Agashe, Sadanand D. (2021) New Formulas and Results for 3-Dimensional Vector Fields. Applied Mathematics, 12 (11). pp. 1058-1096. ISSN 2152-7385
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Abstract
New formulas are derived for once-differentiable 3-dimensional fields, using the operator . This new operator has a property similar to that of the Laplacian operator; however, unlike the Laplacian operator, the new operator requires only once-differentiability. A simpler formula is derived for the classical Helmholtz decomposition. Orthogonality of the solenoidal and irrotational parts of a vector field, the uniqueness of the familiar inverse-square laws, and the existence of solution of a system of first-order PDEs in 3 dimensions are proved. New proofs are given for the Helmholtz Decomposition Theorem and the Divergence theorem. The proofs use the relations between the rectangular-Cartesian and spherical-polar coordinate systems. Finally, an application is made to the study of Maxwell’s equations.
Item Type: | Article |
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Subjects: | Library Eprints > Mathematical Science |
Depositing User: | Managing Editor |
Date Deposited: | 01 Dec 2022 05:17 |
Last Modified: | 11 Jul 2024 04:50 |
URI: | http://news.pacificarchive.com/id/eprint/600 |