A question about pullback of a smooth form.

Question

In Lee book, "Introduction to Smooth Manifold, 2nd edition" there is a proof about the existence and Uniqueness of the Exterior differentiation (page 365). He define the exterior differentiation of a smooth $k$-form $\omega\in \Omega^k(M)$ locally on each smooth chart $(U, \phi)$ by using pullbacks: $$d\omega = \phi^*d(\phi^{-1*}\omega).$$ My question is how do we pullback $\omega$ by $\phi^{-1}$? why I ask because by definition of pullback we could pull back by $\phi^{-1}$ only for forms on $U$, but here $\omega$ is a smooth form on $M$ not $U$. Something is not clear here for me please help.

Answer 1

Properly speaking, Lee is not pulling back $\omega$ but its restriction to $U$. We can define the restriction to $U$ of $\omega$ via $\omega|_U = i^*\omega$ where $i: U \longrightarrow M$ is the inclusion map. So really, this defines $(d\omega)|_U$. You can show that these glue together to properly define $d\omega$.