On $\mathbb{R}^{2n+1}$ use coordinates $z,x^1,y^1,...,x^n,y^n$. Let $\alpha = dz + \sum_ix^idy^i$. I wanted to compute the pull back $f^*\alpha$ for $f\colon \mathbb{R}^{2n+1}\to \mathbb{R}^{2n+1}$ with $f(z,x^1,...,y^n) = (t^2z,tx^1,...,ty^n)$.
Unfortunately, i'm not sure whether my attempt is going in the right direction.
My attempt:
\begin{align*}f^*\alpha = f^*\left(dz + \sum_ix^idy^i\right) = d(t^2z) + \sum_i tx^i d(ty^i)& = 2tzdt+t^2dz + \sum_i tx^i (y^idt+tdy^i) \\ &= 2tzdt+t^2dz + \sum_itx^iy^idt + \sum_itx^itdy^i \\ &= t(2zdt+tdz )+t\left( \sum_ix^iy^idt + \sum_ix^itdy^i\right) \end{align*}
would this be reasonable so far? Am I on the right track? Or did i do something fundamentally wrong? Any help is appreciated. I'm not entirely sure how to further simply and thus would appreciate any feedback.