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.
作者
唐少强
TANG Shaoqiang(Department of Mechanics and Engineering Sciences,Peking University,Beijing 100871,China)
出处
《力学与实践》
北大核心
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。
关键词
张量
客观性原理
线性泛函
对偶空间
内积空间
tensor
principle of objectivity
linear functional
dual space
inner product space
