Abstract: 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 coordi-nate 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.

Key words: general relativity, Lie derivative, vector field, adapted coordinate system