New formulas are derived for once-differentiable 3-dimensional fields, using the operator src="Edit_325b1c1d-8a01-49b4-b4c2-fe0447653ca0.bmp" alt="" />. 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.
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