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I guess last time on Friday we
went over the first half of the
8
00:00:28 --> 00:00:33
class very quickly.
And so today we are going to go
9
00:00:33 --> 00:00:38
over the second half of the
class very quickly.
10
00:00:38 --> 00:00:45
And so that was stuff about
double and triple integrals and
11
00:00:45 --> 00:00:51
vector calculus in the plane and
in space.
12
00:00:51 --> 00:00:55
As usual, what is on the final
is basically what was on the
13
00:00:55 --> 00:00:57
other tests, exactly the same
stuff.
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00:00:57 --> 00:01:03
Well, not the same problem,
unfortunately.
15
00:01:03 --> 00:01:13
The first thing we learned
about was double integrals in
16
00:01:13 --> 00:01:25
the plane and how to set up the
bounds and how to evaluate them.
17
00:01:25 --> 00:01:30
Just to remind you quickly,
the important thing with
18
00:01:30 --> 00:01:34
iterated integrals is when you
integrate a function f of x,
19
00:01:34 --> 00:01:38
y,
say dy dx for example,
20
00:01:38 --> 00:01:41
is that you have to draw a
picture of a region.
21
00:01:41 --> 00:01:45
Unless it is completely obvious
you should really draw some
22
00:01:45 --> 00:01:48
picture of the domain of
integration.
23
00:01:48 --> 00:01:53
And once you have that picture
you can use it to find the
24
00:01:53 --> 00:01:57
bounds.
Remember the general method is
25
00:01:57 --> 00:02:03
that we first look at the inner
integral, here integral of f dy.
26
00:02:03 --> 00:02:07
And in this inner integral the
outer variable here,
27
00:02:07 --> 00:02:10
x, is fixed.
That means we are slicing our
28
00:02:10 --> 00:02:13
region by a vertical line
corresponding to a fixed value
29
00:02:13 --> 00:02:16
of x.
We fix a value of x.
30
00:02:16 --> 00:02:20
And what we have to find out is
the bounds for y,
31
00:02:20 --> 00:02:24
so the value of y at this
point, the value of y at that
32
00:02:24 --> 00:02:28
point.
Let me call that y some bottom
33
00:02:28 --> 00:02:34
of x, in general depends on x.
And this one will be y at that
34
00:02:34 --> 00:02:43
top, and it also depends on x.
And then the bounds for y would
35
00:02:43 --> 00:02:46
be this.
And then, when you look at the
36
00:02:46 --> 00:02:48
outer bound, things are
different.
37
00:02:48 --> 00:02:51
Because there you expect to
have just numbers,
38
00:02:51 --> 00:02:53
no longer functions of
anything.
39
00:02:53 --> 00:02:56
And what you do is look at the
shadow of your region.
40
00:02:56 --> 00:03:00
We are doing it by shadow so
you just project to the x-axis.
41
00:03:00 --> 00:03:04
If you project to the x-axis
your region will look like this.
42
00:03:04 --> 00:03:08
Its shadow is going to be this
integral form,
43
00:03:08 --> 00:03:12
some minimum value of x to some
maximum value of x.
44
00:03:12 --> 00:03:24
And that will give us the
bounds for the outer integral.
45
00:03:24 --> 00:03:27
And then, to evaluate,
we evaluate the usual way.
46
00:03:27 --> 00:03:29
Speaking of evaluation,
what you need to know for the
47
00:03:29 --> 00:03:31
final,
well, essentially the same kind
48
00:03:31 --> 00:03:35
of evaluation techniques that we
were supposed to know for the
49
00:03:35 --> 00:03:40
other tests.
That means the usual functions,
50
00:03:40 --> 00:03:47
substitutions,
basic trig, stuff like that.
51
00:03:47 --> 00:03:51
Well, I don't expect that you
would need integration by parts,
52
00:03:51 --> 00:03:54
although I still hope that some
of you remember it from single
53
00:03:54 --> 00:03:59
variable calculus.
If there is a need to integrate
54
00:03:59 --> 00:04:06
some big power of cosine or sine
then the formula will be given
55
00:04:06 --> 00:04:11
to you the way it is in the
notes.
56
00:04:11 --> 00:04:17
And, of course,
we know also how to set up
57
00:04:17 --> 00:04:23
these integrals in polar
coordinates.
58
00:04:23 --> 00:04:28
And then the area element
becomes r dr d theta.
59
00:04:28 --> 00:04:32
And because you integrate first
over r,
60
00:04:32 --> 00:04:36
well, first of all you should
remember the polar coordinate
61
00:04:36 --> 00:04:42
formulas,
namely x equals r cosine theta
62
00:04:42 --> 00:04:50
and y equals r sine theta.
And second you should remember
63
00:04:50 --> 00:04:53
that what we do,
when we have our region,
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00:04:53 --> 00:04:56
is for a fixed value of theta
we look for the bounds for r,
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00:04:56 --> 00:04:59
just like before,
so the way we are slicing the
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00:04:59 --> 00:05:02
region is now we are actually
shooting rays straight from the
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00:05:02 --> 00:05:04
origin.
And, in a given direction,
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00:05:04 --> 00:05:08
we are asking ourselves how far
does my region go?
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00:05:08 --> 00:05:12
You have to find a bound and
you have to find whatever the
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00:05:12 --> 00:05:15
value of r will be out here as a
function of theta.
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00:05:15 --> 00:05:19
And ways to do that can be
geometric or they can be by
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00:05:19 --> 00:05:22
starting from the x,
y equation of whatever curve
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00:05:22 --> 00:05:26
you have and then expressing it
in terms of r and theta and
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00:05:26 --> 00:05:28
solving for r.
For example,
75
00:05:28 --> 00:05:33
just to illustrate it,
we have seen that one of our
76
00:05:33 --> 00:05:39
classics has been the circle of
radius one centered at one,
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00:05:39 --> 00:05:42
zero.
This guy, you have two
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00:05:42 --> 00:05:45
different ways of getting its
polar coordinate equation.
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00:05:45 --> 00:05:49
One is to argue geometrically
that you have a right angle in
80
00:05:49 --> 00:05:51
here.
And this length is two,
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00:05:51 --> 00:05:54
this angle is theta,
this length is r,
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00:05:54 --> 00:05:59
so the polar equation is r
equals two cosine theta.
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00:05:59 --> 00:06:01
The other way to do it,
if somehow you are missing the
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00:06:01 --> 00:06:03
geometric trick,
is to start from the x,
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00:06:03 --> 00:06:05
y equation.
What is the x,
86
00:06:05 --> 00:06:10
y equation of this guy?
Well, it is x minus one squared
87
00:06:10 --> 00:06:16
plus y squared equals one.
If you expand that you will get
88
00:06:16 --> 00:06:22
x squared minus two x plus one
plus y squared equals one.
89
00:06:22 --> 00:06:27
The ones simplify.
X squared plus y squared
90
00:06:27 --> 00:06:34
becomes r squared minus two x
becomes r cosine theta equals
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00:06:34 --> 00:06:36
zero.
That gives you,
92
00:06:36 --> 00:06:40
when you simplify by r,
r equals two cosine theta.
93
00:06:40 --> 00:06:49
Two ways to get the same polar
equation.
94
00:06:49 --> 00:06:53
I should say this is an
example, in case you were
95
00:06:53 --> 00:07:01
wondering what I was doing.
We have also actually seen how
96
00:07:01 --> 00:07:14
to change variables to more
complicated coordinate systems.
97
00:07:14 --> 00:07:16
Let's say u, v coordinates.
But, of course,
98
00:07:16 --> 00:07:20
you can call them whatever you
want.
99
00:07:20 --> 00:07:24
The main thing to remember is
that you have to look for the
100
00:07:24 --> 00:07:30
Jacobian which will give you the
conversion ratio between dx dy
101
00:07:30 --> 00:07:32
and du dv.
For example,
102
00:07:32 --> 00:07:36
if you know u and v as
functions of x and y then you
103
00:07:36 --> 00:07:42
will write du dv equals absolute
value of the Jacobian partial u,
104
00:07:42 --> 00:07:50
v over partial x, y times dx dy.
Or, if it is easier for you,
105
00:07:50 --> 00:07:52
you can do the Jacobian the
other way around.
106
00:07:52 --> 00:07:56
And this Jacobian,
remember, is the determinant by
107
00:07:56 --> 00:08:00
a two by two matrix that you
obtain by putting the partial
108
00:08:00 --> 00:08:03
derivatives of u and v with
respect to x and y.
109
00:08:03 --> 00:08:08
Then, when we have that,
we can change the integrant,
110
00:08:08 --> 00:08:13
f of x, y, into something
involving u and v possibly.
111
00:08:13 --> 00:08:15
And then we have to find the
bounds.
112
00:08:15 --> 00:08:19
And to find the bounds perhaps
the easiest is to draw a picture
113
00:08:19 --> 00:08:21
of a region in u,
v coordinates.
114
00:08:21 --> 00:08:24
Maybe you have some picture in
the x,
115
00:08:24 --> 00:08:28
y plane that might actually be
really hard to draw and maybe in
116
00:08:28 --> 00:08:32
terms of u and v the picture
will become much simpler.
117
00:08:32 --> 00:08:35
It might just become a
rectangle.
118
00:08:35 --> 00:08:37
Of course, if you see
immediately what the bounds are
119
00:08:37 --> 00:08:39
in terms of u and v,
and they turn out to be very
120
00:08:39 --> 00:08:41
easy,
then maybe you don't even have
121
00:08:41 --> 00:08:45
to draw this picture.
But if it is not completely
122
00:08:45 --> 00:08:49
obvious then that might be a
helpful way of figuring out what
123
00:08:49 --> 00:08:52
the bounds will be when you
switch from x,
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00:08:52 --> 00:08:55
y to u, v.
We have seen some problems like
125
00:08:55 --> 00:09:00
that and there are more in the
notes in case you need more.
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00:09:00 --> 00:09:07
Questions?
Yes?
127
00:09:07 --> 00:09:10
That is the second time you've
asked for something real quick
128
00:09:10 --> 00:09:13
in these review sessions.
You are in a hurry.
129
00:09:13 --> 00:09:23
Take your time.
Partial u, v over partial x,
130
00:09:23 --> 00:09:26
y is just going to be the
determinant of u sub x,
131
00:09:26 --> 00:09:29
u sub y, v sub x,
v sub y.
132
00:09:29 --> 00:09:37
That is the definition.
That is pretty direct.
133
00:09:37 --> 00:09:39
And, of course,
a general common sense thing
134
00:09:39 --> 00:09:43
that applies to actually all the
integrals that we are going to
135
00:09:43 --> 00:09:45
see, there are two things in an
integral.
136
00:09:45 --> 00:09:49
One is whatever you integral is
called the integrant.
137
00:09:49 --> 00:09:52
It could be a function here.
It is a vector field in some of
138
00:09:52 --> 00:09:55
the flux things and so on.
There is another thing which is
139
00:09:55 --> 00:09:57
the region over which you
integrate.
140
00:09:57 --> 00:10:02
And the two have strictly
nothing to do with each other.
141
00:10:02 --> 00:10:05
When you are given a piece of
data in the statement of a
142
00:10:05 --> 00:10:07
problem,
you have to figure out whether
143
00:10:07 --> 00:10:09
that is part of a function to be
integrated or whether that is
144
00:10:09 --> 00:10:11
part of the region of
integration.
145
00:10:11 --> 00:10:14
If it is the region of
integration then it will go into
146
00:10:14 --> 00:10:17
the bounds of the integral and
maybe in the choice of the
147
00:10:17 --> 00:10:19
coordinate system that you use
for integrating.
148
00:10:19 --> 00:10:23
While the function that you are
integrating goes before the dx
149
00:10:23 --> 00:10:26
dy and not into the bounds or
anything like that.
150
00:10:26 --> 00:10:32
I know it sounds kind of silly
but it is a good safety check.
151
00:10:32 --> 00:10:34
Ask yourselves,
when you have a piece of data,
152
00:10:34 --> 00:10:36
where in my formula should this
go.
153
00:10:36 --> 00:10:47
Yes?
I case you want the bounds for
154
00:10:47 --> 00:10:50
this region in polar
coordinates, indeed it would be
155
00:10:50 --> 00:10:53
double integral.
For a fixed theta,
156
00:10:53 --> 00:10:57
r goes from zero to whatever it
is on that curve.
157
00:10:57 --> 00:11:04
So it would be zero to two
cosine theta of whatever the
158
00:11:04 --> 00:11:10
function is r dr d theta.
And the bounds on theta would
159
00:11:10 --> 00:11:15
be from negative pi over two to
pi over two.
160
00:11:15 --> 00:11:20
We have seen that one several
times, so hopefully by now it is
161
00:11:20 --> 00:11:21
clearer.
OK.
162
00:11:21 --> 00:11:28
Let me move on a bit because we
have a lot of other kinds of
163
00:11:28 --> 00:11:32
integrals to see.
Other kinds of integrals we
164
00:11:32 --> 00:11:35
have seen are triple integrals.
And I am not doing things in
165
00:11:35 --> 00:11:39
the order that we did them in
the class just so you can see
166
00:11:39 --> 00:11:43
parallels between stuff in the
plane and in space.
167
00:11:43 --> 00:11:47
When we do triple integrals in
space, well, it is the same kind
168
00:11:47 --> 00:11:50
of story, except now we have,
of course, more coordinate
169
00:11:50 --> 00:11:53
systems.
We have rectangular
170
00:11:53 --> 00:11:59
coordinates, we have cylindrical
coordinates and we have
171
00:11:59 --> 00:12:05
spherical coordinates.
And cylindrical coordinates
172
00:12:05 --> 00:12:09
only mean that we are,
instead of x, y and z,
173
00:12:09 --> 00:12:15
we are replacing x and y by the
polar coordinate in the x,
174
00:12:15 --> 00:12:17
y plane,
so the angle theta and the
175
00:12:17 --> 00:12:21
distance r.
So R is somehow the distance
176
00:12:21 --> 00:12:24
from the z-axis and z is the
height.
177
00:12:24 --> 00:12:28
Usually you don't have to
choose between rectangular and
178
00:12:28 --> 00:12:31
cylindrical until somewhat late
in the process,
179
00:12:31 --> 00:12:33
especially if you integrate
first of all z,
180
00:12:33 --> 00:12:36
because then the choice will
come up mostly when you try to
181
00:12:36 --> 00:12:39
figure out what are the bounds
for the shadow of your region.
182
00:12:39 --> 00:12:43
I mean the z part looks exactly
the same in rectangular and in
183
00:12:43 --> 00:12:45
cylindrical.
Spherical is,
184
00:12:45 --> 00:12:49
on the other hand,
a little bit more annoying
185
00:12:49 --> 00:12:52
because it looks quite
different.
186
00:12:52 --> 00:12:56
You should think of it as doing
polar coordinates not only in
187
00:12:56 --> 00:12:59
the horizontal direction but
also in the vertical direction
188
00:12:59 --> 00:13:02
at the same time.
You have this angle phi.
189
00:13:02 --> 00:13:06
That measures the angle down
from the positive z-axis.
190
00:13:06 --> 00:13:10
And you have rho which is the
distance from the origin.
191
00:13:10 --> 00:13:19
And if I project to the z-axis,
r becomes rho sine phi and z
192
00:13:19 --> 00:13:24
becomes rho cosine phi.
I hope that you all know these
193
00:13:24 --> 00:13:27
two formulas,
but if you ever have a small
194
00:13:27 --> 00:13:31
somehow memory lapse during the
final then you should consider
195
00:13:31 --> 00:13:35
drawing this kind of picture
because it will let you check
196
00:13:35 --> 00:13:42
very quickly which one is sine,
which one is cosine.
197
00:13:42 --> 00:13:44
Now, of course,
we have to have formulas for dv
198
00:13:44 --> 00:13:47
in all these coordinate systems.
Here, for example,
199
00:13:47 --> 00:13:52
that might be dz r dr d theta
or r dr d theta dz or anything
200
00:13:52 --> 00:13:58
like that.
Here it might be rho squared
201
00:13:58 --> 00:14:04
times phi times d rho d phi d
theta.
202
00:14:04 --> 00:14:07
And the general method for
setting up bounds is pretty much
203
00:14:07 --> 00:14:10
the same as in the plane,
just there is one more step.
204
00:14:10 --> 00:14:14
If you are doing rectangular or
cylindrical coordinates with z
205
00:14:14 --> 00:14:16
first, for example,
that is the most common.
206
00:14:16 --> 00:14:22
Well, if you do z first then
you have to actually start by
207
00:14:22 --> 00:14:28
figuring out for a given value
of x and y or r and theta what
208
00:14:28 --> 00:14:31
is the portion of a vertical
line above x,
209
00:14:31 --> 00:14:37
y that lies within my region?
That will go from z on the
210
00:14:37 --> 00:14:44
bottom of my solid which depends
on x and y to z at the top of my
211
00:14:44 --> 00:14:49
solid which also usually will
depend on x and y.
212
00:14:49 --> 00:14:52
And so that will give me the
bounds for dz.
213
00:14:52 --> 00:14:56
And then I will be left with
the shadow of my region in the
214
00:14:56 --> 00:15:02
x, y plane.
And that one I will set up like
215
00:15:02 --> 00:15:09
a double integral over there.
Strictly-speaking,
216
00:15:09 --> 00:15:12
if you are curious,
we could also change to weird
217
00:15:12 --> 00:15:15
coordinate systems using
Jacobian with three variables at
218
00:15:15 --> 00:15:18
the same time.
But we haven't seen that so it
219
00:15:18 --> 00:15:24
won't be on the final.
But it would work just the same
220
00:15:24 --> 00:15:30
way, just with more pictures to
do.
221
00:15:30 --> 00:15:33
And, in fact,
I just wanted to say this rho
222
00:15:33 --> 00:15:37
squared sine phi is actually the
Jacobian for the change of
223
00:15:37 --> 00:15:42
variables for rectangular to
spherical coordinates.
224
00:15:42 --> 00:15:48
OK.
Let's not think too much about
225
00:15:48 --> 00:15:51
that.
Applications.
226
00:15:51 --> 00:15:57
Well, we have seen how to use
double integrals to find the
227
00:15:57 --> 00:16:03
area of a volume of a piece of a
plane or a piece of space,
228
00:16:03 --> 00:16:08
and find also the mass.
Remember, area is just double
229
00:16:08 --> 00:16:12
integral of one dA,
volume is triple integral of
230
00:16:12 --> 00:16:14
one dV.
Sometimes if it's the volume
231
00:16:14 --> 00:16:16
between the x,
y plane and the graph of some
232
00:16:16 --> 00:16:19
function, you can just set it up
directly as a double integral.
233
00:16:19 --> 00:16:23
But there is no harm in doing
it as a triple integral if you
234
00:16:23 --> 00:16:27
feel better about that.
And mass will be double or
235
00:16:27 --> 00:16:31
triple integral,
depending on how many
236
00:16:31 --> 00:16:36
dimensions you have,
of whatever density function
237
00:16:36 --> 00:16:45
you have, dA or dV.
Then there is how to find the
238
00:16:45 --> 00:16:53
average value of some function.
Well, let me do the
239
00:16:53 --> 00:16:58
three-dimensional case.
You will just replace volume by
240
00:16:58 --> 00:17:02
area and dV by dA and so on,
if need be.
241
00:17:02 --> 00:17:09
That would be one over volume
of the solid times the triple
242
00:17:09 --> 00:17:18
integral of f dV,
or if it's a weighted average,
243
00:17:18 --> 00:17:24
one over mass times the triple
integral of a function times
244
00:17:24 --> 00:17:33
density times dV.
If you don't have a density or
245
00:17:33 --> 00:17:44
if the density is constant then
that reduces to that one.
246
00:17:44 --> 00:17:46
In particular,
we have seen the notion of
247
00:17:46 --> 00:17:50
center of mass.
The center of mass is just
248
00:17:50 --> 00:17:54
given by taking the average
values of the coordinates,
249
00:17:54 --> 00:17:56
x bar, y bar,
z bar.
250
00:17:56 --> 00:18:00
It is just this formula but
taking x, y or z as the
251
00:18:00 --> 00:18:08
function.
There are moments of inertia.
252
00:18:08 --> 00:18:12
For example,
the moment of inertia about the
253
00:18:12 --> 00:18:17
z-axis is the triple integral of
x squared plus y squared density
254
00:18:17 --> 00:18:19
dV.
Or, if you have just a
255
00:18:19 --> 00:18:22
two-dimensional object,
it is the same formula,
256
00:18:22 --> 00:18:23
but, of course,
with dA.
257
00:18:23 --> 00:18:26
And then we call that the polar
moment of inertia because we
258
00:18:26 --> 00:18:30
thought of it as rotating the
plane about the origin,
259
00:18:30 --> 00:18:34
but the origin is just where
the z-axis hits the x,
260
00:18:34 --> 00:18:38
y plane so it is really the
same thing.
261
00:18:38 --> 00:18:42
And we have also seen
gravitational attraction in
262
00:18:42 --> 00:18:47
space, and I will let you look
at your notes for that.
263
00:18:47 --> 00:18:57
It is just one formula to
remember.
264
00:18:57 --> 00:19:02
Questions about iterated
integrals, things like that?
265
00:19:02 --> 00:19:18
Yes?
The formula that you should
266
00:19:18 --> 00:19:22
know for gravitational
attraction is that if yu have a
267
00:19:22 --> 00:19:28
point mass at the origin and you
have some solid centered on the
268
00:19:28 --> 00:19:33
z-axis that is attracting it
then the force will be given by
269
00:19:33 --> 00:19:38
G times the mass times the
triple integral of density times
270
00:19:38 --> 00:19:42
cosine phi over rho squared
times dV.
271
00:19:42 --> 00:19:45
And, of course,
you will actually do that in
272
00:19:45 --> 00:19:50
spherical coordinates because it
is easier that way.
273
00:19:50 --> 00:19:55
That is the formula I have in
mind.
274
00:19:55 --> 00:19:59
But, see, all these formulas
just give you examples of things
275
00:19:59 --> 00:20:01
to integrate.
And how to set up the bounds
276
00:20:01 --> 00:20:03
and so on does not depend on
what you are actually
277
00:20:03 --> 00:20:08
integrating.
It is done always using the
278
00:20:08 --> 00:20:22
same methods.
Let's move on to work and line
279
00:20:22 --> 00:20:28
integrals.
We have seen how to do that in
280
00:20:28 --> 00:20:35
the plane and in space.
And it looks very similar
281
00:20:35 --> 00:20:41
somehow.
Remember, you have to know how
282
00:20:41 --> 00:20:48
to set up and evaluate a line
integral of this form.
283
00:20:48 --> 00:20:49
Let me do it in the plane this
time.
284
00:20:49 --> 00:20:53
If you are in the plane you
have two components,
285
00:20:53 --> 00:20:58
and then this becomes the line
integral of M dx plus N dy.
286
00:20:58 --> 00:21:01
If you have a space curve then
you will have a third component
287
00:21:01 --> 00:21:05
here.
You will add that guy times dz.
288
00:21:05 --> 00:21:08
Now, how do we evaluate that?
Well, it is very different from
289
00:21:08 --> 00:21:11
there because here we are just
on a curve so there should be
290
00:21:11 --> 00:21:15
only one degree of freedom.
One variable should be enough
291
00:21:15 --> 00:21:21
to know where we are.
We will have to express x and y
292
00:21:21 --> 00:21:28
in terms of -- Well,
I should and z optionally if
293
00:21:28 --> 00:21:32
there is one,
in terms of a single
294
00:21:32 --> 00:21:36
parameters.
And that might be just one of
295
00:21:36 --> 00:21:38
the coordinates.
If you are told y equals z
296
00:21:38 --> 00:21:41
squared, that is easy.
You just substitute y equals x
297
00:21:41 --> 00:21:44
squared and dy equals two x dx
into everything,
298
00:21:44 --> 00:21:48
and you are left with an
integral over x.
299
00:21:48 --> 00:22:01
Maybe it will be something in
terms of time or in terms of an
300
00:22:01 --> 00:22:05
angle.
We express everything in terms
301
00:22:05 --> 00:22:09
of a single parameter,
and that will give us a usual
302
00:22:09 --> 00:22:14
single integral.
Any questions about that?
303
00:22:14 --> 00:22:23
Yes?
If you cannot parameterize the
304
00:22:23 --> 00:22:27
curve then it is really,
really hard to evaluate the
305
00:22:27 --> 00:22:31
line integral.
Well, you might be able to
306
00:22:31 --> 00:22:35
evaluate it numerically into a
computer, but that is the
307
00:22:35 --> 00:22:40
easiest way to describe a curve.
Indeed it could be that in the
308
00:22:40 --> 00:22:43
plane you have an equation in
terms of x and y given by some
309
00:22:43 --> 00:22:46
completed formulas defining some
curve.
310
00:22:46 --> 00:22:49
Then actually there are ways
you can use basically
311
00:22:49 --> 00:22:52
differentials and constrained
partials to figure out what the
312
00:22:52 --> 00:22:55
tangent vector to the curve is
and so on.
313
00:22:55 --> 00:22:57
But we haven't really seen how
to do that.
314
00:22:57 --> 00:23:00
That would be a really nice
topic for tying together the end
315
00:23:00 --> 00:23:02
of the second unit that we
discussed last time,
316
00:23:02 --> 00:23:04
constrained partials,
with this stuff.
317
00:23:04 --> 00:23:08
But that is not going to be
part of our topics.
318
00:23:08 --> 00:23:11
Basically, all the curves we
have seen in this class,
319
00:23:11 --> 00:23:14
there is a way to express the
position of a point in terms of
320
00:23:14 --> 00:23:23
a parameter.
We haven't seen any curves that
321
00:23:23 --> 00:23:34
are so complicated that you
cannot do that.
322
00:23:34 --> 00:23:37
The other thing we have seen is
that there are some special
323
00:23:37 --> 00:23:40
cases of vector fields where we
don't actually have to compute
324
00:23:40 --> 00:23:43
this thing because maybe we know
that it is the gradient of some
325
00:23:43 --> 00:23:48
potential function.
And then we have a fundamental
326
00:23:48 --> 00:23:55
theorem that gives us a way to
compute this without computing
327
00:23:55 --> 00:24:04
it.
We've seen about gradient
328
00:24:04 --> 00:24:16
fields and path independence.
The thing to check is whether
329
00:24:16 --> 00:24:18
the curl of our vector field is
zero.
330
00:24:18 --> 00:24:27
And remember in the plane that
is one condition,
331
00:24:27 --> 00:24:33
Nx equals My.
In 3D in space that is actually
332
00:24:33 --> 00:24:39
three conditions because you
have to check all the mixed
333
00:24:39 --> 00:24:44
partials of the various
components.
334
00:24:44 --> 00:24:47
If the curl of f is zero that
tells us we are likely to have a
335
00:24:47 --> 00:24:52
gradient field.
Strictly-speaking,
336
00:24:52 --> 00:25:06
I should mention and F is
defined in a simply-connected
337
00:25:06 --> 00:25:16
region.
Then F is a gradient field.
338
00:25:16 --> 00:25:25
That means that we can find a
potential function.
339
00:25:25 --> 00:25:30
You can write F as the gradient
of little f for some potential
340
00:25:30 --> 00:25:39
function little f.
And we have seen how to find
341
00:25:39 --> 00:25:46
the potential.
In fact, we have seen two
342
00:25:46 --> 00:25:50
methods for that.
And we have seen them twice.
343
00:25:50 --> 00:25:53
We have seen them once for
functions of two variables,
344
00:25:53 --> 00:25:55
once for functions of three
variables.
345
00:25:55 --> 00:25:58
They look very much the same.
I encourage you to compare your
346
00:25:58 --> 00:26:03
notes for the two side by side
to see where they differ.
347
00:26:03 --> 00:26:05
Where they differ,
roughly-speaking,
348
00:26:05 --> 00:26:09
well, I never know if it is the
first or the second,
349
00:26:09 --> 00:26:14
but one of the two methods was
to compute a line integral.
350
00:26:14 --> 00:26:18
In the plane,
what we did is we set up and
351
00:26:18 --> 00:26:23
evaluated a line integral along
our favorite path from the
352
00:26:23 --> 00:26:27
origin to a point with
coordinates say x1,
353
00:26:27 --> 00:26:32
y1.
And then we had to evaluate the
354
00:26:32 --> 00:26:38
line integral for the work done
along this path.
355
00:26:38 --> 00:26:43
And that will give us the value
of potential at that point.
356
00:26:43 --> 00:26:46
If we are doing it with three
variables, that method remains
357
00:26:46 --> 00:26:49
very similar.
The only difference is now we
358
00:26:49 --> 00:26:52
have to go also up in space to
some point x1,
359
00:26:52 --> 00:26:55
y1, z1.
And so we actually sum three
360
00:26:55 --> 00:26:59
pieces together.
But on each piece it is the
361
00:26:59 --> 00:27:01
same story, only one variable
changes.
362
00:27:01 --> 00:27:06
Here it is only x that changes,
it is only y that changes,
363
00:27:06 --> 00:27:11
and on the third one only z
would be changing.
364
00:27:11 --> 00:27:18
That is one possibility.
And the other possibility for
365
00:27:18 --> 00:27:23
finding the potential is that we
start with the condition that
366
00:27:23 --> 00:27:28
the first component of our
vector field should be equal to
367
00:27:28 --> 00:27:32
f sub x for the unknown
potential function.
368
00:27:32 --> 00:27:36
What we do is integrate with
respect to x,
369
00:27:36 --> 00:27:40
and we will get our potential
function up to an integration
370
00:27:40 --> 00:27:44
constant.
And that integration constant
371
00:27:44 --> 00:27:48
typically depends on the
remaining variables that might
372
00:27:48 --> 00:27:53
be y or equal in space y and z.
And then what we have to do is
373
00:27:53 --> 00:27:56
take the partial of this with
respect to y and compare it to
374
00:27:56 --> 00:28:00
what we want it to be,
namely the y component of a
375
00:28:00 --> 00:28:02
vector field,
and match them to get some
376
00:28:02 --> 00:28:08
information about this guy.
And if we have three variables
377
00:28:08 --> 00:28:13
then there is a third step
because there you will still
378
00:28:13 --> 00:28:18
have an unknown function of z
that you need to get by
379
00:28:18 --> 00:28:23
comparing the partials with
respect to z.
380
00:28:23 --> 00:28:30
I see a lot of very quiet faces
somehow.
381
00:28:30 --> 00:28:36
Well, hopefully that is because
you know that stuff.
382
00:28:36 --> 00:28:42
If it is because you are
hopelessly confused then please
383
00:28:42 --> 00:28:49
review a lot before the final,
but I really hope that is not
384
00:28:49 --> 00:28:52
the case.
And so,
385
00:28:52 --> 00:29:22
386
00:29:22 --> 00:29:26
in particular,
what we have seen is once we
387
00:29:26 --> 00:29:32
have the potential then we can
use the fundamental theorem of
388
00:29:32 --> 00:29:38
calculus to tell us that if we
have a line integral to compute
389
00:29:38 --> 00:29:44
for work along a curve that goes
from some point P zero to some
390
00:29:44 --> 00:29:49
point P one then the line
integral for the work done by
391
00:29:49 --> 00:29:54
gradient F is actually going
just to be the change in value
392
00:29:54 --> 00:29:58
of a potential.
And, in particular,
393
00:29:58 --> 00:30:02
that does not depend on how we
got from P zero to P one.
394
00:30:02 --> 00:30:10
That is why we say that we have
path independence.
395
00:30:10 --> 00:30:22
Next topic is flux in plane and
space.
396
00:30:22 --> 00:30:26
Flux looks quite different in
the plane and in space because,
397
00:30:26 --> 00:30:29
in the plane,
it is just another kind of line
398
00:30:29 --> 00:30:32
integral,
while in space it is a surface
399
00:30:32 --> 00:30:34
integral.
If you were in four-dimensional
400
00:30:34 --> 00:30:35
space it would be a triple
integral.
401
00:30:35 --> 00:30:43
Generally, you do flux for
something that is somehow a wall
402
00:30:43 --> 00:30:50
that separates regions of space
from each other.
403
00:30:50 --> 00:30:56
In the plane,
the way we do it is we have a
404
00:30:56 --> 00:31:02
curve C and we look at its
tangent vector,
405
00:31:02 --> 00:31:09
let's call that T,
and we rotate it by 90 degrees
406
00:31:09 --> 00:31:13
clockwise.
That is our convention to get a
407
00:31:13 --> 00:31:18
unit normal vector that points
to the right of the curve as we
408
00:31:18 --> 00:31:21
move along the curve.
That is our convention for
409
00:31:21 --> 00:31:24
orienting curves.
And we are always going to be
410
00:31:24 --> 00:31:33
using that one.
N equals T rotated 90 degrees
411
00:31:33 --> 00:31:37
clockwise.
In particular,
412
00:31:37 --> 00:31:44
that means that n ds,
which will be what we integrate
413
00:31:44 --> 00:31:52
against when we try to compute
flux, will just end up being dy,
414
00:31:52 --> 00:31:58
negative dx.
Concretely, when we have to
415
00:31:58 --> 00:32:03
evaluate a line integral of F
dot n ds,
416
00:32:03 --> 00:32:07
geometrically we could try to
take the dot product of our
417
00:32:07 --> 00:32:11
field with the normal vector and
then sum the length element
418
00:32:11 --> 00:32:14
along the curve.
And, in some cases,
419
00:32:14 --> 00:32:16
for example,
if you know that the vector
420
00:32:16 --> 00:32:19
field is tangent to the curve or
if a dot product is constant or
421
00:32:19 --> 00:32:22
things like that then that might
actually give you a very easy
422
00:32:22 --> 00:32:24
answer.
But, in general,
423
00:32:24 --> 00:32:28
the most efficient way to do it
will be to say that if your
424
00:32:28 --> 00:32:31
vector field has components,
I don't know,
425
00:32:31 --> 00:32:36
let's call them P and Q,
then that will be just the line
426
00:32:36 --> 00:32:39
integral of PQ dot dy,
negative dx,
427
00:32:39 --> 00:32:44
which means negative Q dx plus
P dy.
428
00:32:44 --> 00:32:49
And, from that point onward,
you evaluate it exactly the
429
00:32:49 --> 00:32:54
same way as you would for a work
integral.
430
00:32:54 --> 00:32:56
But, of course,
the geometric meaning is very
431
00:32:56 --> 00:32:57
different.
It is the same meaning that we
432
00:32:57 --> 00:33:08
have always seen for flux.
It measures how much a vector
433
00:33:08 --> 00:33:22
field goes across the curve.
Now, if we are in space then
434
00:33:22 --> 00:33:30
you take flux for a surface,
not for a curve.
435
00:33:30 --> 00:33:33
And the way it will work is
that you have to choose an
436
00:33:33 --> 00:33:37
orientation of a surface,
which just means choosing one
437
00:33:37 --> 00:33:40
of the two possible unit normal
vectors.
438
00:33:40 --> 00:33:51
And then you will do a surface
integral for F dot n dS.
439
00:33:51 --> 00:34:00
That is the surface i element.
The setup for this surface
440
00:34:00 --> 00:34:07
integral is that first we have
to express n and dS in some way.
441
00:34:07 --> 00:34:14
One possibility is that we can
express the normal vector n dS
442
00:34:14 --> 00:34:18
geometrically.
That is, for example,
443
00:34:18 --> 00:34:23
what we do when we look at,
say, a horizontal plane or a
444
00:34:23 --> 00:34:27
vertical plane or a sphere or a
cylinder.
445
00:34:27 --> 00:34:31
Then we have some geometric
idea of why the normal vector is
446
00:34:31 --> 00:34:34
what it is and we have some
formula for dS.
447
00:34:34 --> 00:34:39
Or, we can use one of the
standard formulas.
448
00:34:39 --> 00:34:43
Basically, we have seen two
formulas that work in fairly
449
00:34:43 --> 00:34:50
generate situations.
One of them says -- If S is
450
00:34:50 --> 00:35:00
given by an equation z equals
some function of x,
451
00:35:00 --> 00:35:06
y then you can just say n dS
equals minus f sub x,
452
00:35:06 --> 00:35:09
minus f sub y,
one, dx dy.
453
00:35:09 --> 00:35:14
And I need to rewrite that
because I am running out of
454
00:35:14 --> 00:35:16
space.
But, while I erase,
455
00:35:16 --> 00:35:20
I would like to point out the
most important there in here.
456
00:35:20 --> 00:35:24
When I say n dS equals blah,
blah, blah times dx dy,
457
00:35:24 --> 00:35:28
dx dy is not the same thing as
dS at all.
458
00:35:28 --> 00:35:31
If you make that mistake you
are going to get into trouble
459
00:35:31 --> 00:35:34
the next time that you try to
buy real estate in a region
460
00:35:34 --> 00:35:37
which hills or cliffs or things
like that.
461
00:35:37 --> 00:35:40
dS is the area on the slanted
surface.
462
00:35:40 --> 00:35:46
dx dy is the area on the map
that shows the x,
463
00:35:46 --> 00:35:49
y plane.
And these are not the same
464
00:35:49 --> 00:35:51
thing.
In particular,
465
00:35:51 --> 00:36:00
you cannot just take one piece
of it and not the other piece.
466
00:36:00 --> 00:36:06
Let me give you formulas for n
and for dS separately just to
467
00:36:06 --> 00:36:14
convince you.
That way, if you feel that you
468
00:36:14 --> 00:36:22
need them, then you will have
them.
469
00:36:22 --> 00:36:26
N is minus f sub x,
minus f sub y,
470
00:36:26 --> 00:36:31
one, but scaled down to unit
length.
471
00:36:31 --> 00:36:35
This is not a unit vector.
It is actually divided by the
472
00:36:35 --> 00:36:38
length of this guy which is fx
squared plus fy squared plus
473
00:36:38 --> 00:36:46
one.
And dS is that same vector
474
00:36:46 --> 00:36:52
times dx dy.
And so the square roots cancel
475
00:36:52 --> 00:36:56
out when you multiply them
together.
476
00:36:56 --> 00:37:00
But it would be completely
wrong to just say I will replace
477
00:37:00 --> 00:37:03
n dS by minus f sub x,
minus f sub y and one.
478
00:37:03 --> 00:37:07
Then I end up again with the dS
and I do something else with dS.
479
00:37:07 --> 00:37:16
That is a pretty bad conceptual
mistake because it gives you the
480
00:37:16 --> 00:37:20
wrong answer.
Another option more general
481
00:37:20 --> 00:37:24
than that.
If we have not seen how to
482
00:37:24 --> 00:37:32
solve for z, how to express z as
a function of x and y,
483
00:37:32 --> 00:37:40
well, maybe we still know some
normal vector to the surface.
484
00:37:40 --> 00:37:49
Then there is another formula
for n dS which is up to sine,
485
00:37:49 --> 00:37:55
N divided by N dot k dx dy.
And, see, that projection
486
00:37:55 --> 00:37:58
formula works also if you have
to project to another coordinate
487
00:37:58 --> 00:37:59
plane.
For example,
488
00:37:59 --> 00:38:02
if you want to project to the
x, z coordinate plane,
489
00:38:02 --> 00:38:10
the relation between n dS and
dx dz is given by N over N dot
490
00:38:10 --> 00:38:13
j,
because j is the direction
491
00:38:13 --> 00:38:19
perpendicular to the xz plane.
But this one is more useful.
492
00:38:19 --> 00:38:24
What is a good example of that?
If you have a slanted plane
493
00:38:24 --> 00:38:28
given to you,
you can easily find its normal
494
00:38:28 --> 00:38:31
vector.
That is just given by the
495
00:38:31 --> 00:38:33
coefficients of x,
y, z in the equation.
496
00:38:33 --> 00:38:38
Another situation where that
might happen is if your surface
497
00:38:38 --> 00:38:41
is given by an equation of a
form of g of x,
498
00:38:41 --> 00:38:44
y, z equals zero.
If that is the case then you
499
00:38:44 --> 00:38:47
know this is a level set of g.
And we know how to find a
500
00:38:47 --> 00:38:50
normal vector to the level set,
namely the gradient vector is
501
00:38:50 --> 00:38:52
always perpendicular to the
level set.
502
00:38:52 --> 00:38:59
You would take the gradient of
g to be your big N.
503
00:38:59 --> 00:39:06
OK.
Now, these are basically all
504
00:39:06 --> 00:39:10
the integrals we have seen how
to set up.
505
00:39:10 --> 00:39:15
Now we have a bunch of theorems
relating them.
506
00:39:15 --> 00:39:23
Let me think about how I am
going to organize that.
507
00:39:23 --> 00:39:32
Let me try like this.
This part of the board will be
508
00:39:32 --> 00:39:35
work,
this part of the board will be
509
00:39:35 --> 00:39:40
about flux and the left part of
the board will be about things
510
00:39:40 --> 00:39:46
in the plane and the right one
will be about things in space.
511
00:39:46 --> 00:39:53
What have we seen?
Well, we have seen Green's
512
00:39:53 --> 00:39:57
theorem for work.
That doesn't work so well
513
00:39:57 --> 00:39:59
because that is too small,
so I am going to actually use
514
00:39:59 --> 00:40:01
more blackboards to do that.
515
00:40:01 --> 00:40:26
516
00:40:26 --> 00:40:32
This side will be space,
this side will be the plane and
517
00:40:32 --> 00:40:37
we are going to start with
theorems about work.
518
00:40:37 --> 00:40:41
And we will see theorems about
flux pretty soon.
519
00:40:41 --> 00:40:47
We have two theorems about work.
In the plane that is called
520
00:40:47 --> 00:40:51
Green's theorem.
In space that is called Stokes'
521
00:40:51 --> 00:40:56
theorem.
Green's theorem says if I have
522
00:40:56 --> 00:41:03
a closed curve in the plane
going counterclockwise enclosing
523
00:41:03 --> 00:41:11
entirely some region R then the
line integral along C for the
524
00:41:11 --> 00:41:20
work of F is equal to the double
integral of a region inside of
525
00:41:20 --> 00:41:27
the curl of F dA.
Concretely, if my components of
526
00:41:27 --> 00:41:35
F are called M and N that is the
line integral of M dx plus N dy
527
00:41:35 --> 00:41:43
is equal to the double integral
of R of N sub x minus M sub y
528
00:41:43 --> 00:41:46
dA.
This side here is a usual line
529
00:41:46 --> 00:41:49
integral.
This side here is a usual
530
00:41:49 --> 00:41:53
double integral in the plane.
And somehow their values end up
531
00:41:53 --> 00:41:58
being magically related.
Well, not quite magically.
532
00:41:58 --> 00:42:03
We actually have seen how to
prove it.
533
00:42:03 --> 00:42:10
And now the analog of that in
space is Stokes' theorem.
534
00:42:10 --> 00:42:15
Stokes says if I have a closed
curve in space,
535
00:42:15 --> 00:42:20
now I have to decide what kind
of thing it bounds.
536
00:42:20 --> 00:42:23
And the answer is it will have
to bound some surface,
537
00:42:23 --> 00:42:28
but I have a choice of surface.
I choose my favorite surface
538
00:42:28 --> 00:42:31
bounded by C.
I guess I will just draw it
539
00:42:31 --> 00:42:33
like that.
And I have to choose a
540
00:42:33 --> 00:42:36
compatible orientation.
Remember, we have seen this
541
00:42:36 --> 00:42:39
right hand rule for choosing how
to orient the surface.
542
00:42:39 --> 00:42:43
I believe, in this case,
if I take C like that then the
543
00:42:43 --> 00:42:49
normal vector has to go up.
And then it tells me how to
544
00:42:49 --> 00:42:53
compute the work done by F along
C.
545
00:42:53 --> 00:43:01
Namely, that becomes the double
integral over that surface S of
546
00:43:01 --> 00:43:08
curl F, which I will write as
dell cross F dot n dS.
547
00:43:08 --> 00:43:10
This line integral is a usual
line integral,
548
00:43:10 --> 00:43:14
but if for some reason we don't
want to compute it directly we
549
00:43:14 --> 00:43:18
can actually replace it by a
surface integral over any
550
00:43:18 --> 00:43:22
surface bounded by the curve.
It might be that a problem will
551
00:43:22 --> 00:43:23
tell you which surface you have
to consider.
552
00:43:23 --> 00:43:28
It might be that you will be
left to choose the simplest
553
00:43:28 --> 00:43:33
possible surface you can think
of that is somehow having this
554
00:43:33 --> 00:43:36
curve as its boundary.
And so now, remember,
555
00:43:36 --> 00:43:40
curl of a vector field in space
is going to be another vector
556
00:43:40 --> 00:43:42
expression.
It has three components.
557
00:43:42 --> 00:43:45
And the way you compute it is
not by remembering the actual
558
00:43:45 --> 00:43:48
formula,
which is really complicated by,
559
00:43:48 --> 00:43:54
but by instead computing the
cross-product between dell and
560
00:43:54 --> 00:44:01
F.
You set up the cross-product.
561
00:44:01 --> 00:44:03
And, of course,
it is a highly symbolic
562
00:44:03 --> 00:44:07
cross-product.
I mean it is not an
563
00:44:07 --> 00:44:16
cross-product of actual vectors
but it works the same way.
564
00:44:16 --> 00:44:20
Both of these formulas
basically relate work on a curve
565
00:44:20 --> 00:44:24
with what happens to the curl on
the surface that is enclosed by
566
00:44:24 --> 00:44:27
this curve,
that is bounded by this curve.
567
00:44:27 --> 00:44:30
And in this one you have less
freedom of choice because you
568
00:44:30 --> 00:44:34
don't have somehow a z direction
in which you could move your
569
00:44:34 --> 00:44:36
surface.
There is only possible choice
570
00:44:36 --> 00:44:39
of surface.
There is only one thing that is
571
00:44:39 --> 00:44:42
enclosed by this curve in the
plane.
572
00:44:42 --> 00:44:45
In both cases,
these things tell you that you
573
00:44:45 --> 00:44:48
can think of curl as measuring
how much the field fails to be
574
00:44:48 --> 00:44:51
conservative.
See, if your field was
575
00:44:51 --> 00:44:55
conservative -- If a curl was
zero then the right-hand side
576
00:44:55 --> 00:44:57
would just be zero.
And that would be fortunate
577
00:44:57 --> 00:45:01
because if a curl is zero then
your field is less conservative.
578
00:45:01 --> 00:45:02
That means it comes from a
potential.
579
00:45:02 --> 00:45:05
That means when you go along a
closed curve,
580
00:45:05 --> 00:45:09
well, the change of value of a
potential should be zero.
581
00:45:09 --> 00:45:13
Another way to say it is path
independence tells you no work.
582
00:45:13 --> 00:45:16
And, of course,
if you have a vector field that
583
00:45:16 --> 00:45:20
is not a gradient field then the
curl is not necessarily zero and
584
00:45:20 --> 00:45:23
then you get a more interesting
answer.
585
00:45:23 --> 00:45:27
Finally, let's move onto the
theorems about flux.
586
00:45:27 --> 00:45:43
That is Green for flux and that
is the divergence theorem.
587
00:45:43 --> 00:45:54
Flux theorems.
Here I say that will be
588
00:45:54 --> 00:46:01
divergence.
And here it will be Green again.
589
00:46:01 --> 00:46:07
Green's theorem for flux says I
have a closed curve that goes
590
00:46:07 --> 00:46:10
counterclockwise around some
region.
591
00:46:10 --> 00:46:13
In particular,
counterclockwise means that the
592
00:46:13 --> 00:46:16
normal vector will be going out
of the region.
593
00:46:16 --> 00:46:20
And then it tells us that the
flux out of the region,
594
00:46:20 --> 00:46:24
through the curve C,
so that will be the line
595
00:46:24 --> 00:46:30
integral of F dot n ds is equal
to the integral of a region
596
00:46:30 --> 00:46:37
inside of div F dA.
And remember the divergence of
597
00:46:37 --> 00:46:42
M, N is just Mx plus Ny.
This one here,
598
00:46:42 --> 00:46:46
the divergence theorem,
tells you something similar but
599
00:46:46 --> 00:46:49
now for a region of space
bounded by a closed surface.
600
00:46:49 --> 00:46:55
So if you have some region of
space and you call its boundary
601
00:46:55 --> 00:47:00
surface S and you let n be the
normal vector that goes out of
602
00:47:00 --> 00:47:07
the region R.
You orient S outwards.
603
00:47:07 --> 00:47:15
Then the flux out of the region
through S is going to be the
604
00:47:15 --> 00:47:23
same as the triple integral over
the region of divergence F dV.
605
00:47:23 --> 00:47:31
Remember, the divergence of a
vector field with components P,
606
00:47:31 --> 00:47:36
Q, R is Px plus Qy plus Rz.
What do these two theorems say?
607
00:47:36 --> 00:47:39
Well, they say essentially the
same thing.
608
00:47:39 --> 00:47:43
They say the total flux out of
a region is equal to the
609
00:47:43 --> 00:47:46
integral of divergence over
whatever is inside.
610
00:47:46 --> 00:47:49
And the reason for that is,
again, we have seen for a
611
00:47:49 --> 00:47:52
velocity field that divergence
measures how much things are
612
00:47:52 --> 00:47:55
expanding or how much stuff is
being created.
613
00:47:55 --> 00:47:59
It tells you the amount of
sources per unit portion of the
614
00:47:59 --> 00:48:01
region.
When you sum that over
615
00:48:01 --> 00:48:03
everything,
you get the amount of fluid
616
00:48:03 --> 00:48:05
that is being,
you know, the total amount of
617
00:48:05 --> 00:48:08
sources inside here,
and that tells us how much
618
00:48:08 --> 00:48:10
stuff has to go out per unit
time.
619
00:48:10 --> 00:48:12
That is basically the
interpretation.
620
00:48:12 --> 00:48:16
In a way, I would be tempted to
say that this table of four
621
00:48:16 --> 00:48:20
theorems is somehow the crucial
point of 18.02.
622
00:48:20 --> 00:48:23
And you would do well to
remember them.
623
00:48:23 --> 00:48:26
However, I would like also to
point out that these theorems
624
00:48:26 --> 00:48:28
are completely useless if you
don't know how to compute any of
625
00:48:28 --> 00:48:31
the integrals that are in there.
So all the stuff that was
626
00:48:31 --> 00:48:35
around there before is actually
somehow more fundamental.
627
00:48:35 --> 00:48:38
And if you don't know how to
compute the double or triple
628
00:48:38 --> 00:48:41
integrals then this is of little
use to you.
629
00:48:41 --> 00:48:48
That is the end.
I guess I have to wish you
630
00:48:48 --> 00:48:52
happy holidays.
631
00:48:52 --> 00:48:57