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## Multivariable calculus

### Course: Multivariable calculus > Unit 5

Lesson 10: Types of regions in three dimensions# Type II regions in three dimensions

Definition and intuition for Type 2 Regions. Created by Sal Khan.

## Want to join the conversation?

- I'm wondering, is every simply connected region in R^3 a Type 1, Type 2, and a Type 3 shape? If so, can we extend this to n-dimensions and say that for regions in R^n, a simply connected region is a Type 1, 2, ..., n region?(3 votes)
- No.

First, I assume that by "simply connected" you are using the usual topological definition that any closed loop can be contracted continuously through the region to a point. So all the examples that Sal uses are simply connected. But, say, a solid torus is not simply connected - consider a loop going all the way round the ring - you can't pull it tight and contract it to a point without crossing the hole in the doughnut, which is not part of the region.

If that isn't what you mean by "simply connected", then you'll have to explain what you do mean as I'm not aware of any other standard usage of that term.

Alternatively, if you are talking about the surface of the region being simply connected, I don't think that changes the situation. The surfaces of all Sal's examples are simply connected. The surface of the torus is not simply connected.

And my gut feeling is that for any bounded, closed surface in R^3 the region it encloses being simply connected is equivalent to the surface being simply connected (unless the region / surface is in some way pathological - so surfaces that aren't piecewise smooth with a finite number of smooth components, fractal surfaces, Klein bottles and the like might not work) - I'd be happy to be contradicted on this, but can't at the moment see a counterexample!

For an example of a region that is not a type 1, a type 2 or a type 3 region, consider a region formed by 3 dumbbells oriented along the x, y and z axes respectively and crossing at the origin - with large enough. bulges / long enough and narrow enough bars connecting them so that the only places where any two of the dumbbells intersect is in around the origin where the connecting bars cross. Note that this IS a simply connected region!(5 votes)

- Wouldn't the x-function for the hourglass change as you go down the hourglass? It looks like it would be a bunch of varying circle (or ellipse) equations.(2 votes)
- any type of regions that are neither type 1,2 or 3?(1 vote)
- A solid cube with a sphere-shaped hollow somewhere in the middle of it should do the trick.(1 vote)

- Is saying that the dumbbell/hourglass shape isn't a Type 2 region when oriented so it points along the x-axis the same as saying that the transformation from the domain to f_1 and f_2 is not onto?(1 vote)

## Video transcript

Let's now think
about Type 2 regions. And you'll see that they're
kind of very similar definitions and it's really a
question of orientation. Type 2 region is a
region-- I'll call it R2-- that's the set of all
x, y's, and z's in three dimensions such
that-- and now instead of thinking of our domain
in terms of xy-coordinates, we're going to think of them
in terms of yz coordinates, such that our yz pairs are
a member of some domain. I'll call it D2 since we're
talking about Type 2 regions. And x is bounded below
by some function of yz. So I'll call it g1 of yz is
less than or equal to x, which is less than or equal to some
other function of yz, g2 of yz. And so you'll immediately see
a very similar way of thinking about it, but instead of
having z vary between two functions of x and y as
we had in a Type 1 region, we now have x varying between
two functions of y and z. Now let's think about some
of the shapes we explored. We saw that these
two right up here, this sphere and the cylinder,
were Type 1 regions, but this dumbbell, the way
that I oriented it here, was not a Type 1 region. Let's think about which of
these are Type 2 regions and what might not
be a Type 2 region. So first let's think
about the sphere. So I have my axes
right over here. Let me scroll down a little bit. So I got my axes. And so over here, our
domain, we could still construct our sphere,
but our domain is now going to be
in the yz plane. So yz plane is this
business right over here. So this will be our domain. I want to make it more
spherical than that. So our domain is this right
over here in the yz plane. That is our D2. And now the lower
bound, in order to construct the solid region
of the sphere or the globe or whatever you want to call
it, the lower bound on x would be kind of the back half
of the sphere, the one that's away from us right over here. So the lower bound--
so let me see how well I can wire
frame it at first. I can do a better job than that. So with my ghost do
something like that, then do something like that. But this is if the domain
right over here is transparent. But all we might catch-- we'll
just catch a glimpse of it in the back right over here. So it's the side of the sphere
that's facing away from us. And then the upper
bound on x would be the side of the
sphere that's facing us. So if I were to
do some contours, it might look
something like this and then look
something like this. And then we would color in this
entire region right over here. And x can take on all of the
values above that magenta surface and below
this green surface. And essentially,
it would fill up the globe for every yz
point in our domain. So a sphere is both a Type
1 and a Type 2 region. Actually, we're
going to see it's going to be a Type
3 region as well. What about this cylinder
right over here? Can we construct it or
think about it in a way that it would actually
be a Type 2 region? So let's try to do that. So let me paste it. So what if we had
a domain-- what if our domain was
something like this? It was a rectangle
in the yz plane. So this is our domain, a
rectangle in the yz plane. So that would be my D2. And what if the
lower bound was kind of the back side
of the cylinder? So the backside of
the cylinder, try to draw it as good as I can. And so if we just saw
the outside of it, it would look
something like that. It's facing away from
us so we barely see it. If we could see
through the cylinder or see through the little
flat cut of the cylinder, it would look
something like that. So that over there
would be our g1. And then our g2 would be the
front side of the cylinder. The g2 could be the front
side of the cylinder. So let me color it
in as best as I can. So the g2 would be the
front side of cylinder. And x can vary above
g1 and below g2, and it would fill up
this entire cylinder. So we see that this same
cylinder that we also saw was a Type 1 region can
also be a Type 2 region. Now what about this
hourglass thing that we saw could not be a Type 1 region? Can this be a Type 2 region? Well, let's think about it. I'll do it the same way. We can construct a domain. So maybe our domain,
it's in the y-- well, it should be in the yz plane
if we're talking about Type 2 regions or if we want to think
of it as a Type 2 region. So our domain could be this kind
of flat hourglass shape that's in the yz plane. So our domain could be a
region that looks something like this in the yz plane. So this is kind
of flattened out. So this is our domain
right over there. And then the lower
bound on x, g1, could be a surface,
the function of y and z that is kind of the
backside of our hourglass. The backside of our
hourglass you could see. I'll try to show the
contours from the underside right over there. So that could be our g1. And then our g2 could be the
front side of the hourglass. So my best attempt to draw the
front side of the hourglass. And I could color it in. And so the way I somewhat
confusingly drew it just now, you see that this hourglass
oriented the way it is would actually be
a Type 2 region. Now if we were to rotate
it like this-- so let me draw it like this. Edit. So if we were to
make it like this so that the top of my
hourglass is facing us-- try my best to draw it. So let's say the top intersects
the x-axis right over there. This is the bottom of my
hourglass right over there. And then it bends in and then
comes back out like that. For the same reasons that
this was not a Type 1 region, this now would not
be a Type 2 region. For any zy, you can see there
could be multiple x points that are associated with
the different points of this hourglass. You can't just have a
simple lower and upper round functions right over here. So this right over here
is not a Type 2 region. You could show a
rationale or this is going to be a Type 1 region. You could create a region
over here in the xy plane and have an upper and lower
bound functions for z. So you could be Type 1, but
this will not be Type 2.