Classical Physics


Helmholtz's Decomposition Theorem, Etc.

Authors: R. I. Khrapko

The fact is used that electromagnetic fields are covariant (antisymmetric) tensors or contravariant (antisymmetric) tensor densities, which are mutual conjugated. The conjugation allows many-fold specific differentiation of the fields and leads to field chains. An integral operation, named the generation, is considered, which is reverse to the specific differentiation. The double generation yields zero as well as the double differentiation. The Helmholtz decomposition is compared with the Poincare decomposition, and many ways of the Helmholtz decomposition are presented. Laplace operator and the inverse Laplace operator are expressed in terms of the differential and integral operations. All results are illustrated by simple examples.

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Submission history

[v1] 12 May 2008

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