Suppose $\phi: \mathbb{S}^{n-1} \to \mathbb{R}^k\backslash \{0\},$ we set $\psi:= \phi/ |\phi| : \mathbb{S}^{n-1} \to \mathbb{S}^{k-1}.$ The claim, is that for every smooth extension $F$ of $\phi$ insider the unit $n-$ ball, there exists a zero of $F,$ (i.e. a point $x$ such that $F(x) = 0$) if and only if $\psi$ is homotopically nontrivial.
Here's an attempt. let $F$ be a smooth extension of $\phi$ inside the unit ball, such that $F(x) \neq 0$ for any $x$ in the unit ball. Then, one notes that $G: \mathbb{S}^{n-1} \times [0,1] \to \mathbb{S}^{k-1}$ given by $G(x,t) = F(tx)/|F(tx)|,$ yields a homotopy between $\psi$ and a constant.
How does one show the converse?