In the picture below, from the book I'm using, we find a definition of a smooth vector field along $\partial M$.
My doubt rest on why $p\in M$ but the local definition is for $q\in \partial M$.
Does the author mean that for smoothness, the coefficients of the vector field must have an $C^{\infty}$ extension to a neighbourhood of $p$? What's the objective of such a definition?