The book on general relativity I'm reading states that the motivation behind the concept of manifold is to extend the theory of analysis in $ℝ^n$ to curved spaces. As an example of manifold, it gives the 2-sphere $S^2$ \begin{equation} S^2 = \{x,y,z \in ℝ^3 | x^2 + y^2 + z^3 =1 \} \end{equation} and how it is possible to map this set with charts to $ℝ^2$
I don't understand why it is not enough to consider the parametrization of the sphere as:
\begin{equation} \begin{cases} x= R \sin \phi \cos \theta\\ y= R \sin \phi \sin \theta\\ z=R \cos \phi\\ \end{cases} \end{equation}
and use vectorial analysis instead of resorting to manifolds.
If it is a matter of generalization to all possible curved spaces then if it is true that a manifold is a continuous space which "looks" locally like Euclidean open subsets of $ℝ^n$, In the case of curved surfaces shouldn't be possible to approximate locally the surface by that of a spherical cap?
No, it isn't possible to approximate locally a smooth surface with a spherical cap. On a spherical cap, the curves which are the intersections with normal planes at some point have all the same curvature. This is not the case for a generic surface.