For my analysis class, I am going through a differential forms book, and have encountered a problem that has made me super confused. I have attached an image of how the book we are using defines the pullback of differential 1-forms $f^*\mu$.
After this section, there is an exercise: Let $U, V$ be an open subset of $\mathbb{R}^n$, and $f: U \to V$ is a $C^k$ map. Show that for $\phi \in C^{\infty}(V)$ the chain rule for $f$ and $\phi$: $f^*d\phi = df^*\phi$.
This makes no sense to me. Doesn't $f$ need to be $C^{\infty}$ by definition? And if the notation is map then form (as far as order), then why is it $df^*\phi$, which is reversed? Is there something I'm missing? Any help would be greatly appreciated!