vector field of a sphere I have a problem to understand vector fields on manifolds. Theoretically I get it, but I cannot find a concrete example with numbers.
For example, I have the $S^2$ sphere and I take the classic polar parametriazation ($\theta$,$\phi$)$\to$ r($\theta$,$\phi$) in order to get the vector fields r$\theta$,r$\phi$.What are those vector fields?Are they a base of the tangent space?Obviously they are, but why? 
Are they the same with the canonical base $\frac{\partial}{\partial \theta}$, $\frac{\partial}{\partial \phi}$?
 A: The basis of the tangent space is indeed $\frac\partial{\partial\theta},\frac\partial{\partial\phi}$. This is an abstracted form, which is also sometimes written as $\partial_\theta,\partial_\phi$.
We can associate the points on the sphere with the coordinates in the map $(\theta,\phi)$ and then 'read' the basis vectors 'through the coordinate map'. If we do, we get:
$$\frac\partial{\partial\theta}(\theta,\phi)=(1,0)\quad\text{and}\quad\frac\partial{\partial\phi}(\theta,\phi)=(0,1)$$
So we see that they span indeed a 2-dimensional tangent space. The differential map of the coordinate map transforms them to the unit vectors in its coordinate space.
For comparison, if we look at the embedding of $S^2$ in $\mathbb R^3$ with its canonical cartesian coordinate map $(x,y,z)$, we get:
$$\frac\partial{\partial\theta}(\cos\phi\sin\theta,\ \sin\phi\sin\theta,\ \cos\theta)=(\cos\phi\cos\theta,\ \cos\phi\cos\theta,\ -\sin\theta)$$
And if we 'read through the cartesian coordinate map' $(x,y)$ of $S^2$, we get:
$$\frac\partial{\partial\theta}(\cos\phi\sin\theta,\ \sin\phi\sin\theta)=(\cos\phi\cos\theta,\ \cos\phi\cos\theta)$$
which is the representation of $\frac\partial{\partial\theta}$ in the cartesian coordinate map.
