CHRISTINE BREINER: Welcome back to recitation.
In this video, I'd like us to work on the following problem.
So the problem is as follows.
For which of the following vector fields is the domain
where each vector field is defined and continuously
differentiable a simply connected region.
So there's a lot there.
I'm going to break that down first, and then show you the
vector fields.
So we're starting with some different vector fields.
And we want to determine first the domain for each vector
field where it is both defined and continuously
differentiable.
And then once you've determined that domain, the
next object is to determine whether or not that region is
simply connected.
So there are two parts for each of these problems.
So again, the first thing is you want to determine all of
the values for which the vector field is both defined
and continuously differentiable.
You want to look at that region that contains all those
values, and you want to determine if that region is
simply connected.
So there are four different vector fields, and I'll just
point them out here.
They're all in the plane.
So the first one is root x i plus root y j.
The second one is i plus j, divided by the square root of
1 minus x squared minus y squared.
The third one looks fairly similar to the second one, but
it's i plus j, divided by the square root of the quantity x
squared plus y squared minus 1.
And the fourth one is i plus j times the quantity natural log
of r squared-- natural log of x squared plus y squared.
So there are four different vector fields here.
You have do two things for each one.
So find the domain where it's defined and continuously
differentiable, and then determine if that domain is
simply connected.
So why don't you pause the video, work on these, and then
when you're ready to see what I did, you can bring
the video back up.
OK, welcome back.
So we're going to do this problem one
vector field at a time.
We're going to take each vector field, I'm going to
determine where it's defined and differentiable, and then
I'm going to determine if that region is simply connected.
So we're going to do each one separately.
So I'm going to start off with a, which was root x
i plus root y j.
And I want to point out first, for the function f of x is
equal to square root of x, it is defined for all x greater
than or equal to 0, and it is differentiable for x
greater than 0.
I left out the T. There we go.
So it is both defined and differentiable when x is
greater than 0.
And now if I replace this with a y, the same thing is true
for y greater than or equal to 0, and y greater than 0.
So we know the region where this vector field is both
defined and differentiable is when x is greater than 0 and y
is greater than 0.
So we need the region--
let me draw it this way--
the region that is the first quadrant in the xy plane.
And it's not including the x-axis or the y-axis.
So it's this fully shaded region.
OK, that is the region where it's defined and
differentiable.
If I was going to write that precisely, I would say
something like, all (x, y) with x greater than 0 and y
greater than 0.
Right?
You need both.
So it's exactly the (x, y) pairs with x positive and y
positive, and that's the region.
And now the question is, is that region simply connected?
Well, the way we think about simply connectedness from what
you've seen in class, is you want to show that if you take
a closed curve that's contained in the region, that
everything on the interior of that closed curve is also in
the region.
And you notice that is in fact true for this first quadrant.
Any closed curve I draw that's in the region, all the points
on the interior of the curve are also in the region.
So this domain where it's defined and differentiable is
simply connected.
So the first one is simply connected.
All right.
So that's part a.
Part b--
let me rewrite that one so we don't have to zoom over to the
other side--
this was i plus j divided by the function square root of 1
minus x squared minus y squared.
Well, we already know that the square root function is
defined as long as the inside function is greater than or
equal to 0.
Because it's in the denominator, we actually need
this function 1 minus x squared minus y squared to be
greater than 0.
And that's also where it's going to be differentiable.
The differentiable and the defined regions are exactly
the same, and they're both where 1 minus x squared minus
y squared is greater than 0.
Right?
The function is only defined as long as this quantity is
greater than 0, and that's exactly where it's
differentiable as well.
And so what does this correspond to?
Well, if you think about it, this is actually 1 is greater
than x squared plus y squared.
And what are the points that look like this?
Well, the x- and y-points that satisfy this inequality are
the x- and y-values that are on the
interior of the unit circle.
So if I draw a picture of that.
Let me try and dot the unit circle.
It's not containing the boundary, but it's all the
points that are on the interior of the unit circle.
Every point here, when I take the ordered pair (x, y) and
it's on the interior of the unit circle, it satisfies this
inequality.
Those are the only points that do that.
So this is the region that has this vector field both
differentiable and defined.
Right?
Now, is this region simply connected?
It is, again for the same reason.
Because if you take any closed curve here, and you look at
the interior of that closed curve, every point on the
interior of that closed curve is also in the region.
Right?
So it is also simply connected.
OK.
So we had two so far that were simply connected.
They were different looking regions, but ultimately they
both had any closed curve-- the interior of it-- was all
contained in the region that we were interested in.
So now the third one we have is somewhat similar-looking to
part b, except that what's in the square
root is a little different.
So now we can use exactly the same logic as what
we did in part b.
And what we see is by the exact same logic that this
vector field will be defined and differentiable as long as
x squared plus y squared minus 1 is positive.
Because that's where the square root function is
differentiable, and that's also where 1 divided by the
square root of this thing is defined.
So they correspond to exactly the same regions.
And that is x squared plus y squared greater than 1.
So if you think about that, what we did previously was we
had x squared plus y squared less than 1.
So obviously if you want x squared plus y squared greater
than 1, we're taking all the (x, y) pairs that are outside
the unit circle.
So again, we take the unit circle.
We don't include the unit circle, because that's where x
squared plus y squared equals 1.
And then we want all of the values outside of that region.
So this extends off to infinity.
All of the values outside of that region.
Now, is this region simply connected?
It is not.
And the point is that while you do have some curves, that
if I take a closed curve, all of the points on the interior
of that closed curve are in the region.
There are some curves for which that's not true.
For example, if I take the circle of radius--
what does that look like--
2, 1 1/2, something like that?
If I look at all of the points on the
interior of this curve--
I'm going to try and shade it without getting rid of
everything, so you can see still what's behind--
if I look at all those points, notice in particular, there
are a bunch of points-- for instance, this one here, this
one here, and this one here-- all the ones inside the unit
circle, are on the interior of this curve, but they're not in
the region.
Right?
I'll shade it extra dark.
All the points that are in here, that are inside the unit
circle, are actually still on the interior of this curve,
but they're not in the region.
And while there are some curves for which everything on
the inside is in the region, but there are some curves for
which it's not true, and that is what we know about
not-simply connectedness.
So we know this one is not simply connected.
OK.
So now we have one left.
And the last one was i plus j, times natural log of x squared
plus y squared.
So in this vector field, what I'm really interested in is
the behavior of this function natural log of x
squared plus y squared.
And the point I want to make is that natural log is defined
as long as the input value is positive, and it is
differentiable everywhere it's defined.
And so this function will be both defined and
differentiable as long as x squared plus y squared is
greater than 0.
So it's defined for all of these values, and natural log
is differentiable everywhere it's defined.
So it's also differentiable for all of these values.
And so we see that this vector field is defined and
differentiable everywhere except at one point.
And so let me draw--
this is zooming in showing that it's missing that point--
I want to make it extra large.
But it's really only missing one point.
And what point is that?
That point is the origin.
So everywhere except the origin.
Maybe I should make it smaller, because maybe it
looks like it's missing a whole circle.
It's just missing the point.
It's just missing the origin.
But every other point on the xy plane is a place where this
vector field is differentiable and defined.
Right?
So it's only missing that one point, but that still gives us
the fact that this region is not simply connected.
And again, it's exactly the same type of logic as the
previous problem, that I could draw curves, where every point
on the interior is contained in the region.
But there are curves that also fail.
Right?
If I draw a curve that contains the origin, every
point on the interior of this region except the origin, is
contained in the domain of interest, right?
But because the interior of the curve contains the origin,
I know that this region is, in fact, not simply connected.
So if I look at all of R2--
so if I look at all the x- and y-values except x equals 0 and
y equals 0, except the origin--
I get a region that's not simply connected.
And this one is maybe a little tricky, so I'm going to say it
one more time.
OK?
So while there are some curves that we can see some portions
of this region behave like simply connected regions, when
you're around the origin, any curve you take around the
origin is going to contain the origin on its interior.
Right?
But the origin is not in our domain of interest. And
therefore, there are curves that we can take that their
interior contains a point that's not in the region.
And that's what it means to be not simply connected.
OK?
So let me go back to the beginning and just remind you
again real quickly what we did here.
We had these four vector fields.
We wanted to do two things with all of them.
We wanted to first find the regions where they were
defined and differentiable, and then determine if those
regions were simply connected.
And so we had two examples where the
regions were simply connected.
And then I gave you two examples where the regions
were not simply connected.
And so hopefully this was informative for how we can
understand that vector fields are not necessarily always
defined everywhere, but also to understand what this simply
connected region term actually means.
And I guess that's where I'll stop.