关于二阶反对称张量及其对偶矢两者间正向反向关系问题的注记

首先着力阐明,唯独二阶反对称张量Ω才存在着它唯一的对偶矢ω.后者一经找到,立刻可见其表达式实际上就是ω与Ω之间关系,亦即本文正向关系问题之解.接着处理反向关系问题,又求得我们也想要找的Ω与ω之间关系.综合以上二结果,便导致总结论——本工作之成果汇总.另外,文末附带将Lurie[2]给出的,同时反映上述Ω与ω两者特征的一对公式作了推广.

A Note on the Positive-Way and Negative-Way Problems Concerning the Relations between an Antisymmetric Second Order Tensor and Its Dual Vector

Firstly, make certain that as to an antisymmetric tensor Ω of order two, there exists exactly one corresponding vectorω, known as the dual vector ofΩ. Once the latter has been found, we see that its expression is just the relation betweenωandΩ, i.e., the solution for the positive-way problem. Next, let us treat the negative-way problem so as to get the relation between Ω and ω. Synthesizing both results above, we are led to the general conclusion. In addition, a pair of formulas given by Lurie, which reflect simultaneously the features of both Ωandω, are generalized at the end of this paper.