Reduction of a Vector Field in a Manifold to an Incompressible Field PDF

Keywords

vector field, manifold, incompressible field

Abstract

A vector field L in a manifold M is incompressible if its divergence is equal to zero, i.e. if it satisfies the equation divL= 0. The solution of the last equation is well know in the three dimensional Euclidean space. In this work we determine a few expressions for an incompressible field in an n-dimensional Riemannian manifold M. Starting from one of these expressions, we show that every (n-1) functions of class C2 on M define an incompressible field whose lines are determined through setting these function equal to arbitrary constants. It is also shown that to every compressible field K there corresponds an infinite family of incompressible fields that have the same lines of K, but differ from K, as well as from each other, by their integral curves. The concept of "a compressibility removing factor" of a vector field K is introduced; it is defined as any function µ on M such that the field L = µK is incompressible. It is shown that the problem of determining the family of incompressible fields associated with the vector field K is equivalent to determining the family of compressibility removing factors of the field K. The quotient of two compressibility removing factors of field K is proven to be an integral of this field, and a general expression of the compressibility removing factors of the field K is derived. In case of a two-dimensional manifold, the relation between the compressibility removing factors of the field and the integrating factors of the ordinary differential equation associated with the field is found. The theory is illustrated through application to central fields. Finally, it is demonstrated that the set of all incompressible fields in a manifold M forms a sub-algebra of the Lie algebra formed by the set of all C(infinity) vector fields in M, and the sub-algebra of incompressible fields contains the set of Killing fields as a sub-algebra.