let $\alpha,\beta\in\Lambda_1(\Bbb R^3)$ and $\gamma\in\Lambda_2(\Bbb R^3)$ be given by \begin{align*} \alpha &= xdx + ydy + zdz,\\ \beta &= xdy − ydx + dz,\\ \gamma &= z^2dx\wedge dy + x^2dy\wedge dz + y^2dx\wedge dz. \end{align*}
Let $\varphi:\Bbb R^3\to\Bbb R^3$ be given by $\varphi(u, v, w)=(e^w\sin u, e^w \cos u, v^3 − u^3)$ and let $\psi:\Bbb R^2\to\Bbb R^3$ be given by $\psi(s, t)=(s \cos t, s \sin t, t)$.
How to compute $\varphi*\gamma,\psi∗\beta$ and $\psi*\gamma$ ?