I have to calculate the Lie derivative of $\alpha = d x_1$ with respect to the vector field $X = \partial_1$, but I cannot use Cartan's magic formula (which would immediatly show that $L_X\alpha= 0$).
So the flow is given by $\phi_X^t(x) = (x_1+t,x_2,...,x_n)$, and thus $D\phi_X^t(x) = id_n$, right? Then it follows that $$\left.\frac{d}{dt}\right|_{t=0}(\phi_X^t(x))^*\alpha = \left.\frac{d}{dt}\right|_{t=0} \alpha\circ id_n = 0,$$ which should be the right answer... but I am not really sure if my linerarization of the flow is actually correct.