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OK, so remember, 
we've seen Stokes theorem, 

8
00:00:29 --> 00:00:37
which says if I have a closed
curve bounding some surface,

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00:00:37 --> 00:00:40
S,
and I orient the curve and the

10
00:00:40 --> 00:00:44
surface compatible with each
other,

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00:00:44 --> 00:00:53
then I can compute the line
integral along C along my curve

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00:00:53 --> 00:00:57
in terms of,
instead, 

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00:00:57 --> 00:01:04
surface integral for flux of a
different vector field,

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00:01:04 --> 00:01:12
namely, curl f dot n dS.
OK, so that's the statement.

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00:01:12 --> 00:01:18
And, just to clarify a little
bit, so, again,

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00:01:18 --> 00:01:24
we've seen various kinds of
integrals.

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00:01:24 --> 00:01:26
So, line integrals we know how
to evaluate.

18
00:01:26 --> 00:01:30
They take place in a curve.
You express everything in terms

19
00:01:30 --> 00:01:32
of one variable,
and after substituting,

20
00:01:32 --> 00:01:36
you end up with a usual one
variable integral that you know

21
00:01:36 --> 00:01:40
how to evaluate.
And, surface integrals,

22
00:01:40 --> 00:01:44
we know also how to evaluate.
Namely, we've seen various

23
00:01:44 --> 00:01:47
formulas for ndS.
Once you have such a formula,

24
00:01:47 --> 00:01:50
due to the dot product with
this vector field,

25
00:01:50 --> 00:01:52
which is not the same as that
one.

26
00:01:52 --> 00:01:56
But it's a new vector field
that you can build out of f.

27
00:01:56 --> 00:02:00
You do the dot product.
You express everything in terms

28
00:02:00 --> 00:02:03
of your two integration
variables, and then you

29
00:02:03 --> 00:02:06
evaluate.
So, now, what does this have to

30
00:02:06 --> 00:02:12
do with various other things?
So, one thing I want to say has

31
00:02:12 --> 00:02:18
to do with how Stokes helps us
understand path independence,

32
00:02:18 --> 00:02:24
so, how it actually motivates
our criterion for gradient

33
00:02:24 --> 00:02:30
fields,
independence.

34
00:02:30 --> 00:02:35
OK, so, 
we've seen that if we have a

35
00:02:35 --> 00:02:40
vector field defined in a simply
connected region,

36
00:02:40 --> 00:02:43
and its curl is zero, 
then it's a gradient field, 

37
00:02:43 --> 00:02:47
and the line integral is path
independent.

38
00:02:47 --> 00:02:53
So, let me first define for you
when a simply connected region

39
00:02:53 --> 00:03:01
is.
So, we say that a region in

40
00:03:01 --> 00:03:17
space is simply connected -- --
if every closed loop inside this

41
00:03:17 --> 00:03:31
region bounds some surface again
inside this region.

42
00:03:31 --> 00:03:39
OK, so let me just give you
some examples just to clarify.

43
00:03:39 --> 00:03:46
So, for example,
let's say that I have a region

44
00:03:46 --> 00:03:52
that's the entire space with the
origin removed.

45
00:03:52 --> 00:04:00
OK, so space with the origin
removed, OK, you think it's

46
00:04:00 --> 00:04:06
simply connected?
Who thinks it's simply

47
00:04:06 --> 00:04:09
connected?
Who thinks it's not simply

48
00:04:09 --> 00:04:14
connected?
Let's think a little bit harder.

49
00:04:14 --> 00:04:17
Let's say that I take a loop
like this one,

50
00:04:17 --> 00:04:20
OK, it doesn't go through the
origin.

51
00:04:20 --> 00:04:24
Can I find a surface that's
bounded by this loop and that

52
00:04:24 --> 00:04:26
does not pass through the
origin?

53
00:04:26 --> 00:04:30
Yeah, I can take the sphere,
you know, for example,

54
00:04:30 --> 00:04:34
or anything that's just not
quite the disk?

55
00:04:34 --> 00:04:36
So, 
and similarly,

56
00:04:36 --> 00:04:39
if I take any other loop that
avoids the origin,

57
00:04:39 --> 00:04:42
I can find, actually,
a surface bounded by it that

58
00:04:42 --> 00:04:44
does not pass through the
origin.

59
00:04:44 --> 00:04:47
So, actually,
that's kind of a not so obvious

60
00:04:47 --> 00:04:49
theorem to prove,
but maybe intuitively,

61
00:04:49 --> 00:04:52
start by finding any surface.
Well, if that surface passes

62
00:04:52 --> 00:04:54
through the origin,
just wiggle it a little bit,

63
00:04:54 --> 00:04:56
you can make sure it doesn't
pass through the origin anymore.

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00:04:56 --> 00:05:00
Just push it a little bit.
So, in fact,

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00:05:00 --> 00:05:08
this is simply connected.
That was a trick question.

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00:05:08 --> 00:05:13
OK, now on the other hand, 
a good example of something

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00:05:13 --> 00:05:16
that is not simply connected is
if I take space,

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00:05:16 --> 00:05:36
and I remove the z axis -- --
that is not simply connected.

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00:05:36 --> 00:05:39
And, see, the reason is, 
if I look again, say, 

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00:05:39 --> 00:05:44
at the unit circle in the x
axis,

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00:05:44 --> 00:05:47
sorry, unit circle in the xy
plane,

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I mean, in the xy plane, 
so, if I try to find a surface

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00:05:52 --> 00:05:58
whose boundary is this disk,
well, it has to actually cross

74
00:05:58 --> 00:06:03
the z axis somewhere.
There's no way that I can find

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00:06:03 --> 00:06:09
a surface whose only boundary is
this curve, which doesn't hit

76
00:06:09 --> 00:06:13
the z axis anywhere.
Of course, you could try to use

77
00:06:13 --> 00:06:17
the same trick as there,
say, maybe we want to go up,

78
00:06:17 --> 00:06:19
up, up.
You know, let's start with a

79
00:06:19 --> 00:06:21
cylinder.
Well, the problem is you have

80
00:06:21 --> 00:06:24
to go infinitely far because the
z axis goes infinitely far.

81
00:06:24 --> 00:06:27
And, you'll never be able to
actually close your surface.

82
00:06:27 --> 00:06:30
So, the matter what kind of
trick you might want to use,

83
00:06:30 --> 00:06:34
it's actually a theorem in
topology that you cannot find a

84
00:06:34 --> 00:06:39
surface bounded by this disk
without intersecting the z axis.

85
00:06:39 --> 00:06:44
Yes?
Well, a doughnut shape

86
00:06:44 --> 00:06:47
certainly would stay away from
the z axis, but it wouldn't be a

87
00:06:47 --> 00:06:50
surface with boundary just this
guy.

88
00:06:50 --> 00:06:53
Right, it would have to have
either some other boundary.

89
00:06:53 --> 00:06:57
So, maybe what you have in mind
is some sort of doughnut shape

90
00:06:57 --> 00:07:01
like this that curves on itself,
and maybe comes back.

91
00:07:01 --> 00:07:05
Well, if you don't quite close
it all the way around,

92
00:07:05 --> 00:07:07
so I can try to,
indeed, draw some sort of

93
00:07:07 --> 00:07:10
doughnut here.
Well, if I don't quite close

94
00:07:10 --> 00:07:13
it, that it will have another
edge at the other end wherever I

95
00:07:13 --> 00:07:15
started.
If I close it completely,

96
00:07:15 --> 00:07:18
then this curve is no longer
its boundary because my surface

97
00:07:18 --> 00:07:20
lives on both sides of this
curve.

98
00:07:20 --> 00:07:22
See, I want a surface that
stops on this curve,

99
00:07:22 --> 00:07:25
and doesn't go beyond it.
And, nowhere else does it have

100
00:07:25 --> 00:07:28
that kind of behavior.
Everywhere else,

101
00:07:28 --> 00:07:33
it keeps going on.
So, 

102
00:07:33 --> 00:07:36
actually, I mean, 
maybe actually another way to

103
00:07:36 --> 00:07:40
convince yourself is to find a
counter example to the statement

104
00:07:40 --> 00:07:44
I'm going to make about vector
fields with curl zero and simply

105
00:07:44 --> 00:07:48
connected regions always being
conservative.

106
00:07:48 --> 00:07:52
So, what you can do is you can
take the example that we had in

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00:07:52 --> 00:07:55
one of our older problem sets.
That was a vector field in the

108
00:07:55 --> 00:07:58
plane.
But, you can also use it to

109
00:07:58 --> 00:08:02
define a vector field in space
just with no z component.

110
00:08:02 --> 00:08:05
That vector field is actually
defined everywhere except on the

111
00:08:05 --> 00:08:08
z axis, and it violates the
usual theorem that we would

112
00:08:08 --> 00:08:12
expect.
So, that's one way to check

113
00:08:12 --> 00:08:20
just for sure that this thing is
not simply connected.

114
00:08:20 --> 00:08:26
So, what's the statement I want
to make?

115
00:08:26 --> 00:08:40
So, recall we've seen if F is a
gradient field -- -- then its

116
00:08:40 --> 00:08:46
curl is zero.
That's just the fact that the

117
00:08:46 --> 00:08:49
mixed second partial derivatives
are equal.

118
00:08:49 --> 00:08:53
So, now, the converse is the
following theorem.

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00:08:53 --> 00:09:01
It says if the curl of F equals
zero in, sorry,

120
00:09:01 --> 00:09:09
and F is defined -- No,
is not the logical in which to

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00:09:09 --> 00:09:15
say it.
So, if F is defined in a simply

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00:09:15 --> 00:09:30
connected region,
and curl F is zero -- -- then F

123
00:09:30 --> 00:09:45
is a gradient field,
and the line integral for F is

124
00:09:45 --> 00:09:53
path independent -- -- F is
conservative,

125
00:09:53 --> 00:09:55
and so on, 
all the usual consequences.

126
00:09:55 --> 00:09:58
Remember, these are all
equivalent to each other,

127
00:09:58 --> 00:10:01
for example, 
because you can use path

128
00:10:01 --> 00:10:05
independence to define the
potential by doing the line

129
00:10:05 --> 00:10:08
integral of F.
OK, so where do we use the

130
00:10:08 --> 00:10:12
assumption of being defined in a
simply connected region?

131
00:10:12 --> 00:10:17
Well, the way which we will
prove this is to use Stokes

132
00:10:17 --> 00:10:20
theorem.
OK, so the proof,

133
00:10:20 --> 00:10:25
so just going to prove that the
line integral is path

134
00:10:25 --> 00:10:29
independent;
the others work the same way.

135
00:10:29 --> 00:10:34
OK, so let's assume that we
have a vector field whose curl

136
00:10:34 --> 00:10:38
is zero.
And, let's say that we have two

137
00:10:38 --> 00:10:44
curves, C1 and C2,
that go from some point P0 to

138
00:10:44 --> 00:10:49
some point P1,
the same point to the same

139
00:10:49 --> 00:10:54
point.
Well, we'd like to understand

140
00:10:54 --> 00:11:00
the line integral along C1,
say, minus the line integral

141
00:11:00 --> 00:11:04
along C2 to show that this is
zero.

142
00:11:04 --> 00:11:06
That's what we are trying to
prove.

143
00:11:06 --> 00:11:12
So, how will we compute that?
Well, the line integral along

144
00:11:12 --> 00:11:17
C1 minus C2, well,
let's just form a closed curve

145
00:11:17 --> 00:11:24
that is C1 minus C2.
OK, so let's call C,

146
00:11:24 --> 00:11:36
woops -- So that's equal to the
integral along C of f dot dr

147
00:11:36 --> 00:11:44
where C is C1 followed by C2
backwards.

148
00:11:44 --> 00:11:50
Now, C is a closed curve.
So, I can use Stokes theorem.

149
00:11:50 --> 00:11:52
Well, to be able to use Stokes
theorem, I need,

150
00:11:52 --> 00:11:54
actually, to find a surface to
apply it to.

151
00:11:54 --> 00:11:57
And, that's where the
assumption of simply connected

152
00:11:57 --> 00:12:00
is useful.
I know in advance that any

153
00:12:00 --> 00:12:02
closed curve,
so, C in particular,

154
00:12:02 --> 00:12:10
has to bound some surface.
OK, so we can find S,

155
00:12:10 --> 00:12:21
a surface, S,
that bounds C because the

156
00:12:21 --> 00:12:32
region is simply connected.
So, now that tells us we can

157
00:12:32 --> 00:12:38
actually apply Stokes theorem,
except it won't fit here.

158
00:12:38 --> 00:12:40
So, instead,
I will do that on the next

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00:12:40 --> 00:12:45
line.
That's equal by Stokes to the

160
00:12:45 --> 00:12:51
double integral over S of curl F
dot vector dS,

161
00:12:51 --> 00:12:54
or ndS.
But now, the curl is zero.

162
00:12:54 --> 00:12:58
So, if I integrate zero,
I will get zero.

163
00:12:58 --> 00:13:02
OK, so I proved that my two
line integrals along C1 and C2

164
00:13:02 --> 00:13:04
are equal.
But for that,

165
00:13:04 --> 00:13:08
I needed to be able to find a
surface which to apply Stokes

166
00:13:08 --> 00:13:11
theorem.
And that required my region to

167
00:13:11 --> 00:13:14
be simply connected.
If I had a vector field that

168
00:13:14 --> 00:13:17
was defined only outside of the
z axis and I took two paths that

169
00:13:17 --> 00:13:20
went on one side and the other
side of the z axis,

170
00:13:20 --> 00:13:21
I might have obtained,
actually,

171
00:13:21 --> 00:13:27
different values of the line
integral.

172
00:13:27 --> 00:13:35
OK, so anyway,
that's the customary warning

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00:13:35 --> 00:13:43
about simply connected things.
OK, let me just mention very

174
00:13:43 --> 00:13:46
quickly that there's a lot of
interesting topology you can do,

175
00:13:46 --> 00:13:48
actually in space.
So, for example, 

176
00:13:48 --> 00:13:50
this concept of being simply
connected or not,

177
00:13:50 --> 00:13:55
and studying which loops bound
surfaces or not can be used to

178
00:13:55 --> 00:13:58
classify shapes of things inside
space.

179
00:13:58 --> 00:14:07
So, for example, 
one of the founding

180
00:14:07 --> 00:14:15
achievements of topology in the
19th century was to classify

181
00:14:15 --> 00:14:24
surfaces in space -- -- by
trying to look at loops on them.

182
00:14:24 --> 00:14:33
So, what I mean by that is that
if I take the surface of a

183
00:14:33 --> 00:14:39
sphere, well,
I claim the surface of a sphere

184
00:14:39 --> 00:14:44
-- -- is simply connected.
Why is that?

185
00:14:44 --> 00:14:49
Well, let's take my favorite
closed curve on the surface of a

186
00:14:49 --> 00:14:53
sphere.
I can always find a portion of

187
00:14:53 --> 00:14:59
the sphere that's bounded by it.
OK, so that's the definition of

188
00:14:59 --> 00:15:03
the surface of a sphere being
simply connected.

189
00:15:03 --> 00:15:06
On the other hand, 
if I take what's called a

190
00:15:06 --> 00:15:07
torus,
or if you prefer,

191
00:15:07 --> 00:15:10
the surface of a doughnut,
that's more,

192
00:15:10 --> 00:15:21
it's a less technical term, 
but it's -- -- well, 

193
00:15:21 --> 00:15:24
that's not simply connected.
And, in fact,

194
00:15:24 --> 00:15:26
for example,
if you look at this loop here

195
00:15:26 --> 00:15:29
that goes around it,
well, of course it bounds a

196
00:15:29 --> 00:15:32
surface in space.
But, that surface cannot be

197
00:15:32 --> 00:15:35
made to be just a piece of the
donut.

198
00:15:35 --> 00:15:39
You have to go through the hole.
You have to leave the surface

199
00:15:39 --> 00:15:41
of a torus.
In fact, there's another one.

200
00:15:41 --> 00:15:47
See, this one also does not
bound anything that's completely

201
00:15:47 --> 00:15:50
contained in the torus.
And, of course,

202
00:15:50 --> 00:15:53
it bounds this disc,
but inside of a torus.

203
00:15:53 --> 00:15:56
But, that's not a part of the
surface itself.

204
00:15:56 --> 00:16:02
So, in fact,
there's, and topologists would

205
00:16:02 --> 00:16:09
say, there's two independent --
-- loops that don't bound

206
00:16:09 --> 00:16:15
surfaces, that don't bound
anything.

207
00:16:15 --> 00:16:18
And, so this number two is
somehow an invariant that you

208
00:16:18 --> 00:16:20
can associate to this kind of
shape.

209
00:16:20 --> 00:16:23
And then, if you consider more
complicated surfaces with more

210
00:16:23 --> 00:16:24
holes in them,
you can try, somehow, 

211
00:16:24 --> 00:16:27
to count independent loops on
them,

212
00:16:27 --> 00:16:33
and that's the beginning of the
classification of surfaces.

213
00:16:33 --> 00:16:39
Anyway, that's not really an
18.02 topic, but I thought I

214
00:16:39 --> 00:16:45
would mentioned it because it's
kind of a cool idea.

215
00:16:45 --> 00:16:55
OK, let me say a bit more in
the way of fun remarks like

216
00:16:55 --> 00:16:59
that.
So, food for thought:

217
00:16:59 --> 00:17:05
let's say that I want to apply
Stokes theorem to simplify a

218
00:17:05 --> 00:17:08
line integral along the curve
here.

219
00:17:08 --> 00:17:11
So, this curve is maybe not
easy to see in the picture.

220
00:17:11 --> 00:17:17
It kind of goes twice around
the z axis, but spirals up and

221
00:17:17 --> 00:17:20
then down.
OK, so one way to find a

222
00:17:20 --> 00:17:25
surface that's bounded by this
curve is to take what's called

223
00:17:25 --> 00:17:29
the Mobius strip.
OK, so the Mobius strip,

224
00:17:29 --> 00:17:32
it's a one sided strip where
when you go around,

225
00:17:32 --> 00:17:35
you flip one side becomes the
other.

226
00:17:35 --> 00:17:38
So, you just,
if you want to take a band of

227
00:17:38 --> 00:17:41
paper and glue the two sides
with a twist,

228
00:17:41 --> 00:17:44
so, it's a one sided surface.
And, that gives us,

229
00:17:44 --> 00:17:49
actually, serious trouble if we
try to orient it to apply Stokes

230
00:17:49 --> 00:17:53
theorem.
So, see, for example,

231
00:17:53 --> 00:17:58
if I take this Mobius strip,
and I try to find an

232
00:17:58 --> 00:18:04
orientation,
so here it looks like that, 

233
00:18:04 --> 00:18:08
well, let's say that I've
oriented my curve going in this

234
00:18:08 --> 00:18:11
direction.
So, I go around,

235
00:18:11 --> 00:18:13
around, around,
still going this direction.

236
00:18:13 --> 00:18:19
Well, the orientation I should
have for Stokes theorem is that

237
00:18:19 --> 00:18:22
when I, so, curve continues
here.

238
00:18:22 --> 00:18:26
Well, if you look at the
convention around here,

239
00:18:26 --> 00:18:31
it tells us that the normal
vector should be going this way.

240
00:18:31 --> 00:18:35
OK, if we look at it near here,
if we walk along this way,

241
00:18:35 --> 00:18:37
the surface is to our right .
So, we should actually be

242
00:18:37 --> 00:18:40
flipping things upside down.
The normal vector should be

243
00:18:40 --> 00:18:41
going down.
And, in fact,

244
00:18:41 --> 00:18:44
if you try to follow your
normal vector that's pointing

245
00:18:44 --> 00:18:45
up, it's pointing up,
up, up.

246
00:18:45 --> 00:18:49
It will have to go into things,
into, into, down.

247
00:18:49 --> 00:18:53
There's no way to choose
consistently a normal vector for

248
00:18:53 --> 00:19:01
the Mobius strip.
So, that's what we call a

249
00:19:01 --> 00:19:08
non-orientable surface.
And, that just means it has

250
00:19:08 --> 00:19:10
only one side.
And, if it has only one side, 

251
00:19:10 --> 00:19:14
that we cannot speak of flux
for it because we have no way of

252
00:19:14 --> 00:19:17
saying that we'll be counting
things positively one way,

253
00:19:17 --> 00:19:19
negatively the other way, 
because there's only one,

254
00:19:19 --> 00:19:22
you know,
there's no notion of sides.

255
00:19:22 --> 00:19:26
So, you can't define a side
towards which things will be

256
00:19:26 --> 00:19:34
going positively.
So, that's actually a situation

257
00:19:34 --> 00:19:44
where flux cannot be defined.
OK, so as much as Mobius strips

258
00:19:44 --> 00:19:48
and climb-bottles are exciting
and really cool,

259
00:19:48 --> 00:19:51
well, we can't use them in this
class because we can't define

260
00:19:51 --> 00:19:54
flux through them.
So, if we really wanted to

261
00:19:54 --> 00:19:57
apply Stokes theorem,
because I've been telling you

262
00:19:57 --> 00:19:59
that space is simply connected,
and I will always be able to

263
00:19:59 --> 00:20:01
apply Stokes theorem to any
curve,

264
00:20:01 --> 00:20:05
what would I do?
Well, I claim this curve

265
00:20:05 --> 00:20:10
actually bounds another surface
that is orientable.

266
00:20:10 --> 00:20:11
Yeah, that looks
counterintuitive.

267
00:20:11 --> 00:20:16
Well, let's see it.
I claim you can take a

268
00:20:16 --> 00:20:22
hemisphere, and you can take a
small thing and twist it around.

269
00:20:22 --> 00:20:26
So, in case you don't believe
me, let me do it again with the

270
00:20:26 --> 00:20:28
transparency.
Here's my loop,

271
00:20:28 --> 00:20:31
and see, well,
the scale is not exactly the

272
00:20:31 --> 00:20:33
same.
So, it doesn't quite match.

273
00:20:33 --> 00:20:35
But, and it's getting a bit
dark.

274
00:20:35 --> 00:20:41
But, that spherical thing with
a little slit going twisting

275
00:20:41 --> 00:20:46
into it will actually have
boundary my loop.

276
00:20:46 --> 00:20:50
And, that one is orientable.
I mean, I leave it up to you to

277
00:20:50 --> 00:20:56
stare at the picture long enough
to convince yourselves that

278
00:20:56 --> 00:21:00
there's a well-defined up and
down.

279
00:21:00 --> 00:21:11
OK.
So now, I mean,

280
00:21:11 --> 00:21:14
in case you are getting really,
really worried,

281
00:21:14 --> 00:21:18
I mean, there won't be any
Mobius strips on the exam on

282
00:21:18 --> 00:21:24
Tuesday, OK?
It's just to show you some cool

283
00:21:24 --> 00:21:29
stuff.
OK, questions?

284
00:21:29 --> 00:21:34
No?
OK, one last thing I want to

285
00:21:34 --> 00:21:38
show you before we start
reviewing,

286
00:21:38 --> 00:21:41
so one question you might have
about Stokes theorem is,

287
00:21:41 --> 00:21:44
how come we can choose whatever
surface we want?

288
00:21:44 --> 00:21:47
I mean, sure,
it seems to work,

289
00:21:47 --> 00:21:52
but why?
So, I'm going to say a couple

290
00:21:52 --> 00:22:02
of words about surface
independence in Stokes theorem.

291
00:22:02 --> 00:22:08
So, let's say that I have a
curve, C, in space.

292
00:22:08 --> 00:22:11
And, let's say that I want to
apply Stokes theorem.

293
00:22:11 --> 00:22:16
So, then I can choose my
favorite surface bounded by C.

294
00:22:16 --> 00:22:18
So, in a situation like this,
for example,

295
00:22:18 --> 00:22:21
I might want to make my first
choice be this guy,

296
00:22:21 --> 00:22:25
S1, like maybe some sort of
upper half sphere.

297
00:22:25 --> 00:22:28
And, if you pay attention to
the orientation conventions,

298
00:22:28 --> 00:22:31
you'll see that you need to
take it with normal vector

299
00:22:31 --> 00:22:34
pointing up.
Maybe actually I would rather

300
00:22:34 --> 00:22:36
make a different choice.
And actually,

301
00:22:36 --> 00:22:41
I will choose another surface,
S2, that maybe looks like that.

302
00:22:41 --> 00:22:44
And, if I look carefully at the
orientation convention,

303
00:22:44 --> 00:22:47
Stokes theorem tells me that I
have to take the normal vector

304
00:22:47 --> 00:22:52
pointing up again.
So, that's actually into things.

305
00:22:52 --> 00:22:57
So, 
Stokes says that the line

306
00:22:57 --> 00:23:04
integral along C of my favorite
vector field can be computed

307
00:23:04 --> 00:23:09
either as a flux integral for
the curl through S1,

308
00:23:09 --> 00:23:16
or as the same integral, 
but through S2 instead of S1.

309
00:23:16 --> 00:23:21
So, that seems to suggest that
curl F has some sort of surface

310
00:23:21 --> 00:23:24
independence property.
It doesn't really matter which

311
00:23:24 --> 00:23:27
surface I take,
as long as the boundary is this

312
00:23:27 --> 00:23:29
given curve, C.
Why is that?

313
00:23:29 --> 00:23:31
That's a strange property to
have.

314
00:23:31 --> 00:23:36
Where does it come from?
Well, let's think about it for

315
00:23:36 --> 00:23:40
a second.
So, why are these the same?

316
00:23:40 --> 00:23:42
I mean, of course,
they have to be the same

317
00:23:42 --> 00:23:44
because that's what Stokes tell
us.

318
00:23:44 --> 00:23:48
But, why is that OK?
Well, let's think about

319
00:23:48 --> 00:23:53
comparing the flux integral for
S1 and the flux integral for S2.

320
00:23:53 --> 00:23:57
So, if we want to compare them,
we should probably subtract

321
00:23:57 --> 00:24:02
them from each other.
OK, so let's do the flux

322
00:24:02 --> 00:24:09
integral for S1 minus the flux
integral for S2 of the same

323
00:24:09 --> 00:24:12
thing.
Well, let's give a name.

324
00:24:12 --> 00:24:18
Let's call S the surface S1
minus S2.

325
00:24:18 --> 00:24:21
So, what is S?
S is S1 with its given

326
00:24:21 --> 00:24:26
orientation together with S2
with the reversed orientation.

327
00:24:26 --> 00:24:32
So, S is actually this whole
closed surface here.

328
00:24:32 --> 00:24:37
And, the normal vector to S
seems to be pointing outwards

329
00:24:37 --> 00:24:39
everywhere.
OK, so now, if we have a closed

330
00:24:39 --> 00:24:41
surface with a normal vector
pointing outwards,

331
00:24:41 --> 00:24:44
and we want to find a flux
integral for it,

332
00:24:44 --> 00:24:47
well, 
we can replace that with a

333
00:24:47 --> 00:24:57
triple integral.
So, that's the divergence

334
00:24:57 --> 00:25:03
theorem.
So, that's by the divergence

335
00:25:03 --> 00:25:09
theorem using the fact that S is
a closed surface.

336
00:25:09 --> 00:25:13
That's equal to the triple
integral over the region inside.

337
00:25:13 --> 00:25:26
Let me call that region D of
divergence, of curl F dV.

338
00:25:26 --> 00:25:34
OK, and what I'm going to claim
now is that we can actually

339
00:25:34 --> 00:25:41
check that if you take the
divergence of the curl of a

340
00:25:41 --> 00:25:47
vector field,
you always get zero.

341
00:25:47 --> 00:25:50
OK, and so that will tell you
that this integral will always

342
00:25:50 --> 00:25:53
be zero.
And that's why the flux for S1, 

343
00:25:53 --> 00:25:58
and the flux for S2 were the
same a priori and we didn't have

344
00:25:58 --> 00:26:01
to worry about which one we
chose when we did Stokes

345
00:26:01 --> 00:26:06
theorem.
OK, so let's just check quickly

346
00:26:06 --> 00:26:10
that divergence of a curve is
zero.

347
00:26:10 --> 00:26:12
OK, in case you're wondering
why I'm doing all this,

348
00:26:12 --> 00:26:13
well, first I think it's kind
of interesting,

349
00:26:13 --> 00:26:17
and second, it reminds you of a
statement of all these theorems,

350
00:26:17 --> 00:26:19
and all these definitions.
So, in a way,

351
00:26:19 --> 00:26:25
we are already reviewing.
OK, so let's see.

352
00:26:25 --> 00:26:30
If my vector field has
components P,

353
00:26:30 --> 00:26:39
Q, and R, remember that the
curl was defined by this cross

354
00:26:39 --> 00:26:47
product between del and our
given vector field.

355
00:26:47 --> 00:27:04
So, that's Ry - Qz followed by
Pz - Rx, and Qx - Py.

356
00:27:04 --> 00:27:14
So, now, we want to take the
divergence of this.

357
00:27:14 --> 00:27:19
Well, so we have to take the
first component,

358
00:27:19 --> 00:27:23
Ry minus Qz,
and take its partial with

359
00:27:23 --> 00:27:28
respect to x.
Then, take the y component,

360
00:27:28 --> 00:27:35
Pz minus Rx partial with
respect to y plus Qx minus Py

361
00:27:35 --> 00:27:41
partial with respect to z.
And, well, now we should expand

362
00:27:41 --> 00:27:43
this.
But I claim it will always

363
00:27:43 --> 00:27:44
simplify to zero.

364
00:27:44 --> 00:28:11

365
00:28:11 --> 00:28:24
OK, so I think we have over
there, becomes R sub yx minus Q

366
00:28:24 --> 00:28:39
sub zx plus P sub zy minus R sub
xy plus Q sub xz minus P sub yz.

367
00:28:39 --> 00:28:47
Well, let's see.
We have P sub zy minus P sub yz.

368
00:28:47 --> 00:28:54
These two cancel out.
We have R sub yx minus R sub xy.

369
00:28:54 --> 00:28:58
These cancel out.
Q sub zx and Q sub xz,

370
00:28:58 --> 00:29:04
these two also cancel out.
So, indeed, the divergence of a

371
00:29:04 --> 00:29:10
curl is always zero.
OK, so the claim is divergence

372
00:29:10 --> 00:29:17
of curl is always zero.
Del cross F is always zero, 

373
00:29:17 --> 00:29:26
and just a small remark, 
if we had actually real vectors

374
00:29:26 --> 00:29:30
rather than this strange del
guy,

375
00:29:30 --> 00:29:33
indeed we know that if we have
two vectors,

376
00:29:33 --> 00:29:37
U and V, 
and we do u dot u cross v, 

377
00:29:37 --> 00:29:40
what is that?
Well, one way to say it is it's

378
00:29:40 --> 00:29:43
the determinant of u,
u, and v, which is the volume

379
00:29:43 --> 00:29:45
of the box.
But, it's completely flat

380
00:29:45 --> 00:29:47
because u, u,
and v are all in the plane

381
00:29:47 --> 00:29:50
defined by u and v.
The other way to say it is that

382
00:29:50 --> 00:29:53
u cross v is perpendicular to u
and v.

383
00:29:53 --> 00:29:56
Well, if it's perpendicular u,
then its dot product with u

384
00:29:56 --> 00:29:59
will be zero.
So, no matter how you say it,

385
00:29:59 --> 00:30:02
this is always zero.
So, in a way,

386
00:30:02 --> 00:30:09
this reinforces our intuition
that del, even though it's not

387
00:30:09 --> 00:30:15
at all an actual vector
sometimes can be manipulated in

388
00:30:15 --> 00:30:20
the same way.
OK, I think that's it for new

389
00:30:20 --> 00:30:26
topics for today.
And, 

390
00:30:26 --> 00:30:30
so, now I should maybe try to
recap quickly what we've learned

391
00:30:30 --> 00:30:34
in these past three weeks so
that you know,

392
00:30:34 --> 00:30:39
so, the exam is probably going
to be similar in difficulty to

393
00:30:39 --> 00:30:42
the practice exams.
That's my goal.

394
00:30:42 --> 00:30:45
I don't know if I will have
reached that goal or not.

395
00:30:45 --> 00:30:48
We'll only know that after
you've taken the test.

396
00:30:48 --> 00:30:53
But, the idea is it's meant to
be more or less the same level

397
00:30:53 --> 00:30:58
of difficulty.
So, at this point,

398
00:30:58 --> 00:31:06
we've learned about three kinds
of beasts in space.

399
00:31:06 --> 00:31:12
OK, so I'm going to divide my
blackboard into three pieces,

400
00:31:12 --> 00:31:16
and here I will write triple
integrals.

401
00:31:16 --> 00:31:20
We've learned about double
integrals, and we've learned

402
00:31:20 --> 00:31:26
about line integrals.
OK, so triple integrals over a

403
00:31:26 --> 00:31:33
region in space,
we integrate a scalar quantity,

404
00:31:33 --> 00:31:35
dV.
How do we do that?

405
00:31:35 --> 00:31:41
Well, we can do that in
rectangular coordinates where dV

406
00:31:41 --> 00:31:46
becomes something like,
maybe, dz dx dy,

407
00:31:46 --> 00:31:52
or any permutation of these.
We've seen how to do it also in

408
00:31:52 --> 00:31:59
cylindrical coordinates where dV
is maybe dz times r dr d theta

409
00:31:59 --> 00:32:02
or more commonly r dr d theta
dz.

410
00:32:02 --> 00:32:06
But, what I want to emphasize
in this way is that both of

411
00:32:06 --> 00:32:09
these you set up pretty much in
the same way.

412
00:32:09 --> 00:32:12
So, remember,
the main trick here is to find

413
00:32:12 --> 00:32:15
the bounds of integration.
So, when you do it,

414
00:32:15 --> 00:32:18
say, with dz first,
that means for fixed xy,

415
00:32:18 --> 00:32:23
so, for a fixed point in the xy
plane, you have to look at the

416
00:32:23 --> 00:32:25
bounds for z.
So, that means you have to

417
00:32:25 --> 00:32:28
figure out what's the bottom
surface of your solid,

418
00:32:28 --> 00:32:31
and what's the top surface of
your solid?

419
00:32:31 --> 00:32:34
And, you have to find the value
of z at the bottom,

420
00:32:34 --> 00:32:37
the value of z at the top as
functions of x and y.

421
00:32:37 --> 00:32:40
And then, you will put that as
bounds for z.

422
00:32:40 --> 00:32:43
Once you've done that,
you are left with the question

423
00:32:43 --> 00:32:45
of finding bounds for x and y.
Well, for that,

424
00:32:45 --> 00:32:49
you just rotate the picture,
look at your solid from above, 

425
00:32:49 --> 00:32:52
so, look at its projection to
the xy plane,

426
00:32:52 --> 00:32:56
and you set up a double
integral either in rectangular

427
00:32:56 --> 00:33:02
xy coordinates,
or in polar coordinates for x

428
00:33:02 --> 00:33:04
and y.
Of course, you can always do it

429
00:33:04 --> 00:33:08
a different orders.
And, I'll let you figure out

430
00:33:08 --> 00:33:11
again how that goes.
But, if you do dz first,

431
00:33:11 --> 00:33:15
then the inner bounds are given
by bottom and top,

432
00:33:15 --> 00:33:20
and the outer ones are given by
looking at the shadow of the

433
00:33:20 --> 00:33:23
region.
Now, there's also spherical

434
00:33:23 --> 00:33:28
coordinates.
And there, we've seen that dV

435
00:33:28 --> 00:33:32
is rho squared sine phi d rho d
phi d theta.

436
00:33:32 --> 00:33:35
So now, of course, 
if this orgy of Greek letters

437
00:33:35 --> 00:33:39
is confusing you at this point,
then you probably need to first

438
00:33:39 --> 00:33:41
review spherical coordinates for
themselves.

439
00:33:41 --> 00:33:44
Remember that rho is the
distance from the origin.

440
00:33:44 --> 00:33:47
Phi is the angle down from the
z axis.

441
00:33:47 --> 00:33:49
So, it's zero,
and the positive z axis,

442
00:33:49 --> 00:33:53
pi over two in the xy plane,
and increases all the way to pi

443
00:33:53 --> 00:33:59
on the negative z axis.
And, theta is the angle around

444
00:33:59 --> 00:34:02
the z axis.
So, now, when we set up bounds

445
00:34:02 --> 00:34:04
here,
it will look a lot like what

446
00:34:04 --> 00:34:07
you've done in polar coordinates
in the plane because when you

447
00:34:07 --> 00:34:09
look at the inner bound down on
rho,

448
00:34:09 --> 00:34:12
for a fixed phi and theta, 
that means you're shooting a

449
00:34:12 --> 00:34:15
straight ray from the origin in
some direction in space.

450
00:34:15 --> 00:34:17
So, you know,
you're sending a laser beam,

451
00:34:17 --> 00:34:20
and you want to know what part
of your beam is going to be in

452
00:34:20 --> 00:34:23
your given solid.
You want to solve for the value

453
00:34:23 --> 00:34:26
of rho when you enter the solid
and when you leave it.

454
00:34:26 --> 00:34:29
I mean, very often,
if the origin is in your solid,

455
00:34:29 --> 00:34:33
then rho will start at zero.
Then you want to know when you

456
00:34:33 --> 00:34:34
exit.
And, I mean,

457
00:34:34 --> 00:34:38
there's a fairly small list of
kinds of surfaces that we've

458
00:34:38 --> 00:34:41
seen how to set up in spherical
coordinates.

459
00:34:41 --> 00:34:44
So, if you're really upset by
this, go over the problems in

460
00:34:44 --> 00:34:47
the notes.
That will give you a good idea

461
00:34:47 --> 00:34:53
of what kinds of things we've
seen in spherical coordinates.

462
00:34:53 --> 00:34:56
OK, and then evaluation is the
usual way.

463
00:34:56 --> 00:35:01
Questions about this?
No?

464
00:35:01 --> 00:35:08
OK, so, I should say we can do
something bad,

465
00:35:08 --> 00:35:15
but so we've seen,
of course, applications of

466
00:35:15 --> 00:35:19
this.
So, we should know how to use a

467
00:35:19 --> 00:35:24
triple integral to evaluate
things like a mass of a solid,

468
00:35:24 --> 00:35:29
the average value of a
function,

469
00:35:29 --> 00:35:37
the moment of inertia about one
of the coordinate axes,

470
00:35:37 --> 00:35:54
or the gravitational attraction
on a mass at the origin.

471
00:35:54 --> 00:35:58
OK, so these are just formulas
to remember for examples of

472
00:35:58 --> 00:36:01
triple integrals.
It doesn't change conceptually.

473
00:36:01 --> 00:36:04
You always set them up and
evaluate them the same way.

474
00:36:04 --> 00:36:11
It just tells you what to put
there for the integrand.

475
00:36:11 --> 00:36:15
Now, 
double integrals: so, 

476
00:36:15 --> 00:36:18
when we have a surface in
space,

477
00:36:18 --> 00:36:21
well, what we will integrate on
it,

478
00:36:21 --> 00:36:26
at least what we've seen how to
integrate is a vector field

479
00:36:26 --> 00:36:31
dotted with the unit normal
vector times the area element.

480
00:36:31 --> 00:36:38
OK, and this is sometimes
called vector dS.

481
00:36:38 --> 00:36:48
Now, how do we evaluate that?
Well, we've seen formulas for

482
00:36:48 --> 00:36:55
ndS in various settings.
And, once you have a formula

483
00:36:55 --> 00:37:01
for ndS, that will relate ndS to
maybe dx dy, or something else.

484
00:37:01 --> 00:37:07
And then, you will express,
so, for example,

485
00:37:07 --> 00:37:15
ndS equals something dx dy.
And then, it becomes a double

486
00:37:15 --> 00:37:21
integral of something dx dy.
Now, in the integrand,

487
00:37:21 --> 00:37:23
you want to express everything
in terms of x and y.

488
00:37:23 --> 00:37:26
So, if you had a z,
maybe you have a formula for z

489
00:37:26 --> 00:37:28
in terms of x and y.
And, when you set up the

490
00:37:28 --> 00:37:30
bounds, well,
you try to figure out what are

491
00:37:30 --> 00:37:33
the bounds for x and y?
That would be just looking at

492
00:37:33 --> 00:37:35
it from above.
Of course, if you are using

493
00:37:35 --> 00:37:37
other variables,
figure out the bounds for those

494
00:37:37 --> 00:37:40
variables.
And, when you've done that,

495
00:37:40 --> 00:37:44
it becomes just a double
integral in the usual sense.

496
00:37:44 --> 00:37:46
OK, so maybe I should be a bit
more explicit about formulas

497
00:37:46 --> 00:37:52
because there have been a lot.
So, let me tell you about a few

498
00:37:52 --> 00:37:56
of them.
Let me actually do that over

499
00:37:56 --> 00:38:02
here because I don't want to
make this too crowded.

500
00:38:02 --> 00:38:24
 
 

501
00:38:24 --> 00:38:28
OK, so what kinds of formulas
for ndS have we seen?

502
00:38:28 --> 00:38:32
Well, we've seen a formula,
for example,

503
00:38:32 --> 00:38:37
for a horizontal plane,
or for something that's

504
00:38:37 --> 00:38:42
parallel to the yz plane or the
xz plane.

505
00:38:42 --> 00:38:47
Well, let's do just the yz
plane for a quick reminder.

506
00:38:47 --> 00:38:52
So, if I have a surface that's
contained inside the yz plane,

507
00:38:52 --> 00:38:56
then obviously I will express
ds in terms of,

508
00:38:56 --> 00:39:01
well, I will use y and z as my
variables.

509
00:39:01 --> 00:39:05
So, I will say that ds is dy
dz, or dz dy,

510
00:39:05 --> 00:39:11
whatever's most convenient.
Maybe we will even switch to

511
00:39:11 --> 00:39:15
polar coordinates after that if
a problem wants us to.

512
00:39:15 --> 00:39:16
And, what about the normal
vector?

513
00:39:16 --> 00:39:21
Well, the normal vector is
either coming straight at us,

514
00:39:21 --> 00:39:26
or it's maybe going back away
from us depending on which

515
00:39:26 --> 00:39:30
orientation we've chosen.
So, this gives us ndS.

516
00:39:30 --> 00:39:32
We dot our favorite vector
field with it.

517
00:39:32 --> 00:39:38
We integrate,
and we get the answer.

518
00:39:38 --> 00:39:48
OK, we've seen about spheres
and cylinders centered at the

519
00:39:48 --> 00:39:54
origin or centered on the z
axis.

520
00:39:54 --> 00:40:00
So, the normal vector sticks
straight out or straight in,

521
00:40:00 --> 00:40:05
depending on which direction
you do it in.

522
00:40:05 --> 00:40:09
So, for a sphere,
the normal vector is <x,

523
00:40:09 --> 00:40:14
y, z> divided by the radius
of the sphere.

524
00:40:14 --> 00:40:17
For a cylinder,
it's <x, y,

525
00:40:17 --> 00:40:21
0>, divided by the radius of
a cylinder.

526
00:40:21 --> 00:40:25
And, the surface element on a
sphere,

527
00:40:25 --> 00:40:28
so, see, it's very closely
related to the volume element of

528
00:40:28 --> 00:40:31
spherical coordinates except you
don't have a rho anymore.

529
00:40:31 --> 00:40:37
You just plug in a rho equals a.
So, you get a squared sine phi

530
00:40:37 --> 00:40:41
d phi d theta.
And, for a cylinder,

531
00:40:41 --> 00:40:47
it would be a dz d theta.
So, 

532
00:40:47 --> 00:40:51
by the way, just a quick check, 
when you're doing an integral, 

533
00:40:51 --> 00:40:55
if it's the surface integral, 
there should be two integral

534
00:40:55 --> 00:40:56
signs,
and there should be two

535
00:40:56 --> 00:40:59
integration variables.
And, there should be two d

536
00:40:59 --> 00:41:03
somethings.
If you end up with a dx,

537
00:41:03 --> 00:41:10
dy, dz in the surface integral,
something is seriously wrong.

538
00:41:10 --> 00:41:18
OK, now, besides these specific
formulas, we've seen two general

539
00:41:18 --> 00:41:24
formulas that are also useful.
So, one is, 

540
00:41:24 --> 00:41:29
if we know how to express z in
terms of x and y,

541
00:41:29 --> 00:41:33
and just to change notation to
show you that it's not set in

542
00:41:33 --> 00:41:36
stone,
let's say that z is known as a

543
00:41:36 --> 00:41:42
function z of x and y.
So, how do I get ndS in that

544
00:41:42 --> 00:41:45
case?
Well, we've seen a formula that

545
00:41:45 --> 00:41:51
says negative partial z partial
x, negative partial z partial y,

546
00:41:51 --> 00:41:54
one dx dy.
So, this formula relates the

547
00:41:54 --> 00:41:57
volume, sorry,
the surface element on our

548
00:41:57 --> 00:42:01
surface to the area element in
the xy plane.

549
00:42:01 --> 00:42:08
It lets us convert between dS
and dx dy.

550
00:42:08 --> 00:42:11
OK, so we just plug in this,
and we dot with F,

551
00:42:11 --> 00:42:14
and then we substitute
everything in terms of x and y,

552
00:42:14 --> 00:42:17
and we evaluate the integral
over x and y.

553
00:42:17 --> 00:42:22
If we don't really want to find
a way to find z as a function of

554
00:42:22 --> 00:42:29
x and y,
but we have a normal vector

555
00:42:29 --> 00:42:35
given to us,
then we have another formula

556
00:42:35 --> 00:42:39
which says that ndS is,
sorry, I should have said it's

557
00:42:39 --> 00:42:42
always up to sign because we
have a two orientation

558
00:42:42 --> 00:42:45
convention.
We have to decide based on what

559
00:42:45 --> 00:42:48
we are trying to do,
whether we are doing the

560
00:42:48 --> 00:42:51
correct convention or the wrong
one.

561
00:42:51 --> 00:43:02
So, the other formula is n over
n dot k dx dy.

562
00:43:02 --> 00:43:09
Sorry, are they all the same?
Well, if you want,

563
00:43:09 --> 00:43:12
you can put an absolute value
here.

564
00:43:12 --> 00:43:16
But, it doesn't matter because
it's up to sign anyway.

565
00:43:16 --> 00:43:22
So, I mean, this formula is
valid as it is.

566
00:43:22 --> 00:43:24
OK, and, I mean,
if you're in a situation where

567
00:43:24 --> 00:43:26
you can apply more than one
formula,

568
00:43:26 --> 00:43:32
they will all give you the same
answer in the end because it's

569
00:43:32 --> 00:43:37
the same flux integral.
OK, so anyway, 

570
00:43:37 --> 00:43:40
so we have various ways of
computing surface integrals,

571
00:43:40 --> 00:43:44
and probably one of the best
possible things you can do to

572
00:43:44 --> 00:43:48
prepare for the test is actually
to look again at some practice

573
00:43:48 --> 00:43:51
problems from the notes that do
flux integrals over various

574
00:43:51 --> 00:43:55
kinds of surfaces because that's
probably one of the hardest

575
00:43:55 --> 00:43:58
topics in this unit of the
class.

576
00:43:58 --> 00:44:05
OK, anyway, let's move on to
line integrals.

577
00:44:05 --> 00:44:11
So, those are actually a piece
of cake in comparison,

578
00:44:11 --> 00:44:17
OK, because all that this is,
is just integral of P dx Q dy R

579
00:44:17 --> 00:44:24
dz.
And, then all you have to do is

580
00:44:24 --> 00:44:35
parameterize the curve,
C, to express everything in

581
00:44:35 --> 00:44:42
terms of a single variable.
And then, you end up with a

582
00:44:42 --> 00:44:46
usual single integral,
and you can just compute it.

583
00:44:46 --> 00:44:48
So, that one works pretty much
as it did in the plane.

584
00:44:48 --> 00:44:52
So, if you forgotten what we
did in the plane,

585
00:44:52 --> 00:44:56
it's really the same thing.
OK, so now we have three

586
00:44:56 --> 00:44:58
different kinds of integrals,
and really, well,

587
00:44:58 --> 00:45:01
they certainly have in common
that they integrate things

588
00:45:01 --> 00:45:03
somehow.
But, apart from that,

589
00:45:03 --> 00:45:05
they are extremely different in
what they do.

590
00:45:05 --> 00:45:08
I mean, this one involves a
function, a scalar quantity.

591
00:45:08 --> 00:45:11
These involve vector quantities.
They don't involve the same

592
00:45:11 --> 00:45:13
kinds of shapes over which to
integrate.

593
00:45:13 --> 00:45:16
Here, you integrate over a
three-dimensional region.

594
00:45:16 --> 00:45:19
Here, you integrate only over a
two-dimensional surface,

595
00:45:19 --> 00:45:21
and here, only a
one-dimensional curve.

596
00:45:21 --> 00:45:24
So, try not to confuse them.
That's basically the most

597
00:45:24 --> 00:45:27
important advice.
Don't get mistaken.

598
00:45:27 --> 00:45:30
Each of them has a different
way of getting evaluated.

599
00:45:30 --> 00:45:34
Eventually, they will all give
you numbers, but through

600
00:45:34 --> 00:45:36
different processes.
So now, well,

601
00:45:36 --> 00:45:38
I said these guys are
completely different.

602
00:45:38 --> 00:45:40
Well, they are,
but we still have some bridges

603
00:45:40 --> 00:45:42
between them.
OK, so we have two,

604
00:45:42 --> 00:45:46
maybe I should say three,
well, two bridges between these

605
00:45:46 --> 00:45:49
guys.
OK, so we have somehow a

606
00:45:49 --> 00:45:54
connection between these which
is the divergence theorem.

607
00:45:54 --> 00:46:02
We have a connection between
that, which is Stokes theorem.

608
00:46:02 --> 00:46:20
So -- Just to write them again, 
so the divergence theorem says

609
00:46:20 --> 00:46:25
if I have a region in space,
and I call its boundary S, 

610
00:46:25 --> 00:46:27
so, it's going to be a closed
surface,

611
00:46:27 --> 00:46:31
and I orient S with a normal
vector pointing outwards,

612
00:46:31 --> 00:46:35
then whenever I have a surface
integral over S,

613
00:46:35 --> 00:46:40
sorry,
I can replace it by a triple

614
00:46:40 --> 00:46:47
integral over the region inside.
OK, so this guy is a vector

615
00:46:47 --> 00:46:49
field.
And, this guy is a function

616
00:46:49 --> 00:46:52
that somehow relates to the
vector field.

617
00:46:52 --> 00:46:54
I mean, you should know how.
You should know the definition

618
00:46:54 --> 00:46:55
of divergence,
of course.

619
00:46:55 --> 00:46:59
But, what I want to point out
is if you have to compute the

620
00:46:59 --> 00:47:01
two sides separately,
well, this is just,

621
00:47:01 --> 00:47:04
you know, your standard flux
integral.

622
00:47:04 --> 00:47:07
This is just your standard
triple integral over a region in

623
00:47:07 --> 00:47:09
space.
Once you have computed what

624
00:47:09 --> 00:47:13
this guy is, it's really just a
triple integral of the function.

625
00:47:13 --> 00:47:16
So, the way in which you
compute it doesn't see that it

626
00:47:16 --> 00:47:19
came from a divergence.
It's just the same way that you

627
00:47:19 --> 00:47:23
would compute any other triple
integral.

628
00:47:23 --> 00:47:28
The way we compute it doesn't
depend on what actually we are

629
00:47:28 --> 00:47:32
integrating.
Stokes theorem says if I have a

630
00:47:32 --> 00:47:35
curve that's the boundary of a
surface, S,

631
00:47:35 --> 00:47:40
and I orient the two in
compatible manners,

632
00:47:40 --> 00:47:52
then I can replace a line
integral on C by a surface

633
00:47:52 --> 00:47:57
integral on S.
OK, and that surface integral,

634
00:47:57 --> 00:48:00
well, it's not for the same
vector field.

635
00:48:00 --> 00:48:03
This relates a line integral
for one field to a surface

636
00:48:03 --> 00:48:06
integral from another field.
That other field is given from

637
00:48:06 --> 00:48:10
the first one just by taking its
curl So, after you take the

638
00:48:10 --> 00:48:13
curl, you obtain a different
vector field.

639
00:48:13 --> 00:48:17
And, the way in which you would
compute the surface integral is

640
00:48:17 --> 00:48:19
just as with any surface
integral.

641
00:48:19 --> 00:48:23
You just find a formula for ndS
dot product, substitute,

642
00:48:23 --> 00:48:26
evaluate.
The calculation of this thing,

643
00:48:26 --> 00:48:30
once you've computed curl does
not remember that it was a curl.

644
00:48:30 --> 00:48:33
It's the same as with any other
flux integral.

645
00:48:33 --> 00:48:35
OK, and finally,
the last bridge,

646
00:48:35 --> 00:48:38
so this was between two and
three.

647
00:48:38 --> 00:48:41
This was between one and two.
Let me just say,

648
00:48:41 --> 00:48:45
there's a bridge between zero
and one,

649
00:48:45 --> 00:48:53
which is that if you have a
function in its gradient,

650
00:48:53 --> 00:48:57
well, the fundamental theorem
of calculus says that the line

651
00:48:57 --> 00:49:01
integral for the vector field
given by the gradient of a

652
00:49:01 --> 00:49:04
function is actually equal to
the change in value of a

653
00:49:04 --> 00:49:08
function.
That's if you have a curve

654
00:49:08 --> 00:49:11
bounded by P0 and P1.
So in a way, actually, 

655
00:49:11 --> 00:49:15
each of these three theorems
relates a quantity with a

656
00:49:15 --> 00:49:19
certain number of integral signs
to a quantity with one more

657
00:49:19 --> 00:49:22
integral sign.
And, that's actually somehow a

658
00:49:22 --> 00:49:24
fundamental similarity between
them.

659
00:49:24 --> 00:49:27
But maybe it's easier to think
of them as completely different

660
00:49:27 --> 00:49:30
stories.
So now, with this one,

661
00:49:30 --> 00:49:36
we additionally have to
remember another topic is given

662
00:49:36 --> 00:49:41
a vector field,
F, with curl equal to zero,

663
00:49:41 --> 00:49:45
find the potential.
And, we've seen two methods for

664
00:49:45 --> 00:49:47
that, and I'm sure you remember
them.

665
00:49:47 --> 00:49:53
So, if not, then try to
remember them for Tuesday.

666
00:49:53 --> 00:49:54
OK, so anyway,
again, conceptually,

667
00:49:54 --> 00:49:56
we have, really,
three different kinds of

668
00:49:56 --> 00:49:59
integrals.
We evaluated them in completely

669
00:49:59 --> 00:50:01
different ways,
and we have a handful of

670
00:50:01 --> 00:50:03
theorems, connecting them to
each other.

671
00:50:03 --> 00:50:06
But, that doesn't have any
impact on how we actually

672
00:50:06 --> 00:50:08
compute things.
 

673
00:50:08 --> 00:50:13