1
00:00:01,000 --> 00:00:03,000
The following content is
provided under a Creative

2
00:00:03,000 --> 00:00:05,000
Commons license.
Your support will help MIT

3
00:00:05,000 --> 00:00:08,000
OpenCourseWare continue to offer
high quality educational

4
00:00:08,000 --> 00:00:13,000
resources for free.
To make a donation or to view

5
00:00:13,000 --> 00:00:18,000
additional materials from
hundreds of MIT courses,

6
00:00:18,000 --> 00:00:23,000
visit MIT OpenCourseWare at
ocw.mit.edu.

7
00:00:23,000 --> 00:00:29,000
OK, so remember,
we've seen Stokes theorem,

8
00:00:29,000 --> 00:00:37,000
which says if I have a closed
curve bounding some surface,

9
00:00:37,000 --> 00:00:40,000
S,
and I orient the curve and the

10
00:00:40,000 --> 00:00:44,000
surface compatible with each
other,

11
00:00:44,000 --> 00:00:53,000
then I can compute the line
integral along C along my curve

12
00:00:53,000 --> 00:00:57,000
in terms of,
instead,

13
00:00:57,000 --> 00:01:04,000
surface integral for flux of a
different vector field,

14
00:01:04,000 --> 00:01:12,000
namely, curl f dot n dS.
OK, so that's the statement.

15
00:01:12,000 --> 00:01:18,000
And, just to clarify a little
bit, so, again,

16
00:01:18,000 --> 00:01:24,000
we've seen various kinds of
integrals.

17
00:01:24,000 --> 00:01:26,000
So, line integrals we know how
to evaluate.

18
00:01:26,000 --> 00:01:30,000
They take place in a curve.
You express everything in terms

19
00:01:30,000 --> 00:01:32,000
of one variable,
and after substituting,

20
00:01:32,000 --> 00:01:36,000
you end up with a usual one
variable integral that you know

21
00:01:36,000 --> 00:01:40,000
how to evaluate.
And, surface integrals,

22
00:01:40,000 --> 00:01:44,000
we know also how to evaluate.
Namely, we've seen various

23
00:01:44,000 --> 00:01:47,000
formulas for ndS.
Once you have such a formula,

24
00:01:47,000 --> 00:01:50,000
due to the dot product with
this vector field,

25
00:01:50,000 --> 00:01:52,000
which is not the same as that
one.

26
00:01:52,000 --> 00:01:56,000
But it's a new vector field
that you can build out of f.

27
00:01:56,000 --> 00:02:00,000
You do the dot product.
You express everything in terms

28
00:02:00,000 --> 00:02:03,000
of your two integration
variables, and then you

29
00:02:03,000 --> 00:02:06,000
evaluate.
So, now, what does this have to

30
00:02:06,000 --> 00:02:12,000
do with various other things?
So, one thing I want to say has

31
00:02:12,000 --> 00:02:18,000
to do with how Stokes helps us
understand path independence,

32
00:02:18,000 --> 00:02:24,000
so, how it actually motivates
our criterion for gradient

33
00:02:24,000 --> 00:02:30,000
fields,
independence.

34
00:02:30,000 --> 00:02:35,000
OK, so,
we've seen that if we have a

35
00:02:35,000 --> 00:02:40,000
vector field defined in a simply
connected region,

36
00:02:40,000 --> 00:02:43,000
and its curl is zero,
then it's a gradient field,

37
00:02:43,000 --> 00:02:47,000
and the line integral is path
independent.

38
00:02:47,000 --> 00:02:53,000
So, let me first define for you
when a simply connected region

39
00:02:53,000 --> 00:03:01,000
is.
So, we say that a region in

40
00:03:01,000 --> 00:03:17,000
space is simply connected -- --
if every closed loop inside this

41
00:03:17,000 --> 00:03:31,000
region bounds some surface again
inside this region.

42
00:03:31,000 --> 00:03:39,000
OK, so let me just give you
some examples just to clarify.

43
00:03:39,000 --> 00:03:46,000
So, for example,
let's say that I have a region

44
00:03:46,000 --> 00:03:52,000
that's the entire space with the
origin removed.

45
00:03:52,000 --> 00:04:00,000
OK, so space with the origin
removed, OK, you think it's

46
00:04:00,000 --> 00:04:06,000
simply connected?
Who thinks it's simply

47
00:04:06,000 --> 00:04:09,000
connected?
Who thinks it's not simply

48
00:04:09,000 --> 00:04:14,000
connected?
Let's think a little bit harder.

49
00:04:14,000 --> 00:04:17,000
Let's say that I take a loop
like this one,

50
00:04:17,000 --> 00:04:20,000
OK, it doesn't go through the
origin.

51
00:04:20,000 --> 00:04:24,000
Can I find a surface that's
bounded by this loop and that

52
00:04:24,000 --> 00:04:26,000
does not pass through the
origin?

53
00:04:26,000 --> 00:04:30,000
Yeah, I can take the sphere,
you know, for example,

54
00:04:30,000 --> 00:04:34,000
or anything that's just not
quite the disk?

55
00:04:34,000 --> 00:04:36,000
So,
and similarly,

56
00:04:36,000 --> 00:04:39,000
if I take any other loop that
avoids the origin,

57
00:04:39,000 --> 00:04:42,000
I can find, actually,
a surface bounded by it that

58
00:04:42,000 --> 00:04:44,000
does not pass through the
origin.

59
00:04:44,000 --> 00:04:47,000
So, actually,
that's kind of a not so obvious

60
00:04:47,000 --> 00:04:49,000
theorem to prove,
but maybe intuitively,

61
00:04:49,000 --> 00:04:52,000
start by finding any surface.
Well, if that surface passes

62
00:04:52,000 --> 00:04:54,000
through the origin,
just wiggle it a little bit,

63
00:04:54,000 --> 00:04:56,000
you can make sure it doesn't
pass through the origin anymore.

64
00:04:56,000 --> 00:05:00,000
Just push it a little bit.
So, in fact,

65
00:05:00,000 --> 00:05:08,000
this is simply connected.
That was a trick question.

66
00:05:08,000 --> 00:05:13,000
OK, now on the other hand,
a good example of something

67
00:05:13,000 --> 00:05:16,000
that is not simply connected is
if I take space,

68
00:05:16,000 --> 00:05:36,000
and I remove the z axis -- --
that is not simply connected.

69
00:05:36,000 --> 00:05:39,000
And, see, the reason is,
if I look again, say,

70
00:05:39,000 --> 00:05:44,000
at the unit circle in the x
axis,

71
00:05:44,000 --> 00:05:47,000
sorry, unit circle in the xy
plane,

72
00:05:47,000 --> 00:05:52,000
I mean, in the xy plane,
so, if I try to find a surface

73
00:05:52,000 --> 00:05:58,000
whose boundary is this disk,
well, it has to actually cross

74
00:05:58,000 --> 00:06:03,000
the z axis somewhere.
There's no way that I can find

75
00:06:03,000 --> 00:06:09,000
a surface whose only boundary is
this curve, which doesn't hit

76
00:06:09,000 --> 00:06:13,000
the z axis anywhere.
Of course, you could try to use

77
00:06:13,000 --> 00:06:17,000
the same trick as there,
say, maybe we want to go up,

78
00:06:17,000 --> 00:06:19,000
up, up.
You know, let's start with a

79
00:06:19,000 --> 00:06:21,000
cylinder.
Well, the problem is you have

80
00:06:21,000 --> 00:06:24,000
to go infinitely far because the
z axis goes infinitely far.

81
00:06:24,000 --> 00:06:27,000
And, you'll never be able to
actually close your surface.

82
00:06:27,000 --> 00:06:30,000
So, the matter what kind of
trick you might want to use,

83
00:06:30,000 --> 00:06:34,000
it's actually a theorem in
topology that you cannot find a

84
00:06:34,000 --> 00:06:39,000
surface bounded by this disk
without intersecting the z axis.

85
00:06:39,000 --> 00:06:44,000
Yes?
Well, a doughnut shape

86
00:06:44,000 --> 00:06:47,000
certainly would stay away from
the z axis, but it wouldn't be a

87
00:06:47,000 --> 00:06:50,000
surface with boundary just this
guy.

88
00:06:50,000 --> 00:06:53,000
Right, it would have to have
either some other boundary.

89
00:06:53,000 --> 00:06:57,000
So, maybe what you have in mind
is some sort of doughnut shape

90
00:06:57,000 --> 00:07:01,000
like this that curves on itself,
and maybe comes back.

91
00:07:01,000 --> 00:07:05,000
Well, if you don't quite close
it all the way around,

92
00:07:05,000 --> 00:07:07,000
so I can try to,
indeed, draw some sort of

93
00:07:07,000 --> 00:07:10,000
doughnut here.
Well, if I don't quite close

94
00:07:10,000 --> 00:07:13,000
it, that it will have another
edge at the other end wherever I

95
00:07:13,000 --> 00:07:15,000
started.
If I close it completely,

96
00:07:15,000 --> 00:07:18,000
then this curve is no longer
its boundary because my surface

97
00:07:18,000 --> 00:07:20,000
lives on both sides of this
curve.

98
00:07:20,000 --> 00:07:22,000
See, I want a surface that
stops on this curve,

99
00:07:22,000 --> 00:07:25,000
and doesn't go beyond it.
And, nowhere else does it have

100
00:07:25,000 --> 00:07:28,000
that kind of behavior.
Everywhere else,

101
00:07:28,000 --> 00:07:33,000
it keeps going on.
So,

102
00:07:33,000 --> 00:07:36,000
actually, I mean,
maybe actually another way to

103
00:07:36,000 --> 00:07:40,000
convince yourself is to find a
counter example to the statement

104
00:07:40,000 --> 00:07:44,000
I'm going to make about vector
fields with curl zero and simply

105
00:07:44,000 --> 00:07:48,000
connected regions always being
conservative.

106
00:07:48,000 --> 00:07:52,000
So, what you can do is you can
take the example that we had in

107
00:07:52,000 --> 00:07:55,000
one of our older problem sets.
That was a vector field in the

108
00:07:55,000 --> 00:07:58,000
plane.
But, you can also use it to

109
00:07:58,000 --> 00:08:02,000
define a vector field in space
just with no z component.

110
00:08:02,000 --> 00:08:05,000
That vector field is actually
defined everywhere except on the

111
00:08:05,000 --> 00:08:08,000
z axis, and it violates the
usual theorem that we would

112
00:08:08,000 --> 00:08:12,000
expect.
So, that's one way to check

113
00:08:12,000 --> 00:08:20,000
just for sure that this thing is
not simply connected.

114
00:08:20,000 --> 00:08:26,000
So, what's the statement I want
to make?

115
00:08:26,000 --> 00:08:40,000
So, recall we've seen if F is a
gradient field -- -- then its

116
00:08:40,000 --> 00:08:46,000
curl is zero.
That's just the fact that the

117
00:08:46,000 --> 00:08:49,000
mixed second partial derivatives
are equal.

118
00:08:49,000 --> 00:08:53,000
So, now, the converse is the
following theorem.

119
00:08:53,000 --> 00:09:01,000
It says if the curl of F equals
zero in, sorry,

120
00:09:01,000 --> 00:09:09,000
and F is defined -- No,
is not the logical in which to

121
00:09:09,000 --> 00:09:15,000
say it.
So, if F is defined in a simply

122
00:09:15,000 --> 00:09:30,000
connected region,
and curl F is zero -- -- then F

123
00:09:30,000 --> 00:09:45,000
is a gradient field,
and the line integral for F is

124
00:09:45,000 --> 00:09:53,000
path independent -- -- F is
conservative,

125
00:09:53,000 --> 00:09:55,000
and so on,
all the usual consequences.

126
00:09:55,000 --> 00:09:58,000
Remember, these are all
equivalent to each other,

127
00:09:58,000 --> 00:10:01,000
for example,
because you can use path

128
00:10:01,000 --> 00:10:05,000
independence to define the
potential by doing the line

129
00:10:05,000 --> 00:10:08,000
integral of F.
OK, so where do we use the

130
00:10:08,000 --> 00:10:12,000
assumption of being defined in a
simply connected region?

131
00:10:12,000 --> 00:10:17,000
Well, the way which we will
prove this is to use Stokes

132
00:10:17,000 --> 00:10:20,000
theorem.
OK, so the proof,

133
00:10:20,000 --> 00:10:25,000
so just going to prove that the
line integral is path

134
00:10:25,000 --> 00:10:29,000
independent;
the others work the same way.

135
00:10:29,000 --> 00:10:34,000
OK, so let's assume that we
have a vector field whose curl

136
00:10:34,000 --> 00:10:38,000
is zero.
And, let's say that we have two

137
00:10:38,000 --> 00:10:44,000
curves, C1 and C2,
that go from some point P0 to

138
00:10:44,000 --> 00:10:49,000
some point P1,
the same point to the same

139
00:10:49,000 --> 00:10:54,000
point.
Well, we'd like to understand

140
00:10:54,000 --> 00:11:00,000
the line integral along C1,
say, minus the line integral

141
00:11:00,000 --> 00:11:04,000
along C2 to show that this is
zero.

142
00:11:04,000 --> 00:11:06,000
That's what we are trying to
prove.

143
00:11:06,000 --> 00:11:12,000
So, how will we compute that?
Well, the line integral along

144
00:11:12,000 --> 00:11:17,000
C1 minus C2, well,
let's just form a closed curve

145
00:11:17,000 --> 00:11:24,000
that is C1 minus C2.
OK, so let's call C,

146
00:11:24,000 --> 00:11:36,000
woops -- So that's equal to the
integral along C of f dot dr

147
00:11:36,000 --> 00:11:44,000
where C is C1 followed by C2
backwards.

148
00:11:44,000 --> 00:11:50,000
Now, C is a closed curve.
So, I can use Stokes theorem.

149
00:11:50,000 --> 00:11:52,000
Well, to be able to use Stokes
theorem, I need,

150
00:11:52,000 --> 00:11:54,000
actually, to find a surface to
apply it to.

151
00:11:54,000 --> 00:11:57,000
And, that's where the
assumption of simply connected

152
00:11:57,000 --> 00:12:00,000
is useful.
I know in advance that any

153
00:12:00,000 --> 00:12:02,000
closed curve,
so, C in particular,

154
00:12:02,000 --> 00:12:10,000
has to bound some surface.
OK, so we can find S,

155
00:12:10,000 --> 00:12:21,000
a surface, S,
that bounds C because the

156
00:12:21,000 --> 00:12:32,000
region is simply connected.
So, now that tells us we can

157
00:12:32,000 --> 00:12:38,000
actually apply Stokes theorem,
except it won't fit here.

158
00:12:38,000 --> 00:12:40,000
So, instead,
I will do that on the next

159
00:12:40,000 --> 00:12:45,000
line.
That's equal by Stokes to the

160
00:12:45,000 --> 00:12:51,000
double integral over S of curl F
dot vector dS,

161
00:12:51,000 --> 00:12:54,000
or ndS.
But now, the curl is zero.

162
00:12:54,000 --> 00:12:58,000
So, if I integrate zero,
I will get zero.

163
00:12:58,000 --> 00:13:02,000
OK, so I proved that my two
line integrals along C1 and C2

164
00:13:02,000 --> 00:13:04,000
are equal.
But for that,

165
00:13:04,000 --> 00:13:08,000
I needed to be able to find a
surface which to apply Stokes

166
00:13:08,000 --> 00:13:11,000
theorem.
And that required my region to

167
00:13:11,000 --> 00:13:14,000
be simply connected.
If I had a vector field that

168
00:13:14,000 --> 00:13:17,000
was defined only outside of the
z axis and I took two paths that

169
00:13:17,000 --> 00:13:20,000
went on one side and the other
side of the z axis,

170
00:13:20,000 --> 00:13:21,000
I might have obtained,
actually,

171
00:13:21,000 --> 00:13:27,000
different values of the line
integral.

172
00:13:27,000 --> 00:13:35,000
OK, so anyway,
that's the customary warning

173
00:13:35,000 --> 00:13:43,000
about simply connected things.
OK, let me just mention very

174
00:13:43,000 --> 00:13:46,000
quickly that there's a lot of
interesting topology you can do,

175
00:13:46,000 --> 00:13:48,000
actually in space.
So, for example,

176
00:13:48,000 --> 00:13:50,000
this concept of being simply
connected or not,

177
00:13:50,000 --> 00:13:55,000
and studying which loops bound
surfaces or not can be used to

178
00:13:55,000 --> 00:13:58,000
classify shapes of things inside
space.

179
00:13:58,000 --> 00:14:07,000
So, for example,
one of the founding

180
00:14:07,000 --> 00:14:15,000
achievements of topology in the
19th century was to classify

181
00:14:15,000 --> 00:14:24,000
surfaces in space -- -- by
trying to look at loops on them.

182
00:14:24,000 --> 00:14:33,000
So, what I mean by that is that
if I take the surface of a

183
00:14:33,000 --> 00:14:39,000
sphere, well,
I claim the surface of a sphere

184
00:14:39,000 --> 00:14:44,000
-- -- is simply connected.
Why is that?

185
00:14:44,000 --> 00:14:49,000
Well, let's take my favorite
closed curve on the surface of a

186
00:14:49,000 --> 00:14:53,000
sphere.
I can always find a portion of

187
00:14:53,000 --> 00:14:59,000
the sphere that's bounded by it.
OK, so that's the definition of

188
00:14:59,000 --> 00:15:03,000
the surface of a sphere being
simply connected.

189
00:15:03,000 --> 00:15:06,000
On the other hand,
if I take what's called a

190
00:15:06,000 --> 00:15:07,000
torus,
or if you prefer,

191
00:15:07,000 --> 00:15:10,000
the surface of a doughnut,
that's more,

192
00:15:10,000 --> 00:15:21,000
it's a less technical term,
but it's -- -- well,

193
00:15:21,000 --> 00:15:24,000
that's not simply connected.
And, in fact,

194
00:15:24,000 --> 00:15:26,000
for example,
if you look at this loop here

195
00:15:26,000 --> 00:15:29,000
that goes around it,
well, of course it bounds a

196
00:15:29,000 --> 00:15:32,000
surface in space.
But, that surface cannot be

197
00:15:32,000 --> 00:15:35,000
made to be just a piece of the
donut.

198
00:15:35,000 --> 00:15:39,000
You have to go through the hole.
You have to leave the surface

199
00:15:39,000 --> 00:15:41,000
of a torus.
In fact, there's another one.

200
00:15:41,000 --> 00:15:47,000
See, this one also does not
bound anything that's completely

201
00:15:47,000 --> 00:15:50,000
contained in the torus.
And, of course,

202
00:15:50,000 --> 00:15:53,000
it bounds this disc,
but inside of a torus.

203
00:15:53,000 --> 00:15:56,000
But, that's not a part of the
surface itself.

204
00:15:56,000 --> 00:16:02,000
So, in fact,
there's, and topologists would

205
00:16:02,000 --> 00:16:09,000
say, there's two independent --
-- loops that don't bound

206
00:16:09,000 --> 00:16:15,000
surfaces, that don't bound
anything.

207
00:16:15,000 --> 00:16:18,000
And, so this number two is
somehow an invariant that you

208
00:16:18,000 --> 00:16:20,000
can associate to this kind of
shape.

209
00:16:20,000 --> 00:16:23,000
And then, if you consider more
complicated surfaces with more

210
00:16:23,000 --> 00:16:24,000
holes in them,
you can try, somehow,

211
00:16:24,000 --> 00:16:27,000
to count independent loops on
them,

212
00:16:27,000 --> 00:16:33,000
and that's the beginning of the
classification of surfaces.

213
00:16:33,000 --> 00:16:39,000
Anyway, that's not really an
18.02 topic, but I thought I

214
00:16:39,000 --> 00:16:45,000
would mentioned it because it's
kind of a cool idea.

215
00:16:45,000 --> 00:16:55,000
OK, let me say a bit more in
the way of fun remarks like

216
00:16:55,000 --> 00:16:59,000
that.
So, food for thought:

217
00:16:59,000 --> 00:17:05,000
let's say that I want to apply
Stokes theorem to simplify a

218
00:17:05,000 --> 00:17:08,000
line integral along the curve
here.

219
00:17:08,000 --> 00:17:11,000
So, this curve is maybe not
easy to see in the picture.

220
00:17:11,000 --> 00:17:17,000
It kind of goes twice around
the z axis, but spirals up and

221
00:17:17,000 --> 00:17:20,000
then down.
OK, so one way to find a

222
00:17:20,000 --> 00:17:25,000
surface that's bounded by this
curve is to take what's called

223
00:17:25,000 --> 00:17:29,000
the Mobius strip.
OK, so the Mobius strip,

224
00:17:29,000 --> 00:17:32,000
it's a one sided strip where
when you go around,

225
00:17:32,000 --> 00:17:35,000
you flip one side becomes the
other.

226
00:17:35,000 --> 00:17:38,000
So, you just,
if you want to take a band of

227
00:17:38,000 --> 00:17:41,000
paper and glue the two sides
with a twist,

228
00:17:41,000 --> 00:17:44,000
so, it's a one sided surface.
And, that gives us,

229
00:17:44,000 --> 00:17:49,000
actually, serious trouble if we
try to orient it to apply Stokes

230
00:17:49,000 --> 00:17:53,000
theorem.
So, see, for example,

231
00:17:53,000 --> 00:17:58,000
if I take this Mobius strip,
and I try to find an

232
00:17:58,000 --> 00:18:04,000
orientation,
so here it looks like that,

233
00:18:04,000 --> 00:18:08,000
well, let's say that I've
oriented my curve going in this

234
00:18:08,000 --> 00:18:11,000
direction.
So, I go around,

235
00:18:11,000 --> 00:18:13,000
around, around,
still going this direction.

236
00:18:13,000 --> 00:18:19,000
Well, the orientation I should
have for Stokes theorem is that

237
00:18:19,000 --> 00:18:22,000
when I, so, curve continues
here.

238
00:18:22,000 --> 00:18:26,000
Well, if you look at the
convention around here,

239
00:18:26,000 --> 00:18:31,000
it tells us that the normal
vector should be going this way.

240
00:18:31,000 --> 00:18:35,000
OK, if we look at it near here,
if we walk along this way,

241
00:18:35,000 --> 00:18:37,000
the surface is to our right .
So, we should actually be

242
00:18:37,000 --> 00:18:40,000
flipping things upside down.
The normal vector should be

243
00:18:40,000 --> 00:18:41,000
going down.
And, in fact,

244
00:18:41,000 --> 00:18:44,000
if you try to follow your
normal vector that's pointing

245
00:18:44,000 --> 00:18:45,000
up, it's pointing up,
up, up.

246
00:18:45,000 --> 00:18:49,000
It will have to go into things,
into, into, down.

247
00:18:49,000 --> 00:18:53,000
There's no way to choose
consistently a normal vector for

248
00:18:53,000 --> 00:19:01,000
the Mobius strip.
So, that's what we call a

249
00:19:01,000 --> 00:19:08,000
non-orientable surface.
And, that just means it has

250
00:19:08,000 --> 00:19:10,000
only one side.
And, if it has only one side,

251
00:19:10,000 --> 00:19:14,000
that we cannot speak of flux
for it because we have no way of

252
00:19:14,000 --> 00:19:17,000
saying that we'll be counting
things positively one way,

253
00:19:17,000 --> 00:19:19,000
negatively the other way,
because there's only one,

254
00:19:19,000 --> 00:19:22,000
you know,
there's no notion of sides.

255
00:19:22,000 --> 00:19:26,000
So, you can't define a side
towards which things will be

256
00:19:26,000 --> 00:19:34,000
going positively.
So, that's actually a situation

257
00:19:34,000 --> 00:19:44,000
where flux cannot be defined.
OK, so as much as Mobius strips

258
00:19:44,000 --> 00:19:48,000
and climb-bottles are exciting
and really cool,

259
00:19:48,000 --> 00:19:51,000
well, we can't use them in this
class because we can't define

260
00:19:51,000 --> 00:19:54,000
flux through them.
So, if we really wanted to

261
00:19:54,000 --> 00:19:57,000
apply Stokes theorem,
because I've been telling you

262
00:19:57,000 --> 00:19:59,000
that space is simply connected,
and I will always be able to

263
00:19:59,000 --> 00:20:01,000
apply Stokes theorem to any
curve,

264
00:20:01,000 --> 00:20:05,000
what would I do?
Well, I claim this curve

265
00:20:05,000 --> 00:20:10,000
actually bounds another surface
that is orientable.

266
00:20:10,000 --> 00:20:11,000
Yeah, that looks
counterintuitive.

267
00:20:11,000 --> 00:20:16,000
Well, let's see it.
I claim you can take a

268
00:20:16,000 --> 00:20:22,000
hemisphere, and you can take a
small thing and twist it around.

269
00:20:22,000 --> 00:20:26,000
So, in case you don't believe
me, let me do it again with the

270
00:20:26,000 --> 00:20:28,000
transparency.
Here's my loop,

271
00:20:28,000 --> 00:20:31,000
and see, well,
the scale is not exactly the

272
00:20:31,000 --> 00:20:33,000
same.
So, it doesn't quite match.

273
00:20:33,000 --> 00:20:35,000
But, and it's getting a bit
dark.

274
00:20:35,000 --> 00:20:41,000
But, that spherical thing with
a little slit going twisting

275
00:20:41,000 --> 00:20:46,000
into it will actually have
boundary my loop.

276
00:20:46,000 --> 00:20:50,000
And, that one is orientable.
I mean, I leave it up to you to

277
00:20:50,000 --> 00:20:56,000
stare at the picture long enough
to convince yourselves that

278
00:20:56,000 --> 00:21:00,000
there's a well-defined up and
down.

279
00:21:00,000 --> 00:21:11,000
OK.
So now, I mean,

280
00:21:11,000 --> 00:21:14,000
in case you are getting really,
really worried,

281
00:21:14,000 --> 00:21:18,000
I mean, there won't be any
Mobius strips on the exam on

282
00:21:18,000 --> 00:21:24,000
Tuesday, OK?
It's just to show you some cool

283
00:21:24,000 --> 00:21:29,000
stuff.
OK, questions?

284
00:21:29,000 --> 00:21:34,000
No?
OK, one last thing I want to

285
00:21:34,000 --> 00:21:38,000
show you before we start
reviewing,

286
00:21:38,000 --> 00:21:41,000
so one question you might have
about Stokes theorem is,

287
00:21:41,000 --> 00:21:44,000
how come we can choose whatever
surface we want?

288
00:21:44,000 --> 00:21:47,000
I mean, sure,
it seems to work,

289
00:21:47,000 --> 00:21:52,000
but why?
So, I'm going to say a couple

290
00:21:52,000 --> 00:22:02,000
of words about surface
independence in Stokes theorem.

291
00:22:02,000 --> 00:22:08,000
So, let's say that I have a
curve, C, in space.

292
00:22:08,000 --> 00:22:11,000
And, let's say that I want to
apply Stokes theorem.

293
00:22:11,000 --> 00:22:16,000
So, then I can choose my
favorite surface bounded by C.

294
00:22:16,000 --> 00:22:18,000
So, in a situation like this,
for example,

295
00:22:18,000 --> 00:22:21,000
I might want to make my first
choice be this guy,

296
00:22:21,000 --> 00:22:25,000
S1, like maybe some sort of
upper half sphere.

297
00:22:25,000 --> 00:22:28,000
And, if you pay attention to
the orientation conventions,

298
00:22:28,000 --> 00:22:31,000
you'll see that you need to
take it with normal vector

299
00:22:31,000 --> 00:22:34,000
pointing up.
Maybe actually I would rather

300
00:22:34,000 --> 00:22:36,000
make a different choice.
And actually,

301
00:22:36,000 --> 00:22:41,000
I will choose another surface,
S2, that maybe looks like that.

302
00:22:41,000 --> 00:22:44,000
And, if I look carefully at the
orientation convention,

303
00:22:44,000 --> 00:22:47,000
Stokes theorem tells me that I
have to take the normal vector

304
00:22:47,000 --> 00:22:52,000
pointing up again.
So, that's actually into things.

305
00:22:52,000 --> 00:22:57,000
So,
Stokes says that the line

306
00:22:57,000 --> 00:23:04,000
integral along C of my favorite
vector field can be computed

307
00:23:04,000 --> 00:23:09,000
either as a flux integral for
the curl through S1,

308
00:23:09,000 --> 00:23:16,000
or as the same integral,
but through S2 instead of S1.

309
00:23:16,000 --> 00:23:21,000
So, that seems to suggest that
curl F has some sort of surface

310
00:23:21,000 --> 00:23:24,000
independence property.
It doesn't really matter which

311
00:23:24,000 --> 00:23:27,000
surface I take,
as long as the boundary is this

312
00:23:27,000 --> 00:23:29,000
given curve, C.
Why is that?

313
00:23:29,000 --> 00:23:31,000
That's a strange property to
have.

314
00:23:31,000 --> 00:23:36,000
Where does it come from?
Well, let's think about it for

315
00:23:36,000 --> 00:23:40,000
a second.
So, why are these the same?

316
00:23:40,000 --> 00:23:42,000
I mean, of course,
they have to be the same

317
00:23:42,000 --> 00:23:44,000
because that's what Stokes tell
us.

318
00:23:44,000 --> 00:23:48,000
But, why is that OK?
Well, let's think about

319
00:23:48,000 --> 00:23:53,000
comparing the flux integral for
S1 and the flux integral for S2.

320
00:23:53,000 --> 00:23:57,000
So, if we want to compare them,
we should probably subtract

321
00:23:57,000 --> 00:24:02,000
them from each other.
OK, so let's do the flux

322
00:24:02,000 --> 00:24:09,000
integral for S1 minus the flux
integral for S2 of the same

323
00:24:09,000 --> 00:24:12,000
thing.
Well, let's give a name.

324
00:24:12,000 --> 00:24:18,000
Let's call S the surface S1
minus S2.

325
00:24:18,000 --> 00:24:21,000
So, what is S?
S is S1 with its given

326
00:24:21,000 --> 00:24:26,000
orientation together with S2
with the reversed orientation.

327
00:24:26,000 --> 00:24:32,000
So, S is actually this whole
closed surface here.

328
00:24:32,000 --> 00:24:37,000
And, the normal vector to S
seems to be pointing outwards

329
00:24:37,000 --> 00:24:39,000
everywhere.
OK, so now, if we have a closed

330
00:24:39,000 --> 00:24:41,000
surface with a normal vector
pointing outwards,

331
00:24:41,000 --> 00:24:44,000
and we want to find a flux
integral for it,

332
00:24:44,000 --> 00:24:47,000
well,
we can replace that with a

333
00:24:47,000 --> 00:24:57,000
triple integral.
So, that's the divergence

334
00:24:57,000 --> 00:25:03,000
theorem.
So, that's by the divergence

335
00:25:03,000 --> 00:25:09,000
theorem using the fact that S is
a closed surface.

336
00:25:09,000 --> 00:25:13,000
That's equal to the triple
integral over the region inside.

337
00:25:13,000 --> 00:25:26,000
Let me call that region D of
divergence, of curl F dV.

338
00:25:26,000 --> 00:25:34,000
OK, and what I'm going to claim
now is that we can actually

339
00:25:34,000 --> 00:25:41,000
check that if you take the
divergence of the curl of a

340
00:25:41,000 --> 00:25:47,000
vector field,
you always get zero.

341
00:25:47,000 --> 00:25:50,000
OK, and so that will tell you
that this integral will always

342
00:25:50,000 --> 00:25:53,000
be zero.
And that's why the flux for S1,

343
00:25:53,000 --> 00:25:58,000
and the flux for S2 were the
same a priori and we didn't have

344
00:25:58,000 --> 00:26:01,000
to worry about which one we
chose when we did Stokes

345
00:26:01,000 --> 00:26:06,000
theorem.
OK, so let's just check quickly

346
00:26:06,000 --> 00:26:10,000
that divergence of a curve is
zero.

347
00:26:10,000 --> 00:26:12,000
OK, in case you're wondering
why I'm doing all this,

348
00:26:12,000 --> 00:26:13,000
well, first I think it's kind
of interesting,

349
00:26:13,000 --> 00:26:17,000
and second, it reminds you of a
statement of all these theorems,

350
00:26:17,000 --> 00:26:19,000
and all these definitions.
So, in a way,

351
00:26:19,000 --> 00:26:25,000
we are already reviewing.
OK, so let's see.

352
00:26:25,000 --> 00:26:30,000
If my vector field has
components P,

353
00:26:30,000 --> 00:26:39,000
Q, and R, remember that the
curl was defined by this cross

354
00:26:39,000 --> 00:26:47,000
product between del and our
given vector field.

355
00:26:47,000 --> 00:27:04,000
So, that's Ry - Qz followed by
Pz - Rx, and Qx - Py.

356
00:27:04,000 --> 00:27:14,000
So, now, we want to take the
divergence of this.

357
00:27:14,000 --> 00:27:19,000
Well, so we have to take the
first component,

358
00:27:19,000 --> 00:27:23,000
Ry minus Qz,
and take its partial with

359
00:27:23,000 --> 00:27:28,000
respect to x.
Then, take the y component,

360
00:27:28,000 --> 00:27:35,000
Pz minus Rx partial with
respect to y plus Qx minus Py

361
00:27:35,000 --> 00:27:41,000
partial with respect to z.
And, well, now we should expand

362
00:27:41,000 --> 00:27:43,000
this.
But I claim it will always

363
00:27:43,000 --> 00:27:44,000
simplify to zero.

364
00:28:11,000 --> 00:28:24,000
OK, so I think we have over
there, becomes R sub yx minus Q

365
00:28:24,000 --> 00:28:39,000
sub zx plus P sub zy minus R sub
xy plus Q sub xz minus P sub yz.

366
00:28:39,000 --> 00:28:47,000
Well, let's see.
We have P sub zy minus P sub yz.

367
00:28:47,000 --> 00:28:54,000
These two cancel out.
We have R sub yx minus R sub xy.

368
00:28:54,000 --> 00:28:58,000
These cancel out.
Q sub zx and Q sub xz,

369
00:28:58,000 --> 00:29:04,000
these two also cancel out.
So, indeed, the divergence of a

370
00:29:04,000 --> 00:29:10,000
curl is always zero.
OK, so the claim is divergence

371
00:29:10,000 --> 00:29:17,000
of curl is always zero.
Del cross F is always zero,

372
00:29:17,000 --> 00:29:26,000
and just a small remark,
if we had actually real vectors

373
00:29:26,000 --> 00:29:30,000
rather than this strange del
guy,

374
00:29:30,000 --> 00:29:33,000
indeed we know that if we have
two vectors,

375
00:29:33,000 --> 00:29:37,000
U and V,
and we do u dot u cross v,

376
00:29:37,000 --> 00:29:40,000
what is that?
Well, one way to say it is it's

377
00:29:40,000 --> 00:29:43,000
the determinant of u,
u, and v, which is the volume

378
00:29:43,000 --> 00:29:45,000
of the box.
But, it's completely flat

379
00:29:45,000 --> 00:29:47,000
because u, u,
and v are all in the plane

380
00:29:47,000 --> 00:29:50,000
defined by u and v.
The other way to say it is that

381
00:29:50,000 --> 00:29:53,000
u cross v is perpendicular to u
and v.

382
00:29:53,000 --> 00:29:56,000
Well, if it's perpendicular u,
then its dot product with u

383
00:29:56,000 --> 00:29:59,000
will be zero.
So, no matter how you say it,

384
00:29:59,000 --> 00:30:02,000
this is always zero.
So, in a way,

385
00:30:02,000 --> 00:30:09,000
this reinforces our intuition
that del, even though it's not

386
00:30:09,000 --> 00:30:15,000
at all an actual vector
sometimes can be manipulated in

387
00:30:15,000 --> 00:30:20,000
the same way.
OK, I think that's it for new

388
00:30:20,000 --> 00:30:26,000
topics for today.
And,

389
00:30:26,000 --> 00:30:30,000
so, now I should maybe try to
recap quickly what we've learned

390
00:30:30,000 --> 00:30:34,000
in these past three weeks so
that you know,

391
00:30:34,000 --> 00:30:39,000
so, the exam is probably going
to be similar in difficulty to

392
00:30:39,000 --> 00:30:42,000
the practice exams.
That's my goal.

393
00:30:42,000 --> 00:30:45,000
I don't know if I will have
reached that goal or not.

394
00:30:45,000 --> 00:30:48,000
We'll only know that after
you've taken the test.

395
00:30:48,000 --> 00:30:53,000
But, the idea is it's meant to
be more or less the same level

396
00:30:53,000 --> 00:30:58,000
of difficulty.
So, at this point,

397
00:30:58,000 --> 00:31:06,000
we've learned about three kinds
of beasts in space.

398
00:31:06,000 --> 00:31:12,000
OK, so I'm going to divide my
blackboard into three pieces,

399
00:31:12,000 --> 00:31:16,000
and here I will write triple
integrals.

400
00:31:16,000 --> 00:31:20,000
We've learned about double
integrals, and we've learned

401
00:31:20,000 --> 00:31:26,000
about line integrals.
OK, so triple integrals over a

402
00:31:26,000 --> 00:31:33,000
region in space,
we integrate a scalar quantity,

403
00:31:33,000 --> 00:31:35,000
dV.
How do we do that?

404
00:31:35,000 --> 00:31:41,000
Well, we can do that in
rectangular coordinates where dV

405
00:31:41,000 --> 00:31:46,000
becomes something like,
maybe, dz dx dy,

406
00:31:46,000 --> 00:31:52,000
or any permutation of these.
We've seen how to do it also in

407
00:31:52,000 --> 00:31:59,000
cylindrical coordinates where dV
is maybe dz times r dr d theta

408
00:31:59,000 --> 00:32:02,000
or more commonly r dr d theta
dz.

409
00:32:02,000 --> 00:32:06,000
But, what I want to emphasize
in this way is that both of

410
00:32:06,000 --> 00:32:09,000
these you set up pretty much in
the same way.

411
00:32:09,000 --> 00:32:12,000
So, remember,
the main trick here is to find

412
00:32:12,000 --> 00:32:15,000
the bounds of integration.
So, when you do it,

413
00:32:15,000 --> 00:32:18,000
say, with dz first,
that means for fixed xy,

414
00:32:18,000 --> 00:32:23,000
so, for a fixed point in the xy
plane, you have to look at the

415
00:32:23,000 --> 00:32:25,000
bounds for z.
So, that means you have to

416
00:32:25,000 --> 00:32:28,000
figure out what's the bottom
surface of your solid,

417
00:32:28,000 --> 00:32:31,000
and what's the top surface of
your solid?

418
00:32:31,000 --> 00:32:34,000
And, you have to find the value
of z at the bottom,

419
00:32:34,000 --> 00:32:37,000
the value of z at the top as
functions of x and y.

420
00:32:37,000 --> 00:32:40,000
And then, you will put that as
bounds for z.

421
00:32:40,000 --> 00:32:43,000
Once you've done that,
you are left with the question

422
00:32:43,000 --> 00:32:45,000
of finding bounds for x and y.
Well, for that,

423
00:32:45,000 --> 00:32:49,000
you just rotate the picture,
look at your solid from above,

424
00:32:49,000 --> 00:32:52,000
so, look at its projection to
the xy plane,

425
00:32:52,000 --> 00:32:56,000
and you set up a double
integral either in rectangular

426
00:32:56,000 --> 00:33:02,000
xy coordinates,
or in polar coordinates for x

427
00:33:02,000 --> 00:33:04,000
and y.
Of course, you can always do it

428
00:33:04,000 --> 00:33:08,000
a different orders.
And, I'll let you figure out

429
00:33:08,000 --> 00:33:11,000
again how that goes.
But, if you do dz first,

430
00:33:11,000 --> 00:33:15,000
then the inner bounds are given
by bottom and top,

431
00:33:15,000 --> 00:33:20,000
and the outer ones are given by
looking at the shadow of the

432
00:33:20,000 --> 00:33:23,000
region.
Now, there's also spherical

433
00:33:23,000 --> 00:33:28,000
coordinates.
And there, we've seen that dV

434
00:33:28,000 --> 00:33:32,000
is rho squared sine phi d rho d
phi d theta.

435
00:33:32,000 --> 00:33:35,000
So now, of course,
if this orgy of Greek letters

436
00:33:35,000 --> 00:33:39,000
is confusing you at this point,
then you probably need to first

437
00:33:39,000 --> 00:33:41,000
review spherical coordinates for
themselves.

438
00:33:41,000 --> 00:33:44,000
Remember that rho is the
distance from the origin.

439
00:33:44,000 --> 00:33:47,000
Phi is the angle down from the
z axis.

440
00:33:47,000 --> 00:33:49,000
So, it's zero,
and the positive z axis,

441
00:33:49,000 --> 00:33:53,000
pi over two in the xy plane,
and increases all the way to pi

442
00:33:53,000 --> 00:33:59,000
on the negative z axis.
And, theta is the angle around

443
00:33:59,000 --> 00:34:02,000
the z axis.
So, now, when we set up bounds

444
00:34:02,000 --> 00:34:04,000
here,
it will look a lot like what

445
00:34:04,000 --> 00:34:07,000
you've done in polar coordinates
in the plane because when you

446
00:34:07,000 --> 00:34:09,000
look at the inner bound down on
rho,

447
00:34:09,000 --> 00:34:12,000
for a fixed phi and theta,
that means you're shooting a

448
00:34:12,000 --> 00:34:15,000
straight ray from the origin in
some direction in space.

449
00:34:15,000 --> 00:34:17,000
So, you know,
you're sending a laser beam,

450
00:34:17,000 --> 00:34:20,000
and you want to know what part
of your beam is going to be in

451
00:34:20,000 --> 00:34:23,000
your given solid.
You want to solve for the value

452
00:34:23,000 --> 00:34:26,000
of rho when you enter the solid
and when you leave it.

453
00:34:26,000 --> 00:34:29,000
I mean, very often,
if the origin is in your solid,

454
00:34:29,000 --> 00:34:33,000
then rho will start at zero.
Then you want to know when you

455
00:34:33,000 --> 00:34:34,000
exit.
And, I mean,

456
00:34:34,000 --> 00:34:38,000
there's a fairly small list of
kinds of surfaces that we've

457
00:34:38,000 --> 00:34:41,000
seen how to set up in spherical
coordinates.

458
00:34:41,000 --> 00:34:44,000
So, if you're really upset by
this, go over the problems in

459
00:34:44,000 --> 00:34:47,000
the notes.
That will give you a good idea

460
00:34:47,000 --> 00:34:53,000
of what kinds of things we've
seen in spherical coordinates.

461
00:34:53,000 --> 00:34:56,000
OK, and then evaluation is the
usual way.

462
00:34:56,000 --> 00:35:01,000
Questions about this?
No?

463
00:35:01,000 --> 00:35:08,000
OK, so, I should say we can do
something bad,

464
00:35:08,000 --> 00:35:15,000
but so we've seen,
of course, applications of

465
00:35:15,000 --> 00:35:19,000
this.
So, we should know how to use a

466
00:35:19,000 --> 00:35:24,000
triple integral to evaluate
things like a mass of a solid,

467
00:35:24,000 --> 00:35:29,000
the average value of a
function,

468
00:35:29,000 --> 00:35:37,000
the moment of inertia about one
of the coordinate axes,

469
00:35:37,000 --> 00:35:54,000
or the gravitational attraction
on a mass at the origin.

470
00:35:54,000 --> 00:35:58,000
OK, so these are just formulas
to remember for examples of

471
00:35:58,000 --> 00:36:01,000
triple integrals.
It doesn't change conceptually.

472
00:36:01,000 --> 00:36:04,000
You always set them up and
evaluate them the same way.

473
00:36:04,000 --> 00:36:11,000
It just tells you what to put
there for the integrand.

474
00:36:11,000 --> 00:36:15,000
Now,
double integrals: so,

475
00:36:15,000 --> 00:36:18,000
when we have a surface in
space,

476
00:36:18,000 --> 00:36:21,000
well, what we will integrate on
it,

477
00:36:21,000 --> 00:36:26,000
at least what we've seen how to
integrate is a vector field

478
00:36:26,000 --> 00:36:31,000
dotted with the unit normal
vector times the area element.

479
00:36:31,000 --> 00:36:38,000
OK, and this is sometimes
called vector dS.

480
00:36:38,000 --> 00:36:48,000
Now, how do we evaluate that?
Well, we've seen formulas for

481
00:36:48,000 --> 00:36:55,000
ndS in various settings.
And, once you have a formula

482
00:36:55,000 --> 00:37:01,000
for ndS, that will relate ndS to
maybe dx dy, or something else.

483
00:37:01,000 --> 00:37:07,000
And then, you will express,
so, for example,

484
00:37:07,000 --> 00:37:15,000
ndS equals something dx dy.
And then, it becomes a double

485
00:37:15,000 --> 00:37:21,000
integral of something dx dy.
Now, in the integrand,

486
00:37:21,000 --> 00:37:23,000
you want to express everything
in terms of x and y.

487
00:37:23,000 --> 00:37:26,000
So, if you had a z,
maybe you have a formula for z

488
00:37:26,000 --> 00:37:28,000
in terms of x and y.
And, when you set up the

489
00:37:28,000 --> 00:37:30,000
bounds, well,
you try to figure out what are

490
00:37:30,000 --> 00:37:33,000
the bounds for x and y?
That would be just looking at

491
00:37:33,000 --> 00:37:35,000
it from above.
Of course, if you are using

492
00:37:35,000 --> 00:37:37,000
other variables,
figure out the bounds for those

493
00:37:37,000 --> 00:37:40,000
variables.
And, when you've done that,

494
00:37:40,000 --> 00:37:44,000
it becomes just a double
integral in the usual sense.

495
00:37:44,000 --> 00:37:46,000
OK, so maybe I should be a bit
more explicit about formulas

496
00:37:46,000 --> 00:37:52,000
because there have been a lot.
So, let me tell you about a few

497
00:37:52,000 --> 00:37:56,000
of them.
Let me actually do that over

498
00:37:56,000 --> 00:38:02,000
here because I don't want to
make this too crowded.

499
00:38:24,000 --> 00:38:28,000
OK, so what kinds of formulas
for ndS have we seen?

500
00:38:28,000 --> 00:38:32,000
Well, we've seen a formula,
for example,

501
00:38:32,000 --> 00:38:37,000
for a horizontal plane,
or for something that's

502
00:38:37,000 --> 00:38:42,000
parallel to the yz plane or the
xz plane.

503
00:38:42,000 --> 00:38:47,000
Well, let's do just the yz
plane for a quick reminder.

504
00:38:47,000 --> 00:38:52,000
So, if I have a surface that's
contained inside the yz plane,

505
00:38:52,000 --> 00:38:56,000
then obviously I will express
ds in terms of,

506
00:38:56,000 --> 00:39:01,000
well, I will use y and z as my
variables.

507
00:39:01,000 --> 00:39:05,000
So, I will say that ds is dy
dz, or dz dy,

508
00:39:05,000 --> 00:39:11,000
whatever's most convenient.
Maybe we will even switch to

509
00:39:11,000 --> 00:39:15,000
polar coordinates after that if
a problem wants us to.

510
00:39:15,000 --> 00:39:16,000
And, what about the normal
vector?

511
00:39:16,000 --> 00:39:21,000
Well, the normal vector is
either coming straight at us,

512
00:39:21,000 --> 00:39:26,000
or it's maybe going back away
from us depending on which

513
00:39:26,000 --> 00:39:30,000
orientation we've chosen.
So, this gives us ndS.

514
00:39:30,000 --> 00:39:32,000
We dot our favorite vector
field with it.

515
00:39:32,000 --> 00:39:38,000
We integrate,
and we get the answer.

516
00:39:38,000 --> 00:39:48,000
OK, we've seen about spheres
and cylinders centered at the

517
00:39:48,000 --> 00:39:54,000
origin or centered on the z
axis.

518
00:39:54,000 --> 00:40:00,000
So, the normal vector sticks
straight out or straight in,

519
00:40:00,000 --> 00:40:05,000
depending on which direction
you do it in.

520
00:40:05,000 --> 00:40:09,000
So, for a sphere,
the normal vector is <x,

521
00:40:09,000 --> 00:40:14,000
y, z> divided by the radius
of the sphere.

522
00:40:14,000 --> 00:40:17,000
For a cylinder,
it's <x, y,

523
00:40:17,000 --> 00:40:21,000
0>, divided by the radius of
a cylinder.

524
00:40:21,000 --> 00:40:25,000
And, the surface element on a
sphere,

525
00:40:25,000 --> 00:40:28,000
so, see, it's very closely
related to the volume element of

526
00:40:28,000 --> 00:40:31,000
spherical coordinates except you
don't have a rho anymore.

527
00:40:31,000 --> 00:40:37,000
You just plug in a rho equals a.
So, you get a squared sine phi

528
00:40:37,000 --> 00:40:41,000
d phi d theta.
And, for a cylinder,

529
00:40:41,000 --> 00:40:47,000
it would be a dz d theta.
So,

530
00:40:47,000 --> 00:40:51,000
by the way, just a quick check,
when you're doing an integral,

531
00:40:51,000 --> 00:40:55,000
if it's the surface integral,
there should be two integral

532
00:40:55,000 --> 00:40:56,000
signs,
and there should be two

533
00:40:56,000 --> 00:40:59,000
integration variables.
And, there should be two d

534
00:40:59,000 --> 00:41:03,000
somethings.
If you end up with a dx,

535
00:41:03,000 --> 00:41:10,000
dy, dz in the surface integral,
something is seriously wrong.

536
00:41:10,000 --> 00:41:18,000
OK, now, besides these specific
formulas, we've seen two general

537
00:41:18,000 --> 00:41:24,000
formulas that are also useful.
So, one is,

538
00:41:24,000 --> 00:41:29,000
if we know how to express z in
terms of x and y,

539
00:41:29,000 --> 00:41:33,000
and just to change notation to
show you that it's not set in

540
00:41:33,000 --> 00:41:36,000
stone,
let's say that z is known as a

541
00:41:36,000 --> 00:41:42,000
function z of x and y.
So, how do I get ndS in that

542
00:41:42,000 --> 00:41:45,000
case?
Well, we've seen a formula that

543
00:41:45,000 --> 00:41:51,000
says negative partial z partial
x, negative partial z partial y,

544
00:41:51,000 --> 00:41:54,000
one dx dy.
So, this formula relates the

545
00:41:54,000 --> 00:41:57,000
volume, sorry,
the surface element on our

546
00:41:57,000 --> 00:42:01,000
surface to the area element in
the xy plane.

547
00:42:01,000 --> 00:42:08,000
It lets us convert between dS
and dx dy.

548
00:42:08,000 --> 00:42:11,000
OK, so we just plug in this,
and we dot with F,

549
00:42:11,000 --> 00:42:14,000
and then we substitute
everything in terms of x and y,

550
00:42:14,000 --> 00:42:17,000
and we evaluate the integral
over x and y.

551
00:42:17,000 --> 00:42:22,000
If we don't really want to find
a way to find z as a function of

552
00:42:22,000 --> 00:42:29,000
x and y,
but we have a normal vector

553
00:42:29,000 --> 00:42:35,000
given to us,
then we have another formula

554
00:42:35,000 --> 00:42:39,000
which says that ndS is,
sorry, I should have said it's

555
00:42:39,000 --> 00:42:42,000
always up to sign because we
have a two orientation

556
00:42:42,000 --> 00:42:45,000
convention.
We have to decide based on what

557
00:42:45,000 --> 00:42:48,000
we are trying to do,
whether we are doing the

558
00:42:48,000 --> 00:42:51,000
correct convention or the wrong
one.

559
00:42:51,000 --> 00:43:02,000
So, the other formula is n over
n dot k dx dy.

560
00:43:02,000 --> 00:43:09,000
Sorry, are they all the same?
Well, if you want,

561
00:43:09,000 --> 00:43:12,000
you can put an absolute value
here.

562
00:43:12,000 --> 00:43:16,000
But, it doesn't matter because
it's up to sign anyway.

563
00:43:16,000 --> 00:43:22,000
So, I mean, this formula is
valid as it is.

564
00:43:22,000 --> 00:43:24,000
OK, and, I mean,
if you're in a situation where

565
00:43:24,000 --> 00:43:26,000
you can apply more than one
formula,

566
00:43:26,000 --> 00:43:32,000
they will all give you the same
answer in the end because it's

567
00:43:32,000 --> 00:43:37,000
the same flux integral.
OK, so anyway,

568
00:43:37,000 --> 00:43:40,000
so we have various ways of
computing surface integrals,

569
00:43:40,000 --> 00:43:44,000
and probably one of the best
possible things you can do to

570
00:43:44,000 --> 00:43:48,000
prepare for the test is actually
to look again at some practice

571
00:43:48,000 --> 00:43:51,000
problems from the notes that do
flux integrals over various

572
00:43:51,000 --> 00:43:55,000
kinds of surfaces because that's
probably one of the hardest

573
00:43:55,000 --> 00:43:58,000
topics in this unit of the
class.

574
00:43:58,000 --> 00:44:05,000
OK, anyway, let's move on to
line integrals.

575
00:44:05,000 --> 00:44:11,000
So, those are actually a piece
of cake in comparison,

576
00:44:11,000 --> 00:44:17,000
OK, because all that this is,
is just integral of P dx Q dy R

577
00:44:17,000 --> 00:44:24,000
dz.
And, then all you have to do is

578
00:44:24,000 --> 00:44:35,000
parameterize the curve,
C, to express everything in

579
00:44:35,000 --> 00:44:42,000
terms of a single variable.
And then, you end up with a

580
00:44:42,000 --> 00:44:46,000
usual single integral,
and you can just compute it.

581
00:44:46,000 --> 00:44:48,000
So, that one works pretty much
as it did in the plane.

582
00:44:48,000 --> 00:44:52,000
So, if you forgotten what we
did in the plane,

583
00:44:52,000 --> 00:44:56,000
it's really the same thing.
OK, so now we have three

584
00:44:56,000 --> 00:44:58,000
different kinds of integrals,
and really, well,

585
00:44:58,000 --> 00:45:01,000
they certainly have in common
that they integrate things

586
00:45:01,000 --> 00:45:03,000
somehow.
But, apart from that,

587
00:45:03,000 --> 00:45:05,000
they are extremely different in
what they do.

588
00:45:05,000 --> 00:45:08,000
I mean, this one involves a
function, a scalar quantity.

589
00:45:08,000 --> 00:45:11,000
These involve vector quantities.
They don't involve the same

590
00:45:11,000 --> 00:45:13,000
kinds of shapes over which to
integrate.

591
00:45:13,000 --> 00:45:16,000
Here, you integrate over a
three-dimensional region.

592
00:45:16,000 --> 00:45:19,000
Here, you integrate only over a
two-dimensional surface,

593
00:45:19,000 --> 00:45:21,000
and here, only a
one-dimensional curve.

594
00:45:21,000 --> 00:45:24,000
So, try not to confuse them.
That's basically the most

595
00:45:24,000 --> 00:45:27,000
important advice.
Don't get mistaken.

596
00:45:27,000 --> 00:45:30,000
Each of them has a different
way of getting evaluated.

597
00:45:30,000 --> 00:45:34,000
Eventually, they will all give
you numbers, but through

598
00:45:34,000 --> 00:45:36,000
different processes.
So now, well,

599
00:45:36,000 --> 00:45:38,000
I said these guys are
completely different.

600
00:45:38,000 --> 00:45:40,000
Well, they are,
but we still have some bridges

601
00:45:40,000 --> 00:45:42,000
between them.
OK, so we have two,

602
00:45:42,000 --> 00:45:46,000
maybe I should say three,
well, two bridges between these

603
00:45:46,000 --> 00:45:49,000
guys.
OK, so we have somehow a

604
00:45:49,000 --> 00:45:54,000
connection between these which
is the divergence theorem.

605
00:45:54,000 --> 00:46:02,000
We have a connection between
that, which is Stokes theorem.

606
00:46:02,000 --> 00:46:20,000
So -- Just to write them again,
so the divergence theorem says

607
00:46:20,000 --> 00:46:25,000
if I have a region in space,
and I call its boundary S,

608
00:46:25,000 --> 00:46:27,000
so, it's going to be a closed
surface,

609
00:46:27,000 --> 00:46:31,000
and I orient S with a normal
vector pointing outwards,

610
00:46:31,000 --> 00:46:35,000
then whenever I have a surface
integral over S,

611
00:46:35,000 --> 00:46:40,000
sorry,
I can replace it by a triple

612
00:46:40,000 --> 00:46:47,000
integral over the region inside.
OK, so this guy is a vector

613
00:46:47,000 --> 00:46:49,000
field.
And, this guy is a function

614
00:46:49,000 --> 00:46:52,000
that somehow relates to the
vector field.

615
00:46:52,000 --> 00:46:54,000
I mean, you should know how.
You should know the definition

616
00:46:54,000 --> 00:46:55,000
of divergence,
of course.

617
00:46:55,000 --> 00:46:59,000
But, what I want to point out
is if you have to compute the

618
00:46:59,000 --> 00:47:01,000
two sides separately,
well, this is just,

619
00:47:01,000 --> 00:47:04,000
you know, your standard flux
integral.

620
00:47:04,000 --> 00:47:07,000
This is just your standard
triple integral over a region in

621
00:47:07,000 --> 00:47:09,000
space.
Once you have computed what

622
00:47:09,000 --> 00:47:13,000
this guy is, it's really just a
triple integral of the function.

623
00:47:13,000 --> 00:47:16,000
So, the way in which you
compute it doesn't see that it

624
00:47:16,000 --> 00:47:19,000
came from a divergence.
It's just the same way that you

625
00:47:19,000 --> 00:47:23,000
would compute any other triple
integral.

626
00:47:23,000 --> 00:47:28,000
The way we compute it doesn't
depend on what actually we are

627
00:47:28,000 --> 00:47:32,000
integrating.
Stokes theorem says if I have a

628
00:47:32,000 --> 00:47:35,000
curve that's the boundary of a
surface, S,

629
00:47:35,000 --> 00:47:40,000
and I orient the two in
compatible manners,

630
00:47:40,000 --> 00:47:52,000
then I can replace a line
integral on C by a surface

631
00:47:52,000 --> 00:47:57,000
integral on S.
OK, and that surface integral,

632
00:47:57,000 --> 00:48:00,000
well, it's not for the same
vector field.

633
00:48:00,000 --> 00:48:03,000
This relates a line integral
for one field to a surface

634
00:48:03,000 --> 00:48:06,000
integral from another field.
That other field is given from

635
00:48:06,000 --> 00:48:10,000
the first one just by taking its
curl So, after you take the

636
00:48:10,000 --> 00:48:13,000
curl, you obtain a different
vector field.

637
00:48:13,000 --> 00:48:17,000
And, the way in which you would
compute the surface integral is

638
00:48:17,000 --> 00:48:19,000
just as with any surface
integral.

639
00:48:19,000 --> 00:48:23,000
You just find a formula for ndS
dot product, substitute,

640
00:48:23,000 --> 00:48:26,000
evaluate.
The calculation of this thing,

641
00:48:26,000 --> 00:48:30,000
once you've computed curl does
not remember that it was a curl.

642
00:48:30,000 --> 00:48:33,000
It's the same as with any other
flux integral.

643
00:48:33,000 --> 00:48:35,000
OK, and finally,
the last bridge,

644
00:48:35,000 --> 00:48:38,000
so this was between two and
three.

645
00:48:38,000 --> 00:48:41,000
This was between one and two.
Let me just say,

646
00:48:41,000 --> 00:48:45,000
there's a bridge between zero
and one,

647
00:48:45,000 --> 00:48:53,000
which is that if you have a
function in its gradient,

648
00:48:53,000 --> 00:48:57,000
well, the fundamental theorem
of calculus says that the line

649
00:48:57,000 --> 00:49:01,000
integral for the vector field
given by the gradient of a

650
00:49:01,000 --> 00:49:04,000
function is actually equal to
the change in value of a

651
00:49:04,000 --> 00:49:08,000
function.
That's if you have a curve

652
00:49:08,000 --> 00:49:11,000
bounded by P0 and P1.
So in a way, actually,

653
00:49:11,000 --> 00:49:15,000
each of these three theorems
relates a quantity with a

654
00:49:15,000 --> 00:49:19,000
certain number of integral signs
to a quantity with one more

655
00:49:19,000 --> 00:49:22,000
integral sign.
And, that's actually somehow a

656
00:49:22,000 --> 00:49:24,000
fundamental similarity between
them.

657
00:49:24,000 --> 00:49:27,000
But maybe it's easier to think
of them as completely different

658
00:49:27,000 --> 00:49:30,000
stories.
So now, with this one,

659
00:49:30,000 --> 00:49:36,000
we additionally have to
remember another topic is given

660
00:49:36,000 --> 00:49:41,000
a vector field,
F, with curl equal to zero,

661
00:49:41,000 --> 00:49:45,000
find the potential.
And, we've seen two methods for

662
00:49:45,000 --> 00:49:47,000
that, and I'm sure you remember
them.

663
00:49:47,000 --> 00:49:53,000
So, if not, then try to
remember them for Tuesday.

664
00:49:53,000 --> 00:49:54,000
OK, so anyway,
again, conceptually,

665
00:49:54,000 --> 00:49:56,000
we have, really,
three different kinds of

666
00:49:56,000 --> 00:49:59,000
integrals.
We evaluated them in completely

667
00:49:59,000 --> 00:50:01,000
different ways,
and we have a handful of

668
00:50:01,000 --> 00:50:03,000
theorems, connecting them to
each other.

669
00:50:03,000 --> 00:50:06,000
But, that doesn't have any
impact on how we actually

670
00:50:06,000 --> 00:50:08,000
compute things.