Background: Let $S_1,S_2$ be two regular surfaces in $\mathbb{R}^3$, $p\in S$, and $\varphi:U\rightarrow S_2$ be a $C^\infty$ function defined in a neighborhood $U$ around $p$ in $S$.
Suppose that for all $q\in U$, the linear operator $L_q$ on $T_q S_1$ (associated with the quadratic form $Q_q(v)=\langle d\varphi_q(v), d\varphi_q(v)\rangle, \forall v\in T_q S_1$) always has two distinct eigenvalues $k_1(q)>k_2(q)>0.$
Then by linear algebra, there exist orthonormal eigenvectors $e_1(q),e_2(q)$ in $T_q S_1$ such that $L_q(e_1(q))=k_1(q), L_q(e_2(q))=k_2(q)$.
Question: Is the vector field $w$ in $U$ defined by $w(q)=e_1(q),\forall q\in U$, differentiable? For each $q$ there are two choices of $e_1(q),$ I don't care about this choice as long as $w$ can be made differentiable.
My thoughts: I'm sure that $k_1(q)$ is a differentiable function in $U$ since, when written in local coordinates, it is the bigger one of the two solutions of the (quadratic) characteristic polynomial. I wonder if there is a way to show that the unit eigenvector in $T_q S_1$ varies smoothly as $q$ moves.