Define a vector field on $\mathbb{R}^d$ by $X = \frac{\partial}{\partial x_{d}}$. That is a vector field that always points upward along the $x_{d}$-axis. Consider starting at any point $p \in \mathbb{R}^d$ and following the vector field $X$ for all time. This path is the integral curve starting at the point $p$ for the vector field $X$. Now even though $\mathbb{R}^d$ is not compact, it is clear that this defines a global flow, so the integral curves exist for all time.
Now consider a convex polygon $P \subset \mathbb{R}^d$, not necessarily of full dimension. For each point $p \in P$ I can consider the vector $X(p)$. Define a vector field on $P$ by projecting $X(p)$ onto $P$ for every $p \in P$. Denote this vector field by $X|_{P}$. I'd like to know under what conditions is this vector field smooth? I'd also like to know under what conditions can I consider the integral curves of $X|_{P}$? Furthermore what happens to the integral curves as they hit the boundary of $P$? Are they still defined on the boundary?