Stokes' Theorem for the upper half space.

Question

How can I prove Stokes' Theorem $\int_M dω = \int_{∂M} ω$ where $M = \mathbb{H}^ n$, the upper half space.

Answer 1

In John M. Lee's Introduction to Smooth Manifolds there is a proof of Stokes' Theorem (Theorem $14.9$). The proof begins with a proof of Stokes' Theorem for the upper half plane $\mathbb{H}^n$. I recommend checking out the book, it's very good.