# sidht

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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