What are some proofs on why only the third one of the seen subsets of R^3 is viable?
Why does it have to be parabolic?
>>7898189
Learn how to communicate and then come back
>>7898189
define 'viable'
>>7898194
I think he's asking why asymptotes have to be parabolic
>>7898189
>What are some proofs on why only the third one of the seen subsets of R^3 is viable?
What do you mean viable?
>>7898566
What the fuck are you talking about retard. Just explain what you mean by viable
>>7898566
If you weren't so "special", you might have realized that you didn't provide any context whatsoever.
>>7898652
You will understand the context if you aren't too autistic.
>>7898222
But they wouldn't be asymptotic to the cylinder of radius r is they're parabolic. Because of the rotational symmetry, we can just look at the 1D case, and y=x^2 clearly doesn't have a vertical asymptote.
>>7898566
Their complaints are valid - "viable" isn't a technical mathematical term, outside of some obscure applications of dynamical systems, which is clearly not what you're discussing.
Is English not your first language?
>>7899669
>"viable" isn't a technical mathematical term
I knew this when asking the question but didn't expect pointless nitpicking of this level. The question should be understandable to anyone who knows what viable means in general. I'm sure these people actually understand the question but decide to concentrate on a single word that pokes them in their ultra-rigorous eye.
Please post proofs relating to the question.
>>7899753
VIABLE FOR FUCKING WHAT
The hell is a viable subset?
>>7899753
viable for what you brainless fucker?
>>7899753
>The question should be understandable to anyone who knows what viable means in general.
That's point everyone is making. Nobody knows what it means *in general* for a subset of R^3. The point of defining everything in math is to make sure that everyone talking about the same thing.
Now what did you mean ? Convex ? (ie. convex epigraph) Smooth ? Submanifold of R^3 ?
All of these are properties that only the last one has.
>>7899928
>Now what did you mean ? Convex ? (ie. convex epigraph) Smooth ? Submanifold of R^3 ?
Thank you for partly answering. Was I expected to know all these properties? Why are the first two not submanifolds of R^3?
Intuitively they're not but I'd like to see ways of showing this.
Do all convex submanifolds of R^3 blow up? Or is this one specifically an asymptotic one?
>>7900899
>Why are the first two not submanifolds of R^3?
>Intuitively they're not but I'd like to see ways of showing this.
They have singularities (those pointy "cusps") at the origin. See what happens when you try to find Euclidean neighborhoods there.
>>7901013
Are the first two valid at all? Can they blow up if they aren't convex?
How can an OP be this clueless
>the city of OP
>>7901072
Please post the appropriate proofs for clueless OP.
>>7901055
Valid as what? You can certainly define those subsets of R^3; they're just not nice enough to be manifolds. I don't know how important that is to you (or what it is you're looking for).
>>7901860
I was assuming they would not be defined at all. It's good that you made it clear they're defineable.
>>7901072
Dumb weeaboo
>>7900899
>>7901013
>>7901860
The half-cone (picture 2) is in fact a manifold. However it is not a submanifold of R^3.
i.e. Consider [math] {C^ + } = \left\{ {x \in {\mathbb{R}^3}|{x_3}^2 = {x_1}^2 + {x_2}^2\& {x_3} \geqslant 0} \right\}[/math]
We can construct the manifold [math] \left( {{C^ + },{{\left. {{\mathcal{O}_{{\mathbb{R}^3}}}} \right|}_{{C^ + }}},{\mathcal{A}_{{C^ + }}}} \right)[/math]
Where the atlas is [math] {\mathcal{A}_{{C^ + }}} = \left\{ {\left( {U,h} \right)} \right\} [/math] with [math] h\left( x \right) = \left( {{x_1},...,{x_n}} \right) [/math] and [math] {h^{ - 1}}\left( y \right) = \left( {y,\left| y \right|} \right) [/math].
So the half-cone is a manifold despite the singularity, and is diffeomorphic to R^2. However it is NOT a submanifold of R^3.
>>7902156
I meant [math] h\left( x \right) = \left( {{x_1},{x_2}} \right) [/math]
(forgot we weren't talking about general dimension, although the same idea is true)
>>7902670
Guess shit threads just aren't viable
>>7902850
This is a viable thread with a valid answer. Please excuse yourself.
>>7901072
cute gril
>>7902156
What the fuck are you talking about
all three are topological manifolds, they are submanifolds of R^3
The first two are not smooth manifolds and not diffiomorphic to shit. The third is a smooth manifold diffeomorphic to the plane.
I'll give you some hints, without any rigor, you'll have add that yourself, but you'll get the idea :
Since the graph is symetrical regarding to the z-axis, you can just work in a plane and consider for instance the upper right part of the graph (ie For x=0, y and z positive).
Then, replace +infinity with some strictly positive real number a, we'll name A = (0, r, a), and check the angle between the line (OA) and the z-axis and look at what happens when a goes to +infinity.
For the first one, also consider only the graph for x = 0, y and z positive.
I'll assume that part of the graph can be parameterized as y = f(z) for some twice differentiable function f. If f''(z) is strictly positive for all strictly positive z, my guess would be to check that it is not possible for f(z) not to go over r at some point, using some convexity inequalities or something.
Also you should have used "possible" instead of "viable", which would have made a lot more sense.
>>7904224
>they are submanifolds of R^3
They are not
>>7904349
Why do you say this
>>7904356
Definition of a submanifold:
[math]{C^ + }[/math] is a submanifold of [math]{\mathbb{R}^3}[/math] iff [math]\forall x \in {C^ + }[/math] [math] \exists \left( {U,h} \right) \in {\mathcal{A}_{{\mathbb{R}^3}}}[/math] where [math] x \in U [/math] such that [math] h\left( {U \cap {C^ + }} \right) = \left( {{\mathbb{R}^2} \times \left\{ 0 \right\}} \right) \cap h\left( U \right) [/math].
I think it is pretty easy to see that this is not true for x=0 (although slightly harder to prove).
>>7905245
>[math] \exists \left( {U,h} \right) \in {\mathcal{A}_{{\mathbb{R}^3}}}[/math]
Should say: There exists [math]\left( {U,h} \right) \in {A_{{\mathbb{R}^3}}}[/math]
