![]() Simple lower and upper round functions right over here. For any zy, you can see thereĬould be multiple x points that are associated with This was not a Type 1 region, this now would notīe a Type 2 region. And then it bends in and thenĬomes back out like that. Hourglass is facing us- try my best to draw it. It like this- so let me draw it like this. And so the way I somewhatĬonfusingly drew it just now, you see that this hourglassĪ Type 2 region. So my best attempt to draw theįront side of the hourglass. And then our g2 could be theįront side of the hourglass. I'll try to show theĬontours from the underside right over there. The function of y and z that is kind of theīackside of our hourglass. Region that looks something like this in the yz plane. Of flat hourglass shape that's in the yz plane. If we're talking about Type 2 regions or if we want to think It's in the y- well, it should be in the yz plane 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. So we see that this sameĬylinder that we also saw was a Type 1 region canĪlso be a Type 2 region. So the g2 would be theįront side of cylinder. ![]() And then our g2 would be theįront side of the cylinder. ![]() Through the cylinder or see through the little The cylinder, try to draw it as good as I can. Think about it in a way that it would actuallyīe a Type 2 region? So let's try to do that. So a sphere is both a Typeģ region as well. Values above that magenta surface and below And then we would color in thisĮntire region right over here. Just catch a glimpse of it in the back right over here. Something like that, then do something like that. Of the sphere, the one that's away from us right over here. It, the lower bound on x would be kind of the back half Of the sphere or the globe or whatever you want to call And now the lowerīound, in order to construct the solid region And so over here, ourĭomain, we could still construct our sphere,īusiness right over here. These are Type 2 regions and what might not That I oriented it here, was not a Type 1 region. Were Type 1 regions, but this dumbbell, the way Two right up here, this sphere and the cylinder, We had in a Type 1 region, we now have x varying between Having z vary between two functions of x and y as And so you'll immediately seeĪ very similar way of thinking about it, but instead of Less than or equal to x, which is less than or equal to some And x is bounded belowīy some function of yz. In terms of yz coordinates, such that our yz pairs areĪ member of some domain. In terms of xy-coordinates, we're going to think of them That- and now instead of thinking of our domain Region- I'll call it R2- that's the set of all Kind of very similar definitions and it's really a Note that this IS a simply connected region!Ībout Type 2 regions. 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. The surface of the torus is not simply connected.Īnd 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!įor 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. The surfaces of all Sal's examples are simply connected. 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.Īlternatively, if you are talking about the surface of the region being simply connected, I don't think that changes the situation. 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. So all the examples that Sal uses are simply connected. 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.
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