I'm reading Warner. "Foundations of Differentiable Manifolds and Lie Groups." I don't understand what the right hand side of the formula in the following theorem means.
4.9 Stokes' Theorem II $\;$ Let $D$ be a regular domain in an oriented $n$-dimensional manifold $M$, and let $\omega$ be a smooth $(n-1)$ form of compact support. Then $$\int_D d\omega = \int_{\partial D}\omega.$$
The book only gives the definition of $\int_D\omega$, where $M$ is an $n$-dimensional smooth manifold, $D$ is a regular domain of $M$, and $\omega$ is a continuous $n$-form with compact support. So if the right hand side of the formula in the above theorem($\int_{\partial D}\omega$) were to make sense, $\omega$ should be an $(n-1)$ form on $\partial D$ ($\partial D$ can be seen as a smooth $(n-1)$ dimensional manifold). But $\omega$ is an $(n-1)$ form on $M$. So how do I identify $\omega$ with an $(n-1)$ form on $\partial D$?
restricting $\omega$ to $\partial D$ wouldn't make $\omega$ an $(n-1)$ form on $\partial D$ because for each $m\in\partial D$, $\omega(m)$ is an element of $\Lambda_{n-1}(M_m^*)$, not $\Lambda_{n-1}((\partial D)_m^*)$.