Ellipticity of divergence and curl

Let $M$ be a 3-dimensional Riemannian manifold with Riemannian metric $g$. Let $E = S^2 T^* M$ be the vector bundle of symmetric covariant 2-tensors on $M$, and let $F = T^* M \oplus S^2 T^* M$. We consider the linear differential operator $L : \Gamma(E) \to \Gamma(F)$, where $\Gamma(B)$ denotes sections of a bundle $B$, defined by: \begin{equation} L_1(k) = ({\text{div}} k, {\text{curl}} k) \end{equation} and also the related operator \begin{equation} L_2(k) = ({\text{div}} k, \text{S}{\text{curl}} k). \end{equation}