Helmholtz's decomposition theorem, etc.
Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
AbstractSeveral ways of the decomposition of vector fields into the irrotational and solenoidal parts are considered. The fact is used that quantities of electromagnetism are covariant (antisymmetric) tensors or contravariant (antisymmetric) tensor densities. The classes of quantities is related by a conjugation. All differential operations, grad, div, curl, are considered from a single point of view as well as the inverse integral operations are. It is shown that a repeated differential operation or repeated integral operation yields zero, but the conjugation between the two operations leads to infinite or finite chains of quantities. Laplace operator and the inverse Laplace operator are expressed in terms of the differential and integral operations. The results are illustrated by simple examples. The paper is published at www.sciprint.org.