李导数一种新的讲授方式  

A new definition of Lie derivative

作　　者：何孝凯 曹周键 HE Xiao-kai;CAO Zhou-jian(School of Mathematics and Computational Science,Hunan First Normal University,Changsha,Hunan 410205,China;Department of Astronomy,Beijing Normal University,Beijing 100875,China)

机构地区：[1]湖南第一师范学院数学与计算科学学院,湖南长沙410205 [2]北京师范大学天文系,北京100875

出　　处：《大学物理》2021年第7期8-11,共4页College Physics

基　　金：湖南省青年骨干教师专项经费的资助。

摘　　要：李导数是微分流形上的一类重要导数算子,在数学和物理上都有着广泛的应用.本文给出关于李导数一种新的讲授方式.不同于通常教材利用推前和拉回映射定义李导数,本文首先介绍了流形上光滑矢量场的适配坐标系的概念,然后给出了流形上的任意光滑矢量场沿某一给定矢量场的李导数在适配坐标下的定义.在此基础上,我们证明了这一新定义与具体适配坐标系的选择无关.进一步地,我们给出了矢量场李导数在任意坐标系下的表达式,与通常教材中定义的李导数相衔接.与通常的教材引入李导数的方式相比,本文采用的讲授方式更便于初学者理解和掌握.本文对矢量场李导数的处理方式,可以自然推广到流形上的任意张量场李导数.Lie derivative is an important concept of differential geometry. It is widely used in mathematics and physics. The Lie derivative is defined through the push-forward and pull-back maps. In this paper,a new definition of Lie derivative is presented. Instead of the abstract maps,we firstly introduce the concept of an adapted coordinate system of a smooth vector field on a manifold. Then a new definition of Lie derivative is given based on an adapted coordinate. It can be shown easily that the new definition is independent of the specific choice of an adapted coordinate system. After that,we deduce the explicit expression of the Lie derivative in general coordinate system,which admits the same form as the Lie derivative definition in usual textbooks. Compared to the existing definition of Lie derivative in usual textbooks,our new definition is much easier for beginners to understand.

关 键 词：广义相对论 李导数 矢量场 适配坐标系 

分 类 号：O4-1[理学—物理]