general Stokes theorem

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

$${\int }_{M}d\omega ={\int }_{\partial M}\omega .$$