While reading Spivak's CoM, I was wondering how we verify
$$
\int_{\partial c}\omega=\int_{\partial I^k} c^* \omega
$$
for a arbitrary $k$ singular cube $c$, where $w$ is a $k-1$ form. This is used in proving the Euclidean space version of Stokes theorem.
This should be easy but I don't seem to get it right. I tried to decompose the boundary into chain of $k-1$ cubes. Let $\partial c=\sum a_ic_i$. Then $$ \int_{\partial c}\omega=\sum a_i \int_{c_i}\omega =\sum a_i \int _{I^{k-1}_i} c_i^*\omega \overset{?}{=}\int_{\partial I^k} c^* \omega $$ But now there seems to be pull back via different faces of the cube, and I still can't see how to proceed throught that.
Edit: I’m guessing that we can formally sum the $c^*_i$’s but I’m not sure if that’s right.
Edit 2: Can I argue that by definition $c^*_i \omega=(c\circ I^{k-1}_{i})^*\omega=\omega\circ (c\circ I^{k-1}_{i})_*=\omega\circ c_*$?