Let $f\colon \mathbb{R}^n\rightarrow \mathbb{R}^N$ be a parametrisation of a manifold $M$.
Given the form $\displaystyle \omega = \sum_{j=1}^N w^j\mathrm{d}x_j$ where $f_j\in C^\infty (M)$.
The pullback is defined by $\displaystyle f^*\omega =\sum_{j=1}^N (w^j\circ f)\mathrm{d}f_j$
It holds true that $f^*\omega (X)(x)=\omega_{f(x)}(f'(x)X_x)$ where $X$ is a smooth vector field.
I can not reproduce this equivalence. I don't even see the term $\omega_{f(x)}(f'(x)X_x)$ well defined, since $f'(x)$ must be a $N\times n$ matrix.
If I start with the left side I get
$\displaystyle f^*\omega (X)(x)=\sum_{j=1}^N\sum_{i=1}^N (w^j\circ f)(x)a_i(x)\frac{\partial f_j(x)}{\partial x_i}$
I hope at least this is correct.