in his "Introduction to Smooth Manifolds", 2.ed., p.67, John M. Lee poses the following exercise, after he has outlined how a natural chart of the tangent bundle TM is constructed from an interior chart of M:
Exercise 3.19: "Suppose M is a smooth manifold with bounary. Show that TM has a natural topology and smooth structure making it into a smooth manifold with boundary, such that if $ (U,(x^i)) $ is any smooth boundary chart for M, then rearranging the coordinates in the natural chart $ (\pi^{-1}(U),(x^i,v^i))$ for TM yields a boundary chart $ (\pi^{-1}(U),(v^i,x^i))$."
Presumeably the "natural chart $ (\pi^{-1}(U),(x^i,v^i))$" means a natural interior chart of TM.
I have no idea whatsoever why the interchange of coordinates $x^i$ (of the image of the point p) and $v^i$ (of the image of the tangent vector at p) leads to a boundary chart on TM. Can anyone give me a clue? Thanks in advance.