爱因斯坦求和约定的推广

On the extension of Einstein's summation convention

爱因斯坦求和约定在微分几何、张量分析、连续介质力学等学科中,对于表达式和推导的简化,有着十分重要的作用. 本文从两个方面将其加以推广: 当项中括号或量指标重复出现一次时,和当项中i、j、k指标出现一次时,均可按爱因斯坦求和约定,对表达式的简化和推导过程的简化作推广应用.

In differential geometry, tensor analysis, continuum mechanics and other sciences, Einstein's summation convention plays an important role for simplifying the mathematical expressions and derivations. In the paper, it can be extended to the following two cases. The first case is that the repetition of an index of quantity or brackets including several quantities in a term denotes a summation with respect to that index over all coordinates. And the second case is that the cyclic indexes of i, j, k of quantities or brackets in a term denote a summation with respect to cyclic indexes over all coordinates (such as (1, 2, 3)、(2, 3, 1)、(3, 1, 2)). With the above extension, many mathematical expressions and derivations may be accomplished concisely.