Differential of the height function Consider the mapping $f:S^2\rightarrow \mathbb{R}$, with $f(x,y,z)=z$, giving the height of a point on the sphere. I am asked to find the points for which the differential $(df)_p$ is surjective and i have failed so far. It's an exercise on an introductory course on smooth manifolds, so please try to be clear about what theorems you use, i'm just a beginner!
Thanks a lot in advance!
 A: Intuition 
Draw a picture. At a tangent space, visualize tangent vectors and the result of projecting those tangent vectors to the $z$ axis. Do you see what are the (hint: only two) points such that the projection is not surjective?
Computation
We know that $f$ is actually the restriction of the projection map $F:\mathbb{R}^3\to \mathbb{R}$ taking $(x,y,z)\mapsto z$. Compute the differential of $F$, call it $dF$. Don't think too hard about this.
Now recall that the tangent space to $S^2$ at a point $p$ can be identified with the plane of vectors in $\mathbb{R}^3$ perpendicular to $p$. (Why?) Given a $p\in S^2$, compute a basis for $T_pS^2$.
Compute the matrix for $df_p$ by restricting $dF$ to the tangent space (which you know, as you have a basis).
What are the points where this map is not onto?
A: For $u=(x,y,z)\in R^3,v=(a,b,c)\in T_uR^3 df_u(v)=c$.Thus $df_u(v)=0$, i.e $c=0$, if $u\in S^2, T_uS^2\subset T_uS^3$ and $df_u(T_uS^2)$ i.e $T_uS^2=\{(a,b,0),a,b\in R\}$ i.e $u$ is the north pole or the south pole.
