the answer to b is no.

157
00:07:25,930 --> 00:07:27,390
OK, I'm just going
to write that.

158
00:07:27,390 --> 00:07:30,590
But it's no because the
hypotheses aren't satisfied.

159
00:07:30,590 --> 00:07:33,460
OK, so now let's
look at part c.

160
00:07:33,460 --> 00:07:39,570
So part c suggests, we can't
conclude that the flux is 0.

161
00:07:39,570 --> 00:07:42,030
So we still want to know
what the flux is.

162
00:07:42,030 --> 00:07:45,345
That's still an interesting
question, so part c suggests

163
00:07:45,345 --> 00:07:52,080
that maybe you can still use the
extended Gauss' theorem to

164
00:07:52,080 --> 00:07:54,290
compute what this flux is.

165
00:07:54,290 --> 00:07:56,610
So let's think about how
we could do that.

166
00:07:56,610 --> 00:07:59,250
So remember what extended
Gauss' theorem says?

167
00:07:59,250 --> 00:08:01,430
Or extended divergence
theorem.

168
00:08:01,430 --> 00:08:04,860
I'm going to try and just say
Gauss' theorem from now on so

169
00:08:04,860 --> 00:08:06,000
I stop having to say both.

170
00:08:06,000 --> 00:08:07,590
But I mean both.

171
00:08:07,590 --> 00:08:09,240
I mean, they're the same
theorem, right?

172
00:08:09,240 --> 00:08:10,490
OK.

173
00:08:10,490 --> 00:08:13,660

174
00:08:13,660 --> 00:08:17,410
So Gauss' theorem says, when you
have a surface bounding a

175
00:08:17,410 --> 00:08:21,380
region, the flux through the
surface is equal to the triple

176
00:08:21,380 --> 00:08:23,380
integral of divergence over
the region, provided

177
00:08:23,380 --> 00:08:26,850
everything is well-defined
and nice.

178
00:08:26,850 --> 00:08:31,790
Extended Gauss' theorem says,
this is still true if your

179
00:08:31,790 --> 00:08:35,120
region has more than
one boundary.

180
00:08:35,120 --> 00:08:39,980
So for example, if your region
is a hollow something--

181
00:08:39,980 --> 00:08:44,320
so if it's a spherical shell
that has an outside sphere and

182
00:08:44,320 --> 00:08:46,270
an inside sphere--

183
00:08:46,270 --> 00:08:49,160
then extended Gauss' theorem
says, OK, so

184
00:08:49,160 --> 00:08:49,970
you do the same thing.

185
00:08:49,970 --> 00:08:52,650
You take the triple integral
of the divergence over the

186
00:08:52,650 --> 00:08:53,710
solid region.

187
00:08:53,710 --> 00:08:56,810
And then you take the flux, but
you add up the flux over

188
00:08:56,810 --> 00:08:57,970
all of the boundary pieces.

189
00:08:57,970 --> 00:09:02,000
So you add up the flux over the
outside boundary surface,

190
00:09:02,000 --> 00:09:03,540
and also, if there is
one, through any

191
00:09:03,540 --> 00:09:05,340
other boundary surface.

192
00:09:05,340 --> 00:09:07,330
OK?

193
00:09:07,330 --> 00:09:09,420
And those two things
are equal.

194
00:09:09,420 --> 00:09:13,920
So the total flux through all
of the boundary surface is

195
00:09:13,920 --> 00:09:17,240
equal to the integral of
divergence over the whole

196
00:09:17,240 --> 00:09:19,310
region bounded by
those surfaces.

197
00:09:19,310 --> 00:09:20,942
So how are we going
to use this?

198
00:09:20,942 --> 00:09:26,810

199
00:09:26,810 --> 00:09:29,790
We're trying to compute the
flux through a surface.

200
00:09:29,790 --> 00:09:32,210
OK, but we don't want to compute
a double integral if

201
00:09:32,210 --> 00:09:33,220
we can avoid it.

202
00:09:33,220 --> 00:09:35,200
We don't want to compute
the surface integral.

203
00:09:35,200 --> 00:09:38,170
So what we'd like to do is we'd
like to find a convenient

204
00:09:38,170 --> 00:09:44,390
region over which to compute
this integral to put us in a

205
00:09:44,390 --> 00:09:47,030
situation where we can apply
extended Gauss' theorem.

206
00:09:47,030 --> 00:09:50,060
We can't use just the inside of
the cube, so we want some

207
00:09:50,060 --> 00:09:51,160
other region.

208
00:09:51,160 --> 00:09:55,040
So what we're going to do is
we're going to walk over here.

209
00:09:55,040 --> 00:09:56,620
There are many possible things
you could do, but

210
00:09:56,620 --> 00:09:57,970
this is a nice one.

211
00:09:57,970 --> 00:09:59,060
All right.

212
00:09:59,060 --> 00:10:01,530
One thing you could do is you
could take a big sphere.

213
00:10:01,530 --> 00:10:02,290
Take a big sphere.

214
00:10:02,290 --> 00:10:04,440
So we've got our cube here.

215
00:10:04,440 --> 00:10:08,320
This is the point 2, 2, 2, and
this is the point 2, 2, minus

216
00:10:08,320 --> 00:10:09,900
2, and so on.

217
00:10:09,900 --> 00:10:14,800
So we've taken a big sphere of
radius R-- for some big R that

218
00:10:14,800 --> 00:10:20,280
contains our surface S that
we're interested in-- that

219
00:10:20,280 --> 00:10:22,180
completely contains the cube.

220
00:10:22,180 --> 00:10:23,500
OK?

221
00:10:23,500 --> 00:10:24,700
So why have we done that?

222
00:10:24,700 --> 00:10:25,950
Extended Gauss' theorem.

223
00:10:25,950 --> 00:10:39,750

224
00:10:39,750 --> 00:10:42,360
OK, so what does extended Gauss'
theorem say for the

225
00:10:42,360 --> 00:10:45,610
region between the sphere
and this cube.

226
00:10:45,610 --> 00:10:46,120
All right.

227
00:10:46,120 --> 00:10:51,710
So our cube is named S.
Let's call our sphere

228
00:10:51,710 --> 00:10:55,290
S2, because why not?

229
00:10:55,290 --> 00:10:55,770
OK.

230
00:10:55,770 --> 00:11:08,970
And let's call the solid
region between

231
00:11:08,970 --> 00:11:13,710
the cube and sphere--

232
00:11:13,710 --> 00:11:15,620
just for convenience, let's
give it a name--

233
00:11:15,620 --> 00:11:16,730
so--

234
00:11:16,730 --> 00:11:20,100
we often call solid regions
D-- so let's call it D.

235
00:11:20,100 --> 00:11:23,710
So it's this spherical region,
but it has a cubical hole in

236
00:11:23,710 --> 00:11:24,890
the middle of it.

237
00:11:24,890 --> 00:11:26,110
OK.

238
00:11:26,110 --> 00:11:29,070
So what does extended
Gauss' theorem say?

239
00:11:29,070 --> 00:11:34,820
So extended Gauss' theorem says
that the triple integral

240
00:11:34,820 --> 00:11:45,130
over D of the divergence of F dV
is equal to the sum of the

241
00:11:45,130 --> 00:11:48,650
fluxes through each
of the surfaces.

242
00:11:48,650 --> 00:11:52,990
But for this, we want the flux
out of the solid region.

243
00:11:52,990 --> 00:11:55,910

244
00:11:55,910 --> 00:11:59,340
So for the sphere, the flux
out of the inside of the

245
00:11:59,340 --> 00:12:01,290
sphere is the flux out
of the sphere.

246
00:12:01,290 --> 00:12:11,790
So that's integral over S2 of
F dot n, d surface area.

247
00:12:11,790 --> 00:12:16,150
But for the cube, the flux out
of this region is the flux

248
00:12:16,150 --> 00:12:18,220
into the cube.

249
00:12:18,220 --> 00:12:19,170
Right?

250
00:12:19,170 --> 00:12:21,860
Out here, you're living in a
region outside the cube, so

251
00:12:21,860 --> 00:12:24,730
when you leave that region,
you're going into the cube.

252
00:12:24,730 --> 00:12:28,550
So this is the negative of the
flux that we really want.

253
00:12:28,550 --> 00:12:39,290
So this is minus the flux
through the cube of F dot n,

254
00:12:39,290 --> 00:12:40,950
with respect to surface area.

255
00:12:40,950 --> 00:12:43,700
So remember, the signs here
are different, because I'm

256
00:12:43,700 --> 00:12:46,750
taking this normal to be the
outward pointing normal to

257
00:12:46,750 --> 00:12:47,820
both surfaces.

258
00:12:47,820 --> 00:12:49,900
The normal that points
away from the origin.

259
00:12:49,900 --> 00:12:54,940
But the normal pointing away
from the origin on the cube is

260
00:12:54,940 --> 00:12:58,000
the normal that points into the
solid region instead of

261
00:12:58,000 --> 00:12:59,980
the normal that points out
of the solid region.

262
00:12:59,980 --> 00:13:02,200
So that's why this
minus is here.

263
00:13:02,200 --> 00:13:03,040
OK.

264
00:13:03,040 --> 00:13:03,720
Whew.

265
00:13:03,720 --> 00:13:04,970
All right, so what
does this mean?

266
00:13:04,970 --> 00:13:08,080

267
00:13:08,080 --> 00:13:10,910
So first of all, F is
well-defined everywhere in

268
00:13:10,910 --> 00:13:13,000
this region D. The only
place F was badly

269
00:13:13,000 --> 00:13:13,890
behaved was the origin.

270
00:13:13,890 --> 00:13:16,395
And this region doesn't contain
it, which is why this

271
00:13:16,395 --> 00:13:17,920
trick works.

272
00:13:17,920 --> 00:13:21,550
So we've already computed that
the divergence of F is 0

273
00:13:21,550 --> 00:13:21,870
everywhere.

274
00:13:21,870 --> 00:13:22,370
It's defined.

275
00:13:22,370 --> 00:13:24,540
So it's 0 on all of D,
and so this triple

276
00:13:24,540 --> 00:13:26,470
integral is just 0.

277
00:13:26,470 --> 00:13:31,700
So if this triple integral is 0,
that means we can just add

278
00:13:31,700 --> 00:13:34,210
the thing that we're interested
in to both sides,

279
00:13:34,210 --> 00:13:45,470
and we get that the surface
integral over the cube of F

280
00:13:45,470 --> 00:13:49,950
dot n, with respect to surface
area, is equal to the surface

281
00:13:49,950 --> 00:13:55,540
integral over the sphere
of F dot n, with

282
00:13:55,540 --> 00:13:57,340
respect to surface area.

283
00:13:57,340 --> 00:13:58,640
OK.

284
00:13:58,640 --> 00:14:03,710
So we've converted this original
integral-- our flux

285
00:14:03,710 --> 00:14:06,080
integral that we're interested
in-- and we found that it's

286
00:14:06,080 --> 00:14:09,400
equal to this separate
flux integral

287
00:14:09,400 --> 00:14:10,450
over a different surface.

288
00:14:10,450 --> 00:14:12,740
This time over a big sphere.

289
00:14:12,740 --> 00:14:17,310
OK, so that's nice.

290
00:14:17,310 --> 00:14:20,000
Why do we want to do that?

291
00:14:20,000 --> 00:14:23,090
Well, we want to do that because
F is a really nicely

292
00:14:23,090 --> 00:14:25,980
behaved field with respect
to a sphere.

293
00:14:25,980 --> 00:14:28,150
F is a radial field.

294
00:14:28,150 --> 00:14:32,790
So F dot n is really
easy to understand.

295
00:14:32,790 --> 00:14:39,630
F dot n: n is a unit normal
and F is a radial field.

296
00:14:39,630 --> 00:14:45,820
So on a sphere, the normal
is radial, right?

297
00:14:45,820 --> 00:14:48,590
It's parallel to the
position vector.

298
00:14:48,590 --> 00:14:49,800
And F is radial.

299
00:14:49,800 --> 00:14:52,260
So they're both pointing in
exactly the same direction.

300
00:14:52,260 --> 00:14:56,340
So when you take that dot
product, n is the unit vector

301
00:14:56,340 --> 00:14:59,740
in the same direction as F, so
when you dot that with F, you

302
00:14:59,740 --> 00:15:03,440
just get the length of F. OK,
so what does that mean?

303
00:15:03,440 --> 00:15:07,240
That means over here, this
integrand is really easy to

304
00:15:07,240 --> 00:15:08,510
understand.

305
00:15:08,510 --> 00:15:08,710
OK?

306
00:15:08,710 --> 00:15:15,620
This integrand F dot n on the
sphere is just equal to the

307
00:15:15,620 --> 00:15:19,960
length of the vector F.

308
00:15:19,960 --> 00:15:22,120
Now what is the length
of the vector F?

309
00:15:22,120 --> 00:15:23,870
Well, we know what F is.

310
00:15:23,870 --> 00:15:28,140
It's x over rho cubed, i hat
plus y over rho cubed, j hat

311
00:15:28,140 --> 00:15:30,720
plus z over rho cubed, k hat.

312
00:15:30,720 --> 00:15:33,790
So OK, so you compute the length
of that vector, and

313
00:15:33,790 --> 00:15:34,480
what do you get?

314
00:15:34,480 --> 00:15:38,020
Well, it's exactly 1
over rho squared.

315
00:15:38,020 --> 00:15:38,740
OK.

316
00:15:38,740 --> 00:15:40,350
But we said that this
is a sphere.

317
00:15:40,350 --> 00:15:42,650
I guess I didn't
write it down.

318
00:15:42,650 --> 00:15:44,260
Let me write it down
right here.

319
00:15:44,260 --> 00:15:47,910
This is a sphere whose radius
is big R. It doesn't really

320
00:15:47,910 --> 00:15:51,230
matter very much what R we
choose, we just want it to be

321
00:15:51,230 --> 00:15:53,170
big enough so that it contains
the whole cube.

322
00:15:53,170 --> 00:15:56,320
If you said this a sphere
of radius 10, that would

323
00:15:56,320 --> 00:15:57,420
completely do the trick.

324
00:15:57,420 --> 00:15:59,300
That would be totally fine.

325
00:15:59,300 --> 00:16:04,120
OK, so the radius of the sphere
is big R, so the length

326
00:16:04,120 --> 00:16:06,590
of the field-- we said
back over here--

327
00:16:06,590 --> 00:16:09,900
is 1 over R squared.

328
00:16:09,900 --> 00:16:18,330
The length of the vector F. So
this flux integral then, is

329
00:16:18,330 --> 00:16:20,070
the integral over the sphere--

330
00:16:20,070 --> 00:16:20,690
S2--

331
00:16:20,690 --> 00:16:21,940
of a constant.

332
00:16:21,940 --> 00:16:25,100

333
00:16:25,100 --> 00:16:27,020
So it's the integral over
the sphere of 1

334
00:16:27,020 --> 00:16:29,440
over R squared, dS.

335
00:16:29,440 --> 00:16:32,400
But when you integrate a
constant over a surface, what

336
00:16:32,400 --> 00:16:36,060
you get is just that constant
times the surface area.

337
00:16:36,060 --> 00:16:37,370
Well, what's the surface area?

338
00:16:37,370 --> 00:16:38,090
This is a sphere.

339
00:16:38,090 --> 00:16:39,955
It's easy to understand
its surface area.

340
00:16:39,955 --> 00:16:45,690
Its surface area is
4 pi R squared.

341
00:16:45,690 --> 00:16:46,300
Right?

342
00:16:46,300 --> 00:16:51,680
So this is equal to the surface
area, so that's 4 pi R

343
00:16:51,680 --> 00:16:54,160
squared, times whatever
that constant was.

344
00:16:54,160 --> 00:16:56,220
So the constant was
1 over R squared.

345
00:16:56,220 --> 00:16:58,460
And so the R squared's cancel.

346
00:16:58,460 --> 00:16:58,720
Right?

347
00:16:58,720 --> 00:17:01,700
This is why it didn't matter
what R we chose, because

348
00:17:01,700 --> 00:17:03,810
they're just going to cancel
at the end, anyhow.

349
00:17:03,810 --> 00:17:06,630

350
00:17:06,630 --> 00:17:10,240
OK, so those cancel, and
we're left with 4 pi.

351
00:17:10,240 --> 00:17:15,880
So let's just quickly recap what
we did in this part c.

352
00:17:15,880 --> 00:17:18,630
We're looking to compute
the flux over the cube.

353
00:17:18,630 --> 00:17:22,390
But it's a kind of unpleasant
integral we'd have to compute

354
00:17:22,390 --> 00:17:24,980
to total up the fluxes over
these various, different

355
00:17:24,980 --> 00:17:26,150
faces and so on.

356
00:17:26,150 --> 00:17:29,510
So instead, we had this clever
idea that we'll apply the

357
00:17:29,510 --> 00:17:32,410
divergence theorem to replace
the cube with a

358
00:17:32,410 --> 00:17:36,110
more congenial surface.

359
00:17:36,110 --> 00:17:39,160
Because this is a nice
radial vector field,

360
00:17:39,160 --> 00:17:40,020
that's our main hint.

361
00:17:40,020 --> 00:17:42,500
Because there was a rho
involved in the

362
00:17:42,500 --> 00:17:45,370
problem, if you will.

363
00:17:45,370 --> 00:17:49,420
So the surface that we choose
is some big sphere.

364
00:17:49,420 --> 00:17:53,270
And then we apply the extended
Gauss' theorem to the solid

365
00:17:53,270 --> 00:17:56,120
region between the cube
and the sphere.

366
00:17:56,120 --> 00:17:58,760
Outside the cube, but
inside the sphere.

367
00:17:58,760 --> 00:18:03,930
So because the divergence of the
field is 0, the extended

368
00:18:03,930 --> 00:18:07,540
Gauss' theorem tells us
that the two fluxes--

369
00:18:07,540 --> 00:18:11,645
the flux out of the cube and the
flux out of the sphere--

370
00:18:11,645 --> 00:18:14,300
are actually equal
to each other.

371
00:18:14,300 --> 00:18:18,060
But since the fluxes are
actually equal to each other,

372
00:18:18,060 --> 00:18:20,840
in order to compute the flux out
of the cube, it's enough

373
00:18:20,840 --> 00:18:23,270
to compute the flux
out of the sphere.

374
00:18:23,270 --> 00:18:23,580
OK.

375
00:18:23,580 --> 00:18:26,820
But computing the flux out of
the sphere is relatively easy,

376
00:18:26,820 --> 00:18:29,430
because on the sphere,
the integrand F

377
00:18:29,430 --> 00:18:31,870
dot n is just a constant.

378
00:18:31,870 --> 00:18:36,640
And so then we're integrating a
constant over the surface of

379
00:18:36,640 --> 00:18:39,290
a sphere, and that just gives
us the surface area of the

380
00:18:39,290 --> 00:18:42,160
sphere times that constant,
which is 4 pi R squared, times

381
00:18:42,160 --> 00:18:45,060
1 over R squared,
which is 4 pi.

382
00:18:45,060 --> 00:18:50,220
So the flux out of the cube then
is also equal to 4 pi.

383
00:18:50,220 --> 00:18:52,010
I'll stop there.

384
00:18:52,010 --> 00:18:52,199