The Lie derivative of the function with respect to the vector field

A. Ishidori의 저서에서 정의된 Lie Derivative 이다.
실제로는 일반적인 Differential Derivation on paramerized curve에 대한 Derivation 정의와 동일하다.
본 내용은 Differential Geometry에 정의된 내용을 다시 옮겨 적은 것에 불과하지만, 관련 내용에 대한 사전 지식을 정리하기 위해서 기록한다.

Definition of Lie Derivative

Suppose $h(x) \in S(X)$, i.e. $h(x) \in \mathbb{R}$ is a smooth real-valued function on $X$. Note that $\nabla h(x) \in \mathbb{R}^n $ is a form on $X$. Suppose $f \in V(x)$. Then the map

$$
x \mapsto \nabla h(x) \cdot f : X \rightarrow \mathbb{R}$$

or

$$
x \mapsto dh(x) \cdot f : X \rightarrow \mathbb{R}$$

is smooth; it is called the Lie derivative of the function $h(x)$ with the respect to the vector field $f(x)$, and it is denoted by

$$
L_f h(x) = \nabla h(x) \cdot f(x) = dh(x) \cdot f(x)$$

Differntial Geometrical Notation

$$
L_f h(x) = fh(x) = dh(f)(x)$$

Meaning of Lie Derivative

Note that

$$L_f h(x) \in S(X)$$

Now suppose $h(x) \in F(X)$, i.e. $h(x)$ is a form on $X$. Then the map

… 추가 작성해야 할 듯…

The Lie derivative $ L_f h(x)$ can be interpreted as the derivative of $h(x)$ along the integral curves of the vector field $f(x)$ such that

$$
(L_f h)(x) = \lim_{t \rightarrow 0} \frac{1}{t} \left( h(S_{f,t})(x_0) – h(x_0)\right)$$

1. Curve

   $$
   \frac{d}{dt}x(t)=f(x(t)), \;\; x(0) = x_0$$

2. Integral Curve

   $$
   S_{f.t}(x_0) = \int_0^t \frac{d}{ds}x(s)ds |_{x(0)=x_0} = \int_0^t f(x(s)) ds |_{x(0)=x_0}$$