WEBVTT
00:00:00.000 --> 00:00:08.402
CHRISTINE BREINER: Welcome
back to recitation.
00:00:08.402 --> 00:00:12.460
In this video I'd like us to
work on the following problem.
00:00:12.460 --> 00:00:16.010
What values of b will make
this vector field F a gradient
00:00:16.010 --> 00:00:21.020
field, where F is determined by
e to the x plus y times x plus
00:00:21.020 --> 00:00:23.480
b i plus x*j?
00:00:23.480 --> 00:00:27.060
So the e to the x plus y is in
both the i component and the j
00:00:27.060 --> 00:00:28.640
component.
00:00:28.640 --> 00:00:31.240
And then once you've determined
what values b will make that
00:00:31.240 --> 00:00:34.650
a gradient field, for this
b-- or I should've said these
00:00:34.650 --> 00:00:38.380
b's-- find a potential function
f using both methods from
00:00:38.380 --> 00:00:39.590
the lecture.
00:00:39.590 --> 00:00:42.510
So why don't you pause
the video, work on this,
00:00:42.510 --> 00:00:45.390
and then when you are
ready to look at how I
00:00:45.390 --> 00:00:46.690
do it bring the video back up.
00:00:55.200 --> 00:00:55.700
OK.
00:00:55.700 --> 00:00:56.680
Welcome back.
00:00:56.680 --> 00:00:58.534
So I'm going to
start off working
00:00:58.534 --> 00:01:00.200
on the first part of
this problem, which
00:01:00.200 --> 00:01:02.658
is to find the values of b that
will make this vector field
00:01:02.658 --> 00:01:04.310
F a gradient field.
00:01:04.310 --> 00:01:06.550
And to clarify
things for myself,
00:01:06.550 --> 00:01:11.870
I'm going to write down what M
and what N are based on F. So
00:01:11.870 --> 00:01:14.910
just to have it
clear, M is equal to e
00:01:14.910 --> 00:01:25.390
to the x plus y times x plus
b and N is equal to x times
00:01:25.390 --> 00:01:27.170
e to the x plus y.
00:01:27.170 --> 00:01:29.360
So those are our
values for M and N.
00:01:29.360 --> 00:01:31.650
And now if I want f to
be a gradient field, what
00:01:31.650 --> 00:01:36.670
I have to do is I have to
have M sub y equal N sub x.
00:01:36.670 --> 00:01:38.940
So I'm going to
determine M sub y
00:01:38.940 --> 00:01:40.750
and I'm going to
determine N sub x
00:01:40.750 --> 00:01:44.130
and I'm going to compare them
and see what value of b I get.
00:01:44.130 --> 00:01:48.280
So M sub y, fairly
straightforward because this
00:01:48.280 --> 00:01:50.390
is a constant in y.
00:01:50.390 --> 00:01:53.070
And the derivative of this in
terms of y is just this back.
00:01:53.070 --> 00:01:53.570
Right?
00:01:53.570 --> 00:01:56.280
It's an exponential
function with the value
00:01:56.280 --> 00:01:57.620
that it has in y is linear.
00:01:57.620 --> 00:01:59.120
So you get exactly
that thing back.
00:01:59.120 --> 00:02:04.450
So it actually is just e to
the x plus y times x plus b.
00:02:04.450 --> 00:02:07.635
So the derivative of M
sub y is just itself.
00:02:07.635 --> 00:02:09.510
The derivative of M with
respect to y, sorry.
00:02:09.510 --> 00:02:12.140
Not the derivative of M sub y.
00:02:12.140 --> 00:02:12.760
OK.
00:02:12.760 --> 00:02:13.680
That's an x.
00:02:13.680 --> 00:02:15.930
Let me just rewrite that.
00:02:15.930 --> 00:02:19.940
OK, now N sub x is
going to have two parts.
00:02:19.940 --> 00:02:24.720
N sub x, the derivative with
respect to x of this is 1.
00:02:24.720 --> 00:02:26.540
And so I have an
e to the x plus y.
00:02:29.120 --> 00:02:32.110
And the derivative with respect
to x of e to the x plus y
00:02:32.110 --> 00:02:35.230
is just e to the plus
y, for the same reason
00:02:35.230 --> 00:02:38.400
as the derivative with
respect to y was the same.
00:02:38.400 --> 00:02:42.690
So then I'm just going to get
a plus x e to the x plus y.
00:02:42.690 --> 00:02:47.680
So that means if I factor that
out, I get an e to the x plus y
00:02:47.680 --> 00:02:49.360
times 1 plus x.
00:02:49.360 --> 00:02:52.290
And we see that if F is going
to be a gradient field then
00:02:52.290 --> 00:02:53.700
b has to equal 1.
00:02:53.700 --> 00:02:58.830
Because it can only have one
value, and so b has to equal 1.
00:02:58.830 --> 00:03:03.840
To get N sub x to equal M
sub y, b has to equal 1.
00:03:03.840 --> 00:03:06.870
So now what I'm going to do
is I'm going to erase that b,
00:03:06.870 --> 00:03:11.510
put in a 1, so that the rest of
my calculations refer to that.
00:03:11.510 --> 00:03:14.170
So now the second
part said for this
00:03:14.170 --> 00:03:17.180
b find a potential function
f using both methods
00:03:17.180 --> 00:03:18.392
from the lecture.
00:03:18.392 --> 00:03:20.100
So we're going to go
through both methods
00:03:20.100 --> 00:03:23.220
and hopefully we get the
same answer both times.
00:03:23.220 --> 00:03:24.890
So let me come back here.
00:03:28.220 --> 00:03:31.540
The first method is where
I'm integrating along a curve
00:03:31.540 --> 00:03:34.460
from (0, 0) to (x_1, y_1).
00:03:34.460 --> 00:03:37.440
So I'm going to do it
in the following way.
00:03:37.440 --> 00:03:41.446
I'm going to let C_1--
so here's (0, 0)--
00:03:41.446 --> 00:03:46.290
I'm going to let C_1 be the
curve from (0, 0) up to (0,
00:03:46.290 --> 00:03:48.110
y_1).
00:03:48.110 --> 00:03:50.790
And then C_2 be the curve--
so it's parameterized
00:03:50.790 --> 00:03:55.680
in that direction-- C_2 be
the curve from (0, y_1) to
00:03:55.680 --> 00:03:58.600
(x_1, y_1).
00:03:58.600 --> 00:04:00.460
OK?
00:04:00.460 --> 00:04:02.170
So that's what I'm
going to do, and I'm
00:04:02.170 --> 00:04:05.190
going to let C equal
the curve C_1 plus C_2.
00:04:05.190 --> 00:04:07.160
So I'm going to have
C be the full curve.
00:04:07.160 --> 00:04:11.770
And what I'm interested in
doing is finding f of x_1,
00:04:11.770 --> 00:04:21.587
y_1, which will just equal the
integral along C of F dot dr.
00:04:21.587 --> 00:04:23.670
So now we need to figure
out some important things
00:04:23.670 --> 00:04:25.900
about C_1 and C_2.
00:04:25.900 --> 00:04:28.435
What's happening on C_1 and
what's happening on C_2.
00:04:28.435 --> 00:04:30.560
And the first thing I want
to point out-- actually,
00:04:30.560 --> 00:04:33.060
before I do that, let me remind
you that this is going to be
00:04:33.060 --> 00:04:39.260
the integral on C
of M*dx plus N*dy.
00:04:39.260 --> 00:04:41.010
So this will be helpful
to refer back to.
00:04:41.010 --> 00:04:44.830
That's really what
we're also doing here.
00:04:44.830 --> 00:04:47.390
So on C_1, what do I notice?
00:04:47.390 --> 00:04:54.050
On C_1, x is 0 and dx is 0.
00:04:54.050 --> 00:04:56.690
And y goes between 0 and y_1.
00:04:59.280 --> 00:05:05.740
And then on C_2,
y is equal to y_1.
00:05:05.740 --> 00:05:12.770
So dy is equal to 0 and x
is going between 0 and x_1.
00:05:12.770 --> 00:05:16.220
So those are the values
that are important to me.
00:05:16.220 --> 00:05:19.370
So if you notice from
this fact and this fact,
00:05:19.370 --> 00:05:24.550
we see that if we look at
the integral just along C_1,
00:05:24.550 --> 00:05:26.840
there's going to
be no M*dx term.
00:05:26.840 --> 00:05:30.670
And if we look at the
integral along C_2,
00:05:30.670 --> 00:05:33.580
there's going to be no
dy term because of that.
00:05:33.580 --> 00:05:35.962
So let me write down
the terms that do
00:05:35.962 --> 00:05:38.170
exist, and we'll see some
other things drop out also.
00:05:40.990 --> 00:05:44.120
If I look along
first just C_1, I'm
00:05:44.120 --> 00:05:49.110
only going to get-- I said
the dy term, which-- let
00:05:49.110 --> 00:05:50.470
me just make sure-- dx is 0.
00:05:50.470 --> 00:05:53.130
I'm only going to get the
dy term, which is-- well,
00:05:53.130 --> 00:05:54.100
x is 0 there.
00:05:54.100 --> 00:05:59.270
So I'm going to get 0 times
e to the 0 plus y, dy.
00:05:59.270 --> 00:06:00.819
From 0 to y_1.
00:06:00.819 --> 00:06:03.110
Well that's nice and easy to
calculate, thank goodness.
00:06:03.110 --> 00:06:04.620
That's just 0.
00:06:04.620 --> 00:06:08.440
So all I have to do for this
one is just leave it at 0.
00:06:08.440 --> 00:06:10.690
That's everything that
happens along C_1.
00:06:10.690 --> 00:06:12.220
That's what I'm interested in.
00:06:12.220 --> 00:06:13.840
I just get 0 there.
00:06:13.840 --> 00:06:17.940
And if I integrate along
C_2, as I mentioned, dy is 0.
00:06:17.940 --> 00:06:21.490
So we don't have any
component with N*dy.
00:06:21.490 --> 00:06:25.530
We just have the component
M*dx that we're integrating.
00:06:25.530 --> 00:06:29.930
OK, so if I integrate along
C_2, I just have M*dx and M is e
00:06:29.930 --> 00:06:32.610
to the x plus y times x plus 1.
00:06:32.610 --> 00:06:34.660
And y is fixed at y_1.
00:06:34.660 --> 00:06:40.620
So it's e to the x plus
y1 times x plus 1 dx.
00:06:40.620 --> 00:06:44.127
And I'm going from 0 to x_1.
00:06:44.127 --> 00:06:46.210
I'm going to make sure I
didn't make any mistakes.
00:06:46.210 --> 00:06:48.130
I'm going to check my work here.
00:06:48.130 --> 00:06:49.520
Yes, I'm looking good.
00:06:49.520 --> 00:06:50.020
OK.
00:06:50.020 --> 00:06:51.550
So this one is 0.
00:06:51.550 --> 00:06:54.800
So all I have to do is find
an antiderivative of this.
00:06:54.800 --> 00:06:57.340
And the term-- if
I multiply through,
00:06:57.340 --> 00:06:59.930
I see that here I get
exactly the same thing when
00:06:59.930 --> 00:07:01.610
I look for an antiderivative.
00:07:01.610 --> 00:07:04.212
And here I get, I
believe, two terms when
00:07:04.212 --> 00:07:05.420
I look for an antiderivative.
00:07:05.420 --> 00:07:08.180
But I'm going to get
some cancellation.
00:07:08.180 --> 00:07:11.280
And ultimately, when I'm all
done I'm going to get this.
00:07:11.280 --> 00:07:16.660
x e to the x plus y_1
evaluated at 0 and x_1.
00:07:16.660 --> 00:07:18.026
You could do this.
00:07:18.026 --> 00:07:19.900
This is really now a
single variable problem.
00:07:19.900 --> 00:07:22.680
So I'm not going to work
out all the details,
00:07:22.680 --> 00:07:24.520
but you might want
to do an integration
00:07:24.520 --> 00:07:26.740
by parts on that first
part of it, if that helps.
00:07:26.740 --> 00:07:29.260
Or an integration by
parts on the whole thing.
00:07:29.260 --> 00:07:31.820
That would also do the trick.
00:07:31.820 --> 00:07:32.790
So what do I get here?
00:07:32.790 --> 00:07:37.800
Then I get x_1 e to
the x_1 plus y_1.
00:07:37.800 --> 00:07:40.800
And then when I put in 0 for
x here, I get 0, so that's it.
00:07:40.800 --> 00:07:47.430
So this, plus possibly a
constant, is equal to my f.
00:07:47.430 --> 00:07:51.910
So I see that in general I
get f of x, y is equal to x e
00:07:51.910 --> 00:07:55.080
to the x plus y plus a constant.
00:07:55.080 --> 00:07:58.060
So that's what I get
in the first method.
00:07:58.060 --> 00:08:01.594
So now let's use
the second method.
00:08:01.594 --> 00:08:05.770
So I should say f of x_1, y_1.
00:08:05.770 --> 00:08:08.740
In the second method, what
I do is-- M, remember,
00:08:08.740 --> 00:08:10.550
is equal to f sub x.
00:08:13.890 --> 00:08:18.100
So f sub x is equal to
M which is equal to e
00:08:18.100 --> 00:08:26.350
to the x plus y times x plus 1.
00:08:26.350 --> 00:08:28.440
So if I want to find
an antiderivative--
00:08:28.440 --> 00:08:32.230
if I want to find f, I should
take an antiderivative, right?
00:08:32.230 --> 00:08:33.930
With respect to x.
00:08:33.930 --> 00:08:35.820
And so notice I already
did that, actually.
00:08:35.820 --> 00:08:38.450
If I just put this as y,
I already did that here.
00:08:38.450 --> 00:08:40.000
And so I should
get something that
00:08:40.000 --> 00:08:44.360
looks like this: x
e to the x plus y
00:08:44.360 --> 00:08:47.699
plus possibly a function
that only depends on y.
00:08:47.699 --> 00:08:49.990
And the reason is when I take
a derivative with respect
00:08:49.990 --> 00:08:52.347
to x of this, obviously
this would be 0.
00:08:52.347 --> 00:08:53.680
So it doesn't show up over here.
00:08:53.680 --> 00:08:59.660
So this, we make sure that-- oh,
that, I shouldn't write equals.
00:08:59.660 --> 00:09:00.610
Sorry.
00:09:00.610 --> 00:09:02.260
That, I shouldn't write equals.
00:09:02.260 --> 00:09:03.110
OK?
00:09:03.110 --> 00:09:06.030
This would imply that
this is equal to f.
00:09:06.030 --> 00:09:07.310
Sorry about that.
00:09:07.310 --> 00:09:09.790
f sub x was equal to
M was equal to this.
00:09:09.790 --> 00:09:12.530
That implies-- when I take
an antiderivative of an x--
00:09:12.530 --> 00:09:16.520
that x e to the x plus y
plus g of y is equal to f.
00:09:16.520 --> 00:09:17.346
So I apologize.
00:09:17.346 --> 00:09:19.470
That wouldn't have been an
equals because obviously
00:09:19.470 --> 00:09:20.860
those two things are not equal.
00:09:20.860 --> 00:09:23.150
That would imply,
I think-- yeah,
00:09:23.150 --> 00:09:26.530
that would imply something
very bad mathematically.
00:09:26.530 --> 00:09:29.260
So make sure you understand
I put an equals sign where
00:09:29.260 --> 00:09:30.950
I should not have.
00:09:30.950 --> 00:09:32.600
This is actually a
derivative of that.
00:09:32.600 --> 00:09:35.790
So this is
antiderivative of this.
00:09:35.790 --> 00:09:38.514
So now I have a candidate for f.
00:09:38.514 --> 00:09:40.680
And so now I'm going to
take the derivative of that.
00:09:40.680 --> 00:09:44.700
And what's the derivative
of that with respect to y?
00:09:44.700 --> 00:09:51.290
So f sub y based on this
is going to be equal to x e
00:09:51.290 --> 00:09:55.650
to the x plus y
plus g prime of y.
00:09:55.650 --> 00:10:00.060
So the prime here indicates
it's in a derivative in y.
00:10:00.060 --> 00:10:04.940
And now that f sub y
should also equal N.
00:10:04.940 --> 00:10:09.820
And N equals x e
to the x plus y.
00:10:09.820 --> 00:10:11.270
So what do I get here?
00:10:11.270 --> 00:10:13.370
I see x e to the x
plus y has to equal
00:10:13.370 --> 00:10:17.050
x e to the x plus y
plus g prime of y.
00:10:17.050 --> 00:10:21.300
Which means g prime
of y is equal to 0.
00:10:21.300 --> 00:10:24.408
Which means when I take an
antiderivative of that I just
00:10:24.408 --> 00:10:25.033
get a constant.
00:10:27.620 --> 00:10:29.590
That means g of
y was a constant.
00:10:29.590 --> 00:10:34.630
So that implies that this
boxed expression right here
00:10:34.630 --> 00:10:41.350
is f of x, y if g of
y is just a constant.
00:10:41.350 --> 00:10:43.640
So let me go through
that logic one more time.
00:10:43.640 --> 00:10:45.190
I had f sub x.
00:10:45.190 --> 00:10:47.450
I took an
antiderivative to get f
00:10:47.450 --> 00:10:50.960
but it involved a constant in
x that was a function of y.
00:10:50.960 --> 00:10:54.650
I take a derivative
of that in y.
00:10:54.650 --> 00:10:57.890
I compare that to what I
know the derivative is in y.
00:10:57.890 --> 00:10:59.770
That gives me that this is 0.
00:10:59.770 --> 00:11:03.090
So its antiderivative, which
is g of y, is just a constant.
00:11:03.090 --> 00:11:09.670
And so altogether this implies
that f of x, y is equal to x e
00:11:09.670 --> 00:11:12.970
to the x plus y plus a constant.
00:11:12.970 --> 00:11:15.060
Which is exactly
what I got before.
00:11:15.060 --> 00:11:18.410
Fortunately, I got two
answers that are the same.
00:11:18.410 --> 00:11:19.700
So that's it.
00:11:19.700 --> 00:11:21.550
I'll stop there.