Chains of electromagnetic's fields

Mathematics. Physics. Mechanics


Khrapko R. I.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia



A consistent use of the exterior differential forms in the electromagnetism shows that fields of electromagnetism are geometrical quantities of two different types: differential forms and contravariant (antisymmetric) tensor densities. These types are connected with each other by a specific operation, named the conjugation, which is a part of the Hodge operation. Field tubes and families of bisurfaces depict electromagnetic fields. The conjugation allows a many-fold differentiation of the fields and leads to field chains. The Helmholtz’s decomposition of a field is a decomposition into a closed part and a closed after conjugation part of the field. Chains of fields of electric and magnetic dipoles are considered in detail. Concept of boundary of a field is used widely. Laplacian is represented in terms of the conjugation, and its action on isolated chains is considered. A sufficient condition of harmonicity of a field is found.


Differential forms, Helmholtz decomposition, Hodge operation, Laplacian

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