One-Form pullback equation

Question

Let $U \subseteq \mathbb R^n$ and $V \subseteq \mathbb R^m,g: U \to \mathbb V$ continuably differentiable and let be $g$ One-Form on $V$. We define the pull-back $g^*w$. Let $w=\sum_{i=1}^m f_idy_i$. How can one show $g^*w=\sum_{k=1}^m (f_i \circ g)dg_i$.

A little help or hint is much appreciated.

Answer 1

Let $x\in R^n$, $u\in T_xR^n$, $(g^*w)_x(u)=w_{g(x)}dg.u=(f_1dy_1+..+f_mdy_m)_{g(x)}(dg_1(u),..,dg_m(u))$

$=f_1(g(x))dg_1(u)+...+f_m(g(x))dg_m(u)$.