Finding the Lie derivative of a 2-form exercise

Question

Let $\beta=-x dx \wedge dy + ydy \wedge dz$. The vector field is $X=(y,0,z)$.Find the Lie derivative.

I try that

$\begin{align}L_X \beta &=L_X (-x dx \wedge dy + ydy \wedge dz) = -x L_X(dx \wedge dy)+ y L_X(dy \wedge dz) \\ &= -x [ L_X(dx) \wedge dy + dx \wedge L_X(dy) ]+ y [ L_X(dy) \wedge dz + dy \wedge L_X(dz) ] \\ &= -x [ dL_X(x) \wedge dy + dx \wedge dL_X(y) ]+ y [ dL_X(y) \wedge dz + dy \wedge dL_X(z) ] \end{align}$

but how do I evaluate this further. I cannot see what $L_X(x)$ for example would be?

Answer 1

I'm not sure how you are getting the second equality and I don't think it is true. However, the Lie derivative of a function is just the directional derivative so that $L_X(x)=X(x)$, which looks silly. But you have $X=y\frac{\partial}{\partial x}+z\frac{\partial}{\partial z}$, so that $X(x)=y$.

Anyway, a simpler way to solve this problem would be to use Cartan's magic formula $$L_X\beta=\iota_X(d\beta)+d(\iota_X\beta)$$ where $\iota_X$ is the interior product.