Let $(\Sigma,g)$ be a Riemannian manifold. For a differential form $\alpha$, given $d^{*}\alpha=0$, where $d^*$ is the codifferential with respect to $g$, can we rewrite the equation $d^*d\alpha=0$ as a divergence-form strongly elliptic system of equations using local coordinates (just like the case when $\alpha$ is a function) ?