Extension of covariant derivative(Ⅰ): From component form to objective form  

作　　者：Ya-Jun Yin 

机构地区：[1]Department of Engineering Mechanics, School of Aerospace,Tsinghua University

出　　处：《Acta Mechanica Sinica》2015年第1期79-87,共9页力学学报（英文版）

基　　金：supported by the NSFC(11072125 and 11272175);the NSF of Jiangsu Province(SBK201140044);the Specialized Research Fund for Doctoral Program of Higher Education(20130002110044)

摘　　要：This paper extends the covariant derivative un der curved coordinate systems in 3D Euclid space. Based on the axiom of the covariant form invariability, the classical covariant derivative that can only act on components is ex tended to the generalized covariant derivative that can act on any geometric quantity including base vectors, vectors and tensors. Under the axiom, the algebra structure of the gen eralized covariant derivative is proved to be covariant dif ferential ring. Based on the powerful operation capabilities and simple analytical properties of the generalized covariant derivative, the tensor analysis in curved coordinate systems is simplified to a large extent.This paper extends the covariant derivative un der curved coordinate systems in 3D Euclid space. Based on the axiom of the covariant form invariability, the classical covariant derivative that can only act on components is ex tended to the generalized covariant derivative that can act on any geometric quantity including base vectors, vectors and tensors. Under the axiom, the algebra structure of the gen eralized covariant derivative is proved to be covariant dif ferential ring. Based on the powerful operation capabilities and simple analytical properties of the generalized covariant derivative, the tensor analysis in curved coordinate systems is simplified to a large extent.

关 键 词：Tensor analysis Classical covariant derivatives Generalized covariant derivatives The axiom of the covari-ant form invariability Covariant differential ring 

分 类 号：O341[理学—固体力学]