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.
Highlights
In this article, the following new formula is derived, where f : R3 → R is a continuously differentiable function which vanishes at infinity:x ∂f + y ∂f + z ∂f f (a,b, c) = − ∫R3 ( ∂x a)2 + ( ∂y − b)2 ∂z (z − c)2 dxdydz
We prove an extension of Theorem 1 (Theorem 6) for bounded regions, involving volume and surface integrals, with a new, more natural definition of a region bounded by a surface, appropriate for spherical-polar coordinates
We will use a new definition of a surface in spherical-polar coordinates
Summary
The following new formula is derived, where f : R3 → R is a continuously differentiable function which vanishes at infinity:. Blumenthal assumes that the first partial derivatives of the vector field vanish at infinity; he proves uniqueness up to an additive constant function. His proof technique is different from ours, using Green’s theorem, among others All three formulas above immediately prove (a part of) Helmholtz’s Theorem which state that a 3-dimensional vector field is uniquely determined by its divergence, ∇ ⋅ F and curl, ∇ × F. The technique used leads immediately to an extension (Theorem 12) of Theorem 9 It appears to be a better alternative to the usual formula for vector fields over bounded regions.
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