Suppose I have a smooth map $F \colon M \rightarrow N$ and an embedded submanifold $S \subset M$. Suppose that the differential of $F$ is non-singular at a point $z \in M$.
If $z$ also happens to lie in $S$, is it true that the differential of $F\vert_S$ (as a smooth map from $S$ to $N$) is non-singular at $z$?
$\textbf{Edit}$
So it seems this was a bad question to ask. What I'm trying to do is the following:
I have a smooth vector field $\widetilde{V} \colon \Bbb R^3 \rightarrow T\Bbb R^3$ which sends $$(x,y,z) \mapsto (-y,x,x^2+y^2+z^2-1).$$
Its restriction to $\Bbb S^2$ is then a smooth vector field $V$.
I want to somehow see that $dV_p$ is non-singular at the points $p =(0,0,\pm1)$.