发表机构
School of Mechatronics Engineering, Guizhou Minzu University; School of Mathematical Science, Guizhou Normal University（贵州民族大学机电工程学院; 贵州师范大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过向量分析与拓扑工具，揭示有界三维电磁域中调和向量场与贝蒂数的对应关系，并证明诺伊曼与狄利克雷调和场空间的维数，为谐振器直流模式提供拓扑解释。
AI 中文摘要
有界三维域的拓扑结构可以强烈影响电磁场。在拓扑复杂的域中，无旋场可能不具有全局单值标量势，无散场也可能不具有全局向量势。这些拓扑效应自然导致亥姆霍兹分解中的调和向量场。本文利用向量分析、环量积分、切割面、标量拉普拉斯问题、斯托克斯定理和格林恒等式，对这些场进行了几何和构造性研究。第一贝蒂数通过独立的柄状环流来解释，而第二贝蒂数则计数封闭的空腔。文中直接证明了诺伊曼和狄利克雷调和场空间的维数。分析还扩展到各向异性无耗介质。结果为有界电磁谐振器中的调和场和物理直流模式提供了简单的拓扑解释。
英文摘要
The topology of a bounded three-dimensional domain can strongly affect electromagnetic fields. In a topologically complex domain, a curl-free field may not have a globally single-valued scalar potential and a divergence-free field may also fail to have a global vector potential. These topological effects lead naturally to harmonic vector fields in the Helmholtz decomposition. This paper gives a geometric and constructive study of such fields using vector analysis, circulation integrals, cutting surfaces, scalar Laplace problems, Stokes' theorem, and Green's identity. The first Betti number is interpreted through independent handle-type circulations, whereas the second Betti number counts enclosed voids. Direct proofs are given for the dimensions of the Neumann and Dirichlet harmonic-field spaces. The analysis is also extended to anisotropic lossless media. The results provide a simple topological interpretation of harmonic fields and physical DC modes in bounded electromagnetic resonators.
Comments8 pages
