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10h
accepted Is a sphere really a (differentiable) manifold?
10h
comment Is a sphere really a (differentiable) manifold?
Alright, I'll save that for another question.
10h
comment Is a sphere really a (differentiable) manifold?
A moment of thought made me realize that is the answer. But what would happen if one of the domain is not a circle?
10h
comment Is a sphere really a (differentiable) manifold?
I just computed it to see what the map geometrically means.
10h
comment Is a sphere really a (differentiable) manifold?
@Ryam Reich My question is this: The cartesian equation of the map $(\sqrt{1 - u^2 - v^2}, u)$ is $x^2 + y^2 = 1 - v$ right? How do I know that $v < 1$? I know its contained back in the $U_+$, but I can't find a way to make the inequalities work.
10h
comment Is a sphere really a (differentiable) manifold?
@Ryam Reich, also I figured out the transition map traces the curve to $x^2 + y^2 = 1 - v^2$, where this should still be a circle, but how do I make the argument that $v < 1$?
10h
comment Is a sphere really a (differentiable) manifold?
okay, but doesn't this mean the inverse of patches/charts of the form $(x,y,f(x,y))$ will lose its "information" when we collapse down back to two-dimensional? This type of parametrization seems "dangerous"
10h
revised Is a sphere really a (differentiable) manifold?
added 381 characters in body
10h
comment Is a sphere really a (differentiable) manifold?
I mean I set $s = x, t = y, r = \sqrt{1 - x^2 - y^2}$ (so I didn't actually call it $z$)
10h
comment Is a sphere really a (differentiable) manifold?
Some information i missing here. When you collapse down to $\mathbb{R}^2$, what happened to $z = \sqrt{1 - x^2 - y^2}$ in the third component for $\sigma_+^z$?
10h
comment Is a sphere really a (differentiable) manifold?
I meant in spherical coordinates.
11h
asked Is a sphere really a (differentiable) manifold?
21h
comment Cross product of the reals question
Doesn't the cross product on the LHS have two components while the RHS has one? How can it be a subset?
1d
accepted How is a relation defined on ordered sets?
1d
reviewed Approve suggested edit on Increasing sequence and converging to zero.
1d
comment How to show proper set inclusion/exclusion? Please don't give me the solution.
@neuguy, subsets are not subspaces?
1d
asked How to show proper set inclusion/exclusion? Please don't give me the solution.
1d
comment How is a relation defined on ordered sets?
I meant to say that if we define $a \sim b$ as just $a \leq b$ (without $b \leq a$)
1d
comment How is a relation defined on ordered sets?
actually from this question, we see that our set cannot be partially ordered. Maybe this will answered why I was confused in the first place.
1d
comment How is a relation defined on ordered sets?
I see, actually from this we see that our set cannot be partially ordered