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We were looking at vector
fields last time.
00:00:34.000 --> 00:00:45.000
Last time we saw that if a
vector field happens to be a
00:00:45.000 --> 00:00:56.000
gradient field -- -- then the
line integral can be computed
00:00:56.000 --> 00:01:08.000
actually by taking the change in
value of the potential between
00:01:08.000 --> 00:01:19.000
the end point and the starting
point of the curve.
00:01:19.000 --> 00:01:24.000
If we have a curve c,
from a point p0 to a point p1
00:01:24.000 --> 00:01:29.000
then the line integral for work
depends only on the end points
00:01:29.000 --> 00:01:32.000
and not on the actual path we
chose.
00:01:32.000 --> 00:01:43.000
We say that the line integral
is path independent.
00:01:43.000 --> 00:01:49.000
And we also said that the
vector field is conservative
00:01:49.000 --> 00:01:55.000
because of conservation of
energy which tells you if you
00:01:55.000 --> 00:02:02.000
start at a point and you come
back to the same point then you
00:02:02.000 --> 00:02:07.000
haven't gotten any work out of
that force.
00:02:07.000 --> 00:02:15.000
If we have a closed curve then
the line integral for work is
00:02:15.000 --> 00:02:18.000
just zero.
And, basically,
00:02:18.000 --> 00:02:23.000
we say that these properties
are equivalent being a gradient
00:02:23.000 --> 00:02:28.000
field or being path independent
or being conservative.
00:02:28.000 --> 00:02:31.000
And what I promised to you is
that today we would see a
00:02:31.000 --> 00:02:35.000
criterion to decide whether a
vector field is a gradient field
00:02:35.000 --> 00:02:38.000
or not and how to find the
potential function if it is a
00:02:38.000 --> 00:02:47.000
gradient field.
So, that is the topic for today.
00:02:47.000 --> 00:03:00.000
The question is testing whether
a given vector field,
00:03:00.000 --> 00:03:14.000
let's say M and N compliments,
is a gradient field.
00:03:14.000 --> 00:03:16.000
For that, well,
let's start with an
00:03:16.000 --> 00:03:26.000
observation.
Say that it is a gradient field.
00:03:26.000 --> 00:03:31.000
That means that the first
component of a field is just the
00:03:31.000 --> 00:03:35.000
partial of f with respect to
some variable x and the second
00:03:35.000 --> 00:03:40.000
component is the partial of f
with respect to y.
00:03:40.000 --> 00:03:43.000
Now we have seen an interesting
property of the second partial
00:03:43.000 --> 00:03:46.000
derivatives of the function,
which is if you take the
00:03:46.000 --> 00:03:49.000
partial derivative first with
respect to x,
00:03:49.000 --> 00:03:52.000
then with respect to y,
or first with respect to y,
00:03:52.000 --> 00:03:58.000
then with respect to x you get
the same thing.
00:03:58.000 --> 00:04:07.000
We know f sub xy equals f sub
yx, and that means M sub y
00:04:07.000 --> 00:04:12.000
equals N sub x.
If you have a gradient field
00:04:12.000 --> 00:04:14.000
then it should have this
property.
00:04:14.000 --> 00:04:17.000
You take the y component,
take the derivative with
00:04:17.000 --> 00:04:19.000
respect to x,
take the x component,
00:04:19.000 --> 00:04:20.000
differentiate with respect to
y,
00:04:20.000 --> 00:04:31.000
you should get the same answer.
And that is important to know.
00:04:31.000 --> 00:04:37.000
So, I am going to put that in a
box.
00:04:37.000 --> 00:04:43.000
It is a broken box.
The claim that I want to make
00:04:43.000 --> 00:04:45.000
is that there is a converse of
sorts.
00:04:45.000 --> 00:04:47.000
This is actually basically all
we need to check.
00:05:06.000 --> 00:05:18.000
Conversely, if,
and I am going to put here a
00:05:18.000 --> 00:05:33.000
condition, My equals Nx,
then F is a gradient field.
00:05:33.000 --> 00:05:35.000
What is the condition that I
need to put here?
00:05:35.000 --> 00:05:37.000
Well, we will see a more
precise version of that next
00:05:37.000 --> 00:05:44.000
week.
But for now let's just say if
00:05:44.000 --> 00:05:59.000
our vector field is defined and
differentiable everywhere in the
00:05:59.000 --> 00:06:01.000
plane.
We need, actually,
00:06:01.000 --> 00:06:04.000
a vector field that is
well-defined everywhere.
00:06:04.000 --> 00:06:07.000
You are not allowed to have
somehow places where it is not
00:06:07.000 --> 00:06:09.000
well-defined.
Otherwise, actually,
00:06:09.000 --> 00:06:13.000
you have a counter example on
your problem set this week.
00:06:13.000 --> 00:06:16.000
If you look at the last problem
on the problem set this week,
00:06:16.000 --> 00:06:20.000
it gives you a vector field
that satisfies this condition
00:06:20.000 --> 00:06:22.000
everywhere where it is defined.
But, actually,
00:06:22.000 --> 00:06:24.000
there is a point where it is
not defined.
00:06:24.000 --> 00:06:28.000
And that causes it,
actually, to somehow -- I mean
00:06:28.000 --> 00:06:33.000
everything that I am going to
say today breaks down for that
00:06:33.000 --> 00:06:36.000
example because of that.
I mean, we will shed more light
00:06:36.000 --> 00:06:39.000
on this a bit later with the
notion of simply connected
00:06:39.000 --> 00:06:42.000
regions and so on.
But for now let's just say if
00:06:42.000 --> 00:06:47.000
it is defined everywhere and it
satisfies this criterion then it
00:06:47.000 --> 00:06:52.000
is a gradient field.
If you ignore the technical
00:06:52.000 --> 00:06:57.000
condition, being a gradient
field means essentially the same
00:06:57.000 --> 00:07:11.000
thing as having this property.
That is what we need to check.
00:07:11.000 --> 00:07:20.000
Let's look at an example.
Well, one vector field that we
00:07:20.000 --> 00:07:24.000
have been looking at a lot was -
yi xj.
00:07:24.000 --> 00:07:30.000
Remember that was the vector
field that looked like a
00:07:30.000 --> 00:07:35.000
rotation at the unit speed.
I think last time we already
00:07:35.000 --> 00:07:39.000
decided that this guy should not
be allowed to be a gradient
00:07:39.000 --> 00:07:42.000
field and should not be
conservative because if we
00:07:42.000 --> 00:07:45.000
integrate on the unit circle
then we would get a positive
00:07:45.000 --> 00:07:49.000
answer.
But let's check that indeed it
00:07:49.000 --> 00:07:55.000
fails our test.
Well, let's call this M and
00:07:55.000 --> 00:08:01.000
let's call this guy N.
If you look at partial M,
00:08:01.000 --> 00:08:07.000
partial y, that is going to be
a negative one.
00:08:07.000 --> 00:08:11.000
If you take partial N,
partial x, that is going to be
00:08:11.000 --> 00:08:12.000
one.
These are not the same.
00:08:12.000 --> 00:08:17.000
So, indeed, this is not a
gradient field.
00:08:32.000 --> 00:08:53.000
Any questions about that?
Yes?
00:08:53.000 --> 00:08:58.000
Your question is if I have the
property M sub y equals N sub x
00:08:58.000 --> 00:09:03.000
only in a certain part of a
plane for some values of x and
00:09:03.000 --> 00:09:06.000
y,
can I conclude these things?
00:09:06.000 --> 00:09:09.000
And it is a gradient field in
that part of the plane and
00:09:09.000 --> 00:09:13.000
conservative and so on.
The answer for now is,
00:09:13.000 --> 00:09:17.000
in general, no.
And when we spend a bit more
00:09:17.000 --> 00:09:20.000
time on it, actually,
maybe I should move that up.
00:09:20.000 --> 00:09:24.000
Maybe we will talk about it
later this week instead of when
00:09:24.000 --> 00:09:28.000
I had planned.
There is a notion what it means
00:09:28.000 --> 00:09:30.000
for a region to be without
holes.
00:09:30.000 --> 00:09:34.000
Basically, if you have that
kind of property in a region
00:09:34.000 --> 00:09:38.000
that doesn't have any holes
inside it then things will work.
00:09:38.000 --> 00:09:42.000
The problem comes from a vector
field satisfying this criterion
00:09:42.000 --> 00:09:44.000
in a region but it has a hole in
it.
00:09:44.000 --> 00:09:47.000
Because what you don't know is
whether your potential is
00:09:47.000 --> 00:09:51.000
actually well-defined and takes
the same value when you move all
00:09:51.000 --> 00:09:53.000
around the hole.
It might come back to take a
00:09:53.000 --> 00:09:56.000
different value.
If you look carefully and think
00:09:56.000 --> 00:10:00.000
hard about the example in the
problem sets that is exactly
00:10:00.000 --> 00:10:04.000
what happens there.
Again, I will say more about
00:10:04.000 --> 00:10:08.000
that later.
For now we basically need our
00:10:08.000 --> 00:10:11.000
function to be,
I mean,
00:10:11.000 --> 00:10:14.000
I should still say if you have
this property for a vector field
00:10:14.000 --> 00:10:16.000
that is not quite defined
everywhere,
00:10:16.000 --> 00:10:17.000
you are more than welcome,
you know,
00:10:17.000 --> 00:10:20.000
you should probably still try
to look for a potential using
00:10:20.000 --> 00:10:23.000
methods that we will see.
But something might go wrong
00:10:23.000 --> 00:10:30.000
later.
You might end up with a
00:10:30.000 --> 00:10:39.000
potential that is not
well-defined.
00:10:39.000 --> 00:10:53.000
Let's do another example.
Let's say that I give you this
00:10:53.000 --> 00:11:03.000
vector field.
And this a here is a number.
00:11:03.000 --> 00:11:08.000
The question is for which value
of a is this going to be
00:11:08.000 --> 00:11:13.000
possibly a gradient?
If you have your flashcards
00:11:13.000 --> 00:11:17.000
then that is a good time to use
them to vote,
00:11:17.000 --> 00:11:23.000
assuming that the number is
small enough to be made with.
00:11:23.000 --> 00:11:27.000
Let's try to think about it.
We want to call this guy M.
00:11:27.000 --> 00:11:35.000
We want to call that guy N.
And we want to test M sub y
00:11:35.000 --> 00:11:42.000
versus N sub x.
I don't see anyone.
00:11:42.000 --> 00:11:46.000
I see people doing it with
their hands, and that works very
00:11:46.000 --> 00:11:48.000
well.
OK.
00:11:48.000 --> 00:12:04.000
The question is for which value
of a is this a gradient?
00:12:04.000 --> 00:12:10.000
I see various people with the
correct answer.
00:12:10.000 --> 00:12:15.000
OK.
That a strange answer.
00:12:15.000 --> 00:12:20.000
That is a good answer.
OK.
00:12:20.000 --> 00:12:28.000
The vote seems to be for a
equals eight.
00:12:28.000 --> 00:12:35.000
Let's see.
What if I take M sub y?
00:12:35.000 --> 00:12:41.000
That is going to be just ax.
And N sub x?
00:12:41.000 --> 00:12:47.000
That is 8x.
I would like a equals eight.
00:12:47.000 --> 00:12:50.000
By the way, when you set these
two equal to each other,
00:12:50.000 --> 00:12:52.000
they really have to be equal
everywhere.
00:12:52.000 --> 00:12:55.000
You don't want to somehow solve
for x or anything like that.
00:12:55.000 --> 00:12:59.000
You just want these
expressions, in terms of x and
00:12:59.000 --> 00:13:02.000
y, to be the same quantities.
I mean you cannot say if x
00:13:02.000 --> 00:13:07.000
equals z they are always equal.
Yeah, that is true.
00:13:07.000 --> 00:13:13.000
But that is not what we are
asking.
00:13:13.000 --> 00:13:18.000
Now we come to the next logical
question.
00:13:18.000 --> 00:13:20.000
Let's say that we have passed
the test.
00:13:20.000 --> 00:13:23.000
We have put a equals eight in
here.
00:13:23.000 --> 00:13:26.000
Now it should be a gradient
field.
00:13:26.000 --> 00:13:30.000
The question is how do we find
the potential?
00:13:30.000 --> 00:13:36.000
That becomes eight from now on.
The question is how do we find
00:13:36.000 --> 00:13:39.000
the function which has this as
gradient?
00:13:39.000 --> 00:13:43.000
One option is to try to guess.
Actually, quite often you will
00:13:43.000 --> 00:13:47.000
succeed that way.
But that is not a valid method
00:13:47.000 --> 00:13:50.000
on next week's test.
We are going to see two
00:13:50.000 --> 00:13:55.000
different systematic methods.
And you should be using one of
00:13:55.000 --> 00:14:00.000
these because guessing doesn't
always work.
00:14:00.000 --> 00:14:03.000
And, actually,
I can come up with examples
00:14:03.000 --> 00:14:07.000
where if you try to guess you
will surely fail.
00:14:07.000 --> 00:14:15.000
I can come up with trick ones,
but I don't want to put that on
00:14:15.000 --> 00:14:24.000
the test.
The next stage is finding the
00:14:24.000 --> 00:14:30.000
potential.
And let me just emphasize that
00:14:30.000 --> 00:14:36.000
we can only do that if step one
was successful.
00:14:36.000 --> 00:14:41.000
If we have a vector field that
cannot possibly be a gradient
00:14:41.000 --> 00:14:45.000
then we shouldn't try to look
for a potential.
00:14:45.000 --> 00:14:52.000
It is kind of obvious but is
probably worth pointing out.
00:14:52.000 --> 00:15:00.000
There are two methods.
The first method that we will
00:15:00.000 --> 00:15:16.000
see is computing line integrals.
Let's see how that works.
00:15:16.000 --> 00:15:25.000
Let's say that I take some path
that starts at the origin.
00:15:25.000 --> 00:15:26.000
Or, actually,
anywhere you want,
00:15:26.000 --> 00:15:29.000
but let's take the origin.
That is my favorite point.
00:15:29.000 --> 00:15:36.000
And let's go to a point with
coordinates (x1,
00:15:36.000 --> 00:15:40.000
y1).
And let's take my favorite
00:15:40.000 --> 00:15:45.000
curve and compute the line
integral of that field,
00:15:45.000 --> 00:15:49.000
you know, the work done along
the curve.
00:15:49.000 --> 00:15:55.000
Well, by the fundamental
theorem, that should be equal to
00:15:55.000 --> 00:16:02.000
the value of the potential at
the end point minus the value at
00:16:02.000 --> 00:16:09.000
the origin.
That means I can actually write
00:16:09.000 --> 00:16:19.000
f of (x1, y1) equals -- -- that
line integral plus the value at
00:16:19.000 --> 00:16:26.000
the origin.
And that is just a constant.
00:16:26.000 --> 00:16:27.000
We don't know what it is.
And, actually,
00:16:27.000 --> 00:16:30.000
we can choose what it is.
Because if you have a
00:16:30.000 --> 00:16:33.000
potential, say that you have
some potential function.
00:16:33.000 --> 00:16:34.000
And let's say that you add one
to it.
00:16:34.000 --> 00:16:36.000
It is still a potential
function.
00:16:36.000 --> 00:16:38.000
Adding one doesn't change the
gradient.
00:16:38.000 --> 00:16:41.000
You can even add 18 or any
number that you want.
00:16:41.000 --> 00:16:44.000
This is just going to be an
integration constant.
00:16:44.000 --> 00:16:47.000
It is the same thing as,
in one variable calculus,
00:16:47.000 --> 00:16:49.000
when you take the
anti-derivative of a function it
00:16:49.000 --> 00:16:52.000
is only defined up to adding the
constant.
00:16:52.000 --> 00:16:56.000
We have this integration
constant, but apart from that we
00:16:56.000 --> 00:16:59.000
know that we should be able to
get a potential from this.
00:16:59.000 --> 00:17:03.000
And this we can compute using
the definition of the line
00:17:03.000 --> 00:17:06.000
integral.
And we don't know what little f
00:17:06.000 --> 00:17:11.000
is, but we know what the vector
field is so we can compute that.
00:17:11.000 --> 00:17:14.000
Of course, to do the
calculation we probably don't
00:17:14.000 --> 00:17:18.000
want to use this kind of path.
I mean if that is your favorite
00:17:18.000 --> 00:17:21.000
path then that is fine,
but it is not very easy to
00:17:21.000 --> 00:17:24.000
compute the line integral along
this,
00:17:24.000 --> 00:17:28.000
especially since I didn't tell
you what the definition is.
00:17:28.000 --> 00:17:31.000
There are easier favorite paths
to have.
00:17:31.000 --> 00:17:33.000
For example,
you can go on a straight line
00:17:33.000 --> 00:17:37.000
from the origin to that point.
That would be slightly easier.
00:17:37.000 --> 00:17:40.000
But then there is one easier.
The easiest of all,
00:17:40.000 --> 00:17:47.000
probably, is to just go first
along the x-axis to (x1,0) and
00:17:47.000 --> 00:17:51.000
then go up parallel to the
y-axis.
00:17:51.000 --> 00:17:54.000
Why is that easy?
Well, that is because when we
00:17:54.000 --> 00:17:57.000
do the line integral it becomes
M dx N dy.
00:17:57.000 --> 00:18:05.000
And then, on each of these
pieces, one-half just goes away
00:18:05.000 --> 00:18:11.000
because x, y is constant.
Let's try to use that method in
00:18:11.000 --> 00:18:12.000
our example.
00:18:45.000 --> 00:18:56.000
Let's say that I want to go
along this path from the origin,
00:18:56.000 --> 00:19:06.000
first along the x-axis to
(x1,0) and then vertically to
00:19:06.000 --> 00:19:14.000
(x1, y1).
And so I want to compute for
00:19:14.000 --> 00:19:21.000
the line integral along that
curve.
00:19:21.000 --> 00:19:24.000
Let's say I want to do it for
this vector field.
00:19:24.000 --> 00:19:33.000
I want to find the potential
for this vector field.
00:19:33.000 --> 00:19:37.000
Let me copy it because I will
have to erase at some point.
00:19:37.000 --> 00:19:50.000
4x squared plus 8xy and 3y
squared plus 4x squared.
00:19:50.000 --> 00:19:59.000
That will become the integral
of 4x squared plus 8 xy times dx
00:19:59.000 --> 00:20:05.000
plus 3y squared plus 4x squared
times dy.
00:20:05.000 --> 00:20:08.000
To evaluate on this broken
line, I will,
00:20:08.000 --> 00:20:13.000
of course, evaluate separately
on each of the two segments.
00:20:13.000 --> 00:20:20.000
I will start with this segment
that I will call c1 and then I
00:20:20.000 --> 00:20:25.000
will do this one that I will
call c2.
00:20:25.000 --> 00:20:30.000
On c1, how do I evaluate my
integral?
00:20:30.000 --> 00:20:38.000
Well, if I am on c1 then x
varies from zero to x1.
00:20:38.000 --> 00:20:40.000
Well, actually,
I don't know if x1 is positive
00:20:40.000 --> 00:20:41.000
or not so I shouldn't write
this.
00:20:41.000 --> 00:20:48.000
I really should say just x goes
from zero to x1.
00:20:48.000 --> 00:20:54.000
And what about y?
y is just 0.
00:20:54.000 --> 00:21:00.000
I will set y equal to zero and
also dy equal to zero.
00:21:00.000 --> 00:21:08.000
I get that the line integral on
c1 -- Well, a lot of stuff goes
00:21:08.000 --> 00:21:11.000
away.
The entire second term with dy
00:21:11.000 --> 00:21:15.000
goes away because dy is zero.
And, in the first term,
00:21:15.000 --> 00:21:18.000
8xy goes away because y is zero
as well.
00:21:18.000 --> 00:21:27.000
I just have an integral of 4x
squared dx from zero to x1.
00:21:27.000 --> 00:21:31.000
By the way, now you see why I
have been using an x1 and a y1
00:21:31.000 --> 00:21:33.000
for my point and not just x and
y.
00:21:33.000 --> 00:21:36.000
It is to avoid confusion.
I am using x and y as my
00:21:36.000 --> 00:21:41.000
integration variables and x1,
y1 as constants that are
00:21:41.000 --> 00:21:45.000
representing the end point of my
path.
00:21:45.000 --> 00:21:51.000
And so, if I integrate this,
I should get four-thirds x1
00:21:51.000 --> 00:21:54.000
cubed.
That is the first part.
00:21:54.000 --> 00:22:01.000
Next I need to do the second
segment.
00:22:01.000 --> 00:22:09.000
If I am on c2,
y goes from zero to y1.
00:22:09.000 --> 00:22:16.000
And what about x?
x is constant equal to x1 so dx
00:22:16.000 --> 00:22:22.000
becomes just zero.
It is a constant.
00:22:22.000 --> 00:22:30.000
If I take the line integral of
c2, F dot dr then I will get the
00:22:30.000 --> 00:22:37.000
integral from zero to y1.
The entire first term with dx
00:22:37.000 --> 00:22:47.000
goes away and then I have 3y
squared plus 4x1 squared times
00:22:47.000 --> 00:22:52.000
dy.
That integrates to y cubed plus
00:22:52.000 --> 00:23:01.000
4x1 squared y from zero to y1.
Or, if you prefer,
00:23:01.000 --> 00:23:11.000
that is y1 cubed plus 4x1
squared y1.
00:23:11.000 --> 00:23:15.000
Now that we have done both of
them we can just add them
00:23:15.000 --> 00:23:19.000
together, and that will give us
the formula for the potential.
00:23:40.000 --> 00:23:50.000
F of x1 and y1 is four-thirds
x1 cubed plus y1 cubed plus 4x1
00:23:50.000 --> 00:23:57.000
squared y1 plus a constant.
That constant is just the
00:23:57.000 --> 00:24:03.000
integration constant that we had
from the beginning.
00:24:03.000 --> 00:24:05.000
Now you can drop the subscripts
if you prefer.
00:24:05.000 --> 00:24:14.000
You can just say f is
four-thirds x cubed plus y cubed
00:24:14.000 --> 00:24:20.000
plus 4x squared y plus constant.
And you can check.
00:24:20.000 --> 00:24:25.000
If you take the gradient of
this, you should get again this
00:24:25.000 --> 00:24:29.000
vector field over there.
Any questions about this method?
00:24:29.000 --> 00:24:33.000
Yes?
No.
00:24:33.000 --> 00:24:35.000
Well, it depends whether you
are just trying to find one
00:24:35.000 --> 00:24:38.000
potential or if you are trying
to find all the possible
00:24:38.000 --> 00:24:40.000
potentials.
If a problem just says find a
00:24:40.000 --> 00:24:43.000
potential then you don't have to
use the constant.
00:24:43.000 --> 00:24:47.000
This guy without the constant
is a valid potential.
00:24:47.000 --> 00:24:52.000
You just have others.
If your neighbor comes to you
00:24:52.000 --> 00:24:58.000
and say your answer must be
wrong because I got this plus
00:24:58.000 --> 00:25:01.000
18, well, both answers are
correct.
00:25:01.000 --> 00:25:05.000
By the way.
Instead of going first along
00:25:05.000 --> 00:25:08.000
the x-axis vertically,
you could do it the other way
00:25:08.000 --> 00:25:11.000
around.
Of course, start along the
00:25:11.000 --> 00:25:15.000
y-axis and then horizontally.
That is the same level of
00:25:15.000 --> 00:25:19.000
difficulty.
You just exchange roles of x
00:25:19.000 --> 00:25:21.000
and y.
In some cases,
00:25:21.000 --> 00:25:26.000
it is actually even making more
sense maybe to go radially,
00:25:26.000 --> 00:25:30.000
start out from the origin to
your end point.
00:25:30.000 --> 00:25:37.000
But usually this setting is
easier just because each of
00:25:37.000 --> 00:25:43.000
these two guys were very easy to
compute.
00:25:43.000 --> 00:25:46.000
But somehow maybe if you
suspect that polar coordinates
00:25:46.000 --> 00:25:49.000
will be involved somehow in the
answer then maybe it makes sense
00:25:49.000 --> 00:26:01.000
to choose different paths.
Maybe a straight line is better.
00:26:01.000 --> 00:26:13.000
Now we have another method to
look at which is using
00:26:13.000 --> 00:26:19.000
anti-derivatives.
The goal is the same,
00:26:19.000 --> 00:26:21.000
still to find the potential
function.
00:26:21.000 --> 00:26:26.000
And you see that finding the
potential is really the
00:26:26.000 --> 00:26:31.000
multivariable analog of finding
the anti-derivative in the one
00:26:31.000 --> 00:26:34.000
variable.
Here we did it basically by
00:26:34.000 --> 00:26:38.000
hand by computing the integral.
The other thing you could try
00:26:38.000 --> 00:26:39.000
to say is, wait,
I already know how to take
00:26:39.000 --> 00:26:42.000
anti-derivatives.
Let's use that instead of
00:26:42.000 --> 00:26:45.000
computing integrals.
And it works but you have to be
00:26:45.000 --> 00:26:51.000
careful about how you do it.
Let's see how that works.
00:26:51.000 --> 00:26:53.000
Let's still do it with the same
example.
00:26:53.000 --> 00:27:02.000
We want to solve the equations.
We want a function such that f
00:27:02.000 --> 00:27:13.000
sub x is 4x squared plus 8xy and
f sub y is 3y squared plus 4x
00:27:13.000 --> 00:27:16.000
squared.
Let's just look at one of these
00:27:16.000 --> 00:27:20.000
at a time.
If we look at this one,
00:27:20.000 --> 00:27:28.000
well, we know how to solve this
because it is just telling us we
00:27:28.000 --> 00:27:33.000
have to integrate this with
respect to x.
00:27:33.000 --> 00:27:38.000
Well, let's call them one and
two because I will have to refer
00:27:38.000 --> 00:27:43.000
to them again.
Let's start with equation one
00:27:43.000 --> 00:27:48.000
and lets integrate with respect
to x.
00:27:48.000 --> 00:27:51.000
Well, it tells us that f should
be,
00:27:51.000 --> 00:27:55.000
what do I get when I integrate
this with respect to x,
00:27:55.000 --> 00:28:02.000
four-thirds x cubed plus,
when I integrate 8xy,
00:28:02.000 --> 00:28:08.000
y is just a constant,
so I will get 4x squared y.
00:28:08.000 --> 00:28:11.000
And that is not quite the end
to it because there is an
00:28:11.000 --> 00:28:15.000
integration constant.
And here, when I say there is
00:28:15.000 --> 00:28:18.000
an integration constant,
it just means the extra term
00:28:18.000 --> 00:28:21.000
does not depend on x.
That is what it means to be a
00:28:21.000 --> 00:28:25.000
constant in this setting.
But maybe my constant still
00:28:25.000 --> 00:28:28.000
depends on y so it is not
actually a true constant.
00:28:28.000 --> 00:28:30.000
A constant that depends on y is
not really a constant.
00:28:30.000 --> 00:28:38.000
It is actually a function of y.
The good news that we have is
00:28:38.000 --> 00:28:40.000
that this function normally
depends on x.
00:28:40.000 --> 00:28:46.000
We have made some progress.
We have part of the answer and
00:28:46.000 --> 00:28:53.000
we have simplified the problem.
If we have anything that looks
00:28:53.000 --> 00:28:56.000
like this, it will satisfy the
first condition.
00:28:56.000 --> 00:28:59.000
Now we need to look at the
second condition.
00:28:59.000 --> 00:29:12.000
We want f sub y to be that.
But we know what f is,
00:29:12.000 --> 00:29:15.000
so let's compute f sub y from
this.
00:29:15.000 --> 00:29:20.000
From this I get f sub y.
What do I get if I
00:29:20.000 --> 00:29:22.000
differentiate this with respect
to y?
00:29:22.000 --> 00:29:37.000
Well, I get zero plus 4x
squared plus the derivative of
00:29:37.000 --> 00:29:46.000
g.
I would like to match this with
00:29:46.000 --> 00:29:51.000
what I had.
If I match this with equation
00:29:51.000 --> 00:29:55.000
two then that will tell me what
the derivative of g should be.
00:30:15.000 --> 00:30:20.000
If we compare the two things
there, we get 4x squared plus g
00:30:20.000 --> 00:30:26.000
prime of y should be equal to 3y
squared by 4x squared.
00:30:26.000 --> 00:30:31.000
And, of course,
the 4x squares go away.
00:30:31.000 --> 00:30:35.000
That tells you g prime is 3y
squared.
00:30:35.000 --> 00:30:42.000
And that integrates to y cubed
plus constant.
00:30:42.000 --> 00:30:46.000
Now, this time the constant is
a true constant because g did
00:30:46.000 --> 00:30:48.000
not depend on anything other
than y.
00:30:48.000 --> 00:30:54.000
And the constant does not
depend on y so it is a real
00:30:54.000 --> 00:30:58.000
constant now.
Now we just plug this back into
00:30:58.000 --> 00:31:05.000
this guy.
Let's call him star.
00:31:05.000 --> 00:31:13.000
If we plug this into star,
we get f equals four-thirds x
00:31:13.000 --> 00:31:21.000
cubed plus 4x squared y plus y
cubed plus constant.
00:31:21.000 --> 00:31:30.000
I mean, of course,
again, now this constant is
00:31:30.000 --> 00:31:33.000
optional.
The advantage of this method is
00:31:33.000 --> 00:31:35.000
you don't have to write any
integrals.
00:31:35.000 --> 00:31:40.000
The small drawback is you have
to follow this procedure
00:31:40.000 --> 00:31:45.000
carefully.
By the way, one common pitfall
00:31:45.000 --> 00:31:48.000
that is tempting.
After you have done this,
00:31:48.000 --> 00:31:51.000
what is very tempting is to
just say, well,
00:31:51.000 --> 00:31:53.000
let's do the same with this
guy.
00:31:53.000 --> 00:31:55.000
Let's integrate this with
respect to y.
00:31:55.000 --> 00:31:58.000
You will get another expression
for f up to a constant that
00:31:58.000 --> 00:32:01.000
depends on x.
And then let's match them.
00:32:01.000 --> 00:32:04.000
Well, the difficulty is
matching is actually quite
00:32:04.000 --> 00:32:09.000
tricky because you don't know in
advance whether they will be the
00:32:09.000 --> 00:32:13.000
same expression.
It could be you could say let's
00:32:13.000 --> 00:32:16.000
just take the terms that are
here and missing there and
00:32:16.000 --> 00:32:20.000
combine the terms,
you know, take all the terms
00:32:20.000 --> 00:32:23.000
that appear in either one.
That is actually not a good way
00:32:23.000 --> 00:32:25.000
to do it,
because if I put sufficiently
00:32:25.000 --> 00:32:28.000
complicated trig functions in
there then you might not be able
00:32:28.000 --> 00:32:30.000
to see that two terms are the
same.
00:32:30.000 --> 00:32:34.000
Take an easy one.
Let's say that here I have one
00:32:34.000 --> 00:32:40.000
plus tangent square and here I
have a secan square then you
00:32:40.000 --> 00:32:46.000
might not actually notice that
there is a difference.
00:32:46.000 --> 00:32:50.000
But there is no difference.
Whatever.
00:32:50.000 --> 00:32:54.000
Anyway, I am saying do it this
way, don't do it any other way
00:32:54.000 --> 00:32:57.000
because there is a risk of
making a mistake otherwise.
00:32:57.000 --> 00:33:00.000
I mean, on the other hand,
you could start with
00:33:00.000 --> 00:33:03.000
integrating with respect to y
and then differentiate and match
00:33:03.000 --> 00:33:06.000
with respect to x.
But what I am saying is just
00:33:06.000 --> 00:33:09.000
take one of them,
integrate,
00:33:09.000 --> 00:33:12.000
get an answer that involves a
function of the other variable,
00:33:12.000 --> 00:33:18.000
then differentiate that answer
and compare and see what you
00:33:18.000 --> 00:33:21.000
get.
By the way, here,
00:33:21.000 --> 00:33:27.000
of course, after we simplified
there were only y's here.
00:33:27.000 --> 00:33:29.000
There were no x's.
And that is kind of good news.
00:33:29.000 --> 00:33:33.000
I mean, if you had had an x
here in this expression that
00:33:33.000 --> 00:33:36.000
would have told you that
something is going wrong.
00:33:36.000 --> 00:33:39.000
g is a function of y only.
If you get an x here,
00:33:39.000 --> 00:33:42.000
maybe you want to go back and
check whether it is really a
00:33:42.000 --> 00:33:47.000
gradient field.
Yes?
00:33:47.000 --> 00:33:49.000
Yes, this will work with
functions of more than two
00:33:49.000 --> 00:33:51.000
variables.
Both methods work with more
00:33:51.000 --> 00:33:53.000
than two variables.
We are going to see it in the
00:33:53.000 --> 00:33:56.000
case where more than two means
three.
00:33:56.000 --> 00:34:00.000
We are going to see that in two
or three weeks from now.
00:34:00.000 --> 00:34:04.000
I mean, basically starting at
the end of next week,
00:34:04.000 --> 00:34:08.000
we are going to do triple
integrals, line integrals in
00:34:08.000 --> 00:34:10.000
space and so on.
The format is first we do
00:34:10.000 --> 00:34:13.000
everything in two variables.
Then we will do three variables.
00:34:13.000 --> 00:34:20.000
And then what happens with more
than three will be left to your
00:34:20.000 --> 00:34:25.000
imagination.
Any other questions about
00:34:25.000 --> 00:34:29.000
either of these methods?
A quick poll.
00:34:29.000 --> 00:34:34.000
Who prefers the first method?
Who prefers the second method?
00:34:34.000 --> 00:34:41.000
Wow.
OK.
00:34:41.000 --> 00:34:45.000
Anyway, you will get to use
whichever one you want.
00:34:45.000 --> 00:34:47.000
And I would agree with you,
but the second method is
00:34:47.000 --> 00:34:50.000
slightly more effective in that
you are writing less stuff.
00:34:50.000 --> 00:34:54.000
You don't have to set up all
these line integrals.
00:34:54.000 --> 00:35:03.000
On the other hand,
it does require a little bit
00:35:03.000 --> 00:35:19.000
more attention.
Let's move on a bit.
00:35:19.000 --> 00:35:24.000
Let me start by actually doing
a small recap.
00:35:24.000 --> 00:35:38.000
We said we have various notions.
One is to say that the vector
00:35:38.000 --> 00:35:48.000
field is a gradient in a certain
region of a plane.
00:35:48.000 --> 00:35:54.000
And we have another notion
which is being conservative.
00:35:54.000 --> 00:36:06.000
It says that the line integral
is zero along any closed curve.
00:36:06.000 --> 00:36:10.000
Actually, let me introduce a
new piece of notation.
00:36:10.000 --> 00:36:14.000
To remind ourselves that we are
doing it along a closed curve,
00:36:14.000 --> 00:36:18.000
very often we put just a circle
for the integral to tell us this
00:36:18.000 --> 00:36:21.000
is a curve that closes on
itself.
00:36:21.000 --> 00:36:25.000
It ends where it started.
I mean it doesn't change
00:36:25.000 --> 00:36:28.000
anything concerning the
definition or how you compute it
00:36:28.000 --> 00:36:31.000
or anything.
It just reminds you that you
00:36:31.000 --> 00:36:34.000
are doing it on a closed curve.
It is actually useful for
00:36:34.000 --> 00:36:37.000
various physical applications.
And also, when you state
00:36:37.000 --> 00:36:41.000
theorems in that way,
it reminds you,oh..
00:36:41.000 --> 00:36:45.000
I need to be on a closed curve
to do it.
00:36:45.000 --> 00:36:51.000
And so we have said these two
things are equivalent.
00:36:51.000 --> 00:37:00.000
Now we have a third thing which
is N sub x equals M sub y at
00:37:00.000 --> 00:37:03.000
every point.
Just to summarize the
00:37:03.000 --> 00:37:06.000
discussion.
We have said if we have a
00:37:06.000 --> 00:37:09.000
gradient field then we have
this.
00:37:09.000 --> 00:37:18.000
And the converse is true in
suitable regions.
00:37:18.000 --> 00:37:32.000
We have a converse if F is
defined in the entire plane.
00:37:32.000 --> 00:37:43.000
Or, as we will see soon,
in a simply connected region.
00:37:43.000 --> 00:37:45.000
I guess some of you cannot see
what I am writing here,
00:37:45.000 --> 00:37:48.000
but it doesn't matter because
you are not officially supposed
00:37:48.000 --> 00:37:53.000
to know it yet.
That will be next week.
00:37:53.000 --> 00:37:57.000
Anyway,
I said the fact that Nx equals
00:37:57.000 --> 00:38:01.000
My implies that we have a
gradient field and is only if a
00:38:01.000 --> 00:38:06.000
vector field is defined in the
entire plane or in a region that
00:38:06.000 --> 00:38:12.000
is called simply connected.
And more about that later.
00:38:12.000 --> 00:38:17.000
Now let me just introduce a
quantity that probably a lot of
00:38:17.000 --> 00:38:22.000
you have heard about in physics
that measures precisely fairly
00:38:22.000 --> 00:38:26.000
ought to be conservative.
That is called the curl of a
00:38:26.000 --> 00:38:27.000
vector field.
00:39:06.000 --> 00:39:19.000
For the definition we say that
the curl of F is the quantity N
00:39:19.000 --> 00:39:27.000
sub x - M sub y.
It is just replicating the
00:39:27.000 --> 00:39:35.000
information we had but in a way
that is a single quantity.
00:39:35.000 --> 00:39:43.000
In this new language,
the conditions that we had over
00:39:43.000 --> 00:39:50.000
there, this condition says curl
F equals zero.
00:39:50.000 --> 00:39:56.000
That is the new version of Nx
equals My.
00:39:56.000 --> 00:40:06.000
It measures failure of a vector
field to be conservative.
00:40:06.000 --> 00:40:21.000
The test for conservativeness
is that the curl of F should be
00:40:21.000 --> 00:40:25.000
zero.
I should probably tell you a
00:40:25.000 --> 00:40:29.000
little bit about what the curl
is, what it measures and what it
00:40:29.000 --> 00:40:34.000
does because that is something
that is probably useful.
00:40:34.000 --> 00:40:37.000
It is a very strange quantity
if you put it in that form.
00:40:37.000 --> 00:40:42.000
Yes?
I think it is the same as the
00:40:42.000 --> 00:40:45.000
physics one, but I haven't
checked the physics textbook.
00:40:45.000 --> 00:40:49.000
I believe it is the same.
Yes, I think it is the same as
00:40:49.000 --> 00:40:53.000
the physics one.
It is not the opposite this
00:40:53.000 --> 00:40:55.000
time.
Of course, in physics maybe you
00:40:55.000 --> 00:40:59.000
have seen curl in space.
We are going to see curl in
00:40:59.000 --> 00:41:07.000
space in two or three weeks.
Yes?
00:41:07.000 --> 00:41:11.000
Yes. Well, you can also use it.
If you fail this test then you
00:41:11.000 --> 00:41:14.000
know for sure that you are not
gradient field so you might as
00:41:14.000 --> 00:41:18.000
well do that.
If you satisfy the test but you
00:41:18.000 --> 00:41:24.000
are not defined everywhere then
there is still a bit of
00:41:24.000 --> 00:41:29.000
ambiguity and you don't know for
sure.
00:41:29.000 --> 00:41:40.000
OK.
Let's try to see a little bit
00:41:40.000 --> 00:41:48.000
what the curl measures.
Just to give you some
00:41:48.000 --> 00:41:55.000
intuition, let's first think
about a velocity field.
00:41:55.000 --> 00:42:10.000
The curl measures the rotation
component of a motion.
00:42:10.000 --> 00:42:13.000
If you want a fancy word,
it measures the vorticity of a
00:42:13.000 --> 00:42:16.000
motion.
It tells you how much twisting
00:42:16.000 --> 00:42:19.000
is taking place at a given
point.
00:42:19.000 --> 00:42:24.000
For example,
if I take a constant vector
00:42:24.000 --> 00:42:32.000
field where my fluid is just all
moving in the same direction
00:42:32.000 --> 00:42:37.000
where this is just constants
then,
00:42:37.000 --> 00:42:41.000
of course, the curl is zero.
Because if you take the
00:42:41.000 --> 00:42:43.000
partials you get zero.
And, indeed,
00:42:43.000 --> 00:42:46.000
that is not what you would call
swirling.
00:42:46.000 --> 00:42:58.000
There is no vortex in here.
Let's do another one where this
00:42:58.000 --> 00:43:02.000
is still nothing going on.
Let's say that I take the
00:43:02.000 --> 00:43:06.000
radial vector field where
everything just flows away from
00:43:06.000 --> 00:43:11.000
the origin.
That is f equals x, y.
00:43:11.000 --> 00:43:16.000
Well, if I take the curl,
I have to take partial over
00:43:16.000 --> 00:43:18.000
partial x of the second
component,
00:43:18.000 --> 00:43:21.000
which is y,
minus partial over partial y of
00:43:21.000 --> 00:43:22.000
the first component,
which is x.
00:43:22.000 --> 00:43:25.000
I will get zero.
And, indeed,
00:43:25.000 --> 00:43:29.000
if you think about what is
going on here,
00:43:29.000 --> 00:43:32.000
there is no rotation involved.
On the other hand,
00:43:32.000 --> 00:43:45.000
if you consider our favorite
rotation vector field -- --
00:43:45.000 --> 00:44:00.000
negative y and x then this curl
is going to be N sub x minus M
00:44:00.000 --> 00:44:08.000
sub y,
one plus one equals two.
00:44:08.000 --> 00:44:13.000
That corresponds to the fact
that we are rotating.
00:44:13.000 --> 00:44:16.000
Actually, we are rotating at
unit angular speed.
00:44:16.000 --> 00:44:20.000
The curl actually measures
twice the angular speed of a
00:44:20.000 --> 00:44:24.000
rotation part of a motion at any
given point.
00:44:24.000 --> 00:44:26.000
Now, if you have an actual
motion,
00:44:26.000 --> 00:44:30.000
a more complicated field than
these then no matter where you
00:44:30.000 --> 00:44:34.000
are you can think of a motion as
a combination of translation
00:44:34.000 --> 00:44:37.000
effects,
maybe dilation effects,
00:44:37.000 --> 00:44:43.000
maybe rotation effects,
possibly other things like that.
00:44:43.000 --> 00:44:48.000
And what a curl will measure is
how intense the rotation effect
00:44:48.000 --> 00:44:52.000
is at that particular point.
I am not going to try to make a
00:44:52.000 --> 00:44:55.000
much more precise statement.
A precise statement is what a
00:44:55.000 --> 00:44:58.000
curl measures is really this
quantity up there.
00:44:58.000 --> 00:45:01.000
But the intuition you should
have is it measures how much
00:45:01.000 --> 00:45:04.000
rotation is taking place at any
given point.
00:45:04.000 --> 00:45:06.000
And, of course,
in a complicated motion you
00:45:06.000 --> 00:45:09.000
might have more rotation at some
point than at some others,
00:45:09.000 --> 00:45:12.000
which is why the curl will
depend on x and y.
00:45:12.000 --> 00:45:20.000
It is not just a constant
because how much you rotate
00:45:20.000 --> 00:45:26.000
depends on where you are.
If you are looking at actual
00:45:26.000 --> 00:45:30.000
wind velocities in weather
prediction then the regions with
00:45:30.000 --> 00:45:33.000
high curl tend to be hurricanes
or tornadoes or things like
00:45:33.000 --> 00:45:37.000
that.
They are not very pleasant
00:45:37.000 --> 00:45:40.000
things.
And the sign of a curl tells
00:45:40.000 --> 00:45:43.000
you whether you are going
clockwise or counterclockwise.
00:46:09.000 --> 00:46:27.000
Curl measures twice the angular
velocity of the rotation
00:46:27.000 --> 00:46:41.000
component of a velocity field.
Now, what about a force field?
00:46:41.000 --> 00:46:44.000
Because, after all,
how we got to this was coming
00:46:44.000 --> 00:46:47.000
from and trying to understand
forces and the work they do.
00:46:47.000 --> 00:46:50.000
So I should tell you what it
means for a force.
00:46:50.000 --> 00:47:10.000
Well, the curl of a force field
-- -- measures the torque
00:47:10.000 --> 00:47:29.000
exerted on a test object that
you put at any point.
00:47:29.000 --> 00:47:36.000
Remember, torque is the
rotational analog of the force.
00:47:36.000 --> 00:47:41.000
We had this analogy about
velocity versus angular velocity
00:47:41.000 --> 00:47:45.000
and mass versus moment of
inertia.
00:47:45.000 --> 00:47:49.000
And then, in that analogy,
force divided by the mass is
00:47:49.000 --> 00:47:53.000
what will cause acceleration,
which is the derivative of
00:47:53.000 --> 00:47:56.000
velocity.
Torque divided by moment of
00:47:56.000 --> 00:47:59.000
inertia is what will cause the
angular acceleration,
00:47:59.000 --> 00:48:02.000
namely the derivative of
angular velocity.
00:48:02.000 --> 00:48:04.000
Maybe I should write that down.
00:48:18.000 --> 00:48:31.000
Torque divided by moment of
inertia is going to be d over dt
00:48:31.000 --> 00:48:38.000
of angular velocity.
I leave it up to your physics
00:48:38.000 --> 00:48:41.000
teachers to decide what letters
to use for all these things.
00:48:41.000 --> 00:48:49.000
That is the analog of force
divided by mass equals
00:48:49.000 --> 00:48:56.000
acceleration,
which is d over dt of velocity.
00:48:56.000 --> 00:49:03.000
And so now you see if the curl
of a velocity field measure the
00:49:03.000 --> 00:49:07.000
angular velocity of its rotation
then,
00:49:07.000 --> 00:49:13.000
by this analogy,
the curl of a force field
00:49:13.000 --> 00:49:24.000
should measure the torque it
exerts on a mass per unit moment
00:49:24.000 --> 00:49:28.000
of inertia.
Concretely, if you imagine that
00:49:28.000 --> 00:49:29.000
you are putting something in
there,
00:49:29.000 --> 00:49:32.000
you know, if you are in a
velocity field the curl will
00:49:32.000 --> 00:49:35.000
tell you how fast your guy is
spinning at a given time.
00:49:35.000 --> 00:49:37.000
If you put something that
floats, for example,
00:49:37.000 --> 00:49:40.000
in your fluid,
something very light then it is
00:49:40.000 --> 00:49:44.000
going to start spinning.
And the curl of a velocity
00:49:44.000 --> 00:49:48.000
field tells you how fast it is
spinning at any given time up to
00:49:48.000 --> 00:49:51.000
a factor of two.
And the curl of a force field
00:49:51.000 --> 00:49:55.000
tells you how quickly the
angular velocity is going to
00:49:55.000 --> 00:50:01.000
increase or decrease.
OK.
00:50:01.000 --> 00:50:04.000
Well, next time we are going to
see Green's theorem which is
00:50:04.000 --> 00:50:08.000
actually going to tell us a lot
more about curl and failure of
00:50:08.000 --> 00:50:11.000
conservativeness.