If $M^n$ is a smooth manifold with boundary and $p\in\partial M$, then $T_pM$ is the disjoint union of inward and outward point vectors, and $T_p\partial M$. If $(U,(x^i))$ is a smooth boundary chart at $p$, then the inward pointing vectors are those with positive $x^n$ coordinate, the outward pointing vectors are those with negative $x^n$ coordinate, and $T_p\partial M$ is those with zero $x^n$ coordinate.
What's confusing me is that the coordinate map $(x^i)$ is on $M$, and $T_pM$ is a vector space, not a manifold. So how can you talk of the $x^n$ coordinate of a tangent vector in the tangent space of an abstract manifold?
Best Answer
It's probably more accurate to say that inward-pointing vectors are those with positive $x^n$ component. In the given boundary chart, each element of $T_pM$ can be written $v = v^i\partial/\partial x^i$ (using the summation convention), and $v$ is inward-pointing if and only if $v^n>0$.