Given a manifold $M$ and open neighborhood $U \subset M$:
Map: $x: U \rightarrow \Omega \subset \mathbb{R}^m$
Local parametrization: $F := x^{-1} : \Omega \rightarrow U \subset M$
If I now consider a sphere as a manifold, i.e. $M = \mathbb{S}^2$, then we can write:
$$M = [(x,y,z)\ | \ \sqrt{x^2+y^2+z^2} = R]$$
A sphere can be described with spherical coordinates, means:
$$x_1 = R \ \cos\theta\ \cos\varphi$$
$$x_2 = R \ \cos\theta\ \sin\varphi$$
$$x_3 = R \ \sin\theta$$
Here $\varphi \in [0,2\pi)$ and $\theta \in [0,\pi)$. But if now
$$x(\varphi,\theta) = \begin{pmatrix} x_1 \\\ x_2 \\\ x_3 \end{pmatrix}$$
is the map, then $M = [0,2\pi) \times [0,\pi)$ and $\Omega \subset \mathbb{R}^3$ describes the sphere. In addition, the local parametrization is
$$ \varphi = \text{arctan2} (x_1,x_2), \hspace{25pt} \theta = \arcsin \frac{x_3}{R}$$
However, this does not seem to be correct. What is my thinking error?