Show that the height function on a sphere has 2 critical points

Question

Let $f(x_1,..,x_k) = x_k$ on the sphere $S^{k-1}$ and need to show that the poles are the 2 critical points.

My question is elementary to begin with. Isn't $df = 1.dx_k$? How can $df_x = 0$ in such a case for the point $x$ to be critical?

Answer 1

You have to restric $df$ to the tangent space of $x$, $T_xS^{k-1}$ and if the tangent space is parallel to $x_k=0$, the restriction of $df$ to $T_xS^{k-1}$ is zero as it is for $(0,..,1)$ and $(0,..,-1)$.