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Gauss, C.F. (1840) General Propositions Relating to Attractive and Repulsive Forces Acting in the Inverse Ratio of the Square of the Distance. In: Taylor, R., Ed., Scientific Memoirs: Selected from the Transactions of Foreign Academies of Science and from Foreign Journals, Vol. III, R. and J. E. Taylor, New York, 153-196.

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• TITLE: New Formulas and Results for 3-Dimensional Vector Fields

  AUTHORS: Sadanand D. Agashe

  KEYWORDS: Spherical-Polar Coordinates, Helmholtz Decomposition, Divergence Theorem, Orthogonality, Maxwell’s Equations

  JOURNAL NAME: Applied Mathematics, Vol.12 No.11, November 30, 2021

  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.