general Stokes theorem

Let $M$ be an oriented $r$ -dimensional differentiable manifold with a piecewise differentiable boundary $\partial M$ . Further, let $\partial M$ have the orientation induced by $M$ . If $\omega$ is an $(r-1)$ -form on $M$ with compact support, whose components have continuous first partial derivatives in any coordinate chart, then

\int_{M}\mathrm{d}\omega=\int_{\partial M}\omega.