一般光滑曲面上的二类微分算子(英文)  

On Two Kinds of Differential Operators on General Smooth Surfaces

作　　者：谢锡麟[1] 

机构地区：[1]复旦大学力学与工程科学系,上海200433

出　　处：《复旦学报（自然科学版）》2013年第5期688-711,共24页Journal of Fudan University：Natural Science

基　　金：Projects support by National Nature Science Foundation of China(11172069);undergraduate key reform project Curriculum system of theories and applications of modern continuum mechanics issued by Shanghai Municipal Education Commission in 2011

摘　　要：Euclid空间中一般光滑曲面上可以定义二类微分算子,一类称为曲面梯度算子,另一类称为Levi-Civita算子.曲面梯度算子的定义源于定义于曲面上的张量场的可微性.理论研究了若干曲面梯度算子的积分及微分恒等式,这些恒等式在研究几何形态为曲面的连续介质力学以及流体与可变形边界的相互作用中具有重要意义.LeviCivita梯度算子的定义基于一般Riemann流形上的Levi-Civita联络.基于Levi-Civita梯度算子可以建立一些内蕴/坐标无关的微分恒等式,这些恒等式为建立固定光滑曲面上二维流动的涡量动力学理论奠定了基础.Two kinds of differential operators that can be generally defined on an arbitrary smooth surface in a finite dimensional Euclid space are studied, one is termed as surface gradient and the other one as Levi-Civita gradient. The surface gradient operator is originated from the differentiability of a tensor field defined on the surface. Some integral and differential identities have been theoretically studied that play the important role in the studies on continuous mediums whose geometrical configurations can be taken as surfaces and on interactions between fluids and deformahle boundaries. The definition of Levi-Civita gradient operator is based on Levi-Civita connections generally defined on Riemann manifolds. It can be used to set up some differential identities in the intrinsic/ coordiantes-independent form that play the essential role in the theory of vorticity dynamics for two dimensional flows on general fixed smooth surfaces.

关 键 词：曲面梯度算子 Levi-Civita梯度算子 内蕴形式广义Stokes公式 可变形边界上的流固耦合 曲面变形理论 固定光滑曲面上二维流动 

分 类 号：O331[理学—一般力学与力学基础]