由反对称化操作探究Levi-Civita与Kronecker符号的性质

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

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。

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.