Definition of integration of forms over manifolds, Spivak.

Question

Let $c$ be an orientation preserving k-cube in $M$( k dimensional manifold with boundary with orientation $\mu$) such that $c_{(k,0)} $ lies in $\partial M$ and is the only face that has any interior points in $\partial M$.
$c_{(i,\alpha)}=c\circ (I^n_{(i,\alpha)})$ and $I^n_{(i,\alpha)}=(x^1,\cdots , x^{i-1},\alpha,x^i,\cdots,x^{n-1})$
$\omega$ is a k-1 form on $M$ which is $0$ outside of $c([0,1]^k)$.

1) In $\int_{c_{(k,0)}} \omega =(-1)^k\int _{\partial M}\omega$ How did we get $(-1)^k$ ?
2) In $ \int _{\partial c}\omega=\int_{(-1)^kc_{(k,0)}}\omega=(-1)^k\int_{c_{(k,0)}} \omega =\int _{\partial M}\omega $ I am concerned with only the first equality and included the others just to give sense of what he is trying to show. Shouldn't it be $\sum_{i,\alpha} (-1)^k\int_{c_{(i,\alpha)}}\omega$ ? Why other faces not appear in the integration as a sum?

Answer 1

The answer to (2) is based on the fact that a continuous function that is zero outside a set must also be zero on the boundary of the set. This applies since $c$ is orientation preserving (see Spivak page 122) so that $c$ takes boundary points to boundary points; thus $\omega$ is zero on all the faces other than the $k$ face.

Answer 2

(1) The boundary $\partial M$ admits the induced orientation and (under the orientation-preserving $c$) it differs by $(-1)^k$ from the orientation defined on $M$.

(2) The support of $\omega$ is contained in $c$ but contains $c_{(k, 0)}$.