Incompressible Fields in Riemannian Manifolds

Incompressible fields are of special importance in elec-trodynamics, fluid mechanics, and quantum mechanics. We shall derive a few expressions for such fields in a Riemannian manifold, and show how to generate an incompressible field from an arbitrary set of scalar differentiable functions. The concept of compressibility removing factors of an arbitrary vector field is introduced and utilized to obtain from an arbitrary vector field an incom-pressible one that has the same vector surfaces as the original field. A general expression for compressibility removing factors of a vector field is derived. The method is applied to central fields. 1. Introduction The equation which expresses that a vector field L has a zero divergence, divL = 0, appears in a number of areas of physics such as electrodynamics, fluid mechanics, and quantum mechanics. In fact two of Maxwell's equations, namely div