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Studying for a test? Prepare with these 11 lessons on Green's, Stokes', and the divergence theorems.

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# Type III regions in three dimensions

Video transcript

After going through type 1
and type 2 region definitions, you can probably guess what a
type 3 region is going to be. So a type 3 is a region
in three dimensions. Since we called the other
the type 2 region R sub 2 and the type 1
region R sub 1, I'll call this region R sub
3-- R with a subscript 3. It's going to be the set of
all points in three dimensions. The set of all x, y and z's
such that the x, z pairs are member of a domain, I'll
call this domain D sub 3. And y is going to vary
between two surfaces that are functions of x and z. So y is going to be greater than
or equal to the surface-- I'll call it h1 of x, z is going
to be less than or equal to y. y is going to be bounded
from above by the surface h2 of x, z. And once again, let's
close our set notation. So let's think about whether
some of these regions that we already saw were type 1 and
type 2, whether they're type 3. And then think about what
would not be a type 3 region. So let's go to this sphere. What could be our domain? Well the domain
is a set of x, z. It's just going to
be the in x, z plane. So over here our domain
could be this region right over here in the x, z plane. So I'll color it in. It could be that region right
over there in the x, z plane. And then the lower
bound on y will be the part that is behind
the sphere in this direction right over here. It's the stuff-- actually,
this might be a little bit hard to visualize since I'm
redrawing on top of it. And then the upper
bound on y is going to be this side right over here. So all of this is now
going to be green. Let me redraw this sphere
just to make it clear. Let me just draw
another coordinate axes. So another coordinate axes. The backside of this
sphere in the y direction. So I guess let's
think of it this way. This is the
hemisphere mark, this is the halfway
point for my sphere. And once again, that's kind
of the boundary of our domain. And then the backside-- let
me do the front side first. The front side y is upper
bound, that would be h2, would be all of this
business right over here. So this would be h2
that I'm coloring in. h2 would be the side that
is facing in that direction. Let me see how well
I can color it. That didn't do a good thing. So h2 is all of this stuff out
on this side of the sphere. And then h1 was the
lower bound on y. So it's going to be that
side right over there. And I could draw it probably
a little bit neater. But hopefully you get the point,
it would be all of that side. And then y can vary
between those two and essentially
fill up the region. Make the exact same
argument with the cylinder. The cylinder can be
defined the same way. So first of all, the
sphere is a type 1, type 2, and type 3 region. It meets all of the constraints. The cylinder, at
least the way it was oriented there,
actually any cylinder, will also be a type 3 region. Exact same argument. So let me draw my axes again. And here to make the cylinder
that we've been making, our domain could be a
rectangle in the x, z plane. So our domain can be
a rectangular region in the x, z plane,
just like that. And then the lower
bound on y could be that side of the cylinder. The side facing in that
direction right over there, that is the lower bound. And then the upper
bound on y could be the side facing
in that direction. So once again, this is
also a type 3 region. By the same logic, this
one right over here, this hourglass, can
be a type 3 region. The front side would be this
front side right over here, all of this, including
that stuff that I-- And then the backside when
you think of it in terms of y, the backside would be that
part right over there. And once again, this could
also be a type 3 region. The boundary of the domain
would be this cross section right over here. So the boundary of
our domain could be that cross section
right over there. The lower bound on y
would be the back half, the back half of
this hour glass. And the upper bound on y
would be the front half. Let me do that magenta color
because I've been using that. Or actually that green color. So the upper bound on y
would be this right half right over here. So what would not be
a type three region? Well, if we just rotated
this around like that. So let me just draw something
that is not a type 3, just to show you that
this definition does not include everything. So something that would
not be a type 3 region-- for the same argument as
we've seen before for a type 2 and type 1 region
is an hourglass where it's along the
y-axis or at least it's oriented this way. It actually does not have to be
right centered on the y-axis. But an hour glass that looks
like this, now all of a sudden the y can't be just expressed as
being between two surfaces that are functions of x and z. You would have to break
this up in order to do that. But you could break
up this region into two types 3 regions. But the whole thing itself
is not a type 3 region. So not type 3.