Is it true that for a k-manifold with boundary $M \subset \mathbb{R}^n$, and subset $A \subset M$, and k-form $w$ on $M$, $\int_M w = \int_{M - A} w + \int_A w$? (Call this Equation 1)
I'm not sure here, because on the one hand, the integral $\int_M w$ is defined to be $\int_c w$ where $c:W \rightarrow \mathbb{R}^n$ is a k-cube which is also a coordinate system. The fact that $c$ is a coordinate system means that $c$ is injective. Injectivity leads me to believe that Equation 1 is correct, since the above equation makes sense if I think about it pulled back to $[0,1]^k$.
On the other hand, if I let $A = \partial M$, then $\int_M w = \int_{M - \partial M} w + \int_{\partial M} w$. But the last term $\int_{\partial M} w$ doesn't make sense to me, since $w$ is a $k$-form but $\partial M$ is a $k-1$ manifold. So I'm tempted to also say that Equation 1 doesn't hold
Which understanding is correct?
