Commutative Rule involving the Laplace Operator

Commutation rules between the Laplace operator and other basic operators (include the divergence, the curl, and the gradient operator) are established. These rules were seldom noticed in the past. However, it is shown that they lead to an alternative derivation of the electromagnetic fields due to an arbitrary current distribution in a homogeneous environment without referring to the vector potential concept. The derived mathematical result can still be linked to show the physical insight of the original problem.