I have a simple problem in differential geometry. I need to study differential dynamical systems and vector fields on the closed hypercube $I^n = [0,1]^n$ (where $n \in \mathbb{N}$) and I don't know how can I do it properly. What I mean is that $I^n$ is not a smooth submanifold of $\mathbb{R}^n$ (neither a smooth submanifold with boundary of $\mathbb{R}^n$), so: how can I define correctly vector fields on $I^n$ ?
For example, I know that as $I^n$ is homeomorphic to the unit disk $D^{n-1}$, we can define a smooth structure on $I^n$ using the fact that $D^{n-1}$ is a smooth submanifold with boundary of $\mathbb{R}^n$. First, I am wondering why this way of doing should be chosen rather than another one and what differences does this induces on vector fields on $I^n$ (compare to another choice of smooth structure) ? Second, in this example, $I^n$ becomes a smooth manifold with boundary: does this affects (in any manner) the result about vector fields (e.g. existence of solutions of the associated ODE) ?
Little precision: I am not used to study dynamical systems and ODE, so please be indulgent. Thank you !