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comment Explain “homotopy” to me
@RickyDemer some texts adopt the convention that map is a continuous function. For instance, Bredon.
Feb
10
comment Does stereographic projection preserve or reverse orientation?
The Jacobian matrix you are getting seems like that because you are looking at a map $f: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ by looking at the sphere inside $\mathbb{R}^3$, which you shouldn't. You should take the charts for your sphere to compute the determinant if you want to think like that, after all you are on a manifold.
Feb
10
comment Explain “homotopy” to me
Sidenote: In my personal opinion, "trivial" examples are commonly more confusing than enlightening ones, mainly for students. If you want to show an example of a structure, start by showing a interesting and simple one, not a trivial one, since it may not illustrate the structure. I think this happened to you. It is similar to presenting as first example of a path a constant path, instead of a straight line.
Feb
10
answered Explain “homotopy” to me
Feb
6
asked Is there any result on the “counting” of minimal atlas?
Feb
5
comment Tangent space to a surface at boundary points
This is really a matter of definitions, in my opinion. For instance, if you use the "equivalence class on charts" definition of tangent space, then you get the full vector space. Sometimes this definition is useful, for instance when talking about Stoke's Theorem.
Feb
4
comment Why does $y(s)$ continuous imply that $f(s)$ with $f_l (s) = \frac{s_l + \max\{0,z_l(s)\}}{1+\sum \max\{0,z_l(s)\}}$ is continuous?
Projections are continuous, $\max$ between two continuous functions is continuous, sums of continuous functions are continuous, quotients of continuous functions are continuous and constant functions are continuous.
Feb
4
answered If $f: \mathbb{R^n} \rightarrow \mathbb{R^m}$ be a function with $\nabla f(x) = 0$ for all $x$, then $f$ is constant.
Feb
3
comment Can the concept of orientability be applied to more general spaces?
Maybe (I don't know) you can adapt the definition of orientability via the orientation bundle on homology manifolds.
Feb
3
comment Advantage of Lebesgue sigma-algebra over Borel?
Lebesgue is complete.
Jan
28
comment Using a Direct Proof to show that two integers of same parity have an even sum?
Just as a comment for future reference, note that this follows easily from the fact that $2=0 \mod 2$. That is, $2x=0 \mod 2$ for $x=1$ or $x=0$.
Jan
25
answered If every closed and bounded subset of a metric space $M$ is compact, does it follow that $M$ is complete?
Jan
24
revised This exercise is solved using the fundamental theorem of calculus?
added 159 characters in body
Jan
24
answered This exercise is solved using the fundamental theorem of calculus?
Jan
24
awarded  Nice Question
Jan
22
comment Why does equation $a · x = 0$ always has $n − 1$ linearly independent solutions for $x$ and never has $n$ linearly independent solutions?
Hint: Use the rank-nullity theorem and the fact that $x \mapsto \langle a, x\rangle$ is a linear functional.
Jan
22
answered Is $‎‎‎\sqrt[3]{y^3}‎‎‎$ or $\frac{x^2}{x}$ a polynomial?
Jan
22
awarded  Yearling
Jan
20
answered Is $z+\overline{z}$ an Analytic function?