中文摘要：

这份报纸在 3D 欧几里得空间在弄弯的坐标系统下面扩大 covariant 衍生物。基于 covariant 形式不变的公理，能仅仅对部件起作用的古典 covariant 衍生物被扩大到能包括基础向量，向量和张肌对任何几何数量起作用的概括 covariant 衍生物。在公理下面，概括 covariant 衍生物的代数学结构被证明是 covariant 微分戒指。基于强大的操作能力和概括 covariant 衍生物的简单分析性质，在弄弯的坐标系统的张肌分析被简化到大程度。

英文摘要：

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.