I've just started reading differential geometry, however i was trying to solve one of the examples from Nonholonomic mechanics and control by Bloch. Chapter 2, problem 2.6-1.
Consider a two-form $\beta=xdy \wedge dz +ydx \wedge dz+ zdx \wedge dy$ and the vector field $X = x \frac{\partial}{\partial x}+y \frac{\partial}{\partial y} + z \frac{\partial}{\partial z}$
part-c of the question asks to compute the Flow of X.
For which my answer is $F_t =(x_0e^t, y_0e^t, z_0e^t)$ Since $\frac{dx}{dt}=x;\frac{dy}{dt}=y; \frac{dz}{dt}=z$
The Question is to compute $\frac{d{F_t^*\beta}}{dt}$
I'm completely lost on approaching this problem. since $\beta$ is given in $x,y,z$, as far as i understand this is the pull-back of the vector field, the book First Steps in Differential Geometry notes its just a substitution of $F_t$ in $\beta$ when i try that since $dx = x_0 e^tdt$ and $dy=y_0 e^tdt $ making $dx \wedge dy = 0$.
Any help is very much appreciated. Thanks.