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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
11
00:01:08 --> 00:01:19
the end point and the starting
point of the curve.
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00:01:19 --> 00:01:24
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
14
00:01:29 --> 00:01:32
and not on the actual path we
chose.
15
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
17
00:01:49 --> 00:01:55
because of conservation of
energy which tells you if you
18
00:01:55 --> 00:02:02
start at a point and you come
back to the same point then you
19
00:02:02 --> 00:02:07
haven't gotten any work out of
that force.
20
00:02:07 --> 00:02:15
If we have a closed curve then
the line integral for work is
21
00:02:15 --> 00:02:18
just zero.
And, basically,
22
00:02:18 --> 00:02:23
we say that these properties
are equivalent being a gradient
23
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field or being path independent
or being conservative.
24
00:02:28 --> 00:02:31
And what I promised to you is
that today we would see a
25
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
27
00:02:38 --> 00:02:47
gradient field.
So, that is the topic for today.
28
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
31
00:03:16 --> 00:03:26
observation.
Say that it is a gradient field.
32
00:03:26 --> 00:03:31
That means that the first
component of a field is just the
33
00:03:31 --> 00:03:35
partial of f with respect to
some variable x and the second
34
00:03:35 --> 00:03:40
component is the partial of f
with respect to y.
35
00:03:40 --> 00:03:43
Now we have seen an interesting
property of the second partial
36
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,
39
00:03:52 --> 00:03:58
then with respect to x you get
the same thing.
40
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.
43
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.
47
00:04:31 --> 00:04:37
So, I am going to put that in a
box.
48
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.
51
00:04:47 --> 00:05:06
52
00:05:06 --> 00:05:18
Conversely, if,
and I am going to put here a
53
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?
55
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
57
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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well-defined.
Otherwise, actually,
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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
80
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.
89
00:08:12 --> 00:08:17
So, indeed, this is not a
gradient field.
90
00:08:17 --> 00:08:32
91
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
122
00:10:53 --> 00:11:03
vector field.
And this a here is a number.
123
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
162
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.
168
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