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We were looking at vector
fields last time.
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00:00:34,000 --> 00:00:45,000
Last time we saw that if a
vector field happens to be a
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gradient field -- -- then the
line integral can be computed
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00:00:56,000 --> 00:01:08,000
actually by taking the change in
value of the potential between
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00:01:08,000 --> 00:01:19,000
the end point and the starting
point of the curve.
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00:01:19,000 --> 00:01:24,000
If we have a curve c,
from a point p0 to a point p1
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00:01:24,000 --> 00:01:29,000
then the line integral for work
depends only on the end points
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00:01:29,000 --> 00:01:32,000
and not on the actual path we
chose.
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00:01:32,000 --> 00:01:43,000
We say that the line integral
is path independent.
17
00:01:43,000 --> 00:01:49,000
And we also said that the
vector field is conservative
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00:01:49,000 --> 00:01:55,000
because of conservation of
energy which tells you if you
19
00:01:55,000 --> 00:02:02,000
start at a point and you come
back to the same point then you
20
00:02:02,000 --> 00:02:07,000
haven't gotten any work out of
that force.
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00:02:07,000 --> 00:02:15,000
If we have a closed curve then
the line integral for work is
22
00:02:15,000 --> 00:02:18,000
just zero.
And, basically,
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00:02:18,000 --> 00:02:23,000
we say that these properties
are equivalent being a gradient
24
00:02:23,000 --> 00:02:28,000
field or being path independent
or being conservative.
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00:02:28,000 --> 00:02:31,000
And what I promised to you is
that today we would see a
26
00:02:31,000 --> 00:02:35,000
criterion to decide whether a
vector field is a gradient field
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00:02:35,000 --> 00:02:38,000
or not and how to find the
potential function if it is a
28
00:02:38,000 --> 00:02:47,000
gradient field.
So, that is the topic for today.
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00:02:47,000 --> 00:03:00,000
The question is testing whether
a given vector field,
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00:03:00,000 --> 00:03:14,000
let's say M and N compliments,
is a gradient field.
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00:03:14,000 --> 00:03:16,000
For that, well,
let's start with an
32
00:03:16,000 --> 00:03:26,000
observation.
Say that it is a gradient field.
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00:03:26,000 --> 00:03:31,000
That means that the first
component of a field is just the
34
00:03:31,000 --> 00:03:35,000
partial of f with respect to
some variable x and the second
35
00:03:35,000 --> 00:03:40,000
component is the partial of f
with respect to y.
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00:03:40,000 --> 00:03:43,000
Now we have seen an interesting
property of the second partial
37
00:03:43,000 --> 00:03:46,000
derivatives of the function,
which is if you take the
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00:03:46,000 --> 00:03:49,000
partial derivative first with
respect to x,
39
00:03:49,000 --> 00:03:52,000
then with respect to y,
or first with respect to y,
40
00:03:52,000 --> 00:03:58,000
then with respect to x you get
the same thing.
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00:03:58,000 --> 00:04:07,000
We know f sub xy equals f sub
yx, and that means M sub y
42
00:04:07,000 --> 00:04:12,000
equals N sub x.
If you have a gradient field
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00:04:12,000 --> 00:04:14,000
then it should have this
property.
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00:04:14,000 --> 00:04:17,000
You take the y component,
take the derivative with
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00:04:17,000 --> 00:04:19,000
respect to x,
take the x component,
46
00:04:19,000 --> 00:04:20,000
differentiate with respect to
y,
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00:04:20,000 --> 00:04:31,000
you should get the same answer.
And that is important to know.
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00:04:31,000 --> 00:04:37,000
So, I am going to put that in a
box.
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00:04:37,000 --> 00:04:43,000
It is a broken box.
The claim that I want to make
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00:04:43,000 --> 00:04:45,000
is that there is a converse of
sorts.
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00:04:45,000 --> 00:04:47,000
This is actually basically all
we need to check.
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00:04:47,000 --> 00:05:06,000
53
00:05:06,000 --> 00:05:18,000
Conversely, if,
and I am going to put here a
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00:05:18,000 --> 00:05:33,000
condition, My equals Nx,
then F is a gradient field.
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00:05:33,000 --> 00:05:35,000
What is the condition that I
need to put here?
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00:05:35,000 --> 00:05:37,000
Well, we will see a more
precise version of that next
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00:05:37,000 --> 00:05:44,000
week.
But for now let's just say if
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00:05:44,000 --> 00:05:59,000
our vector field is defined and
differentiable everywhere in the
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00:05:59,000 --> 00:06:01,000
plane.
We need, actually,
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00:06:01,000 --> 00:06:04,000
a vector field that is
well-defined everywhere.
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00:06:04,000 --> 00:06:07,000
You are not allowed to have
somehow places where it is not
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00:06:07,000 --> 00:06:09,000
well-defined.
Otherwise, actually,
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00:06:09,000 --> 00:06:13,000
you have a counter example on
your problem set this week.
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00:06:13,000 --> 00:06:16,000
If you look at the last problem
on the problem set this week,
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00:06:16,000 --> 00:06:20,000
it gives you a vector field
that satisfies this condition
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00:06:20,000 --> 00:06:22,000
everywhere where it is defined.
But, actually,
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00:06:22,000 --> 00:06:24,000
there is a point where it is
not defined.
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00:06:24,000 --> 00:06:28,000
And that causes it,
actually, to somehow -- I mean
69
00:06:28,000 --> 00:06:33,000
everything that I am going to
say today breaks down for that
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00:06:33,000 --> 00:06:36,000
example because of that.
I mean, we will shed more light
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00:06:36,000 --> 00:06:39,000
on this a bit later with the
notion of simply connected
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00:06:39,000 --> 00:06:42,000
regions and so on.
But for now let's just say if
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00:06:42,000 --> 00:06:47,000
it is defined everywhere and it
satisfies this criterion then it
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00:06:47,000 --> 00:06:52,000
is a gradient field.
If you ignore the technical
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00:06:52,000 --> 00:06:57,000
condition, being a gradient
field means essentially the same
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00:06:57,000 --> 00:07:11,000
thing as having this property.
That is what we need to check.
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00:07:11,000 --> 00:07:20,000
Let's look at an example.
Well, one vector field that we
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00:07:20,000 --> 00:07:24,000
have been looking at a lot was -
yi xj.
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00:07:24,000 --> 00:07:30,000
Remember that was the vector
field that looked like a
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00:07:30,000 --> 00:07:35,000
rotation at the unit speed.
I think last time we already
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00:07:35,000 --> 00:07:39,000
decided that this guy should not
be allowed to be a gradient
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00:07:39,000 --> 00:07:42,000
field and should not be
conservative because if we
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00:07:42,000 --> 00:07:45,000
integrate on the unit circle
then we would get a positive
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00:07:45,000 --> 00:07:49,000
answer.
But let's check that indeed it
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00:07:49,000 --> 00:07:55,000
fails our test.
Well, let's call this M and
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00:07:55,000 --> 00:08:01,000
let's call this guy N.
If you look at partial M,
87
00:08:01,000 --> 00:08:07,000
partial y, that is going to be
a negative one.
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00:08:07,000 --> 00:08:11,000
If you take partial N,
partial x, that is going to be
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00:08:11,000 --> 00:08:12,000
one.
These are not the same.
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00:08:12,000 --> 00:08:17,000
So, indeed, this is not a
gradient field.
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00:08:17,000 --> 00:08:32,000
92
00:08:32,000 --> 00:08:53,000
Any questions about that?
Yes?
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00:08:53,000 --> 00:08:58,000
Your question is if I have the
property M sub y equals N sub x
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00:08:58,000 --> 00:09:03,000
only in a certain part of a
plane for some values of x and
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00:09:03,000 --> 00:09:06,000
y,
can I conclude these things?
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00:09:06,000 --> 00:09:09,000
And it is a gradient field in
that part of the plane and
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00:09:09,000 --> 00:09:13,000
conservative and so on.
The answer for now is,
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00:09:13,000 --> 00:09:17,000
in general, no.
And when we spend a bit more
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00:09:17,000 --> 00:09:20,000
time on it, actually,
maybe I should move that up.
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00:09:20,000 --> 00:09:24,000
Maybe we will talk about it
later this week instead of when
101
00:09:24,000 --> 00:09:28,000
I had planned.
There is a notion what it means
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00:09:28,000 --> 00:09:30,000
for a region to be without
holes.
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00:09:30,000 --> 00:09:34,000
Basically, if you have that
kind of property in a region
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00:09:34,000 --> 00:09:38,000
that doesn't have any holes
inside it then things will work.
105
00:09:38,000 --> 00:09:42,000
The problem comes from a vector
field satisfying this criterion
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00:09:42,000 --> 00:09:44,000
in a region but it has a hole in
it.
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00:09:44,000 --> 00:09:47,000
Because what you don't know is
whether your potential is
108
00:09:47,000 --> 00:09:51,000
actually well-defined and takes
the same value when you move all
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00:09:51,000 --> 00:09:53,000
around the hole.
It might come back to take a
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00:09:53,000 --> 00:09:56,000
different value.
If you look carefully and think
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00:09:56,000 --> 00:10:00,000
hard about the example in the
problem sets that is exactly
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00:10:00,000 --> 00:10:04,000
what happens there.
Again, I will say more about
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00:10:04,000 --> 00:10:08,000
that later.
For now we basically need our
114
00:10:08,000 --> 00:10:11,000
function to be,
I mean,
115
00:10:11,000 --> 00:10:14,000
I should still say if you have
this property for a vector field
116
00:10:14,000 --> 00:10:16,000
that is not quite defined
everywhere,
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00:10:16,000 --> 00:10:17,000
you are more than welcome,
you know,
118
00:10:17,000 --> 00:10:20,000
you should probably still try
to look for a potential using
119
00:10:20,000 --> 00:10:23,000
methods that we will see.
But something might go wrong
120
00:10:23,000 --> 00:10:30,000
later.
You might end up with a
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00:10:30,000 --> 00:10:39,000
potential that is not
well-defined.
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00:10:39,000 --> 00:10:53,000
Let's do another example.
Let's say that I give you this
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00:10:53,000 --> 00:11:03,000
vector field.
And this a here is a number.
124
00:11:03,000 --> 00:11:08,000
The question is for which value
of a is this going to be
125
00:11:08,000 --> 00:11:13,000
possibly a gradient?
If you have your flashcards
126
00:11:13,000 --> 00:11:17,000
then that is a good time to use
them to vote,
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00:11:17,000 --> 00:11:23,000
assuming that the number is
small enough to be made with.
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00:11:23,000 --> 00:11:27,000
Let's try to think about it.
We want to call this guy M.
129
00:11:27,000 --> 00:11:35,000
We want to call that guy N.
And we want to test M sub y
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00:11:35,000 --> 00:11:42,000
versus N sub x.
I don't see anyone.
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00:11:42,000 --> 00:11:46,000
I see people doing it with
their hands, and that works very
132
00:11:46,000 --> 00:11:48,000
well.
OK.
133
00:11:48,000 --> 00:12:04,000
The question is for which value
of a is this a gradient?
134
00:12:04,000 --> 00:12:10,000
I see various people with the
correct answer.
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00:12:10,000 --> 00:12:15,000
OK.
That a strange answer.
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00:12:15,000 --> 00:12:20,000
That is a good answer.
OK.
137
00:12:20,000 --> 00:12:28,000
The vote seems to be for a
equals eight.
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00:12:28,000 --> 00:12:35,000
Let's see.
What if I take M sub y?
139
00:12:35,000 --> 00:12:41,000
That is going to be just ax.
And N sub x?
140
00:12:41,000 --> 00:12:47,000
That is 8x.
I would like a equals eight.
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00:12:47,000 --> 00:12:50,000
By the way, when you set these
two equal to each other,
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00:12:50,000 --> 00:12:52,000
they really have to be equal
everywhere.
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00:12:52,000 --> 00:12:55,000
You don't want to somehow solve
for x or anything like that.
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00:12:55,000 --> 00:12:59,000
You just want these
expressions, in terms of x and
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00:12:59,000 --> 00:13:02,000
y, to be the same quantities.
I mean you cannot say if x
146
00:13:02,000 --> 00:13:07,000
equals z they are always equal.
Yeah, that is true.
147
00:13:07,000 --> 00:13:13,000
But that is not what we are
asking.
148
00:13:13,000 --> 00:13:18,000
Now we come to the next logical
question.
149
00:13:18,000 --> 00:13:20,000
Let's say that we have passed
the test.
150
00:13:20,000 --> 00:13:23,000
We have put a equals eight in
here.
151
00:13:23,000 --> 00:13:26,000
Now it should be a gradient
field.
152
00:13:26,000 --> 00:13:30,000
The question is how do we find
the potential?
153
00:13:30,000 --> 00:13:36,000
That becomes eight from now on.
The question is how do we find
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00:13:36,000 --> 00:13:39,000
the function which has this as
gradient?
155
00:13:39,000 --> 00:13:43,000
One option is to try to guess.
Actually, quite often you will
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00:13:43,000 --> 00:13:47,000
succeed that way.
But that is not a valid method
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00:13:47,000 --> 00:13:50,000
on next week's test.
We are going to see two
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00:13:50,000 --> 00:13:55,000
different systematic methods.
And you should be using one of
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00:13:55,000 --> 00:14:00,000
these because guessing doesn't
always work.
160
00:14:00,000 --> 00:14:03,000
And, actually,
I can come up with examples
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00:14:03,000 --> 00:14:07,000
where if you try to guess you
will surely fail.
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00:14:07,000 --> 00:14:15,000
I can come up with trick ones,
but I don't want to put that on
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00:14:15,000 --> 00:14:24,000
the test.
The next stage is finding the
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00:14:24,000 --> 00:14:30,000
potential.
And let me just emphasize that
165
00:14:30,000 --> 00:14:36,000
we can only do that if step one
was successful.
166
00:14:36,000 --> 00:14:41,000
If we have a vector field that
cannot possibly be a gradient
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00:14:41,000 --> 00:14:45,000
then we shouldn't try to look
for a potential.
168
00:14:45,000 --> 00:14:52,000
It is kind of obvious but is
probably worth pointing out.
169
00:14:52,000 --> 00:15:00,000
There are two methods.
The first method that we will
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00:15:00,000 --> 00:15:16,000
see is computing line integrals.
Let's see how that works.
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00:15:16,000 --> 00:15:25,000
Let's say that I take some path
that starts at the origin.
172
00:15:25,000 --> 00:15:26,000
Or, actually,
anywhere you want,
173
00:15:26,000 --> 00:15:29,000
but let's take the origin.
That is my favorite point.
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00:15:29,000 --> 00:15:36,000
And let's go to a point with
coordinates (x1,
175
00:15:36,000 --> 00:15:40,000
y1).
And let's take my favorite
176
00:15:40,000 --> 00:15:45,000
curve and compute the line
integral of that field,
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00:15:45,000 --> 00:15:49,000
you know, the work done along
the curve.
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00:15:49,000 --> 00:15:55,000
Well, by the fundamental
theorem, that should be equal to
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00:15:55,000 --> 00:16:02,000
the value of the potential at
the end point minus the value at
180
00:16:02,000 --> 00:16:09,000
the origin.
That means I can actually write
181
00:16:09,000 --> 00:16:19,000
f of (x1, y1) equals -- -- that
line integral plus the value at
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00:16:19,000 --> 00:16:26,000
the origin.
And that is just a constant.
183
00:16:26,000 --> 00:16:27,000
We don't know what it is.
And, actually,
184
00:16:27,000 --> 00:16:30,000
we can choose what it is.
Because if you have a
185
00:16:30,000 --> 00:16:33,000
potential, say that you have
some potential function.
186
00:16:33,000 --> 00:16:34,000
And let's say that you add one
to it.
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00:16:34,000 --> 00:16:36,000
It is still a potential
function.
188
00:16:36,000 --> 00:16:38,000
Adding one doesn't change the
gradient.
189
00:16:38,000 --> 00:16:41,000
You can even add 18 or any
number that you want.
190
00:16:41,000 --> 00:16:44,000
This is just going to be an
integration constant.
191
00:16:44,000 --> 00:16:47,000
It is the same thing as,
in one variable calculus,
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00:16:47,000 --> 00:16:49,000
when you take the
anti-derivative of a function it
193
00:16:49,000 --> 00:16:52,000
is only defined up to adding the
constant.
194
00:16:52,000 --> 00:16:56,000
We have this integration
constant, but apart from that we
195
00:16:56,000 --> 00:16:59,000
know that we should be able to
get a potential from this.
196
00:16:59,000 --> 00:17:03,000
And this we can compute using
the definition of the line
197
00:17:03,000 --> 00:17:06,000
integral.
And we don't know what little f
198
00:17:06,000 --> 00:17:11,000
is, but we know what the vector
field is so we can compute that.
199
00:17:11,000 --> 00:17:14,000
Of course, to do the
calculation we probably don't
200
00:17:14,000 --> 00:17:18,000
want to use this kind of path.
I mean if that is your favorite
201
00:17:18,000 --> 00:17:21,000
path then that is fine,
but it is not very easy to
202
00:17:21,000 --> 00:17:24,000
compute the line integral along
this,
203
00:17:24,000 --> 00:17:28,000
especially since I didn't tell
you what the definition is.
204
00:17:28,000 --> 00:17:31,000
There are easier favorite paths
to have.
205
00:17:31,000 --> 00:17:33,000
For example,
you can go on a straight line
206
00:17:33,000 --> 00:17:37,000
from the origin to that point.
That would be slightly easier.
207
00:17:37,000 --> 00:17:40,000
But then there is one easier.
The easiest of all,
208
00:17:40,000 --> 00:17:47,000
probably, is to just go first
along the x-axis to (x1,0) and
209
00:17:47,000 --> 00:17:51,000
then go up parallel to the
y-axis.
210
00:17:51,000 --> 00:17:54,000
Why is that easy?
Well, that is because when we
211
00:17:54,000 --> 00:17:57,000
do the line integral it becomes
M dx N dy.
212
00:17:57,000 --> 00:18:05,000
And then, on each of these
pieces, one-half just goes away
213
00:18:05,000 --> 00:18:11,000
because x, y is constant.
Let's try to use that method in
214
00:18:11,000 --> 00:18:12,000
our example.
215
00:18:12,000 --> 00:18:45,000
216
00:18:45,000 --> 00:18:56,000
Let's say that I want to go
along this path from the origin,
217
00:18:56,000 --> 00:19:06,000
first along the x-axis to
(x1,0) and then vertically to
218
00:19:06,000 --> 00:19:14,000
(x1, y1).
And so I want to compute for
219
00:19:14,000 --> 00:19:21,000
the line integral along that
curve.
220
00:19:21,000 --> 00:19:24,000
Let's say I want to do it for
this vector field.
221
00:19:24,000 --> 00:19:33,000
I want to find the potential
for this vector field.
222
00:19:33,000 --> 00:19:37,000
Let me copy it because I will
have to erase at some point.
223
00:19:37,000 --> 00:19:50,000
4x squared plus 8xy and 3y
squared plus 4x squared.
224
00:19:50,000 --> 00:19:59,000
That will become the integral
of 4x squared plus 8 xy times dx
225
00:19:59,000 --> 00:20:05,000
plus 3y squared plus 4x squared
times dy.
226
00:20:05,000 --> 00:20:08,000
To evaluate on this broken
line, I will,
227
00:20:08,000 --> 00:20:13,000
of course, evaluate separately
on each of the two segments.
228
00:20:13,000 --> 00:20:20,000
I will start with this segment
that I will call c1 and then I
229
00:20:20,000 --> 00:20:25,000
will do this one that I will
call c2.
230
00:20:25,000 --> 00:20:30,000
On c1, how do I evaluate my
integral?
231
00:20:30,000 --> 00:20:38,000
Well, if I am on c1 then x
varies from zero to x1.
232
00:20:38,000 --> 00:20:40,000
Well, actually,
I don't know if x1 is positive
233
00:20:40,000 --> 00:20:41,000
or not so I shouldn't write
this.
234
00:20:41,000 --> 00:20:48,000
I really should say just x goes
from zero to x1.
235
00:20:48,000 --> 00:20:54,000
And what about y?
y is just 0.
236
00:20:54,000 --> 00:21:00,000
I will set y equal to zero and
also dy equal to zero.
237
00:21:00,000 --> 00:21:08,000
I get that the line integral on
c1 -- Well, a lot of stuff goes
238
00:21:08,000 --> 00:21:11,000
away.
The entire second term with dy
239
00:21:11,000 --> 00:21:15,000
goes away because dy is zero.
And, in the first term,
240
00:21:15,000 --> 00:21:18,000
8xy goes away because y is zero
as well.
241
00:21:18,000 --> 00:21:27,000
I just have an integral of 4x
squared dx from zero to x1.
242
00:21:27,000 --> 00:21:31,000
By the way, now you see why I
have been using an x1 and a y1
243
00:21:31,000 --> 00:21:33,000
for my point and not just x and
y.
244
00:21:33,000 --> 00:21:36,000
It is to avoid confusion.
I am using x and y as my
245
00:21:36,000 --> 00:21:41,000
integration variables and x1,
y1 as constants that are
246
00:21:41,000 --> 00:21:45,000
representing the end point of my
path.
247
00:21:45,000 --> 00:21:51,000
And so, if I integrate this,
I should get four-thirds x1
248
00:21:51,000 --> 00:21:54,000
cubed.
That is the first part.
249
00:21:54,000 --> 00:22:01,000
Next I need to do the second
segment.
250
00:22:01,000 --> 00:22:09,000
If I am on c2,
y goes from zero to y1.
251
00:22:09,000 --> 00:22:16,000
And what about x?
x is constant equal to x1 so dx
252
00:22:16,000 --> 00:22:22,000
becomes just zero.
It is a constant.
253
00:22:22,000 --> 00:22:30,000
If I take the line integral of
c2, F dot dr then I will get the
254
00:22:30,000 --> 00:22:37,000
integral from zero to y1.
The entire first term with dx
255
00:22:37,000 --> 00:22:47,000
goes away and then I have 3y
squared plus 4x1 squared times
256
00:22:47,000 --> 00:22:52,000
dy.
That integrates to y cubed plus
257
00:22:52,000 --> 00:23:01,000
4x1 squared y from zero to y1.
Or, if you prefer,
258
00:23:01,000 --> 00:23:11,000
that is y1 cubed plus 4x1
squared y1.
259
00:23:11,000 --> 00:23:15,000
Now that we have done both of
them we can just add them
260
00:23:15,000 --> 00:23:19,000
together, and that will give us
the formula for the potential.
261
00:23:19,000 --> 00:23:40,000
262
00:23:40,000 --> 00:23:50,000
F of x1 and y1 is four-thirds
x1 cubed plus y1 cubed plus 4x1
263
00:23:50,000 --> 00:23:57,000
squared y1 plus a constant.
That constant is just the
264
00:23:57,000 --> 00:24:03,000
integration constant that we had
from the beginning.
265
00:24:03,000 --> 00:24:05,000
Now you can drop the subscripts
if you prefer.
266
00:24:05,000 --> 00:24:14,000
You can just say f is
four-thirds x cubed plus y cubed
267
00:24:14,000 --> 00:24:20,000
plus 4x squared y plus constant.
And you can check.
268
00:24:20,000 --> 00:24:25,000
If you take the gradient of
this, you should get again this
269
00:24:25,000 --> 00:24:29,000
vector field over there.
Any questions about this method?
270
00:24:29,000 --> 00:24:33,000
Yes?
No.
271
00:24:33,000 --> 00:24:35,000
Well, it depends whether you
are just trying to find one
272
00:24:35,000 --> 00:24:38,000
potential or if you are trying
to find all the possible
273
00:24:38,000 --> 00:24:40,000
potentials.
If a problem just says find a
274
00:24:40,000 --> 00:24:43,000
potential then you don't have to
use the constant.
275
00:24:43,000 --> 00:24:47,000
This guy without the constant
is a valid potential.
276
00:24:47,000 --> 00:24:52,000
You just have others.
If your neighbor comes to you
277
00:24:52,000 --> 00:24:58,000
and say your answer must be
wrong because I got this plus
278
00:24:58,000 --> 00:25:01,000
18, well, both answers are
correct.
279
00:25:01,000 --> 00:25:05,000
By the way.
Instead of going first along
280
00:25:05,000 --> 00:25:08,000
the x-axis vertically,
you could do it the other way
281
00:25:08,000 --> 00:25:11,000
around.
Of course, start along the
282
00:25:11,000 --> 00:25:15,000
y-axis and then horizontally.
That is the same level of
283
00:25:15,000 --> 00:25:19,000
difficulty.
You just exchange roles of x
284
00:25:19,000 --> 00:25:21,000
and y.
In some cases,
285
00:25:21,000 --> 00:25:26,000
it is actually even making more
sense maybe to go radially,
286
00:25:26,000 --> 00:25:30,000
start out from the origin to
your end point.
287
00:25:30,000 --> 00:25:37,000
But usually this setting is
easier just because each of
288
00:25:37,000 --> 00:25:43,000
these two guys were very easy to
compute.
289
00:25:43,000 --> 00:25:46,000
But somehow maybe if you
suspect that polar coordinates
290
00:25:46,000 --> 00:25:49,000
will be involved somehow in the
answer then maybe it makes sense
291
00:25:49,000 --> 00:26:01,000
to choose different paths.
Maybe a straight line is better.
292
00:26:01,000 --> 00:26:13,000
Now we have another method to
look at which is using
293
00:26:13,000 --> 00:26:19,000
anti-derivatives.
The goal is the same,
294
00:26:19,000 --> 00:26:21,000
still to find the potential
function.
295
00:26:21,000 --> 00:26:26,000
And you see that finding the
potential is really the
296
00:26:26,000 --> 00:26:31,000
multivariable analog of finding
the anti-derivative in the one
297
00:26:31,000 --> 00:26:34,000
variable.
Here we did it basically by
298
00:26:34,000 --> 00:26:38,000
hand by computing the integral.
The other thing you could try
299
00:26:38,000 --> 00:26:39,000
to say is, wait,
I already know how to take
300
00:26:39,000 --> 00:26:42,000
anti-derivatives.
Let's use that instead of
301
00:26:42,000 --> 00:26:45,000
computing integrals.
And it works but you have to be
302
00:26:45,000 --> 00:26:51,000
careful about how you do it.
Let's see how that works.
303
00:26:51,000 --> 00:26:53,000
Let's still do it with the same
example.
304
00:26:53,000 --> 00:27:02,000
We want to solve the equations.
We want a function such that f
305
00:27:02,000 --> 00:27:13,000
sub x is 4x squared plus 8xy and
f sub y is 3y squared plus 4x
306
00:27:13,000 --> 00:27:16,000
squared.
Let's just look at one of these
307
00:27:16,000 --> 00:27:20,000
at a time.
If we look at this one,
308
00:27:20,000 --> 00:27:28,000
well, we know how to solve this
because it is just telling us we
309
00:27:28,000 --> 00:27:33,000
have to integrate this with
respect to x.
310
00:27:33,000 --> 00:27:38,000
Well, let's call them one and
two because I will have to refer
311
00:27:38,000 --> 00:27:43,000
to them again.
Let's start with equation one
312
00:27:43,000 --> 00:27:48,000
and lets integrate with respect
to x.
313
00:27:48,000 --> 00:27:51,000
Well, it tells us that f should
be,
314
00:27:51,000 --> 00:27:55,000
what do I get when I integrate
this with respect to x,
315
00:27:55,000 --> 00:28:02,000
four-thirds x cubed plus,
when I integrate 8xy,
316
00:28:02,000 --> 00:28:08,000
y is just a constant,
so I will get 4x squared y.
317
00:28:08,000 --> 00:28:11,000
And that is not quite the end
to it because there is an
318
00:28:11,000 --> 00:28:15,000
integration constant.
And here, when I say there is
319
00:28:15,000 --> 00:28:18,000
an integration constant,
it just means the extra term
320
00:28:18,000 --> 00:28:21,000
does not depend on x.
That is what it means to be a
321
00:28:21,000 --> 00:28:25,000
constant in this setting.
But maybe my constant still
322
00:28:25,000 --> 00:28:28,000
depends on y so it is not
actually a true constant.
323
00:28:28,000 --> 00:28:30,000
A constant that depends on y is
not really a constant.
324
00:28:30,000 --> 00:28:38,000
It is actually a function of y.
The good news that we have is
325
00:28:38,000 --> 00:28:40,000
that this function normally
depends on x.
326
00:28:40,000 --> 00:28:46,000
We have made some progress.
We have part of the answer and
327
00:28:46,000 --> 00:28:53,000
we have simplified the problem.
If we have anything that looks
328
00:28:53,000 --> 00:28:56,000
like this, it will satisfy the
first condition.
329
00:28:56,000 --> 00:28:59,000
Now we need to look at the
second condition.
330
00:28:59,000 --> 00:29:12,000
We want f sub y to be that.
But we know what f is,
331
00:29:12,000 --> 00:29:15,000
so let's compute f sub y from
this.
332
00:29:15,000 --> 00:29:20,000
From this I get f sub y.
What do I get if I
333
00:29:20,000 --> 00:29:22,000
differentiate this with respect
to y?
334
00:29:22,000 --> 00:29:37,000
Well, I get zero plus 4x
squared plus the derivative of
335
00:29:37,000 --> 00:29:46,000
g.
I would like to match this with
336
00:29:46,000 --> 00:29:51,000
what I had.
If I match this with equation
337
00:29:51,000 --> 00:29:55,000
two then that will tell me what
the derivative of g should be.
338
00:29:55,000 --> 00:30:15,000
339
00:30:15,000 --> 00:30:20,000
If we compare the two things
there, we get 4x squared plus g
340
00:30:20,000 --> 00:30:26,000
prime of y should be equal to 3y
squared by 4x squared.
341
00:30:26,000 --> 00:30:31,000
And, of course,
the 4x squares go away.
342
00:30:31,000 --> 00:30:35,000
That tells you g prime is 3y
squared.
343
00:30:35,000 --> 00:30:42,000
And that integrates to y cubed
plus constant.
344
00:30:42,000 --> 00:30:46,000
Now, this time the constant is
a true constant because g did
345
00:30:46,000 --> 00:30:48,000
not depend on anything other
than y.
346
00:30:48,000 --> 00:30:54,000
And the constant does not
depend on y so it is a real
347
00:30:54,000 --> 00:30:58,000
constant now.
Now we just plug this back into
348
00:30:58,000 --> 00:31:05,000
this guy.
Let's call him star.
349
00:31:05,000 --> 00:31:13,000
If we plug this into star,
we get f equals four-thirds x
350
00:31:13,000 --> 00:31:21,000
cubed plus 4x squared y plus y
cubed plus constant.
351
00:31:21,000 --> 00:31:30,000
I mean, of course,
again, now this constant is
352
00:31:30,000 --> 00:31:33,000
optional.
The advantage of this method is
353
00:31:33,000 --> 00:31:35,000
you don't have to write any
integrals.
354
00:31:35,000 --> 00:31:40,000
The small drawback is you have
to follow this procedure
355
00:31:40,000 --> 00:31:45,000
carefully.
By the way, one common pitfall
356
00:31:45,000 --> 00:31:48,000
that is tempting.
After you have done this,
357
00:31:48,000 --> 00:31:51,000
what is very tempting is to
just say, well,
358
00:31:51,000 --> 00:31:53,000
let's do the same with this
guy.
359
00:31:53,000 --> 00:31:55,000
Let's integrate this with
respect to y.
360
00:31:55,000 --> 00:31:58,000
You will get another expression
for f up to a constant that
361
00:31:58,000 --> 00:32:01,000
depends on x.
And then let's match them.
362
00:32:01,000 --> 00:32:04,000
Well, the difficulty is
matching is actually quite
363
00:32:04,000 --> 00:32:09,000
tricky because you don't know in
advance whether they will be the
364
00:32:09,000 --> 00:32:13,000
same expression.
It could be you could say let's
365
00:32:13,000 --> 00:32:16,000
just take the terms that are
here and missing there and
366
00:32:16,000 --> 00:32:20,000
combine the terms,
you know, take all the terms
367
00:32:20,000 --> 00:32:23,000
that appear in either one.
That is actually not a good way
368
00:32:23,000 --> 00:32:25,000
to do it,
because if I put sufficiently
369
00:32:25,000 --> 00:32:28,000
complicated trig functions in
there then you might not be able
370
00:32:28,000 --> 00:32:30,000
to see that two terms are the
same.
371
00:32:30,000 --> 00:32:34,000
Take an easy one.
Let's say that here I have one
372
00:32:34,000 --> 00:32:40,000
plus tangent square and here I
have a secan square then you
373
00:32:40,000 --> 00:32:46,000
might not actually notice that
there is a difference.
374
00:32:46,000 --> 00:32:50,000
But there is no difference.
Whatever.
375
00:32:50,000 --> 00:32:54,000
Anyway, I am saying do it this
way, don't do it any other way
376
00:32:54,000 --> 00:32:57,000
because there is a risk of
making a mistake otherwise.
377
00:32:57,000 --> 00:33:00,000
I mean, on the other hand,
you could start with
378
00:33:00,000 --> 00:33:03,000
integrating with respect to y
and then differentiate and match
379
00:33:03,000 --> 00:33:06,000
with respect to x.
But what I am saying is just
380
00:33:06,000 --> 00:33:09,000
take one of them,
integrate,
381
00:33:09,000 --> 00:33:12,000
get an answer that involves a
function of the other variable,
382
00:33:12,000 --> 00:33:18,000
then differentiate that answer
and compare and see what you
383
00:33:18,000 --> 00:33:21,000
get.
By the way, here,
384
00:33:21,000 --> 00:33:27,000
of course, after we simplified
there were only y's here.
385
00:33:27,000 --> 00:33:29,000
There were no x's.
And that is kind of good news.
386
00:33:29,000 --> 00:33:33,000
I mean, if you had had an x
here in this expression that
387
00:33:33,000 --> 00:33:36,000
would have told you that
something is going wrong.
388
00:33:36,000 --> 00:33:39,000
g is a function of y only.
If you get an x here,
389
00:33:39,000 --> 00:33:42,000
maybe you want to go back and
check whether it is really a
390
00:33:42,000 --> 00:33:47,000
gradient field.
Yes?
391
00:33:47,000 --> 00:33:49,000
Yes, this will work with
functions of more than two
392
00:33:49,000 --> 00:33:51,000
variables.
Both methods work with more
393
00:33:51,000 --> 00:33:53,000
than two variables.
We are going to see it in the
394
00:33:53,000 --> 00:33:56,000
case where more than two means
three.
395
00:33:56,000 --> 00:34:00,000
We are going to see that in two
or three weeks from now.
396
00:34:00,000 --> 00:34:04,000
I mean, basically starting at
the end of next week,
397
00:34:04,000 --> 00:34:08,000
we are going to do triple
integrals, line integrals in
398
00:34:08,000 --> 00:34:10,000
space and so on.
The format is first we do
399
00:34:10,000 --> 00:34:13,000
everything in two variables.
Then we will do three variables.
400
00:34:13,000 --> 00:34:20,000
And then what happens with more
than three will be left to your
401
00:34:20,000 --> 00:34:25,000
imagination.
Any other questions about
402
00:34:25,000 --> 00:34:29,000
either of these methods?
A quick poll.
403
00:34:29,000 --> 00:34:34,000
Who prefers the first method?
Who prefers the second method?
404
00:34:34,000 --> 00:34:41,000
Wow.
OK.
405
00:34:41,000 --> 00:34:45,000
Anyway, you will get to use
whichever one you want.
406
00:34:45,000 --> 00:34:47,000
And I would agree with you,
but the second method is
407
00:34:47,000 --> 00:34:50,000
slightly more effective in that
you are writing less stuff.
408
00:34:50,000 --> 00:34:54,000
You don't have to set up all
these line integrals.
409
00:34:54,000 --> 00:35:03,000
On the other hand,
it does require a little bit
410
00:35:03,000 --> 00:35:19,000
more attention.
Let's move on a bit.
411
00:35:19,000 --> 00:35:24,000
Let me start by actually doing
a small recap.
412
00:35:24,000 --> 00:35:38,000
We said we have various notions.
One is to say that the vector
413
00:35:38,000 --> 00:35:48,000
field is a gradient in a certain
region of a plane.
414
00:35:48,000 --> 00:35:54,000
And we have another notion
which is being conservative.
415
00:35:54,000 --> 00:36:06,000
It says that the line integral
is zero along any closed curve.
416
00:36:06,000 --> 00:36:10,000
Actually, let me introduce a
new piece of notation.
417
00:36:10,000 --> 00:36:14,000
To remind ourselves that we are
doing it along a closed curve,
418
00:36:14,000 --> 00:36:18,000
very often we put just a circle
for the integral to tell us this
419
00:36:18,000 --> 00:36:21,000
is a curve that closes on
itself.
420
00:36:21,000 --> 00:36:25,000
It ends where it started.
I mean it doesn't change
421
00:36:25,000 --> 00:36:28,000
anything concerning the
definition or how you compute it
422
00:36:28,000 --> 00:36:31,000
or anything.
It just reminds you that you
423
00:36:31,000 --> 00:36:34,000
are doing it on a closed curve.
It is actually useful for
424
00:36:34,000 --> 00:36:37,000
various physical applications.
And also, when you state
425
00:36:37,000 --> 00:36:41,000
theorems in that way,
it reminds you,oh..
426
00:36:41,000 --> 00:36:45,000
I need to be on a closed curve
to do it.
427
00:36:45,000 --> 00:36:51,000
And so we have said these two
things are equivalent.
428
00:36:51,000 --> 00:37:00,000
Now we have a third thing which
is N sub x equals M sub y at
429
00:37:00,000 --> 00:37:03,000
every point.
Just to summarize the
430
00:37:03,000 --> 00:37:06,000
discussion.
We have said if we have a
431
00:37:06,000 --> 00:37:09,000
gradient field then we have
this.
432
00:37:09,000 --> 00:37:18,000
And the converse is true in
suitable regions.
433
00:37:18,000 --> 00:37:32,000
We have a converse if F is
defined in the entire plane.
434
00:37:32,000 --> 00:37:43,000
Or, as we will see soon,
in a simply connected region.
435
00:37:43,000 --> 00:37:45,000
I guess some of you cannot see
what I am writing here,
436
00:37:45,000 --> 00:37:48,000
but it doesn't matter because
you are not officially supposed
437
00:37:48,000 --> 00:37:53,000
to know it yet.
That will be next week.
438
00:37:53,000 --> 00:37:57,000
Anyway,
I said the fact that Nx equals
439
00:37:57,000 --> 00:38:01,000
My implies that we have a
gradient field and is only if a
440
00:38:01,000 --> 00:38:06,000
vector field is defined in the
entire plane or in a region that
441
00:38:06,000 --> 00:38:12,000
is called simply connected.
And more about that later.
442
00:38:12,000 --> 00:38:17,000
Now let me just introduce a
quantity that probably a lot of
443
00:38:17,000 --> 00:38:22,000
you have heard about in physics
that measures precisely fairly
444
00:38:22,000 --> 00:38:26,000
ought to be conservative.
That is called the curl of a
445
00:38:26,000 --> 00:38:27,000
vector field.
446
00:38:27,000 --> 00:39:06,000
447
00:39:06,000 --> 00:39:19,000
For the definition we say that
the curl of F is the quantity N
448
00:39:19,000 --> 00:39:27,000
sub x - M sub y.
It is just replicating the
449
00:39:27,000 --> 00:39:35,000
information we had but in a way
that is a single quantity.
450
00:39:35,000 --> 00:39:43,000
In this new language,
the conditions that we had over
451
00:39:43,000 --> 00:39:50,000
there, this condition says curl
F equals zero.
452
00:39:50,000 --> 00:39:56,000
That is the new version of Nx
equals My.
453
00:39:56,000 --> 00:40:06,000
It measures failure of a vector
field to be conservative.
454
00:40:06,000 --> 00:40:21,000
The test for conservativeness
is that the curl of F should be
455
00:40:21,000 --> 00:40:25,000
zero.
I should probably tell you a
456
00:40:25,000 --> 00:40:29,000
little bit about what the curl
is, what it measures and what it
457
00:40:29,000 --> 00:40:34,000
does because that is something
that is probably useful.
458
00:40:34,000 --> 00:40:37,000
It is a very strange quantity
if you put it in that form.
459
00:40:37,000 --> 00:40:42,000
Yes?
I think it is the same as the
460
00:40:42,000 --> 00:40:45,000
physics one, but I haven't
checked the physics textbook.
461
00:40:45,000 --> 00:40:49,000
I believe it is the same.
Yes, I think it is the same as
462
00:40:49,000 --> 00:40:53,000
the physics one.
It is not the opposite this
463
00:40:53,000 --> 00:40:55,000
time.
Of course, in physics maybe you
464
00:40:55,000 --> 00:40:59,000
have seen curl in space.
We are going to see curl in
465
00:40:59,000 --> 00:41:07,000
space in two or three weeks.
Yes?
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00:41:07,000 --> 00:41:11,000
Yes. Well, you can also use it.
If you fail this test then you
467
00:41:11,000 --> 00:41:14,000
know for sure that you are not
gradient field so you might as
468
00:41:14,000 --> 00:41:18,000
well do that.
If you satisfy the test but you
469
00:41:18,000 --> 00:41:24,000
are not defined everywhere then
there is still a bit of
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00:41:24,000 --> 00:41:29,000
ambiguity and you don't know for
sure.
471
00:41:29,000 --> 00:41:40,000
OK.
Let's try to see a little bit
472
00:41:40,000 --> 00:41:48,000
what the curl measures.
Just to give you some
473
00:41:48,000 --> 00:41:55,000
intuition, let's first think
about a velocity field.
474
00:41:55,000 --> 00:42:10,000
The curl measures the rotation
component of a motion.
475
00:42:10,000 --> 00:42:13,000
If you want a fancy word,
it measures the vorticity of a
476
00:42:13,000 --> 00:42:16,000
motion.
It tells you how much twisting
477
00:42:16,000 --> 00:42:19,000
is taking place at a given
point.
478
00:42:19,000 --> 00:42:24,000
For example,
if I take a constant vector
479
00:42:24,000 --> 00:42:32,000
field where my fluid is just all
moving in the same direction
480
00:42:32,000 --> 00:42:37,000
where this is just constants
then,
481
00:42:37,000 --> 00:42:41,000
of course, the curl is zero.
Because if you take the
482
00:42:41,000 --> 00:42:43,000
partials you get zero.
And, indeed,
483
00:42:43,000 --> 00:42:46,000
that is not what you would call
swirling.
484
00:42:46,000 --> 00:42:58,000
There is no vortex in here.
Let's do another one where this
485
00:42:58,000 --> 00:43:02,000
is still nothing going on.
Let's say that I take the
486
00:43:02,000 --> 00:43:06,000
radial vector field where
everything just flows away from
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00:43:06,000 --> 00:43:11,000
the origin.
That is f equals x, y.
488
00:43:11,000 --> 00:43:16,000
Well, if I take the curl,
I have to take partial over
489
00:43:16,000 --> 00:43:18,000
partial x of the second
component,
490
00:43:18,000 --> 00:43:21,000
which is y,
minus partial over partial y of
491
00:43:21,000 --> 00:43:22,000
the first component,
which is x.
492
00:43:22,000 --> 00:43:25,000
I will get zero.
And, indeed,
493
00:43:25,000 --> 00:43:29,000
if you think about what is
going on here,
494
00:43:29,000 --> 00:43:32,000
there is no rotation involved.
On the other hand,
495
00:43:32,000 --> 00:43:45,000
if you consider our favorite
rotation vector field -- --
496
00:43:45,000 --> 00:44:00,000
negative y and x then this curl
is going to be N sub x minus M
497
00:44:00,000 --> 00:44:08,000
sub y,
one plus one equals two.
498
00:44:08,000 --> 00:44:13,000
That corresponds to the fact
that we are rotating.
499
00:44:13,000 --> 00:44:16,000
Actually, we are rotating at
unit angular speed.
500
00:44:16,000 --> 00:44:20,000
The curl actually measures
twice the angular speed of a
501
00:44:20,000 --> 00:44:24,000
rotation part of a motion at any
given point.
502
00:44:24,000 --> 00:44:26,000
Now, if you have an actual
motion,
503
00:44:26,000 --> 00:44:30,000
a more complicated field than
these then no matter where you
504
00:44:30,000 --> 00:44:34,000
are you can think of a motion as
a combination of translation
505
00:44:34,000 --> 00:44:37,000
effects,
maybe dilation effects,
506
00:44:37,000 --> 00:44:43,000
maybe rotation effects,
possibly other things like that.
507
00:44:43,000 --> 00:44:48,000
And what a curl will measure is
how intense the rotation effect
508
00:44:48,000 --> 00:44:52,000
is at that particular point.
I am not going to try to make a
509
00:44:52,000 --> 00:44:55,000
much more precise statement.
A precise statement is what a
510
00:44:55,000 --> 00:44:58,000
curl measures is really this
quantity up there.
511
00:44:58,000 --> 00:45:01,000
But the intuition you should
have is it measures how much
512
00:45:01,000 --> 00:45:04,000
rotation is taking place at any
given point.
513
00:45:04,000 --> 00:45:06,000
And, of course,
in a complicated motion you
514
00:45:06,000 --> 00:45:09,000
might have more rotation at some
point than at some others,
515
00:45:09,000 --> 00:45:12,000
which is why the curl will
depend on x and y.
516
00:45:12,000 --> 00:45:20,000
It is not just a constant
because how much you rotate
517
00:45:20,000 --> 00:45:26,000
depends on where you are.
If you are looking at actual
518
00:45:26,000 --> 00:45:30,000
wind velocities in weather
prediction then the regions with
519
00:45:30,000 --> 00:45:33,000
high curl tend to be hurricanes
or tornadoes or things like
520
00:45:33,000 --> 00:45:37,000
that.
They are not very pleasant
521
00:45:37,000 --> 00:45:40,000
things.
And the sign of a curl tells
522
00:45:40,000 --> 00:45:43,000
you whether you are going
clockwise or counterclockwise.
523
00:45:43,000 --> 00:46:09,000
524
00:46:09,000 --> 00:46:27,000
Curl measures twice the angular
velocity of the rotation
525
00:46:27,000 --> 00:46:41,000
component of a velocity field.
Now, what about a force field?
526
00:46:41,000 --> 00:46:44,000
Because, after all,
how we got to this was coming
527
00:46:44,000 --> 00:46:47,000
from and trying to understand
forces and the work they do.
528
00:46:47,000 --> 00:46:50,000
So I should tell you what it
means for a force.
529
00:46:50,000 --> 00:47:10,000
Well, the curl of a force field
-- -- measures the torque
530
00:47:10,000 --> 00:47:29,000
exerted on a test object that
you put at any point.
531
00:47:29,000 --> 00:47:36,000
Remember, torque is the
rotational analog of the force.
532
00:47:36,000 --> 00:47:41,000
We had this analogy about
velocity versus angular velocity
533
00:47:41,000 --> 00:47:45,000
and mass versus moment of
inertia.
534
00:47:45,000 --> 00:47:49,000
And then, in that analogy,
force divided by the mass is
535
00:47:49,000 --> 00:47:53,000
what will cause acceleration,
which is the derivative of
536
00:47:53,000 --> 00:47:56,000
velocity.
Torque divided by moment of
537
00:47:56,000 --> 00:47:59,000
inertia is what will cause the
angular acceleration,
538
00:47:59,000 --> 00:48:02,000
namely the derivative of
angular velocity.
539
00:48:02,000 --> 00:48:04,000
Maybe I should write that down.
540
00:48:04,000 --> 00:48:18,000
541
00:48:18,000 --> 00:48:31,000
Torque divided by moment of
inertia is going to be d over dt
542
00:48:31,000 --> 00:48:38,000
of angular velocity.
I leave it up to your physics
543
00:48:38,000 --> 00:48:41,000
teachers to decide what letters
to use for all these things.
544
00:48:41,000 --> 00:48:49,000
That is the analog of force
divided by mass equals
545
00:48:49,000 --> 00:48:56,000
acceleration,
which is d over dt of velocity.
546
00:48:56,000 --> 00:49:03,000
And so now you see if the curl
of a velocity field measure the
547
00:49:03,000 --> 00:49:07,000
angular velocity of its rotation
then,
548
00:49:07,000 --> 00:49:13,000
by this analogy,
the curl of a force field
549
00:49:13,000 --> 00:49:24,000
should measure the torque it
exerts on a mass per unit moment
550
00:49:24,000 --> 00:49:28,000
of inertia.
Concretely, if you imagine that
551
00:49:28,000 --> 00:49:29,000
you are putting something in
there,
552
00:49:29,000 --> 00:49:32,000
you know, if you are in a
velocity field the curl will
553
00:49:32,000 --> 00:49:35,000
tell you how fast your guy is
spinning at a given time.
554
00:49:35,000 --> 00:49:37,000
If you put something that
floats, for example,
555
00:49:37,000 --> 00:49:40,000
in your fluid,
something very light then it is
556
00:49:40,000 --> 00:49:44,000
going to start spinning.
And the curl of a velocity
557
00:49:44,000 --> 00:49:48,000
field tells you how fast it is
spinning at any given time up to
558
00:49:48,000 --> 00:49:51,000
a factor of two.
And the curl of a force field
559
00:49:51,000 --> 00:49:55,000
tells you how quickly the
angular velocity is going to
560
00:49:55,000 --> 00:50:01,000
increase or decrease.
OK.
561
00:50:01,000 --> 00:50:04,000
Well, next time we are going to
see Green's theorem which is
562
00:50:04,000 --> 00:50:08,000
actually going to tell us a lot
more about curl and failure of
563
00:50:08,000 --> 00:50:11,000
conservativeness.
564
00:50:11,000 --> 00:50:16,000