Calculating $\phi^{*}(\omega)$ and $\phi^{*}(d\omega)$

Question

Let $\omega = xdx +2zdy-ydz$ and $\phi:R^2 \to R^3, (u,v) \to (uv,u^2,3u+v)$

How do I determine $\phi^*(\omega)$ and $\phi^*(d\omega)$?

I calculated $d\omega= -xdx \wedge dy -3 dy \wedge dz$

Thanks for any hint or an example for a part of the term

Answer 1

$\phi^*(\omega) = uv d(uv) + 2(3u +v)d(u^2) - u^2d(3u +v)$ Hope now you can solve this.

Also $\phi^*(d \omega) = d(\phi^*(\omega)) $ .So this can help you to solve the second one too by using the first one.

Answer 2

Actually $\phi^*(\omega)= (u^3v^2+9u^2+4uv)du +(u^4v-u^2)dv$