WEBVTT
00:00:00.000 --> 00:00:07.470
JOEL LEWIS: Hi.
00:00:07.470 --> 00:00:08.990
Welcome back to recitation.
00:00:08.990 --> 00:00:11.540
In lecture, you've been
learning about line integrals
00:00:11.540 --> 00:00:15.080
and computing them around
curves and closed curves
00:00:15.080 --> 00:00:17.140
and in various different ways.
00:00:17.140 --> 00:00:21.820
So here I have some problems
on line integrals for you.
00:00:21.820 --> 00:00:27.070
So in all cases I want C to
be the circle of radius b.
00:00:27.070 --> 00:00:30.090
So b is some constant,
some positive constant.
00:00:30.090 --> 00:00:32.320
It's the circle of radius
b centered at the origin,
00:00:32.320 --> 00:00:34.322
and I want to orient
it counterclockwise.
00:00:34.322 --> 00:00:35.780
And then what I'd
like you to do is
00:00:35.780 --> 00:00:38.360
for each of the following
vector fields F,
00:00:38.360 --> 00:00:43.160
I'd like you to compute the line
integral around C of F dot dr.
00:00:43.160 --> 00:00:47.850
So in the first case,
where F is x*i plus y*j.
00:00:47.850 --> 00:00:52.200
In the second, where F is g
of x, y times x*i plus y*j.
00:00:52.200 --> 00:00:55.680
So here g of x, y is
some scalar function.
00:00:55.680 --> 00:00:59.810
But you don't know a
formula for this function.
00:00:59.810 --> 00:01:02.445
So your answer might be in
terms of g, for example.
00:01:06.550 --> 00:01:08.550
You can assume it's a
continuous, differentiable
00:01:08.550 --> 00:01:10.440
nice function.
00:01:10.440 --> 00:01:14.630
And then the third one,
F is minus y*i plus x*j.
00:01:14.630 --> 00:01:16.690
Now before you start,
I want to give you
00:01:16.690 --> 00:01:18.570
a little suggestion,
which is often
00:01:18.570 --> 00:01:21.650
when we're given a line
integral like this,
00:01:21.650 --> 00:01:26.250
the first thing you want
to do is jump in and do
00:01:26.250 --> 00:01:28.200
a parameterization right
away for the curve,
00:01:28.200 --> 00:01:32.450
and then you get a normal
single variable integral.
00:01:32.450 --> 00:01:34.790
So what I'd like you to
do for these problems
00:01:34.790 --> 00:01:37.600
is to think about the
setup and think about
00:01:37.600 --> 00:01:40.240
whether you can do this
without ever parameterizing
00:01:40.240 --> 00:01:46.415
C, so without ever substituting
in cosine and sine or whatever.
00:01:46.415 --> 00:01:48.040
So for all three
parts of this problem.
00:01:48.040 --> 00:01:50.349
So if you can use some
sort of geometric reasoning
00:01:50.349 --> 00:01:51.890
to save yourself a
little bit of work
00:01:51.890 --> 00:01:53.960
without ever going to
the parameterization.
00:01:53.960 --> 00:01:56.614
So why don't you pause the
video, spend some time,
00:01:56.614 --> 00:01:59.030
work that out, come back, and
we can work it out together.
00:02:06.940 --> 00:02:09.190
Hopefully you had some luck
working on these problems.
00:02:09.190 --> 00:02:11.540
Let's get started.
00:02:11.540 --> 00:02:14.070
So let's do the
first problem first.
00:02:14.070 --> 00:02:17.660
Let's think about what this
vector field F looks like.
00:02:17.660 --> 00:02:19.570
This first vector field.
00:02:19.570 --> 00:02:23.360
So let me just draw a
little picture over here.
00:02:23.360 --> 00:02:28.330
So here's our
circle of radius b.
00:02:28.330 --> 00:02:32.535
And this vector field F
given by x*i plus y*j.
00:02:32.535 --> 00:02:41.250
At every point (x,
y), the vector F
00:02:41.250 --> 00:02:44.110
is the same as the position
vector of that point.
00:02:44.110 --> 00:02:47.220
So over here the
vector's like that.
00:02:47.220 --> 00:02:51.310
Over here, the
vector's like that.
00:02:51.310 --> 00:02:54.700
Up here, the vector
is like that.
00:02:54.700 --> 00:02:58.589
So these are just a
few little values of F
00:02:58.589 --> 00:02:59.630
that I've drawn in there.
00:02:59.630 --> 00:03:06.520
And so down here,
say, F is like that.
00:03:06.520 --> 00:03:09.820
So in particular, so that's
just sort of-- you know,
00:03:09.820 --> 00:03:13.590
if you wanted, you could
draw in some more vectors,
00:03:13.590 --> 00:03:17.050
get a full vector field picture.
00:03:17.050 --> 00:03:20.600
So the thing to observe
here is that a circle
00:03:20.600 --> 00:03:23.040
is a really nice curve.
00:03:23.040 --> 00:03:27.010
So the circle has the
property that the position
00:03:27.010 --> 00:03:30.900
vector at a point is orthogonal
to the tangent vector
00:03:30.900 --> 00:03:31.640
to the circle.
00:03:31.640 --> 00:03:34.180
At every point on the
circle, the tangent vector
00:03:34.180 --> 00:03:37.440
to the circle is perpendicular
to the position vector.
00:03:37.440 --> 00:03:43.190
So that means it's perpendicular
to F, because F is the same,
00:03:43.190 --> 00:03:45.960
in fact, but is parallel
to the position vector.
00:03:45.960 --> 00:03:53.175
So in Part a, you
have that F dot
00:03:53.175 --> 00:03:56.580
the tangent vector
to your curve is
00:03:56.580 --> 00:04:00.490
equal to zero at every
point on the entire curve.
00:04:00.490 --> 00:04:01.290
All right?
00:04:01.290 --> 00:04:06.530
So your field F dot your
tangent vector is always zero.
00:04:06.530 --> 00:04:13.030
So that means that the
integral around C of F dot dr,
00:04:13.030 --> 00:04:17.610
well, we know that dr is T ds.
00:04:17.610 --> 00:04:20.090
So this is F dot T ds.
00:04:20.090 --> 00:04:21.420
But that's just zero.
00:04:21.420 --> 00:04:24.649
It's just an integral and the
integrand is zero everywhere.
00:04:24.649 --> 00:04:27.190
And whenever you take a definite
integral of something that's
00:04:27.190 --> 00:04:29.040
zero everywhere, you get zero.
00:04:29.040 --> 00:04:30.632
So this is just zero right away.
00:04:30.632 --> 00:04:32.840
We didn't have to parameterize
the curve or anything.
00:04:32.840 --> 00:04:34.600
We just had to look
at this picture
00:04:34.600 --> 00:04:37.110
to sort of understand
that this kind of field,
00:04:37.110 --> 00:04:38.870
it's called a
radial vector field,
00:04:38.870 --> 00:04:43.830
where the vector F is always
pointed directly outwards.
00:04:43.830 --> 00:04:45.600
When you integrate a
radial vector field
00:04:45.600 --> 00:04:47.680
around a circle
centered at the origin,
00:04:47.680 --> 00:04:50.160
you get zero, because the
contribution at every point
00:04:50.160 --> 00:04:51.720
is zero.
00:04:51.720 --> 00:04:52.930
So that's Part a.
00:04:52.930 --> 00:04:56.050
Part b is actually
exactly the same.
00:04:56.050 --> 00:04:59.030
If we look back at our
formula over here in Part b,
00:04:59.030 --> 00:05:03.370
we have that F is given by
some function g of x, y times
00:05:03.370 --> 00:05:05.040
x i hat plus y j hat.
00:05:05.040 --> 00:05:07.230
Well, what is this
g of x, y doing?
00:05:07.230 --> 00:05:08.660
It's just rescaling.
00:05:08.660 --> 00:05:11.180
It's telling you at
every point you can scale
00:05:11.180 --> 00:05:13.530
that vector by some amount.
00:05:13.530 --> 00:05:17.130
So if we looked over at this
picture, maybe over here
00:05:17.130 --> 00:05:19.350
you would scale some of
these vectors to be longer,
00:05:19.350 --> 00:05:20.849
and over here they
might be shorter,
00:05:20.849 --> 00:05:22.680
or you might switch
them to be negative,
00:05:22.680 --> 00:05:25.930
but you don't change the
direction of any vector
00:05:25.930 --> 00:05:27.970
in the field from Part a.
00:05:27.970 --> 00:05:29.630
You just change their length.
00:05:29.630 --> 00:05:31.640
So you still have a
radial vector field.
00:05:31.640 --> 00:05:34.250
And you still have the
property that at every point
00:05:34.250 --> 00:05:36.760
on our curve, the tangent
vector to the curve
00:05:36.760 --> 00:05:40.150
is orthogonal to the vector
F. So the tangent vector
00:05:40.150 --> 00:05:42.510
is orthogonal to F, so
that means you again
00:05:42.510 --> 00:05:44.890
have F dot T is equal to zero.
00:05:44.890 --> 00:05:49.220
And so F dot dr is
also equal to 0 ds,
00:05:49.220 --> 00:05:51.380
and so when you integrate
that, you just get zero.
00:05:51.380 --> 00:05:53.560
So that's also what
happens in Part b.
00:05:53.560 --> 00:05:58.000
So Part b, I'm just
going to write ditto.
00:05:58.000 --> 00:06:01.600
The exact same reasoning applies
in Part b as applied in Part a.
00:06:01.600 --> 00:06:05.614
And you also get
zero as your integral
00:06:05.614 --> 00:06:07.530
without having to
parameterize, without having
00:06:07.530 --> 00:06:09.890
to do any tricky
calculations at all.
00:06:09.890 --> 00:06:10.390
All right.
00:06:10.390 --> 00:06:12.600
So let's now look at Part c.
00:06:12.600 --> 00:06:16.330
I'm going to draw
another little picture.
00:06:16.330 --> 00:06:20.620
So in Part c,
there's your curve.
00:06:20.620 --> 00:06:25.270
At the point (x,y)-- so I'm
going to draw some choices of F
00:06:25.270 --> 00:06:26.370
again.
00:06:26.370 --> 00:06:30.500
So in Part c, at
the point (x,y),
00:06:30.500 --> 00:06:42.600
your vector field F is
minus y i hat plus x j hat.
00:06:42.600 --> 00:06:44.590
Now if you draw
that on the picture
00:06:44.590 --> 00:06:49.480
here, over there
that's that vector.
00:06:49.480 --> 00:06:52.270
Over here, so at the
point (0,1), say,
00:06:52.270 --> 00:06:54.970
that gives you the
vector [-1, 0].
00:06:54.970 --> 00:06:57.569
So that's horizontal
to the left.
00:06:57.569 --> 00:06:58.360
Here are some more.
00:06:58.360 --> 00:07:01.520
There's one there,
there's one there.
00:07:01.520 --> 00:07:04.450
There's another one
over here and so on.
00:07:04.450 --> 00:07:08.460
In fact, what you'll notice
is that this vector F is just
00:07:08.460 --> 00:07:12.280
parallel to the tangent vector
of the circle everywhere.
00:07:12.280 --> 00:07:15.360
This field is a
tangential field.
00:07:15.360 --> 00:07:18.236
It's always pointing
parallel to the curve.
00:07:18.236 --> 00:07:18.735
OK?
00:07:22.270 --> 00:07:24.020
It's perpendicular to
the position vector.
00:07:24.020 --> 00:07:26.350
It's in the same direction
as the tangent vector
00:07:26.350 --> 00:07:27.050
at every point.
00:07:27.050 --> 00:07:29.300
So this is something that
you've seen before, I think.
00:07:29.300 --> 00:07:31.750
That this vector
field is giving you
00:07:31.750 --> 00:07:35.160
a sort of nice rotating motion.
00:07:35.160 --> 00:07:39.640
You know, at every point it's
circulating counterclockwise.
00:07:39.640 --> 00:07:40.610
So what does that mean?
00:07:40.610 --> 00:07:44.490
Well, again, it's not exactly
the same as Part a and b,
00:07:44.490 --> 00:07:47.060
but again we'll be able
to compute this integral
00:07:47.060 --> 00:07:48.280
without parameterizing.
00:07:48.280 --> 00:07:48.960
Why?
00:07:48.960 --> 00:07:54.170
Because F dot T in this
case-- well, so, let's see.
00:07:54.170 --> 00:07:57.070
What is the norm of F?
00:07:57.070 --> 00:07:59.530
The magnitude of F is
just the square root of
00:07:59.530 --> 00:08:00.940
(x squared plus y squared).
00:08:00.940 --> 00:08:04.850
So on our circle of radius b,
that means the magnitude of F
00:08:04.850 --> 00:08:05.940
is b.
00:08:05.940 --> 00:08:10.020
And the magnitude of T, the
unit tangent vector, is 1,
00:08:10.020 --> 00:08:11.562
and they point in
the same direction.
00:08:11.562 --> 00:08:13.186
So when you have two
vectors that point
00:08:13.186 --> 00:08:15.090
in the same direction,
their dot product
00:08:15.090 --> 00:08:18.190
is just the product
of their magnitudes.
00:08:18.190 --> 00:08:21.340
So that means F dot
T is equal to b.
00:08:21.340 --> 00:08:23.630
This is a constant.
00:08:23.630 --> 00:08:25.050
F dot T is equal to b.
00:08:25.050 --> 00:08:32.970
So when you integrate
around the circle, F dot dr,
00:08:32.970 --> 00:08:38.150
well, this is equal
to the integral
00:08:38.150 --> 00:08:43.760
around a circle of F dot the
tangent vector with respect
00:08:43.760 --> 00:08:44.900
to arc length.
00:08:44.900 --> 00:08:47.810
But this integrand, F
dot the tangent vector,
00:08:47.810 --> 00:08:49.250
is this constant b.
00:08:49.250 --> 00:08:54.210
So you're integrating
over the curve b ds.
00:08:54.210 --> 00:08:56.210
And when you integrate
a constant ds,
00:08:56.210 --> 00:08:58.180
well, that just gives
you the total arc length.
00:08:58.180 --> 00:09:01.730
So this is b times
the total arc length.
00:09:01.730 --> 00:09:04.310
And this is a
circle of radius b.
00:09:04.310 --> 00:09:12.060
So that's b times 2 pi b, which
we could also write as 2 pi
00:09:12.060 --> 00:09:13.734
b squared.
00:09:13.734 --> 00:09:14.400
So there you go.
00:09:14.400 --> 00:09:17.890
So in this third case, you have
a nice tangential vector field.
00:09:17.890 --> 00:09:20.700
So that means the
integrand actually
00:09:20.700 --> 00:09:23.630
works out to be constant.
00:09:23.630 --> 00:09:25.226
Because the integrand
is constant,
00:09:25.226 --> 00:09:27.100
we don't ever have to
parameterize the curve.
00:09:27.100 --> 00:09:28.766
We can just use the
fact that we already
00:09:28.766 --> 00:09:32.170
know its arc length in order
to compute this integral.
00:09:32.170 --> 00:09:35.430
Again, we could do
all of these integrals
00:09:35.430 --> 00:09:37.660
if we wanted by
parameterizing the circle,
00:09:37.660 --> 00:09:42.110
by x equals b cosine
t, y equals b sine t,
00:09:42.110 --> 00:09:45.190
and going through and writing
this as an integral from t
00:09:45.190 --> 00:09:47.510
equals 0 to 2 pi, and so on.
00:09:47.510 --> 00:09:50.687
But these are
examples of problems
00:09:50.687 --> 00:09:53.020
where it's helpful to think
about what's going on first,
00:09:53.020 --> 00:09:56.320
see if you can understand the
geometry of your situation.
00:09:56.320 --> 00:10:00.190
And sometimes you'll have
a problem like this where
00:10:00.190 --> 00:10:05.080
you'll-- either in this class
or elsewhere in your life--
00:10:05.080 --> 00:10:08.000
where something that might
seem complicated has a simple
00:10:08.000 --> 00:10:09.410
geometric explanation.
00:10:09.410 --> 00:10:11.010
And so when that
does happen, it's
00:10:11.010 --> 00:10:12.640
nice when you can
take advantage of it.
00:10:12.640 --> 00:10:14.639
Sometimes that won't
happen and sometimes you'll
00:10:14.639 --> 00:10:17.020
have to do the parameterization
and the computation.
00:10:17.020 --> 00:10:20.000
But in these cases we have
these nice three examples
00:10:20.000 --> 00:10:23.240
where with a radial
vector field,
00:10:23.240 --> 00:10:25.550
you get that the
integrand is always zero,
00:10:25.550 --> 00:10:29.120
or with a tangential
vector field, you have
00:10:29.120 --> 00:10:30.690
that the integrand is constant.
00:10:30.690 --> 00:10:31.190
All right.
00:10:31.190 --> 00:10:33.134
So, I'll stop there.