New Formulas and Results for 3-Dimensional Vector Fields

作     者：Sadanand D. Agashe Sadanand D. Agashe

作者机构：2-B Anjaneya Society Mumbai India 

出 版 物：《Applied Mathematics》 (应用数学（英文）)

年 卷 期：2021年第12卷第11期

页      面：1058-1096页

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Spherical-Polar Coordinates Helmholtz Decomposition Divergence Theorem Orthogonality Maxwell’s Equations 

摘      要：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.