I consider $f:S^n\rightarrow \mathbb{RP}^n$, the restriction to $n$-sphere $S^n$ of the canonical projection $\pi :\mathbb{R}^{n+1}\setminus \{0\} \rightarrow\mathbb{RP}^n $.
I have to prove that $f$ is a local diffeomorphism.
I have proved that $\pi$ is differentiable so I should check that the differential $df_p$ is invertible $\forall p \in S^n$.
Here I have problems in writing the matrix associated to the differential.
I have found that I should do the jacobian associated to this application:
$$y=(y_1,...,y_n) \rightarrow (\dfrac{y_1}{\sqrt{1-|y|^2}},...,\dfrac{y_n}{\sqrt{1-|y|^2}}).$$
Since it is quite complicated I consider an alternative approach.
I know that, if $\sigma:(-\epsilon, \epsilon) \rightarrow S^n$ is a differentiable curve s.t. $\sigma(0)=p$ and $\sigma'(O)=v$ with $v\in T_p S^n$, then: $$df_p(v)=(f \circ \sigma)'(0).$$ I would like to use this fact considering $\sigma$ a great circle of $S^n$ passing through $p$. The problem is that I don't know how to write the parametrization of one of these circles.
Is this way of proceeding correct? How can I do? Thanks for the help.
The great circle you ask about is parametrized by $$ \sigma(t) = \cos(\|v\|t)\, p + \sin(\|v\|t)\, \frac{v}{\|v\|}. $$ However, your original strategy of picking local coordinates for the sphere and projective space looks easier; it may simply be that your coordinates for $S^{n}$ (graph coordinates for a hemisphere) are not as judicious as they could be. :)
The hyperplane $P = \{x_{n+1} = 1\}$ is tangent to the sphere at the "north pole" $N = (0, \dots, 0, 1)$, and the Cartesian coordinates $(x_{1}, \dots, x_{n})$ on $P$ define local coordinates on the "upper hemisphere" $H = \{x_{n+1} > 0\}$ by radial projection from the origin: $$ x = (x_{1}, \dots, x_{n}, 1) \in P \mapsto \frac{x}{\|x\|} \in H. $$ This map is clearly smooth with smooth inverse $$ (y_{1}, \dots, y_{n}, y_{n+1}) \in H \mapsto \frac{1}{y_{n+1}}(y_{1}, \dots, y_{n}) \in P, $$ hence a diffeomorphism. As a fringe benefit, in these coordinates the covering projection $f$ is the identity map. (Naturally you'll need multiple coordinate systems to cover the entire sphere, but that's just a matter of notation.)