Stokes Theorem for Compact Oriented Smooth Manifolds with Boundary

Let $n\in$ $\mathbb{N}$ with $n\ge 2$, and let $M$ be an oriented smooth manifold with boundary of dimension $n$ that is compact as defined in Smooth Atlas and Smooth Manifold with Boundary.

Let $\omega$ be a smooth differential $(n-1)$-form on $M$ and let $d\omega$ denote the exterior derivative of $\omega$.

Let $\partial M$ be the boundary of $M$, regarded as a compact oriented smooth manifold (without boundary) of dimension $n-1$ with the induced boundary orientation, and let $\iota^{*}\omega$ denote the restriction of $\omega$ to $\partial M$ where $\iota:\partial M\to M$ is the inclusion map.

Then

$$\int_{M} d\omega \;=\; \int_{\partial M} \iota^{*}\omega,$$

where both sides are integrals of smooth top-degree forms over compact oriented smooth manifolds. If $\partial M=\varnothing$, then the right-hand side is understood to be $0$.