Conceptual doubt about pullback and local coordinates

I'm studying differential forms in manifolds through the book: Differential Geometry and Topology With a View to Dynamical System, by Keith Burns.

I have a conceptual question and I would like you to answer me, if possible. (I'm studying Analysis in $$\mathbb{R}^n$$ and this part of differential forms I have to follow with a book on differential geometry.)

Let $$M$$ be a smooth manifold and $$\psi : U \subseteq \mathbb{R}^n \longrightarrow V \subseteq M$$ be chart. If $$\omega$$ is a differential form in $$M$$, then I'm wrong to say that $$\omega$$ can be written in local coordinates as $$\omega = \sum_{j_1 < \dots < j_k} f(x)\,dx_{j_1} \wedge \cdots \wedge dx_{j_k}$$ is the same thing to calculate $$\psi^*\omega?$$

• Looks a bit like you have it backwards. A chart is a map $\phi:M\to\mathbb{R}^n$ for some $n$. What you have is a parametrization $\psi=\phi^{-1}$ (judging by your naming and symbols). Nitpicky, but important. You can not „directly“ write this without a bit more knowledge, only in local coordinates. However, you can embed your manifold in $\mathbb{R}^m$ for large enough $m$ and then write the diff-form on it, taking the pullback then amounts to what I think you are asking. Commented Aug 1 at 15:37