Definition of an inward-pointing tangent vector for smooth manifolds with corners Recall first the definition of an inward-pointing tangent vector for a smooth manifold with boundary.
Let $M$ be a smooth manifold with boundary.
If $p \in \partial M$, a vector $v \in T_pM - T_p\partial M$ is said to be inward-pointing if for some $\varepsilon > 0$, there exists a smooth curve $\gamma: [0,\varepsilon) \to M$ such that $\gamma(0) = p$ and $\gamma'(0) = v$ (John M. Lee - Introduction to Smooth Manifolds - p.118).
How can we extend this definition to smooth manifolds with corners ?
In particular, how do we define an inward-pointing tangent vector at a corner of the unit square $\square = [0,1]^2$ in $\mathbb{R}^2$ ?
 A: Here is a natural way of how to define them. I don't know if this definition is used in literature, but I came up with it and used it in my notes before.
Let $M$ be a manifold (possibly with corners), $p$ a point in $M$, $v$ a member of $T_pM$. We say $v$ is "realizable" iff there exists a smooth path $\gamma:[0,\epsilon[\rightarrow M$ such that $\gamma(0)=p$ and $\gamma'(0)=v$.
Next we define the set of "inward" tangents at $p$ as the topological interior of the set of realizabe tangets at $p$ as a subspace of $T_pM$. Obviously, this is an intrinsic  definition of what it means for a tangent to be inward.
Here is an equivalent definition of inward tangents using coordinates:
Let $M$ be a manifold (possibly with corners), $p$ a point in $M$, $v$ a member of $T_pM$, $\phi:U\rightarrow M$ be a local parameteization around $p$ ($U$ is open subset of $Q^n$). We say $v$ is inward tangent at $p$ iff: for every $i\in[k]$ such that $Pr_i(\phi^{-1}(p))=0$ we have that $Pr_i(D(\phi^{-1})|_pv)>0$.(Where $Pr_i$ is the natural projection to the $i$-th component)
It's an exercise to check that the above definition is  equivalent to the intrinsic definition given at the beginning of my answer and so it is
independent of the choice of the local parameteization $\phi$
