张量紧说  

TENSOR COMPACTLY EXPLAINED

作　　者：唐少强[1] TANG Shaoqiang(Department of Mechanics and Engineering Sciences,Peking University,Beijing 100871,China)

机构地区：[1]北京大学力学与工程科学系,北京100871

出　　处：《力学与实践》2023年第1期150-156,共7页Mechanics in Engineering

基　　金：北京大学基础学科拔尖人才培养计划2.0项目,国家自然科学基金(11988102,11832001)资助。

摘　　要：本文从线性空间出发,介绍了线性泛函与对偶空间,在此基础上以双线性型为例引出了张量的数学定义-多线性泛函;另一方面,内积作为一种特殊的双线性型给予向量另一种身份:余向量,由此通过限定定义张量的线性空间都取为Rn(n=2,3),就得到了(连续介质)力学中常用的张量,即在基变换下坐标(或称分量)按照给定规则变化的量。进一步给出了张量的张量积、缩并、点积和双点积运算。Linear space, linear functional and dual space are introduced. Based on this, tensor is defined mathematically as a multilinear functional, with bilinear form as an example. In addition, as a special case of bilinear form, inner product renders a co-vector interpretation of vector itself. Now setting all the linear spaces that vectors and co-vectors reside as Rn(n = 2, 3)the theory of continuum, viz, an entity that changes its coordinates/components according to certain given rules under coordinate transform of Rn. Tensor product, contraction, dot product and double dot product are also explained.

关 键 词：张量 客观性原理 线性泛函 对偶空间 内积空间 

分 类 号：O302[理学—力学]