WEBVTT
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CHRISTINE BREINER: Welcome
back to recitation.
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In this video, I'd like us to
work on the following problem.
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So the problem is as follows.
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For which of the
following vector fields
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is the domain where
each vector field
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is defined and
continuously differentiable
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a simply connected region.
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So there's a lot there.
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I'm going to break that
down first, and then
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show you the vector fields.
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So we're starting with some
different vector fields.
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And we want to determine first
the domain for each vector
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field where it is both
defined and continuously
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differentiable.
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And then once you've
determined that domain,
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the next object is to determine
whether or not that region is
00:00:45.930 --> 00:00:47.550
simply connected.
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So there are two parts for
each of these problems.
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So again, the first
thing is you want
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to determine all of the values
for which the vector field is
00:00:55.130 --> 00:00:57.900
both defined and
continuously differentiable.
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You want to look at that region
that contains all those values,
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and you want to determine
if that region is simply
00:01:03.100 --> 00:01:04.270
connected.
00:01:04.270 --> 00:01:06.140
So there are four
different vector fields,
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and I'll just point
them out here.
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They're all in the plane.
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So the first one is
root x i plus root y j.
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The second one is i plus j,
divided by the square root of 1
00:01:18.180 --> 00:01:21.100
minus x squared minus y squared.
00:01:21.100 --> 00:01:23.810
The third one looks fairly
similar to the second one,
00:01:23.810 --> 00:01:26.050
but it's i plus j,
divided by the square root
00:01:26.050 --> 00:01:29.470
of the quantity x squared
plus y squared minus 1.
00:01:29.470 --> 00:01:34.180
And the fourth one is i plus j
times the quantity natural log
00:01:34.180 --> 00:01:38.312
of r squared-- natural log
of x squared plus y squared.
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So there are four different
vector fields here.
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You have do two
things for each one.
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So find the domain where
it's defined and continuously
00:01:44.730 --> 00:01:46.512
differentiable,
and then determine
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if that domain is
simply connected.
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So why don't you pause
the video, work on these,
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and then when you're
ready to see what I did,
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you can bring the video back up.
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OK, welcome back.
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So we're going to do
this problem one vector
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field at a time.
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We're going to take
each vector field,
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I'm going to determine
where it's defined
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and differentiable,
and then I'm going
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to determine if that
region is simply connected.
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So we're going to do
each one separately.
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So I'm going to start
off with a, which
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was root x i plus root y j.
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And I want to point out
first, for the function
00:02:33.600 --> 00:02:40.060
f of x is equal to square root
of x, it is defined for all x
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greater than or equal to 0,
and it is differentiable for x
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greater than 0.
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I left out the T. There we go.
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So it is both defined
and differentiable
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when x is greater than 0.
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And now if I replace
this with a y,
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the same thing is true for y
greater than or equal to 0,
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and y greater than 0.
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So we know the region
where this vector field is
00:03:14.950 --> 00:03:18.100
both defined and differentiable
is when x is greater
00:03:18.100 --> 00:03:20.490
than 0 and y is greater than 0.
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So we need the region-- let me
draw it this way-- the region
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that is the first
quadrant in the xy-plane.
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And it's not including
the x-axis or the y-axis.
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So it's this full shaded region.
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OK, that is the
region where it's
00:03:40.350 --> 00:03:41.100
defined and differentiable.
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If I was going to
write that precisely,
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I would say something like, all
(x, y) with x greater than 0
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and y greater than 0.
00:03:55.181 --> 00:03:55.680
Right?
00:03:55.680 --> 00:03:57.220
You need both.
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So it's exactly the (x, y) pairs
with x positive and y positive,
00:04:00.370 --> 00:04:01.506
and that's the region.
00:04:01.506 --> 00:04:03.880
And now the question is, is
that region simply connected?
00:04:03.880 --> 00:04:05.410
Well, the way we
think about simply
00:04:05.410 --> 00:04:08.810
connectedness, from what
you've seen in class, is you
00:04:08.810 --> 00:04:12.290
want to show that if you
take a closed curve that's
00:04:12.290 --> 00:04:14.070
contained in the
region, that everything
00:04:14.070 --> 00:04:16.790
on the interior of that closed
curve is also in the region.
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And you notice that is in fact
true for this first quadrant.
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Any closed curve I draw
that's in the region, all
00:04:23.766 --> 00:04:25.390
the points on the
interior of the curve
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are also in the region.
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So this domain where it's
defined and differentiable
00:04:31.410 --> 00:04:32.390
is simply connected.
00:04:35.137 --> 00:04:36.845
So the first one, it
is simply connected.
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All right.
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So that's part a.
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Part b-- let me
rewrite that one so
00:04:48.010 --> 00:04:58.890
we don't have to zoom
over to the other side--
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this was i plus j divided by
the function square root of 1
00:05:02.240 --> 00:05:03.970
minus x squared minus y squared.
00:05:03.970 --> 00:05:08.530
Well, we already know that
the square root function
00:05:08.530 --> 00:05:12.800
is defined as long as the
inside function is greater
00:05:12.800 --> 00:05:13.787
than or equal to 0.
00:05:13.787 --> 00:05:15.620
Because it's in the
denominator, we actually
00:05:15.620 --> 00:05:18.265
need this function 1
minus x squared minus y
00:05:18.265 --> 00:05:20.040
squared to be greater than 0.
00:05:20.040 --> 00:05:22.290
And that's also where it's
going to be differentiable.
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The differentiable and
the defined regions
00:05:29.540 --> 00:05:30.930
are exactly the
same, and they're
00:05:30.930 --> 00:05:34.110
both where 1 minus x
squared minus y squared
00:05:34.110 --> 00:05:35.231
is greater than 0.
00:05:35.231 --> 00:05:35.730
Right?
00:05:35.730 --> 00:05:38.224
The function is only defined
as long as this quantity is
00:05:38.224 --> 00:05:39.890
greater than 0, and
that's exactly where
00:05:39.890 --> 00:05:41.400
it's differentiable as well.
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And so what does
this correspond to?
00:05:42.900 --> 00:05:44.275
Well, if you think
about it, this
00:05:44.275 --> 00:05:47.250
is actually 1 is greater than
x squared plus y squared.
00:05:47.250 --> 00:05:49.340
And what are the points
that look like this?
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Well, the x- and y-points
that satisfy this inequality
00:05:52.890 --> 00:05:56.560
are the x- and y-values that
are on the interior of the unit
00:05:56.560 --> 00:05:57.330
circle.
00:05:57.330 --> 00:05:58.630
So if I draw a picture of that.
00:06:01.430 --> 00:06:05.520
Let me try and dot
the unit circle.
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It's not containing
the boundary,
00:06:07.440 --> 00:06:09.960
but it's all the points that
are on the interior of the unit
00:06:09.960 --> 00:06:10.460
circle.
00:06:10.460 --> 00:06:15.777
Every point here, when I
take the ordered pair (x, y)
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and it's on the interior
of the unit circle,
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it satisfies this inequality.
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Those are the only
points that do that.
00:06:21.130 --> 00:06:23.960
So this is the region
that has this vector
00:06:23.960 --> 00:06:28.540
field both differentiable
and defined.
00:06:28.540 --> 00:06:29.040
Right?
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Now, is this region
simply connected?
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It is, again for
the same reason.
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Because if you take
any closed curve here,
00:06:37.060 --> 00:06:41.040
and you look at the interior
of that closed curve,
00:06:41.040 --> 00:06:43.350
every point on the interior
of that closed curve
00:06:43.350 --> 00:06:45.060
is also in the region.
00:06:45.060 --> 00:06:45.560
Right?
00:06:45.560 --> 00:06:46.995
So it is also simply connected.
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OK.
00:06:55.290 --> 00:06:58.460
So we had two so far that
were simply connected.
00:06:58.460 --> 00:07:00.120
They were
different-looking regions,
00:07:00.120 --> 00:07:03.730
but ultimately they both
had any closed curve,
00:07:03.730 --> 00:07:06.460
the interior of it was all
contained in the region
00:07:06.460 --> 00:07:08.920
that we were interested in.
00:07:08.920 --> 00:07:11.040
So now the third one
we have is somewhat
00:07:11.040 --> 00:07:16.790
similar-looking to
part b, except that
00:07:16.790 --> 00:07:21.940
what's in the square root
is a little different.
00:07:21.940 --> 00:07:24.170
So now we can use
exactly the same logic
00:07:24.170 --> 00:07:25.900
as what we did in part b.
00:07:25.900 --> 00:07:28.350
And what we see is by
the exact same logic
00:07:28.350 --> 00:07:31.830
that this vector field will
be defined and differentiable
00:07:31.830 --> 00:07:38.050
as long as x squared plus y
squared minus 1 is positive.
00:07:38.050 --> 00:07:40.570
Because that's where the
square root function is
00:07:40.570 --> 00:07:42.560
differentiable, and
that's also where
00:07:42.560 --> 00:07:46.170
1 divided by the square root
of this thing is defined.
00:07:46.170 --> 00:07:49.510
So they correspond to
exactly the same regions.
00:07:49.510 --> 00:07:54.000
And that is x squared plus
y squared greater than 1.
00:07:54.000 --> 00:07:56.090
So if you think about that,
what we did previously
00:07:56.090 --> 00:07:58.384
was we had x squared plus
y squared less than 1.
00:07:58.384 --> 00:07:59.800
So obviously if
you want x squared
00:07:59.800 --> 00:08:03.130
plus y squared greater than
1, we're taking all the (x, y)
00:08:03.130 --> 00:08:06.940
pairs that are outside
the unit circle.
00:08:06.940 --> 00:08:10.620
So again, we take
the unit circle.
00:08:10.620 --> 00:08:12.180
We don't include
the unit circle,
00:08:12.180 --> 00:08:15.280
because that's where x squared
plus y squared equals 1.
00:08:15.280 --> 00:08:21.910
And then we want all of the
values outside of that region.
00:08:21.910 --> 00:08:24.800
So this extends off to infinity.
00:08:24.800 --> 00:08:26.790
All of the values
outside of that region.
00:08:26.790 --> 00:08:28.980
Now, is this region
simply connected?
00:08:28.980 --> 00:08:30.230
It is not.
00:08:30.230 --> 00:08:33.990
And the point is that while
you do have some curves,
00:08:33.990 --> 00:08:36.830
that if I take a
closed curve, all
00:08:36.830 --> 00:08:39.980
of the points on the
interior of that closed curve
00:08:39.980 --> 00:08:42.790
are in the region, there are
some curves for which that's
00:08:42.790 --> 00:08:43.870
not true.
00:08:43.870 --> 00:08:47.510
For example, if
I take the circle
00:08:47.510 --> 00:08:51.510
of radius-- what does that
look like-- 2, 1 and 1/2,
00:08:51.510 --> 00:08:53.870
something like that?
00:08:53.870 --> 00:08:57.690
If I look at all of the points
on the interior of this curve--
00:08:57.690 --> 00:09:00.560
I'm going to try and shade
it without getting rid
00:09:00.560 --> 00:09:02.040
of everything, so
you can see still
00:09:02.040 --> 00:09:05.480
what's behind-- if I look
at all those points, notice
00:09:05.480 --> 00:09:07.510
in particular, there
are a bunch of points--
00:09:07.510 --> 00:09:09.510
for instance, this one
here, this one here,
00:09:09.510 --> 00:09:12.070
and this one here-- all the
ones inside the unit circle,
00:09:12.070 --> 00:09:14.360
are on the interior
of this curve,
00:09:14.360 --> 00:09:16.970
but they're not in the region.
00:09:16.970 --> 00:09:19.570
Right?
00:09:19.570 --> 00:09:21.270
I'll shade it extra dark.
00:09:21.270 --> 00:09:24.930
All the points that
are in here, that
00:09:24.930 --> 00:09:29.140
are inside the unit
circle, are actually still
00:09:29.140 --> 00:09:32.320
on the interior of this curve,
but they're not in the region.
00:09:32.320 --> 00:09:35.540
And while there are some
curves for which everything
00:09:35.540 --> 00:09:38.020
on the inside is in
the region, but there
00:09:38.020 --> 00:09:40.200
are some curves for
which it's not true,
00:09:40.200 --> 00:09:44.430
and that is what we know about
not-simply connectedness.
00:09:44.430 --> 00:09:46.425
So we know this one is
not simply connected.
00:09:52.460 --> 00:09:52.960
OK.
00:09:55.840 --> 00:09:58.730
So now we have one left.
00:09:58.730 --> 00:10:05.030
And the last one was i
plus j, times natural log
00:10:05.030 --> 00:10:07.465
of x squared plus y squared.
00:10:10.100 --> 00:10:13.640
So in this vector field,
what I'm really interested in
00:10:13.640 --> 00:10:16.140
is the behavior of this function
natural log of x squared
00:10:16.140 --> 00:10:17.540
plus y squared.
00:10:17.540 --> 00:10:21.660
And the point I want to make
is that natural log is defined
00:10:21.660 --> 00:10:25.120
as long as the input
value is positive,
00:10:25.120 --> 00:10:28.220
and it is differentiable
everywhere it's defined.
00:10:28.220 --> 00:10:32.410
And so this function will be
both defined and differentiable
00:10:32.410 --> 00:10:38.230
as long as x squared plus y
squared is greater than 0.
00:10:38.230 --> 00:10:40.709
So it's defined for
all of these values,
00:10:40.709 --> 00:10:42.750
and natural log is
differentiable everywhere it's
00:10:42.750 --> 00:10:43.250
defined.
00:10:43.250 --> 00:10:46.290
So it's also differentiable
for all of these values.
00:10:46.290 --> 00:10:49.250
And so we see that
this vector field
00:10:49.250 --> 00:10:53.620
is defined and differentiable
everywhere except at one point.
00:10:53.620 --> 00:10:58.760
And so let me draw--
this is a zooming
00:10:58.760 --> 00:11:00.524
in of that it's
missing that point-- I
00:11:00.524 --> 00:11:01.690
want to make it extra large.
00:11:01.690 --> 00:11:03.922
But it's really only
missing one point.
00:11:03.922 --> 00:11:04.880
And what point is that?
00:11:04.880 --> 00:11:06.950
That point is the origin.
00:11:06.950 --> 00:11:09.600
So everywhere except the origin.
00:11:09.600 --> 00:11:11.600
Maybe I should make it
smaller, because maybe it
00:11:11.600 --> 00:11:12.920
looks like it's
missing a whole circle.
00:11:12.920 --> 00:11:14.260
It's just missing the point.
00:11:14.260 --> 00:11:17.410
It's just missing the origin.
00:11:17.410 --> 00:11:21.190
But every other
point on the xy-plane
00:11:21.190 --> 00:11:24.020
is a place where
this vector field
00:11:24.020 --> 00:11:26.970
is differentiable and defined.
00:11:26.970 --> 00:11:27.950
Right?
00:11:27.950 --> 00:11:30.600
So it's only missing
that one point,
00:11:30.600 --> 00:11:34.160
but that still gives us the
fact that this region is not
00:11:34.160 --> 00:11:35.340
simply connected.
00:11:35.340 --> 00:11:37.420
And again, it's
exactly the same type
00:11:37.420 --> 00:11:39.220
of logic as the
previous problem,
00:11:39.220 --> 00:11:43.960
that I could draw curves, where
every point on the interior
00:11:43.960 --> 00:11:45.450
is contained in the region.
00:11:45.450 --> 00:11:48.240
But there are curves
that also fail.
00:11:48.240 --> 00:11:48.740
Right?
00:11:48.740 --> 00:11:52.930
If I draw a curve that contains
the origin, every point
00:11:52.930 --> 00:11:55.870
on the interior of this
region except the origin,
00:11:55.870 --> 00:11:59.180
is contained in the
domain of interest, right?
00:11:59.180 --> 00:12:04.570
But because the interior of
the curve contains the origin,
00:12:04.570 --> 00:12:08.080
I know that this region is,
in fact, not simply connected.
00:12:08.080 --> 00:12:12.177
So if I look at all of R^2--
so if I look at all the (x,
00:12:12.177 --> 00:12:16.120
y)-values except x
equals 0 and y equals 0--
00:12:16.120 --> 00:12:18.781
except the origin-- I get
a region that's not simply
00:12:18.781 --> 00:12:19.280
connected.
00:12:22.847 --> 00:12:24.430
And this one is maybe
a little tricky,
00:12:24.430 --> 00:12:27.020
so I'm going to say
it one more time.
00:12:27.020 --> 00:12:27.740
OK?
00:12:27.740 --> 00:12:30.190
So while there are
some curves that we
00:12:30.190 --> 00:12:32.600
can see some portions
of this region
00:12:32.600 --> 00:12:36.850
behave like simply connected
regions, when you're
00:12:36.850 --> 00:12:39.990
around the origin, any curve
you take around the origin
00:12:39.990 --> 00:12:42.650
is going to contain the
origin on its interior.
00:12:42.650 --> 00:12:43.350
Right?
00:12:43.350 --> 00:12:47.190
But the origin is not in
our domain of interest.
00:12:47.190 --> 00:12:49.610
And therefore, there
are curves that we
00:12:49.610 --> 00:12:52.330
can take that their interior
contains a point that's
00:12:52.330 --> 00:12:53.480
not in the region.
00:12:53.480 --> 00:12:56.541
And that's what it means
to be not simply connected.
00:12:56.541 --> 00:12:57.040
OK?
00:12:57.040 --> 00:12:58.340
So let me go back
to the beginning
00:12:58.340 --> 00:13:00.673
and just remind you again
real quickly what we did here.
00:13:03.440 --> 00:13:05.530
We had these four vector fields.
00:13:05.530 --> 00:13:07.610
We wanted to do two
things with all of them.
00:13:07.610 --> 00:13:11.010
We wanted to first find the
regions where they were defined
00:13:11.010 --> 00:13:14.560
and differentiable, and then
determine if those regions were
00:13:14.560 --> 00:13:15.890
simply connected.
00:13:15.890 --> 00:13:18.210
And so we had two
examples where the regions
00:13:18.210 --> 00:13:19.670
were simply connected.
00:13:19.670 --> 00:13:22.300
And then I gave you two examples
where the regions were not
00:13:22.300 --> 00:13:23.720
simply connected.
00:13:23.720 --> 00:13:25.900
And so hopefully this
was informative for how
00:13:25.900 --> 00:13:29.080
we can understand that vector
fields are not necessarily
00:13:29.080 --> 00:13:31.330
always defined
everywhere, but also
00:13:31.330 --> 00:13:35.730
to understand what this simply
connected region term actually
00:13:35.730 --> 00:13:37.120
means.
00:13:37.120 --> 00:13:39.600
And I guess that's
where I'll stop.