Generalized Covariant Derivative with Respect to Time in Flat Space(Ⅰ):Eulerian Description

This paper reports a new derivative in the Eulerian description in flat space-the generalized covariant derivative with respect to time. The following contents are included:(a) the restricted covariant derivative with respect to time for Eulerian component is defined;(b) the postulate of the covariant form invariability in time field is set up;(c) the generalized covariant derivative with respect to time for generalized Eulerian component is defined;(d) the algebraic structure of the generalized covariant derivative with respect to time is made clear;(e) the covariant differential transformation group in time filed is derived. These progresses reveal the covariant form invariability of Eulerian space and time.

Generalized Covariant Derivative with Respect to Time in Flat Space(Ⅱ):Lagrangian Description

The previous paper reported a new derivative in the Eulerian description in flat space—the generalized covariant derivative of generalized Eulerian component with respect to time. This paper extends the thought from the Eulerian description to the Lagrangian description:on the basis of the postulate of covariant form invariability in time field, we define a new derivative in the Lagrangian description in flat space—the generalized covariant derivative of generalized Lagrangian component with respect to time. Besides, the covariant differential transformation group is set up. The covariant form invariability of Lagrangian space-time is ascertained.

从空间到时间--张量的协变微分学及协变性思想的拓展

本文聚焦于张量分析学中两类对称性破缺现象。一是基础性概念的对称性破缺:我们有张量对坐标的协变微分(简称空间协变微分),然而,没有明确定义过如下概念--张量对时间的协变微分(简称时间协变微分)。二是基本理论的对称性破缺:我们有空间域上张量的协变微分学,但没有时间域上张量的协变微分学。本文致力于弥补破缺的对称性。基于近年来的研究进展,回顾协变性思想的拓展历史,展示张量的时间协变微分概念的抽象过程,综述时间域上张量的协变微分学的构建历程,揭示空间域上的协变微分学与时间域上的协变微分学之间的对称性。

联络及其在现代物理中的解释

本文从曲线坐标、曲面上向量平移入手,导入了联络,继而引入协变微分、短程线及曲率张量,最后指明联络在广义相对论中的意义。