A quaternionic treatment of the inhomogeneous $\operatorname{div}$-$\operatorname{rot}$ system

In this paper we study the inhomogeneous div-rot system ($\operatorname{div}\vec f=g_0$, $\operatorname{rot}\vec f=\vec g$) where the datum $(g_0,\vec g)$ consists of a continuous scalar and a continuous vector field, respectively. We embed the system in its appropriate quaternionic setting, and by using the right inverse of the Moisil–Teodorescu operator, we provide a necessary and sufficient condition for the solvability of the system and we describe its general solution. As a byproduct we obtain an explicit integral expression for the right inverse for the operators $\operatorname{div}$ and $\operatorname{rot}$. Finally, we show how the same problem could have been studied using algebraic analysis, and we use this different approach to obtain some additional results.