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Helmholtz decomposition theorem and Blumenthal's extension by regularization

Condensed Matter Physics. 2017;20(1):13002 DOI 10.5488/CMP.20.13002


Journal Homepage

Journal Title: Condensed Matter Physics

ISSN: 1607-324X (Print); 2224-9079 (Online)

Publisher: Institute for Condensed Matter Physics

Society/Institution: National Academy of Science of Ukraine

LCC Subject Category: Science: Physics

Country of publisher: Ukraine

Language of fulltext: English

Full-text formats available: PDF



D. Petrascheck

R. Folk


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Time From Submission to Publication: 15 weeks


Abstract | Full Text

Helmholtz decomposition theorem for vector fields is usually presented with too strong restrictions on the fields and only for time independent fields. Blumenthal showed in 1905 that decomposition is possible for any asymptotically weakly decreasing vector field. He used a regularization method in his proof which can be extended to prove the theorem even for vector fields asymptotically increasing sublinearly. Blumenthal's result is then applied to the time-dependent fields of the dipole radiation and an artificial sublinearly increasing field.