Let $S^2\;$ be the unit sphere on $\mathbb R^3\;$ and the function $f:S^2\rightarrow S^2\;$ with $f(x,y,z)=(z,-y,-x)$
Prove $f$ is differentiable in $S^2\;$ and for $p=(0,-1,0) \in S^2\;$ find the matrix of $(Df)_p : T_p S^2 \rightarrow T_{f(p)} S^2\;$ relative to the appropriate bases of tangent spaces.
For the differentiability:
Let the atlas of $S^2\;$ to be $\{ (U_N,φ_Ν)\;,\;(U_S,φ_S) \}\;$ where
$φ_N: U_N=S^2-\{(0,0,1)\} \rightarrow \mathbb R^2\;$ with $φ_N(x,y,z)=(\frac{x}{1-z},\frac{y}{1-z})\;$ and
$φ_S: U_S=S^2-\{(0,0,-1)\} \rightarrow \mathbb R^2\;$ with $φ_S(x,y,z)=(\frac{x}{1+z},\frac{y}{1+z})\;$
We know by definition $f\in C^{\infty}\;$ in $S^2$ if and only if $φ_S \circ f \circ {φ_N}^{-1} : φ_N(U_N) \rightarrow φ_S(U_S)\;$ is differentiable at every $φ_N(x,y,z)\; \in S^2$
$φ_S \circ f \circ {φ_N}^{-1}(x,y)=φ_S \circ f(\frac{2x}{x^2+y^2+1},\frac{2y}{x^2+y^2+1}, ,\frac{ x^2+y^2-1}{x^2+y^2+1})=φ_S(\frac{ x^2+y^2-1}{x^2+y^2+1},-\frac{2y}{x^2+y^2+1},-\frac{2x}{x^2+y^2+1})=(\frac{ x^2+y^2-1}{(x-1)^2+y^2},-\frac{2y}{(x-1)^2+y^2})$
Now my questions is:
How can I be sure $(1,0)\;$ isn't in domain of $φ_S \circ f \circ {φ_N}^{-1}\;$?
Furthermore, I'm lacking good ideas for how to approach the second part of the exercise. Is this matrix equal to the Jacobi or am I wrong?
I would appreciate any help! Hints or other solutions than this are of course welcome. Thanks in advance