Abstract: This is the question asked in class: is it possible to obtain the covariant derivative of natural base vectors? To answer this question, the idea of axiomazitaion is introduced, the concepts of the generalized component and the generalized covariant derivative are defined. Based on these new concepts, the classical covariance is developed into the generalized covariance, and the classical covariant differentiation is developed into the generalized covariant differentiation. This paper summarizes the main difficulties and the important points of the above explorations, and shows the abstractions of the generalized covariant derivatives and the advances of the generalized covaraibilities.