On pg. 417 of Lee's Introduction to Smooth Manifolds, the author discusses integration on a manifold with corners.
The text states:
... for Stokes's theorem, we need to integrate a differential form over a smooth manifold with corners. Since the boundary (of a smooth manifold with corners) is not itself a smooth manifold with corners, this requires a separate definition. Let $M$ be an oriented smooth $n$-manifold with corners, and suppose $\omega$ is an $(n-1)$-form on $\partial M$...
I am perplexed. How can we talk about an $(n-1)$-form on $\partial M$ when $\partial M$ may not even be a smooth manifold? Am I missing something obvious?