中山大学学报自然科学版 ›› 2019, Vol. 58 ›› Issue (5): 26-32.doi: 10.13471/j.cnki.acta.snus.2019.05.004

• 论文 • 上一篇    下一篇

由反对称化操作探究Levi-Civita与Kronecker符号的性质

彭俊金1,2,雷良建1   

  1. 1.贵州师范大学物理与电子科学学院,贵州 贵阳 550001;
    2. 贵州省射电天文数据处理重点实验室,贵州 贵阳 550001
  • 收稿日期:2018-08-25 出版日期:2019-09-25 发布日期:2019-09-25

Exploring the properties of Levi-Civita and Kronecker symbols through antisymmetrization  

PENG Junjin1,2, LEI Liangjian1   

  1. 1. School of Physics and Electronic Science,Guizhou Normal University,Guiyang 550001,China)
    2. Guizhou Provincial Key Laboratory of Radio Astronomy and Data Processing, Guiyang 550001,China
  • Received:2018-08-25 Online:2019-09-25 Published:2019-09-25

摘要:

从度规张量的行列式出发,应用通过反对称化Kronecker符号与逆变Levi-Civita张量的乘积所得到的恒等式,证明了Levi-Civita符号(张量)的一个重要性质,即:两个Levi-Civita符号(张量)的部分或全部指标缩并后可由(推广的)Kronecker delta符号表示。并在此基础上,导出了两个重要推论,且借助行列式与矩阵的性质给出二者的又一证明。与此同时,给出了推广的Kronecker delta符号的若干重要性质。

关键词: Levi-Civita符号, Kronecker符号, 度规张量, 广义相对论

Abstract:

In this paper, on basis of the determinant for the metric tensor, together with the identities got via anti-symmetrizing the Kronecker delta symbol and Levi-Civita tensor, we prove one of the properties of the Levi-Civita symbol (tensor), which shows that two Levi-Civita symbols (tensors) with the contraction for some or total indices can be expressed by a series of Kronecker delta symbols or the generalized Kronecker delta symbol. We further present two important deductions from the property of the Levi-Civita tensor, which are proved in other ways. Besides, we analyze several important properties for the generalized Kronecker delta symbol.

Key words: Levi-Civita symbol, Kronecker symbol, metric tensor, general relativity

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