In the Wikipedia article on Lie derivative we read that $\mathcal{L}\_X \omega$ (where $X$ is a vector field and $\omega$ is a 1-form) evaluates the change of $\omega$ along the flow defined by the vector field $X$. Then $\mathcal{L}\_X \omega = 0$ would mean that $\omega$ remains constant along the flow defined by the vector field $X$.

Suppose $X=\sum \_i x\_i \frac{\partial}{\partial x\_i}$ and $\eta$ is a 1-form such, that $\mathcal{L}\_X \eta = 0$. What is the meaning of this...? Is $X$ actually the position vector? Then it would mean $\eta$ doesn't change anywhere in $\mathbb R^3$? Does it mean $\eta=\mbox{const}$ in $\mathbb R^3$ ?

The question is: 

-What is the meaning of the field $X$? How would you describe it?

-What does it mean for $\eta$ to be an invariant?