WEBVTT
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CHRISTINE BREINER: Welcome
back to recitation.
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In this video, I'd like us to
consider the following problem.
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The first part is I'd like
to know, for what values of b
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is this vector field
F conservative?
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And F is defined as y*i plus
the quantity x plus b*y*z j plus
00:00:24.800 --> 00:00:26.930
the quantity y
squared plus 1, k.
00:00:26.930 --> 00:00:28.729
So as you can see,
the only thing
00:00:28.729 --> 00:00:30.520
we're allowed to
manipulate in this problem
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is b. b will be
some real number.
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And I want to know, what real
numbers can I put in for b so
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that this vector
field is conservative.
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The second part
of this problem is
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for each b-value you
determined from one,
00:00:46.180 --> 00:00:47.680
find a potential function.
00:00:47.680 --> 00:00:52.200
So fix the b-value for one of
the ones that is acceptable,
00:00:52.200 --> 00:00:55.990
based on number one, and then
find the potential function.
00:00:55.990 --> 00:00:59.880
And then the third part
says that you should explain
00:00:59.880 --> 00:01:02.750
why F dot dr is exact,
and this is obviously
00:01:02.750 --> 00:01:04.990
for the b-values
determined from one.
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So the second and third part
are once you know the b-values.
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And you're only going to
use those b-values that
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make F conservative, because
that's the place where we can
00:01:13.770 --> 00:01:15.760
talk about finding a
potential function,
00:01:15.760 --> 00:01:18.540
and that's where we can talk
about F dot dr being exact,
00:01:18.540 --> 00:01:20.560
are exactly those values.
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OK.
00:01:21.250 --> 00:01:23.650
So why don't you pause the
video, work on these three
00:01:23.650 --> 00:01:26.070
problems, and then when you're
feeling good about them,
00:01:26.070 --> 00:01:28.153
bring the video back up,
I'll show you what I did.
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OK, welcome back.
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Again, we're interested in doing
three things with this vector
00:01:40.711 --> 00:01:41.210
field.
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The first thing we
want to do is to find
00:01:43.290 --> 00:01:45.880
the values of b that make this
vector field conservative.
00:01:45.880 --> 00:01:47.920
So I will start with that part.
00:01:47.920 --> 00:01:50.170
And as we know from
the lecture, the thing
00:01:50.170 --> 00:01:53.750
I ultimately need to do is I
need to find the curl of F. OK,
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so the curl of F is going
to measure how far F
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is from being conservative.
00:01:57.190 --> 00:01:59.380
So if the curl of
F is 0, I'm going
00:01:59.380 --> 00:02:01.190
to have F being conservative.
00:02:01.190 --> 00:02:04.560
So that's really what I'm
interested in doing first.
00:02:04.560 --> 00:02:09.320
So I'm actually just going
to rewrite what the curl of F
00:02:09.320 --> 00:02:10.570
actually is.
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So I'm going to let F-- I'm
going to denote in our usual
00:02:14.210 --> 00:02:19.410
way by P, Q, R.
And in this case,
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that's specifically y comma x
plus b*y*z comma y squared plus
00:02:27.130 --> 00:02:29.600
1.
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OK, that's my P, Q, R. So
P is y, Q is x plus b*y*z,
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and R is y squared plus 1.
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And now the curl of F-- which
was found in the lecture,
00:02:45.580 --> 00:02:48.020
so I'm not going to show
you again how to find it,
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I'm just going to write the
formula for it-- is exactly
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the following vector.
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It's the derivative of the R-th
component with respect to y,
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minus the Q-th component
with respect to z.
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That's the i-value.
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The j-value is P sub
z minus R sub x, j.
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The k-th value is Q
sub x minus P sub y, k.
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OK, so there are
three components here.
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And let's just start figuring
out what these values are,
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and then we'll see what kind
of restrictions we have on b.
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So let's start doing this.
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So again, this is P,
this is Q, and this
00:03:28.420 --> 00:03:32.020
is R. So R sub y is the
derivative of this component
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with respect to y.
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That's just 2y.
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Q sub z is the derivative
of this with respect to z.
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Well, this is 0,
and this is b*y.
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OK.
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Let's look at the rest first.
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P sub z: the derivative of
this with respect to z is 0.
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The derivative of R
with respect to x is 0.
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So that doesn't have
any b's in it at all.
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And then Q sub x minus P sub y.
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Q is the middle one.
00:04:02.440 --> 00:04:03.970
Q sub x is 1.
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P is the first one.
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P sub y is 1.
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OK, so what do we get here?
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I should have written
equals there, maybe.
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OK, so the j-th component is 0.
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And the k-th component is 0.
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So all I'm left with
is 2y minus b*y, i.
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And if I want F to
be conservative,
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this quantity has
to be 0, so I see
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there's only one b-value
that's going to work,
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and that is b is equal to 2.
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OK?
00:04:40.820 --> 00:04:43.740
So I know in part one,
the answer to the question
00:04:43.740 --> 00:04:47.280
is just for b equals
2, is F conservative.
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That was, maybe, poorly phrased.
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F is conservative
only when b is 2.
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OK.
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And that's because the curl
of F is 0 only when b is 2.
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All right.
00:04:56.310 --> 00:04:58.300
So now we can move on
to the second part.
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And the second part is for
this particular value of b,
00:05:01.230 --> 00:05:02.980
find a potential function.
00:05:02.980 --> 00:05:04.610
And our strategy
for that is going
00:05:04.610 --> 00:05:06.720
to be one of the
methods from lecture.
00:05:06.720 --> 00:05:08.761
And it's going to be the
method from lecture that
00:05:08.761 --> 00:05:11.820
in three dimensions is
much easier than the other.
00:05:11.820 --> 00:05:15.140
So the one method in lecture
that's easy in three dimensions
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is where you start
at the origin,
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and you integrate F dot
dr along a curve that's
00:05:20.680 --> 00:05:21.950
made up of line segments.
00:05:21.950 --> 00:05:25.130
So this strategy I've done
before in two dimensions,
00:05:25.130 --> 00:05:27.170
in one of the problems
in recitation.
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Now, we'll see it
in three dimensions.
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So what we're going
to do is we're
00:05:30.460 --> 00:05:34.930
going to integrate along
a certain curve F dot dr.
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And this curve is going
to go from the origin to
00:05:38.630 --> 00:05:41.500
(x_1, y_1, z_1).
00:05:41.500 --> 00:05:49.950
And that will give us
f of (x_1, y_1, z_1).
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OK.
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So this is a sort
of general strategy,
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and now we'll talk
about it specifically.
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This will actually be f of
(x_1, y_1, z_1) plus a constant,
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but we'll deal with that
part right at the end.
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OK, so C in this case is going
to be made up of three curves.
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And I'm going to draw
them, in a picture,
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and then we're going
to describe them.
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So I'm going to
start at (0, 0, 0).
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My first curve will go
out to x_1 comma 0, 0,
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and that's going to
be the curve C_1.
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Oops.
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I want that to go the other way.
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That way.
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OK, C_1 is going to go from
the origin to x_1 comma 0, 0.
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So the y- and z-values
are going to be
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0 and 0 all the way along, and
the x-value is going to change.
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My next one-- I'm
going to make it long
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so I can have enough
room to write-- that's
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going to be my C_2.
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And that's going to be
x_1 comma y_1 comma 0.
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So in the end, what I've done
is I've taken my x-value,
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I've kept it fixed all
the way along here,
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but I'm varying the
y-value out to y_1.
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And then the last one is
going to go straight up.
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Right there.
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And so it's going to be with
the x-value and y-value fixed.
00:07:05.500 --> 00:07:10.020
And at the end, I will
be at x_1, y_1, z_1.
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And this is C3.
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So those are my three
curves-- And this one,
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I'm going to move
in this direction.
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Those are my three curves.
00:07:17.000 --> 00:07:19.450
And I want to point
out that in order
00:07:19.450 --> 00:07:22.190
to understand how to
simplify this problem,
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I'm going to have to remind
myself what F dot dr is.
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OK.
00:07:25.530 --> 00:07:37.310
So F dot dr is P*dx
plus Q*dy plus R*dz.
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Right?
00:07:38.120 --> 00:07:39.550
That's what F dot dr is.
00:07:39.550 --> 00:07:41.100
And so what I'm
interested in, I'm
00:07:41.100 --> 00:07:44.630
going to integrate each of these
things along C_1, C_2, C_3.
00:07:44.630 --> 00:07:48.700
But let's notice what
happens along certain numbers
00:07:48.700 --> 00:07:51.500
of these curves.
00:07:51.500 --> 00:07:52.660
If we come back over here.
00:07:52.660 --> 00:07:56.220
On C_1, y is fixed
and z as fixed.
00:07:56.220 --> 00:07:58.630
So dy and dz are both 0.
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So on C1, I only have
to integrate P. OK,
00:08:02.490 --> 00:08:04.240
so I'm going to
keep track of that.
00:08:04.240 --> 00:08:06.770
On C_1-- which,
C_1 is parametrized
00:08:06.770 --> 00:08:10.420
by x, 0 to x-- I
only need to worry
00:08:10.420 --> 00:08:14.380
about the P. P of x, 0, 0, dx.
00:08:14.380 --> 00:08:17.200
This is my C_1 component, and
there's nothing else there,
00:08:17.200 --> 00:08:19.680
because these two are both 0.
00:08:19.680 --> 00:08:20.180
Right?
00:08:20.180 --> 00:08:24.470
Now let's consider
what happens on C_2.
00:08:24.470 --> 00:08:25.930
If I look here.
00:08:25.930 --> 00:08:29.400
On C_2, the x-value is fixed
and the z-value is fixed.
00:08:29.400 --> 00:08:32.340
x is fixed at x_1
and z is fixed at 0.
00:08:32.340 --> 00:08:36.590
And so dx and dz are both
0, because x and z are not
00:08:36.590 --> 00:08:37.190
changing.
00:08:37.190 --> 00:08:39.560
So there's only a dy component.
00:08:39.560 --> 00:08:45.120
So on C_2, which is parametrized
in y-- from 0 to y_1--
00:08:45.120 --> 00:08:52.300
I'm only interested in Q
at x_1 comma y comma 0, dy.
00:08:52.300 --> 00:08:57.730
Again, this component is
0 on C_2 because dx is 0.
00:08:57.730 --> 00:09:00.770
And this component is 0
on C_2 because dz is 0.
00:09:00.770 --> 00:09:03.200
And this component--
I'm evaluating it--
00:09:03.200 --> 00:09:06.440
x is fixed at x_1,
z is fixed at 0,
00:09:06.440 --> 00:09:09.070
and the y is varying
from 0 to y1.
00:09:09.070 --> 00:09:10.932
And then there's
one more component,
00:09:10.932 --> 00:09:12.640
and I'm going to write
it below, and then
00:09:12.640 --> 00:09:14.250
we'll do the rest over here.
00:09:14.250 --> 00:09:16.850
And the third component
is the C_3 component.
00:09:16.850 --> 00:09:20.200
Now, not surprisingly--
if I come back over here--
00:09:20.200 --> 00:09:23.290
because x and y are fixed
all along the C3 component,
00:09:23.290 --> 00:09:25.130
the only thing
that's changing is z.
00:09:25.130 --> 00:09:29.340
So dx and dy are 0, so I'm
only worried about the dz part.
00:09:29.340 --> 00:09:29.840
OK.
00:09:29.840 --> 00:09:32.852
So again, as happened
before, I only
00:09:32.852 --> 00:09:34.810
had P in the first one
and Q in the second one,
00:09:34.810 --> 00:09:37.000
and now I have R,
only, in the third one.
00:09:37.000 --> 00:09:39.840
And it's parametrized
in z, from 0 to z_1.
00:09:39.840 --> 00:09:41.730
That's what z varies over.
00:09:41.730 --> 00:09:48.830
And it's going to be R at
x_1 comma y_1 comma z dz.
00:09:48.830 --> 00:09:52.390
Because the x's are fixed at
x_1, the y is fixed at y_1,
00:09:52.390 --> 00:09:54.415
but z is varying from 0 to z_1.
00:09:54.415 --> 00:09:55.020
All right.
00:09:55.020 --> 00:09:57.050
So I have these
three parts, and now
00:09:57.050 --> 00:10:00.830
I just have to fill them in with
the vector field that I have.
00:10:00.830 --> 00:10:03.390
I want to find what
P is at (x, 0, 0),
00:10:03.390 --> 00:10:07.870
what Q is at (x_1, y_1, 0),
and what R is at (x_1, y_1, z).
00:10:07.870 --> 00:10:09.080
And then integrate.
00:10:09.080 --> 00:10:11.360
So I have two steps left.
00:10:11.360 --> 00:10:13.770
One is plugging in
and one is evaluating.
00:10:13.770 --> 00:10:17.940
So let me remind us what P, Q,
and R actually are, and then
00:10:17.940 --> 00:10:20.950
we'll see what we get.
00:10:20.950 --> 00:10:22.260
Let me write it again.
00:10:22.260 --> 00:10:23.630
Maybe here.
00:10:23.630 --> 00:10:31.870
[P, Q, R] was equal to
y comma x plus 2y*z--
00:10:31.870 --> 00:10:34.860
I'll put it here so you don't
have to look and I don't have
00:10:34.860 --> 00:10:38.410
to look-- and then
y squared plus 1.
00:10:38.410 --> 00:10:39.270
OK.
00:10:39.270 --> 00:10:44.380
So P at x comma 0 comma 0.
00:10:44.380 --> 00:10:48.450
Well, if I plug in
0 for y, P is 0.
00:10:48.450 --> 00:10:53.160
So P at (x, 0, 0) is equal to 0.
00:10:53.160 --> 00:10:55.250
So I get nothing to
integrate in the first part.
00:10:55.250 --> 00:10:56.340
That's nice.
00:10:56.340 --> 00:10:57.040
OK.
00:10:57.040 --> 00:11:02.320
Now, what is Q at
x_1 comma y comma 0?
00:11:02.320 --> 00:11:04.600
Well, that would be an x_1 here.
00:11:04.600 --> 00:11:06.550
0 for y makes this term go away.
00:11:06.550 --> 00:11:09.130
So it's just equal to x_1.
00:11:09.130 --> 00:11:09.930
Right?
00:11:09.930 --> 00:11:18.660
And then R at x_1 comma y_1
comma z is y_1 squared plus 1.
00:11:21.480 --> 00:11:23.087
So now I'm going
to substitute these
00:11:23.087 --> 00:11:24.170
into what I'm integrating.
00:11:24.170 --> 00:11:27.192
So in the first one,
there's nothing there.
00:11:27.192 --> 00:11:28.820
Let me just write it right here.
00:11:28.820 --> 00:11:30.330
The Q is going to
be the integral
00:11:30.330 --> 00:11:34.150
from 0 to y_1 of x_1 dy.
00:11:34.150 --> 00:11:39.160
And the R part is going to
be the integral from 0 to z_1
00:11:39.160 --> 00:11:43.090
of y_1 squared plus 1 dz.
00:11:43.090 --> 00:11:46.060
OK, so the P part
was disappeared.
00:11:46.060 --> 00:11:47.780
This is the Q part
evaluated where
00:11:47.780 --> 00:11:50.580
I needed it to be evaluated.
00:11:50.580 --> 00:11:52.310
It's just x_1 dy.
00:11:52.310 --> 00:11:55.200
And the R part evaluated
at (x_1, y_1, z)
00:11:55.200 --> 00:11:57.130
is just y_1 squared plus 1.
00:11:57.130 --> 00:11:59.500
And so I integrate that in z.
00:11:59.500 --> 00:12:03.910
So if I integrate this in y,
all I get is x_1*y evaluated 0
00:12:03.910 --> 00:12:04.720
and y1.
00:12:04.720 --> 00:12:08.070
So here I just get an x_1*y_1.
00:12:08.070 --> 00:12:09.510
Right?
00:12:09.510 --> 00:12:12.800
And then here, if I integrate
this in z, I just get a z--
00:12:12.800 --> 00:12:14.600
and so I evaluate
that at z_1 and 0--
00:12:14.600 --> 00:12:18.890
I just get z_1 times
y1 squared plus 1.
00:12:18.890 --> 00:12:19.400
OK.
00:12:19.400 --> 00:12:22.330
So this is actually
my potential function.
00:12:22.330 --> 00:12:23.932
And so let me write it formally.
00:12:23.932 --> 00:12:25.890
I should actually say,
this is my final answer.
00:12:25.890 --> 00:12:26.760
Right?
00:12:26.760 --> 00:12:27.920
I was integrating.
00:12:27.920 --> 00:12:29.730
This is actually what I get.
00:12:29.730 --> 00:12:31.180
And so what I was
trying to find,
00:12:31.180 --> 00:12:33.429
if you remember-- I'm going
to come back here and just
00:12:33.429 --> 00:12:34.190
mention it again.
00:12:34.190 --> 00:12:36.740
What I was doing was I was
integrating along a curve F dot
00:12:36.740 --> 00:12:39.210
dr, to give me f
of (x_1, y_1, z_1).
00:12:39.210 --> 00:12:40.010
Right?
00:12:40.010 --> 00:12:41.530
So now I've found it.
00:12:41.530 --> 00:12:43.650
The only thing I said
is we also have to allow
00:12:43.650 --> 00:12:44.941
for there to be a constant.
00:12:44.941 --> 00:12:45.440
OK.
00:12:45.440 --> 00:12:50.000
So the potential function is
actually exactly this function
00:12:50.000 --> 00:12:52.990
plus a constant.
00:12:52.990 --> 00:12:53.950
OK.
00:12:53.950 --> 00:12:56.280
So this is f of (x_1, y_1, z_1).
00:12:56.280 --> 00:12:57.905
And since I don't
have much room above,
00:12:57.905 --> 00:12:59.373
I'll just write it below.
00:13:03.450 --> 00:13:05.330
This is f of x_1, y_1, z_1.
00:13:05.330 --> 00:13:10.810
So that's my potential function
for this vector field, capital
00:13:10.810 --> 00:13:12.920
F, when it is conservative.
00:13:12.920 --> 00:13:15.290
So when b is equal to 2.
00:13:15.290 --> 00:13:17.470
OK, and there was one last
part to this question.
00:13:17.470 --> 00:13:17.970
Right?
00:13:17.970 --> 00:13:19.920
So if we come all
the way back over,
00:13:19.920 --> 00:13:22.790
we're reminded of one last part.
00:13:22.790 --> 00:13:26.160
It was explain why F dot dr
is exact for the b values
00:13:26.160 --> 00:13:27.790
determined from number one.
00:13:27.790 --> 00:13:32.410
And the reason is exactly
because of the following thing.
00:13:32.410 --> 00:13:36.360
F is conservative based
on the fact that b is 2.
00:13:36.360 --> 00:13:42.240
And so when we talk about
when F dot dr is exact,
00:13:42.240 --> 00:13:47.080
the simplest case is
capital F is conservative,
00:13:47.080 --> 00:13:49.810
and I'm on a simply
connected domain.
00:13:49.810 --> 00:13:50.440
OK.
00:13:50.440 --> 00:13:57.320
And if you notice, capital F is
defined for all values x, y, z,
00:13:57.320 --> 00:13:59.900
and is differentiable for
all values of x, y, z.
00:13:59.900 --> 00:14:03.250
So F is defined and
differentiable everywhere
00:14:03.250 --> 00:14:04.550
on R^3.
00:14:04.550 --> 00:14:06.040
R^3 is simply connected.
00:14:06.040 --> 00:14:08.120
So we have a
conservative vector field
00:14:08.120 --> 00:14:09.650
on a simply connected region.
00:14:09.650 --> 00:14:10.700
And that's what it means.
00:14:10.700 --> 00:14:14.100
That's one way that we have
of knowing F dot dr is exact.
00:14:14.100 --> 00:14:17.910
And so that actually answers
the third part of the question.
00:14:17.910 --> 00:14:20.010
So again, let me just
remind you what we did.
00:14:20.010 --> 00:14:22.640
We started with
a vector field F,
00:14:22.640 --> 00:14:26.320
we found values for b that made
that vector field conservative,
00:14:26.320 --> 00:14:27.915
and then we used one
of the techniques
00:14:27.915 --> 00:14:31.950
in class to find a potential
function for that value of b.
00:14:31.950 --> 00:14:34.350
So there were a number
of steps involved,
00:14:34.350 --> 00:14:37.240
but ultimately, again, it's
the same type of problem
00:14:37.240 --> 00:14:41.750
you've seen before, when F was a
vector field in two dimensions.
00:14:41.750 --> 00:14:45.110
So it shouldn't be feeling
too different from some
00:14:45.110 --> 00:14:47.120
of the stuff you saw earlier.
00:14:47.120 --> 00:14:49.520
OK, I think that's
where I'll stop.