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Mie表示完全可以借助于R~3中的初等矢量分析进行讨论,但要包含着冗长的计算,而且引入的许多数学构造,除了它们自身以外,很难有任何领域的明确目的,在另一个极端,Mie理论可看作是微分形式的Hodge[1952]理论在曲面上的特定情形,在本文中,将采用中间的领域,而理论将基于三个面算子:面梯度算子Ⅴ_s、面旋度算子A_s和面拉普拉斯算子▽_s~2=▽_s·V_s,这一节就来描述这些算子,并利用它们把Helmholtz定理,即平面内的任一矢量场是二维梯度和二维旋度的矢量和,推广到曲面上,这里仅对球面给出了细节。
Mie said that it is entirely possible to discuss with elementary vector analysis in R ~ 3, but to include lengthy calculations and that many of the mathematical constructs introduced have little if any clear intent beyond their own, at the other extreme , The Mie theory can be thought of as a special case of the Hodge [1952] theory of differential form on a surface. In this paper, the middle field will be used and the theory will be based on three facets: the gradient operator ν_s, Degree operator A_s and plane Laplacian ▽ _s ~ 2 = ▽ _s · V_s, we will use these to describe Helmholtz’s theorem, that is, any vector field in the plane is two-dimensional The vector sum of the gradient and the two-dimensional curl is generalized to the surface, where details are given only for the sphere.