基张量和张量基的概念及其分析

On the concepts and analyses of base tensors and tensor bases

通常张量有整体表示式和分量(指标)表示式,但这二者尚不是全面而完整的. 因此,须将矢量中的基矢量和矢量基的概念推广到基张量和张量基. 为此,需以基矢量及张量外积为基础,导出基张量和张量基,任一张量可表达为诸基张量的线性和. 据此,可以阐明并矢正是二矢量的外积,并建立了其与张量的对应关系. 这样的认识对于物理定律的张量方程及其应用有着重要的意义.

Generally, there are two expressions of tensors: the integral and the component expressions. However they are both not enough comprehensive and perfect. It is beneficial to extend the concepts of base vectors and vector bases to the concepts of base tensors and tensor bases. These concepts can be derived based on the concept of base vectors and the law of tensor product (outer product) and any tensor can be expressed by the linear sum of the corresponding base tensors. It is also shown that the dyad is just the outer product of two vectors, and the relationship between dyads and tensors can be accordingly set up. Such recognition is of great value for understanding and applying tensor equations of physical laws.