On algebraic and uniqueness properties of 3d harmonic quaternion fields

Let $\mathrm{\Omega }$ be a smooth compact oriented 3-dimensional Riemannian manifold with boundary. A quaternion field is a pair $q=\{\alpha ,u\}$ of a function $\alpha$ and a vector field $u$ on $\mathrm{\Omega }$. A field $q$ is {\it harmonic} if $\alpha ,u$ are continuous in $\mathrm{\Omega }$ and $\nabla\alpha={\rm rot,}u,,{\rm div,}u=0$holdsinto$\Omega$\mathrm{.}Thespace${\mathscr Q}(\Omega)$ of harmonic fields is a subspace of the Banach algebra $\mathcal{C}\left(\mathrm{\Omega }\right)$ of continuous quaternion fields with the point-wise multiplication $q{q}^{\mathrm{\prime }}=\{\alpha {\alpha }^{\mathrm{\prime }}-u\cdot u^{\mathrm{\prime }},\alpha u^{\mathrm{\prime }}+{\alpha }^{\mathrm{\prime }}u+u\wedge u^{\mathrm{\prime }}\}$. We prove a Stone-Weierstrass type theorem: the subalgebra $\vee \mathcal{Q}(\mathrm{\Omega })$ generated by harmonic fields is dense in $\mathcal{C}\left(\mathrm{\Omega }\right)$. Some results on 2-jets of harmonic functions and the uniqueness sets of harmonic fields are provided.