1. ## Lie derivative

Hello,

I want to show this equation of the Lie-derivative:

$\displaystyle L_X (i_Y \alpha)=i_{[X,Y]} \alpha+i_Y (L_X \alpha)$

whereas L_X is the Lie derivative and X,Y are vector fields.

I try to understand the equation, but i'm hopelessly.

Can you help me by my task.

Regards

2. How do you define the Lie derivative? For any of the (equivalent) definitions, Lie derivative has the property that it commutes with contraction. And $\displaystyle i_Y(\alpha)$ is the contraction of Y and $\displaystyle \alpha$.

Let $\displaystyle C(Y \otimes \alpha)$ denote the contraction $\displaystyle i_Y(\alpha)$, we have
$\displaystyle L_X(i_Y(\alpha))$
= $\displaystyle L_X(C(Y \otimes \alpha))$
commutes with contraction = $\displaystyle C(L_X(Y \otimes \alpha))$
Leibniz rule = $\displaystyle C(L_X(Y) \otimes \alpha + Y \otimes L_X(\alpha))$
linearity = $\displaystyle C(L_X(Y) \otimes \alpha) + C(Y \otimes L_X(\alpha))$
perform contraction = $\displaystyle i_{L_X(Y)}(\alpha) + i_Y (L_X(\alpha))$
= $\displaystyle i_{[X,Y]}(\alpha) + i_Y (L_X(\alpha))$