A RIGOROUS AND COMPLETED STATEMENT ON HELMHOLTZ THEOREM
A RIGOROUS AND COMPLETED STATEMENT ON HELMHOLTZ THEOREM
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关于亥姆霍兹定理的严格且完整的陈述
DOI：
10.2528/pier06123101
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发表时间：
2007
影响因子：
6.7
通讯作者：
W. Dou
中科院分区：
文献类型：
--
作者：
Y. Gui;W. Dou
经典的Helmholtz定理虽然有广泛的应用，但它的表述也有一定的局限性。实际上，它只适用于单边界面的单连通区域，对场函数不连续的区域没有给出任何结论。然而，在实际应用中经常会遇到不连续的情况，例如面源的位置或两种介质的界面。同时，现有的Helmholtz定理的大多数版本都在一定程度上是不精确和不完善的。本文不仅用数学语言的精确度和逻辑的严密性对经典的Helmholtz定理给出了精确的表述和严格的证明，而且把它推广到多连通区域的情形，得到了定义在三维欧氏空间上的一个广义的Helmholtz定理。同时，我们的证明和推理更加充分和完善。作为广义Helmholtz定理的一个重要应用，重点介绍了无旋向量函数和螺线向量函数的概念。推广的Helmholtz定理和本文的结论在包括矢量分析在内的电磁学等学科中具有重要的参考价值。
There are some limitations on the statement of classic Helmholtz theorem although it has broad applications. Actually, it only applies to simply connected domain with single boundary surface and does not provide any conclusion about the domain where discontinuities of field function exist. However, discontinuity is often encountered in practice, for example, the location of surface sources or interface of two kinds of medium. Meanwhile, most existing versions of Helmholtz theorem are imprecise and imperfect to some extent. This paper not only tries to present a precise statement and rigorous proof on classic Helmholtz theorem with the accuracy of mathematical language and logical strictness, but also generalizes it to the case of multiply connected domain and obtains a generalized Helmholtz theorem in the sense of Lebesgue measure and Lebesgue integral defined on three-dimensional Euclidean space. Meanwhile, our proof and reasoning are more sufficient and perfect. As an important application of the generalized Helmholtz theorem, the concepts of irrotational and solenoidal vector function are emphasized. The generalized Helmholtz theorem and the present conclusion should have important reference value in disciplines including vector analysis such as electromagnetics.