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