On the Laplacian vector fields theory in domains with rectifiable boundary

Given a domain (omega)in R3 with rectifiable boundary, we consider main integral, and some other, theorems for the theory of Laplacian (sometimes called solenoidal and irrotational, or harmonic) vector fields paying a special attention to the problem of decomposing a continuous vector field, with an additional condition, u on the boundary (gamma)of (omega)(subset of)R3 into a sum u = u++u- were u± are boundary values of vector fields which are Laplacian in (omega)and its complement respectively. Our proofs are based on the intimate relations between Laplacian vector fields theory and quaternionic analysis for the Moisil-Theodorescu operator.