In Calculus on Manifolds, the following definition of manifold-with-boundary is given.
I'm now trying to prove that the boundary of a k-manifold is itself a k-1 -dimensional manifold, but I'm starting to doubt the definition given in the book. The proof would be easy if we required in the definition (M') that $h^k(y)=0$ for all $y\in U\cap \partial M$, but instead we only have $h^k(x)=0$. (Why do we do this?) I can't prove the theorem I want to prove with the definition given, and (this is problem 5-1 in the book) solutions in online solution manuals seem to fail as far as I can see.