I am going through Munkres' book Analysis on Manifolds. Let $M\subset \mathbb{R}^n$ be a smooth $m$-dimensional submanifold. He defines a differential form of order $k$ on $M$ to be a smooth map $\omega$ where for each point $p\in M$, $\omega(p)$ is an alternating $k$-covector on the tangent space $T_pM$.
He states that "any differential form on $M$ can be extended to a differential form defined on an open set containing M, but the proof is decidedly nontrivial." That is if $\omega$ is a differential $k$-form on $M$ then there is a set $A$ which is open in $\mathbb{R^n}$ and $M\subset A$ and a differential k-form $\bar{\omega}$ defined on all of $A$ such that $\forall p\in M$, $\omega(p)=\bar{\omega}(p)$.
How would I go about proving this? I have looked elsewhere for a proof, but I cannot find anything. Maybe we could do it locally and patch together with a partition of unity?
With this result, many of his cluttered statements of theorems would become cleaner. For example, he states Stokes' theorem as: If $M\subset \mathbb{R}^n$ is a compact oriented $k$-submanifold with boundary and $\omega$ is a $k-1$-form defined on an open set of $\mathbb{R^n}$ containing $M$, then ... which seems messy. Using the above (unproven) result, Stokes' theorem becomes: If $M\subset \mathbb{R}^n$ is a compact oriented $k$-submanifold with boundary and $\omega$ is a $k-1$-form on $M$...