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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
12
00:00:53 --> 00:00:57
in terms of,
instead,
13
00:00:57 --> 00:01:04
surface integral for flux of a
different vector field,
14
00:01:04 --> 00:01:12
namely, curl f dot n dS.
OK, so that's the statement.
15
00:01:12 --> 00:01:18
And, just to clarify a little
bit, so, again,
16
00:01:18 --> 00:01:24
we've seen various kinds of
integrals.
17
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.
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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,
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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,
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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,
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00:04:49 --> 00:04:52
start by finding any surface.
Well, if that surface passes
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00:04:52 --> 00:04:54
through the origin,
just wiggle it a little bit,
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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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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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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
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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
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00:06:09 --> 00:06:13
the z axis anywhere.
Of course, you could try to use
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00:06:13 --> 00:06:17
the same trick as there,
say, maybe we want to go up,
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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
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00:06:21 --> 00:06:24
to go infinitely far because the
z axis goes infinitely far.
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00:06:24 --> 00:06:27
And, you'll never be able to
actually close your surface.
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00:06:27 --> 00:06:30
So, the matter what kind of
trick you might want to use,
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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.
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00:06:39 --> 00:06:44
Yes?
Well, a doughnut shape
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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
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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
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00:07:40 --> 00:07:44
I'm going to make about vector
fields with curl zero and simply
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00:07:44 --> 00:07:48
connected regions always being
conservative.
106
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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.
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00:08:20 --> 00:08:26
So, what's the statement I want
to make?
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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.
119
00:08:53 --> 00:09:01
It says if the curl of F equals
zero in, sorry,
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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
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00:09:30 --> 00:09:45
is a gradient field,
and the line integral for F is
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00:09:45 --> 00:09:53
path independent -- -- F is
conservative,
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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
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00:10:17 --> 00:10:20
theorem.
OK, so the proof,
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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
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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
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00:10:54 --> 00:11:00
the line integral along C1,
say, minus the line integral
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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
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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
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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
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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,
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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 00:40:14
y, z> divided by the radius
of the sphere.
524
00:40:14 --> 00:40:17
For a cylinder,
it's 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