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We were looking at vector
fields last time.

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00:00:34 --> 00:00:45
Last time we saw that if a
vector field happens to be a

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00:00:45 --> 00:00:56
gradient field -- -- then the
line integral can be computed

10
00:00:56 --> 00:01:08
actually by taking the change in
value of the potential between

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the end point and the starting
point of the curve.

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If we have a curve c, 
from a point p0 to a point p1

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00:01:24 --> 00:01:29
then the line integral for work
depends only on the end points

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00:01:29 --> 00:01:32
and not on the actual path we
chose.

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00:01:32 --> 00:01:43
We say that the line integral
is path independent.

16
00:01:43 --> 00:01:49
And we also said that the
vector field is conservative

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00:01:49 --> 00:01:55
because of conservation of
energy which tells you if you

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00:01:55 --> 00:02:02
start at a point and you come
back to the same point then you

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00:02:02 --> 00:02:07
haven't gotten any work out of
that force.

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00:02:07 --> 00:02:15
If we have a closed curve then
the line integral for work is

21
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just zero.
And, basically,

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00:02:18 --> 00:02:23
we say that these properties
are equivalent being a gradient

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00:02:23 --> 00:02:28
field or being path independent
or being conservative.

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00:02:28 --> 00:02:31
And what I promised to you is
that today we would see a

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00:02:31 --> 00:02:35
criterion to decide whether a
vector field is a gradient field

26
00:02:35 --> 00:02:38
or not and how to find the
potential function if it is a

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00:02:38 --> 00:02:47
gradient field.
So, that is the topic for today.

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00:02:47 --> 00:03:00
The question is testing whether
a given vector field,

29
00:03:00 --> 00:03:14
let's say M and N compliments,
is a gradient field.

30
00:03:14 --> 00:03:16
For that, well,
let's start with an

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00:03:16 --> 00:03:26
observation.
Say that it is a gradient field.

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00:03:26 --> 00:03:31
That means that the first
component of a field is just the

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00:03:31 --> 00:03:35
partial of f with respect to
some variable x and the second

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00:03:35 --> 00:03:40
component is the partial of f
with respect to y.

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00:03:40 --> 00:03:43
Now we have seen an interesting
property of the second partial

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00:03:43 --> 00:03:46
derivatives of the function,
which is if you take the

37
00:03:46 --> 00:03:49
partial derivative first with
respect to x,

38
00:03:49 --> 00:03:52
then with respect to y, 
or first with respect to y, 

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00:03:52 --> 00:03:58
then with respect to x you get
the same thing.

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00:03:58 --> 00:04:07
We know f sub xy equals f sub
yx, and that means M sub y

41
00:04:07 --> 00:04:12
equals N sub x.
If you have a gradient field

42
00:04:12 --> 00:04:14
then it should have this
property.

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00:04:14 --> 00:04:17
You take the y component, 
take the derivative with

44
00:04:17 --> 00:04:19
respect to x,
take the x component, 

45
00:04:19 --> 00:04:20
differentiate with respect to
y,

46
00:04:20 --> 00:04:31
you should get the same answer.
And that is important to know.

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00:04:31 --> 00:04:37
So, I am going to put that in a
box.

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00:04:37 --> 00:04:43
It is a broken box.
The claim that I want to make

49
00:04:43 --> 00:04:45
is that there is a converse of
sorts.

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00:04:45 --> 00:04:47
This is actually basically all
we need to check.

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00:04:47 --> 00:05:06
 
 

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00:05:06 --> 00:05:18
Conversely, if,
and I am going to put here a

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00:05:18 --> 00:05:33
condition, My equals Nx,
then F is a gradient field.

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00:05:33 --> 00:05:35
What is the condition that I
need to put here?

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00:05:35 --> 00:05:37
Well, we will see a more
precise version of that next

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00:05:37 --> 00:05:44
week.
But for now let's just say if

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00:05:44 --> 00:05:59
our vector field is defined and
differentiable everywhere in the

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plane.
We need, actually,

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00:06:01 --> 00:06:04
a vector field that is
well-defined everywhere.

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00:06:04 --> 00:06:07
You are not allowed to have
somehow places where it is not

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00:06:07 --> 00:06:09
well-defined.
Otherwise, actually,

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00:06:09 --> 00:06:13
you have a counter example on
your problem set this week.

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00:06:13 --> 00:06:16
If you look at the last problem
on the problem set this week,

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00:06:16 --> 00:06:20
it gives you a vector field
that satisfies this condition

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00:06:20 --> 00:06:22
everywhere where it is defined.
But, actually,

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00:06:22 --> 00:06:24
there is a point where it is
not defined.

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00:06:24 --> 00:06:28
And that causes it,
actually, to somehow -- I mean

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00:06:28 --> 00:06:33
everything that I am going to
say today breaks down for that

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00:06:33 --> 00:06:36
example because of that.
I mean, we will shed more light

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00:06:36 --> 00:06:39
on this a bit later with the
notion of simply connected

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00:06:39 --> 00:06:42
regions and so on.
But for now let's just say if

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00:06:42 --> 00:06:47
it is defined everywhere and it
satisfies this criterion then it

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00:06:47 --> 00:06:52
is a gradient field.
If you ignore the technical

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00:06:52 --> 00:06:57
condition, being a gradient
field means essentially the same

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00:06:57 --> 00:07:11
thing as having this property.
That is what we need to check.

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00:07:11 --> 00:07:20
Let's look at an example.
Well, one vector field that we

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00:07:20 --> 00:07:24
have been looking at a lot was -
yi xj.

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00:07:24 --> 00:07:30
Remember that was the vector
field that looked like a

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00:07:30 --> 00:07:35
rotation at the unit speed.
I think last time we already

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00:07:35 --> 00:07:39
decided that this guy should not
be allowed to be a gradient

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00:07:39 --> 00:07:42
field and should not be
conservative because if we

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00:07:42 --> 00:07:45
integrate on the unit circle
then we would get a positive

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00:07:45 --> 00:07:49
answer.
But let's check that indeed it

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00:07:49 --> 00:07:55
fails our test.
Well, let's call this M and

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00:07:55 --> 00:08:01
let's call this guy N.
If you look at partial M,

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00:08:01 --> 00:08:07
partial y, that is going to be
a negative one.

87
00:08:07 --> 00:08:11
If you take partial N,
partial x, that is going to be

88
00:08:11 --> 00:08:12
one.
These are not the same.

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00:08:12 --> 00:08:17
So, indeed, this is not a
gradient field.

90
00:08:17 --> 00:08:32
 
 

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00:08:32 --> 00:08:53
Any questions about that?
Yes?

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00:08:53 --> 00:08:58
Your question is if I have the
property M sub y equals N sub x

93
00:08:58 --> 00:09:03
only in a certain part of a
plane for some values of x and

94
00:09:03 --> 00:09:06
y,
can I conclude these things?

95
00:09:06 --> 00:09:09
And it is a gradient field in
that part of the plane and

96
00:09:09 --> 00:09:13
conservative and so on.
The answer for now is,

97
00:09:13 --> 00:09:17
in general, no.
And when we spend a bit more

98
00:09:17 --> 00:09:20
time on it, actually,
maybe I should move that up.

99
00:09:20 --> 00:09:24
Maybe we will talk about it
later this week instead of when

100
00:09:24 --> 00:09:28
I had planned.
There is a notion what it means

101
00:09:28 --> 00:09:30
for a region to be without
holes.

102
00:09:30 --> 00:09:34
Basically, if you have that
kind of property in a region

103
00:09:34 --> 00:09:38
that doesn't have any holes
inside it then things will work.

104
00:09:38 --> 00:09:42
The problem comes from a vector
field satisfying this criterion

105
00:09:42 --> 00:09:44
in a region but it has a hole in
it.

106
00:09:44 --> 00:09:47
Because what you don't know is
whether your potential is

107
00:09:47 --> 00:09:51
actually well-defined and takes
the same value when you move all

108
00:09:51 --> 00:09:53
around the hole.
It might come back to take a

109
00:09:53 --> 00:09:56
different value.
If you look carefully and think

110
00:09:56 --> 00:10:00
hard about the example in the
problem sets that is exactly

111
00:10:00 --> 00:10:04
what happens there.
Again, I will say more about

112
00:10:04 --> 00:10:08
that later.
For now we basically need our

113
00:10:08 --> 00:10:11
function to be,
I mean, 

114
00:10:11 --> 00:10:14
I should still say if you have
this property for a vector field

115
00:10:14 --> 00:10:16
that is not quite defined
everywhere,

116
00:10:16 --> 00:10:17
you are more than welcome,
you know,

117
00:10:17 --> 00:10:20
you should probably still try
to look for a potential using

118
00:10:20 --> 00:10:23
methods that we will see.
But something might go wrong

119
00:10:23 --> 00:10:30
later.
You might end up with a

120
00:10:30 --> 00:10:39
potential that is not
well-defined.

121
00:10:39 --> 00:10:53
Let's do another example.
Let's say that I give you this

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00:10:53 --> 00:11:03
vector field.
And this a here is a number.

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00:11:03 --> 00:11:08
The question is for which value
of a is this going to be

124
00:11:08 --> 00:11:13
possibly a gradient?
If you have your flashcards

125
00:11:13 --> 00:11:17
then that is a good time to use
them to vote,

126
00:11:17 --> 00:11:23
assuming that the number is
small enough to be made with.

127
00:11:23 --> 00:11:27
Let's try to think about it.
We want to call this guy M.

128
00:11:27 --> 00:11:35
We want to call that guy N.
And we want to test M sub y

129
00:11:35 --> 00:11:42
versus N sub x.
I don't see anyone.

130
00:11:42 --> 00:11:46
I see people doing it with
their hands, and that works very

131
00:11:46 --> 00:11:48
well.
OK.

132
00:11:48 --> 00:12:04
The question is for which value
of a is this a gradient?

133
00:12:04 --> 00:12:10
I see various people with the
correct answer.

134
00:12:10 --> 00:12:15
OK.
That a strange answer.

135
00:12:15 --> 00:12:20
That is a good answer.
OK.

136
00:12:20 --> 00:12:28
The vote seems to be for a
equals eight.

137
00:12:28 --> 00:12:35
Let's see.
What if I take M sub y?

138
00:12:35 --> 00:12:41
That is going to be just ax.
And N sub x?

139
00:12:41 --> 00:12:47
That is 8x.
I would like a equals eight.

140
00:12:47 --> 00:12:50
By the way, when you set these
two equal to each other,

141
00:12:50 --> 00:12:52
they really have to be equal
everywhere.

142
00:12:52 --> 00:12:55
You don't want to somehow solve
for x or anything like that.

143
00:12:55 --> 00:12:59
You just want these
expressions, in terms of x and

144
00:12:59 --> 00:13:02
y, to be the same quantities.
I mean you cannot say if x

145
00:13:02 --> 00:13:07
equals z they are always equal.
Yeah, that is true.

146
00:13:07 --> 00:13:13
But that is not what we are
asking.

147
00:13:13 --> 00:13:18
Now we come to the next logical
question.

148
00:13:18 --> 00:13:20
Let's say that we have passed
the test.

149
00:13:20 --> 00:13:23
We have put a equals eight in
here.

150
00:13:23 --> 00:13:26
Now it should be a gradient
field.

151
00:13:26 --> 00:13:30
The question is how do we find
the potential?

152
00:13:30 --> 00:13:36
That becomes eight from now on.
The question is how do we find

153
00:13:36 --> 00:13:39
the function which has this as
gradient?

154
00:13:39 --> 00:13:43
One option is to try to guess.
Actually, quite often you will

155
00:13:43 --> 00:13:47
succeed that way.
But that is not a valid method

156
00:13:47 --> 00:13:50
on next week's test.
We are going to see two

157
00:13:50 --> 00:13:55
different systematic methods.
And you should be using one of

158
00:13:55 --> 00:14:00
these because guessing doesn't
always work.

159
00:14:00 --> 00:14:03
And, actually,
I can come up with examples

160
00:14:03 --> 00:14:07
where if you try to guess you
will surely fail.

161
00:14:07 --> 00:14:15
I can come up with trick ones,
but I don't want to put that on

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00:14:15 --> 00:14:24
the test.
The next stage is finding the

163
00:14:24 --> 00:14:30
potential.
And let me just emphasize that

164
00:14:30 --> 00:14:36
we can only do that if step one
was successful.

165
00:14:36 --> 00:14:41
If we have a vector field that
cannot possibly be a gradient

166
00:14:41 --> 00:14:45
then we shouldn't try to look
for a potential.

167
00:14:45 --> 00:14:52
It is kind of obvious but is
probably worth pointing out.

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00:14:52 --> 00:15:00
There are two methods.
The first method that we will

169
00:15:00 --> 00:15:16
see is computing line integrals.
Let's see how that works.

170
00:15:16 --> 00:15:25
Let's say that I take some path
that starts at the origin.

171
00:15:25 --> 00:15:26
Or, actually,
anywhere you want,

172
00:15:26 --> 00:15:29
but let's take the origin.
That is my favorite point.

173
00:15:29 --> 00:15:36
And let's go to a point with
coordinates (x1,

174
00:15:36 --> 00:15:40
y1).
And let's take my favorite

175
00:15:40 --> 00:15:45
curve and compute the line
integral of that field,

176
00:15:45 --> 00:15:49
you know, the work done along
the curve.

177
00:15:49 --> 00:15:55
Well, by the fundamental
theorem, that should be equal to

178
00:15:55 --> 00:16:02
the value of the potential at
the end point minus the value at

179
00:16:02 --> 00:16:09
the origin.
That means I can actually write

180
00:16:09 --> 00:16:19
f of (x1, y1) equals -- -- that
line integral plus the value at

181
00:16:19 --> 00:16:26
the origin.
And that is just a constant.

182
00:16:26 --> 00:16:27
We don't know what it is.
And, actually,

183
00:16:27 --> 00:16:30
we can choose what it is.
Because if you have a

184
00:16:30 --> 00:16:33
potential, say that you have
some potential function.

185
00:16:33 --> 00:16:34
And let's say that you add one
to it.

186
00:16:34 --> 00:16:36
It is still a potential
function.

187
00:16:36 --> 00:16:38
Adding one doesn't change the
gradient.

188
00:16:38 --> 00:16:41
You can even add 18 or any
number that you want.

189
00:16:41 --> 00:16:44
This is just going to be an
integration constant.

190
00:16:44 --> 00:16:47
It is the same thing as,
in one variable calculus,

191
00:16:47 --> 00:16:49
when you take the
anti-derivative of a function it

192
00:16:49 --> 00:16:52
is only defined up to adding the
constant.

193
00:16:52 --> 00:16:56
We have this integration
constant, but apart from that we

194
00:16:56 --> 00:16:59
know that we should be able to
get a potential from this.

195
00:16:59 --> 00:17:03
And this we can compute using
the definition of the line

196
00:17:03 --> 00:17:06
integral.
And we don't know what little f

197
00:17:06 --> 00:17:11
is, but we know what the vector
field is so we can compute that.

198
00:17:11 --> 00:17:14
Of course, to do the
calculation we probably don't

199
00:17:14 --> 00:17:18
want to use this kind of path.
I mean if that is your favorite

200
00:17:18 --> 00:17:21
path then that is fine,
but it is not very easy to

201
00:17:21 --> 00:17:24
compute the line integral along
this,

202
00:17:24 --> 00:17:28
especially since I didn't tell
you what the definition is.

203
00:17:28 --> 00:17:31
There are easier favorite paths
to have.

204
00:17:31 --> 00:17:33
For example,
you can go on a straight line

205
00:17:33 --> 00:17:37
from the origin to that point.
That would be slightly easier.

206
00:17:37 --> 00:17:40
But then there is one easier.
The easiest of all,

207
00:17:40 --> 00:17:47
probably, is to just go first
along the x-axis to (x1,0) and

208
00:17:47 --> 00:17:51
then go up parallel to the
y-axis.

209
00:17:51 --> 00:17:54
Why is that easy?
Well, that is because when we

210
00:17:54 --> 00:17:57
do the line integral it becomes
M dx N dy.

211
00:17:57 --> 00:18:05
And then, on each of these
pieces, one-half just goes away

212
00:18:05 --> 00:18:11
because x, y is constant.
Let's try to use that method in

213
00:18:11 --> 00:18:12
our example.

214
00:18:12 --> 00:18:45

215
00:18:45 --> 00:18:56
Let's say that I want to go
along this path from the origin,

216
00:18:56 --> 00:19:06
first along the x-axis to
(x1,0) and then vertically to

217
00:19:06 --> 00:19:14
(x1, y1).
And so I want to compute for

218
00:19:14 --> 00:19:21
the line integral along that
curve.

219
00:19:21 --> 00:19:24
Let's say I want to do it for
this vector field.

220
00:19:24 --> 00:19:33
I want to find the potential
for this vector field.

221
00:19:33 --> 00:19:37
Let me copy it because I will
have to erase at some point.

222
00:19:37 --> 00:19:50
4x squared plus 8xy and 3y
squared plus 4x squared.

223
00:19:50 --> 00:19:59
That will become the integral
of 4x squared plus 8 xy times dx

224
00:19:59 --> 00:20:05
plus 3y squared plus 4x squared
times dy.

225
00:20:05 --> 00:20:08
To evaluate on this broken
line, I will,

226
00:20:08 --> 00:20:13
of course, evaluate separately
on each of the two segments.

227
00:20:13 --> 00:20:20
I will start with this segment
that I will call c1 and then I

228
00:20:20 --> 00:20:25
will do this one that I will
call c2.

229
00:20:25 --> 00:20:30
On c1, how do I evaluate my
integral?

230
00:20:30 --> 00:20:38
Well, if I am on c1 then x
varies from zero to x1.

231
00:20:38 --> 00:20:40
Well, actually,
I don't know if x1 is positive

232
00:20:40 --> 00:20:41
or not so I shouldn't write
this.

233
00:20:41 --> 00:20:48
I really should say just x goes
from zero to x1.

234
00:20:48 --> 00:20:54
And what about y?
y is just 0.

235
00:20:54 --> 00:21:00
I will set y equal to zero and
also dy equal to zero.

236
00:21:00 --> 00:21:08
I get that the line integral on
c1 -- Well, a lot of stuff goes

237
00:21:08 --> 00:21:11
away.
The entire second term with dy

238
00:21:11 --> 00:21:15
goes away because dy is zero.
And, in the first term,

239
00:21:15 --> 00:21:18
8xy goes away because y is zero
as well.

240
00:21:18 --> 00:21:27
I just have an integral of 4x
squared dx from zero to x1.

241
00:21:27 --> 00:21:31
By the way, now you see why I
have been using an x1 and a y1

242
00:21:31 --> 00:21:33
for my point and not just x and
y.

243
00:21:33 --> 00:21:36
It is to avoid confusion.
I am using x and y as my

244
00:21:36 --> 00:21:41
integration variables and x1,
y1 as constants that are

245
00:21:41 --> 00:21:45
representing the end point of my
path.

246
00:21:45 --> 00:21:51
And so, if I integrate this,
I should get four-thirds x1

247
00:21:51 --> 00:21:54
cubed.
That is the first part.

248
00:21:54 --> 00:22:01
Next I need to do the second
segment.

249
00:22:01 --> 00:22:09
If I am on c2,
y goes from zero to y1.

250
00:22:09 --> 00:22:16
And what about x?
x is constant equal to x1 so dx

251
00:22:16 --> 00:22:22
becomes just zero.
It is a constant.

252
00:22:22 --> 00:22:30
If I take the line integral of
c2, F dot dr then I will get the

253
00:22:30 --> 00:22:37
integral from zero to y1.
The entire first term with dx

254
00:22:37 --> 00:22:47
goes away and then I have 3y
squared plus 4x1 squared times

255
00:22:47 --> 00:22:52
dy.
That integrates to y cubed plus

256
00:22:52 --> 00:23:01
4x1 squared y from zero to y1.
Or, if you prefer,

257
00:23:01 --> 00:23:11
that is y1 cubed plus 4x1
squared y1.

258
00:23:11 --> 00:23:15
Now that we have done both of
them we can just add them

259
00:23:15 --> 00:23:19
together, and that will give us
the formula for the potential.

260
00:23:19 --> 00:23:40
 
 

261
00:23:40 --> 00:23:50
F of x1 and y1 is four-thirds
x1 cubed plus y1 cubed plus 4x1

262
00:23:50 --> 00:23:57
squared y1 plus a constant.
That constant is just the

263
00:23:57 --> 00:24:03
integration constant that we had
from the beginning.

264
00:24:03 --> 00:24:05
Now you can drop the subscripts
if you prefer.

265
00:24:05 --> 00:24:14
You can just say f is
four-thirds x cubed plus y cubed

266
00:24:14 --> 00:24:20
plus 4x squared y plus constant.
And you can check.

267
00:24:20 --> 00:24:25
If you take the gradient of
this, you should get again this

268
00:24:25 --> 00:24:29
vector field over there.
Any questions about this method?

269
00:24:29 --> 00:24:33
Yes?
No.

270
00:24:33 --> 00:24:35
Well, it depends whether you
are just trying to find one

271
00:24:35 --> 00:24:38
potential or if you are trying
to find all the possible

272
00:24:38 --> 00:24:40
potentials.
If a problem just says find a

273
00:24:40 --> 00:24:43
potential then you don't have to
use the constant.

274
00:24:43 --> 00:24:47
This guy without the constant
is a valid potential.

275
00:24:47 --> 00:24:52
You just have others.
If your neighbor comes to you

276
00:24:52 --> 00:24:58
and say your answer must be
wrong because I got this plus

277
00:24:58 --> 00:25:01
18, well, both answers are
correct.

278
00:25:01 --> 00:25:05
By the way.
Instead of going first along

279
00:25:05 --> 00:25:08
the x-axis vertically,
you could do it the other way

280
00:25:08 --> 00:25:11
around.
Of course, start along the

281
00:25:11 --> 00:25:15
y-axis and then horizontally.
That is the same level of

282
00:25:15 --> 00:25:19
difficulty.
You just exchange roles of x

283
00:25:19 --> 00:25:21
and y.
In some cases,

284
00:25:21 --> 00:25:26
it is actually even making more
sense maybe to go radially,

285
00:25:26 --> 00:25:30
start out from the origin to
your end point.

286
00:25:30 --> 00:25:37
But usually this setting is
easier just because each of

287
00:25:37 --> 00:25:43
these two guys were very easy to
compute.

288
00:25:43 --> 00:25:46
But somehow maybe if you
suspect that polar coordinates

289
00:25:46 --> 00:25:49
will be involved somehow in the
answer then maybe it makes sense

290
00:25:49 --> 00:26:01
to choose different paths.
Maybe a straight line is better.

291
00:26:01 --> 00:26:13
Now we have another method to
look at which is using

292
00:26:13 --> 00:26:19
anti-derivatives.
The goal is the same,

293
00:26:19 --> 00:26:21
still to find the potential
function.

294
00:26:21 --> 00:26:26
And you see that finding the
potential is really the

295
00:26:26 --> 00:26:31
multivariable analog of finding
the anti-derivative in the one

296
00:26:31 --> 00:26:34
variable.
Here we did it basically by

297
00:26:34 --> 00:26:38
hand by computing the integral.
The other thing you could try

298
00:26:38 --> 00:26:39
to say is, wait,
I already know how to take

299
00:26:39 --> 00:26:42
anti-derivatives.
Let's use that instead of

300
00:26:42 --> 00:26:45
computing integrals.
And it works but you have to be

301
00:26:45 --> 00:26:51
careful about how you do it.
Let's see how that works.

302
00:26:51 --> 00:26:53
Let's still do it with the same
example.

303
00:26:53 --> 00:27:02
We want to solve the equations.
We want a function such that f

304
00:27:02 --> 00:27:13
sub x is 4x squared plus 8xy and
f sub y is 3y squared plus 4x

305
00:27:13 --> 00:27:16
squared.
Let's just look at one of these

306
00:27:16 --> 00:27:20
at a time.
If we look at this one,

307
00:27:20 --> 00:27:28
well, we know how to solve this
because it is just telling us we

308
00:27:28 --> 00:27:33
have to integrate this with
respect to x.

309
00:27:33 --> 00:27:38
Well, let's call them one and
two because I will have to refer

310
00:27:38 --> 00:27:43
to them again.
Let's start with equation one

311
00:27:43 --> 00:27:48
and lets integrate with respect
to x.

312
00:27:48 --> 00:27:51
Well, it tells us that f should
be,

313
00:27:51 --> 00:27:55
what do I get when I integrate
this with respect to x,

314
00:27:55 --> 00:28:02
four-thirds x cubed plus, 
when I integrate 8xy, 

315
00:28:02 --> 00:28:08
y is just a constant, 
so I will get 4x squared y.

316
00:28:08 --> 00:28:11
And that is not quite the end
to it because there is an

317
00:28:11 --> 00:28:15
integration constant.
And here, when I say there is

318
00:28:15 --> 00:28:18
an integration constant,
it just means the extra term

319
00:28:18 --> 00:28:21
does not depend on x.
That is what it means to be a

320
00:28:21 --> 00:28:25
constant in this setting.
But maybe my constant still

321
00:28:25 --> 00:28:28
depends on y so it is not
actually a true constant.

322
00:28:28 --> 00:28:30
A constant that depends on y is
not really a constant.

323
00:28:30 --> 00:28:38
It is actually a function of y.
The good news that we have is

324
00:28:38 --> 00:28:40
that this function normally
depends on x.

325
00:28:40 --> 00:28:46
We have made some progress.
We have part of the answer and

326
00:28:46 --> 00:28:53
we have simplified the problem.
If we have anything that looks

327
00:28:53 --> 00:28:56
like this, it will satisfy the
first condition.

328
00:28:56 --> 00:28:59
Now we need to look at the
second condition.

329
00:28:59 --> 00:29:12
We want f sub y to be that.
But we know what f is,

330
00:29:12 --> 00:29:15
so let's compute f sub y from
this.

331
00:29:15 --> 00:29:20
From this I get f sub y.
What do I get if I

332
00:29:20 --> 00:29:22
differentiate this with respect
to y?

333
00:29:22 --> 00:29:37
Well, I get zero plus 4x
squared plus the derivative of

334
00:29:37 --> 00:29:46
g.
I would like to match this with

335
00:29:46 --> 00:29:51
what I had.
If I match this with equation

336
00:29:51 --> 00:29:55
two then that will tell me what
the derivative of g should be.

337
00:29:55 --> 00:30:15
 
 

338
00:30:15 --> 00:30:20
If we compare the two things
there, we get 4x squared plus g

339
00:30:20 --> 00:30:26
prime of y should be equal to 3y
squared by 4x squared.

340
00:30:26 --> 00:30:31
And, of course,
the 4x squares go away.

341
00:30:31 --> 00:30:35
That tells you g prime is 3y
squared.

342
00:30:35 --> 00:30:42
And that integrates to y cubed
plus constant.

343
00:30:42 --> 00:30:46
Now, this time the constant is
a true constant because g did

344
00:30:46 --> 00:30:48
not depend on anything other
than y.

345
00:30:48 --> 00:30:54
And the constant does not
depend on y so it is a real

346
00:30:54 --> 00:30:58
constant now.
Now we just plug this back into

347
00:30:58 --> 00:31:05
this guy.
Let's call him star.

348
00:31:05 --> 00:31:13
If we plug this into star,
we get f equals four-thirds x

349
00:31:13 --> 00:31:21
cubed plus 4x squared y plus y
cubed plus constant.

350
00:31:21 --> 00:31:30
I mean, of course,
again, now this constant is

351
00:31:30 --> 00:31:33
optional.
The advantage of this method is

352
00:31:33 --> 00:31:35
you don't have to write any
integrals.

353
00:31:35 --> 00:31:40
The small drawback is you have
to follow this procedure

354
00:31:40 --> 00:31:45
carefully.
By the way, one common pitfall

355
00:31:45 --> 00:31:48
that is tempting.
After you have done this,

356
00:31:48 --> 00:31:51
what is very tempting is to
just say, well,

357
00:31:51 --> 00:31:53
let's do the same with this
guy.

358
00:31:53 --> 00:31:55
Let's integrate this with
respect to y.

359
00:31:55 --> 00:31:58
You will get another expression
for f up to a constant that

360
00:31:58 --> 00:32:01
depends on x.
And then let's match them.

361
00:32:01 --> 00:32:04
Well, the difficulty is
matching is actually quite

362
00:32:04 --> 00:32:09
tricky because you don't know in
advance whether they will be the

363
00:32:09 --> 00:32:13
same expression.
It could be you could say let's

364
00:32:13 --> 00:32:16
just take the terms that are
here and missing there and

365
00:32:16 --> 00:32:20
combine the terms,
you know, take all the terms

366
00:32:20 --> 00:32:23
that appear in either one.
That is actually not a good way

367
00:32:23 --> 00:32:25
to do it,
because if I put sufficiently

368
00:32:25 --> 00:32:28
complicated trig functions in
there then you might not be able

369
00:32:28 --> 00:32:30
to see that two terms are the
same.

370
00:32:30 --> 00:32:34
Take an easy one.
Let's say that here I have one

371
00:32:34 --> 00:32:40
plus tangent square and here I
have a secan square then you

372
00:32:40 --> 00:32:46
might not actually notice that
there is a difference.

373
00:32:46 --> 00:32:50
But there is no difference.
Whatever.

374
00:32:50 --> 00:32:54
Anyway, I am saying do it this
way, don't do it any other way

375
00:32:54 --> 00:32:57
because there is a risk of
making a mistake otherwise.

376
00:32:57 --> 00:33:00
I mean, on the other hand,
you could start with

377
00:33:00 --> 00:33:03
integrating with respect to y
and then differentiate and match

378
00:33:03 --> 00:33:06
with respect to x.
But what I am saying is just

379
00:33:06 --> 00:33:09
take one of them,
integrate, 

380
00:33:09 --> 00:33:12
get an answer that involves a
function of the other variable,

381
00:33:12 --> 00:33:18
then differentiate that answer
and compare and see what you

382
00:33:18 --> 00:33:21
get.
By the way, here,

383
00:33:21 --> 00:33:27
of course, after we simplified
there were only y's here.

384
00:33:27 --> 00:33:29
There were no x's.
And that is kind of good news.

385
00:33:29 --> 00:33:33
I mean, if you had had an x
here in this expression that

386
00:33:33 --> 00:33:36
would have told you that
something is going wrong.

387
00:33:36 --> 00:33:39
g is a function of y only.
If you get an x here,

388
00:33:39 --> 00:33:42
maybe you want to go back and
check whether it is really a

389
00:33:42 --> 00:33:47
gradient field.
Yes?

390
00:33:47 --> 00:33:49
Yes, this will work with
functions of more than two

391
00:33:49 --> 00:33:51
variables.
Both methods work with more

392
00:33:51 --> 00:33:53
than two variables.
We are going to see it in the

393
00:33:53 --> 00:33:56
case where more than two means
three.

394
00:33:56 --> 00:34:00
We are going to see that in two
or three weeks from now.

395
00:34:00 --> 00:34:04
I mean, basically starting at
the end of next week,

396
00:34:04 --> 00:34:08
we are going to do triple
integrals, line integrals in

397
00:34:08 --> 00:34:10
space and so on.
The format is first we do

398
00:34:10 --> 00:34:13
everything in two variables.
Then we will do three variables.

399
00:34:13 --> 00:34:20
And then what happens with more
than three will be left to your

400
00:34:20 --> 00:34:25
imagination.
Any other questions about

401
00:34:25 --> 00:34:29
either of these methods?
A quick poll.

402
00:34:29 --> 00:34:34
Who prefers the first method?
Who prefers the second method?

403
00:34:34 --> 00:34:41
Wow.
OK.

404
00:34:41 --> 00:34:45
Anyway, you will get to use
whichever one you want.

405
00:34:45 --> 00:34:47
And I would agree with you,
but the second method is

406
00:34:47 --> 00:34:50
slightly more effective in that
you are writing less stuff.

407
00:34:50 --> 00:34:54
You don't have to set up all
these line integrals.

408
00:34:54 --> 00:35:03
On the other hand,
it does require a little bit

409
00:35:03 --> 00:35:19
more attention.
Let's move on a bit.

410
00:35:19 --> 00:35:24
Let me start by actually doing
a small recap.

411
00:35:24 --> 00:35:38
We said we have various notions.
One is to say that the vector

412
00:35:38 --> 00:35:48
field is a gradient in a certain
region of a plane.

413
00:35:48 --> 00:35:54
And we have another notion
which is being conservative.

414
00:35:54 --> 00:36:06
It says that the line integral
is zero along any closed curve.

415
00:36:06 --> 00:36:10
Actually, let me introduce a
new piece of notation.

416
00:36:10 --> 00:36:14
To remind ourselves that we are
doing it along a closed curve,

417
00:36:14 --> 00:36:18
very often we put just a circle
for the integral to tell us this

418
00:36:18 --> 00:36:21
is a curve that closes on
itself.

419
00:36:21 --> 00:36:25
It ends where it started.
I mean it doesn't change

420
00:36:25 --> 00:36:28
anything concerning the
definition or how you compute it

421
00:36:28 --> 00:36:31
or anything.
It just reminds you that you

422
00:36:31 --> 00:36:34
are doing it on a closed curve.
It is actually useful for

423
00:36:34 --> 00:36:37
various physical applications.
And also, when you state

424
00:36:37 --> 00:36:41
theorems in that way,
it reminds you,oh..

425
00:36:41 --> 00:36:45
I need to be on a closed curve
to do it.

426
00:36:45 --> 00:36:51
And so we have said these two
things are equivalent.

427
00:36:51 --> 00:37:00
Now we have a third thing which
is N sub x equals M sub y at

428
00:37:00 --> 00:37:03
every point.
Just to summarize the

429
00:37:03 --> 00:37:06
discussion.
We have said if we have a

430
00:37:06 --> 00:37:09
gradient field then we have
this.

431
00:37:09 --> 00:37:18
And the converse is true in
suitable regions.

432
00:37:18 --> 00:37:32
We have a converse if F is
defined in the entire plane.

433
00:37:32 --> 00:37:43
Or, as we will see soon,
in a simply connected region.

434
00:37:43 --> 00:37:45
I guess some of you cannot see
what I am writing here,

435
00:37:45 --> 00:37:48
but it doesn't matter because
you are not officially supposed

436
00:37:48 --> 00:37:53
to know it yet.
That will be next week.

437
00:37:53 --> 00:37:57
Anyway, 
I said the fact that Nx equals

438
00:37:57 --> 00:38:01
My implies that we have a
gradient field and is only if a

439
00:38:01 --> 00:38:06
vector field is defined in the
entire plane or in a region that

440
00:38:06 --> 00:38:12
is called simply connected.
And more about that later.

441
00:38:12 --> 00:38:17
Now let me just introduce a
quantity that probably a lot of

442
00:38:17 --> 00:38:22
you have heard about in physics
that measures precisely fairly

443
00:38:22 --> 00:38:26
ought to be conservative.
That is called the curl of a

444
00:38:26 --> 00:38:27
vector field.

445
00:38:27 --> 00:39:06

446
00:39:06 --> 00:39:19
For the definition we say that
the curl of F is the quantity N

447
00:39:19 --> 00:39:27
sub x - M sub y.
It is just replicating the

448
00:39:27 --> 00:39:35
information we had but in a way
that is a single quantity.

449
00:39:35 --> 00:39:43
In this new language,
the conditions that we had over

450
00:39:43 --> 00:39:50
there, this condition says curl
F equals zero.

451
00:39:50 --> 00:39:56
That is the new version of Nx
equals My.

452
00:39:56 --> 00:40:06
It measures failure of a vector
field to be conservative.

453
00:40:06 --> 00:40:21
The test for conservativeness
is that the curl of F should be

454
00:40:21 --> 00:40:25
zero.
I should probably tell you a

455
00:40:25 --> 00:40:29
little bit about what the curl
is, what it measures and what it

456
00:40:29 --> 00:40:34
does because that is something
that is probably useful.

457
00:40:34 --> 00:40:37
It is a very strange quantity
if you put it in that form.

458
00:40:37 --> 00:40:42
Yes?
I think it is the same as the

459
00:40:42 --> 00:40:45
physics one, but I haven't
checked the physics textbook.

460
00:40:45 --> 00:40:49
I believe it is the same.
Yes, I think it is the same as

461
00:40:49 --> 00:40:53
the physics one.
It is not the opposite this

462
00:40:53 --> 00:40:55
time.
Of course, in physics maybe you

463
00:40:55 --> 00:40:59
have seen curl in space.
We are going to see curl in

464
00:40:59 --> 00:41:07
space in two or three weeks.
Yes?

465
00:41:07 --> 00:41:11
Yes. Well, you can also use it.
If you fail this test then you

466
00:41:11 --> 00:41:14
know for sure that you are not
gradient field so you might as

467
00:41:14 --> 00:41:18
well do that.
If you satisfy the test but you

468
00:41:18 --> 00:41:24
are not defined everywhere then
there is still a bit of

469
00:41:24 --> 00:41:29
ambiguity and you don't know for
sure.

470
00:41:29 --> 00:41:40
OK.
Let's try to see a little bit

471
00:41:40 --> 00:41:48
what the curl measures.
Just to give you some

472
00:41:48 --> 00:41:55
intuition, let's first think
about a velocity field.

473
00:41:55 --> 00:42:10
The curl measures the rotation
component of a motion.

474
00:42:10 --> 00:42:13
If you want a fancy word,
it measures the vorticity of a

475
00:42:13 --> 00:42:16
motion.
It tells you how much twisting

476
00:42:16 --> 00:42:19
is taking place at a given
point.

477
00:42:19 --> 00:42:24
For example, 
if I take a constant vector

478
00:42:24 --> 00:42:32
field where my fluid is just all
moving in the same direction

479
00:42:32 --> 00:42:37
where this is just constants
then,

480
00:42:37 --> 00:42:41
of course, the curl is zero.
Because if you take the

481
00:42:41 --> 00:42:43
partials you get zero.
And, indeed,

482
00:42:43 --> 00:42:46
that is not what you would call
swirling.

483
00:42:46 --> 00:42:58
There is no vortex in here.
Let's do another one where this

484
00:42:58 --> 00:43:02
is still nothing going on.
Let's say that I take the

485
00:43:02 --> 00:43:06
radial vector field where
everything just flows away from

486
00:43:06 --> 00:43:11
the origin.
That is f equals x, y.

487
00:43:11 --> 00:43:16
Well, if I take the curl, 
I have to take partial over

488
00:43:16 --> 00:43:18
partial x of the second
component,

489
00:43:18 --> 00:43:21
which is y, 
minus partial over partial y of

490
00:43:21 --> 00:43:22
the first component,
which is x.

491
00:43:22 --> 00:43:25
I will get zero.
And, indeed,

492
00:43:25 --> 00:43:29
if you think about what is
going on here,

493
00:43:29 --> 00:43:32
there is no rotation involved.
On the other hand, 

494
00:43:32 --> 00:43:45
if you consider our favorite
rotation vector field -- --

495
00:43:45 --> 00:44:00
negative y and x then this curl
is going to be N sub x minus M

496
00:44:00 --> 00:44:08
sub y,
one plus one equals two.

497
00:44:08 --> 00:44:13
That corresponds to the fact
that we are rotating.

498
00:44:13 --> 00:44:16
Actually, we are rotating at
unit angular speed.

499
00:44:16 --> 00:44:20
The curl actually measures
twice the angular speed of a

500
00:44:20 --> 00:44:24
rotation part of a motion at any
given point.

501
00:44:24 --> 00:44:26
Now, if you have an actual
motion,

502
00:44:26 --> 00:44:30
a more complicated field than
these then no matter where you

503
00:44:30 --> 00:44:34
are you can think of a motion as
a combination of translation

504
00:44:34 --> 00:44:37
effects,
maybe dilation effects, 

505
00:44:37 --> 00:44:43
maybe rotation effects, 
possibly other things like that.

506
00:44:43 --> 00:44:48
And what a curl will measure is
how intense the rotation effect

507
00:44:48 --> 00:44:52
is at that particular point.
I am not going to try to make a

508
00:44:52 --> 00:44:55
much more precise statement.
A precise statement is what a

509
00:44:55 --> 00:44:58
curl measures is really this
quantity up there.

510
00:44:58 --> 00:45:01
But the intuition you should
have is it measures how much

511
00:45:01 --> 00:45:04
rotation is taking place at any
given point.

512
00:45:04 --> 00:45:06
And, of course,
in a complicated motion you

513
00:45:06 --> 00:45:09
might have more rotation at some
point than at some others,

514
00:45:09 --> 00:45:12
which is why the curl will
depend on x and y.

515
00:45:12 --> 00:45:20
It is not just a constant
because how much you rotate

516
00:45:20 --> 00:45:26
depends on where you are.
If you are looking at actual

517
00:45:26 --> 00:45:30
wind velocities in weather
prediction then the regions with

518
00:45:30 --> 00:45:33
high curl tend to be hurricanes
or tornadoes or things like

519
00:45:33 --> 00:45:37
that.
They are not very pleasant

520
00:45:37 --> 00:45:40
things.
And the sign of a curl tells

521
00:45:40 --> 00:45:43
you whether you are going
clockwise or counterclockwise.

522
00:45:43 --> 00:46:09
 
 

523
00:46:09 --> 00:46:27
Curl measures twice the angular
velocity of the rotation

524
00:46:27 --> 00:46:41
component of a velocity field.
Now, what about a force field?

525
00:46:41 --> 00:46:44
Because, after all,
how we got to this was coming

526
00:46:44 --> 00:46:47
from and trying to understand
forces and the work they do.

527
00:46:47 --> 00:46:50
So I should tell you what it
means for a force.

528
00:46:50 --> 00:47:10
Well, the curl of a force field
-- -- measures the torque

529
00:47:10 --> 00:47:29
exerted on a test object that
you put at any point.

530
00:47:29 --> 00:47:36
Remember, torque is the
rotational analog of the force.

531
00:47:36 --> 00:47:41
We had this analogy about
velocity versus angular velocity

532
00:47:41 --> 00:47:45
and mass versus moment of
inertia.

533
00:47:45 --> 00:47:49
And then, in that analogy,
force divided by the mass is

534
00:47:49 --> 00:47:53
what will cause acceleration,
which is the derivative of

535
00:47:53 --> 00:47:56
velocity.
Torque divided by moment of

536
00:47:56 --> 00:47:59
inertia is what will cause the
angular acceleration,

537
00:47:59 --> 00:48:02
namely the derivative of
angular velocity.

538
00:48:02 --> 00:48:04
Maybe I should write that down.

539
00:48:04 --> 00:48:18

540
00:48:18 --> 00:48:31
Torque divided by moment of
inertia is going to be d over dt

541
00:48:31 --> 00:48:38
of angular velocity.
I leave it up to your physics

542
00:48:38 --> 00:48:41
teachers to decide what letters
to use for all these things.

543
00:48:41 --> 00:48:49
That is the analog of force
divided by mass equals

544
00:48:49 --> 00:48:56
acceleration,
which is d over dt of velocity.

545
00:48:56 --> 00:49:03
And so now you see if the curl
of a velocity field measure the

546
00:49:03 --> 00:49:07
angular velocity of its rotation
then,

547
00:49:07 --> 00:49:13
by this analogy, 
the curl of a force field

548
00:49:13 --> 00:49:24
should measure the torque it
exerts on a mass per unit moment

549
00:49:24 --> 00:49:28
of inertia.
Concretely, if you imagine that

550
00:49:28 --> 00:49:29
you are putting something in
there,

551
00:49:29 --> 00:49:32
you know, if you are in a
velocity field the curl will

552
00:49:32 --> 00:49:35
tell you how fast your guy is
spinning at a given time.

553
00:49:35 --> 00:49:37
If you put something that
floats, for example,

554
00:49:37 --> 00:49:40
in your fluid,
something very light then it is

555
00:49:40 --> 00:49:44
going to start spinning.
And the curl of a velocity

556
00:49:44 --> 00:49:48
field tells you how fast it is
spinning at any given time up to

557
00:49:48 --> 00:49:51
a factor of two.
And the curl of a force field

558
00:49:51 --> 00:49:55
tells you how quickly the
angular velocity is going to

559
00:49:55 --> 00:50:01
increase or decrease.
OK.

560
00:50:01 --> 00:50:04
Well, next time we are going to
see Green's theorem which is

561
00:50:04 --> 00:50:08
actually going to tell us a lot
more about curl and failure of

562
00:50:08 --> 00:50:11
conservativeness.
 

563
00:50:11 --> 00:50:16