Proving stokes theorem on chains (spivak) 
I am having troubles with the formula after "note that":
I don't see how the formula is $0$ if $j \not= i$. If I calculate $(I^k_{(j,\alpha)})^{\star} (fdx^1\wedge ... \wedge \hat{dx^i} \wedge ... \wedge dx^k)$   I will get $$f(I^k_{(j,\alpha)}(x^1,...,x^{k-1}))(dx^1\wedge ... \wedge \hat{dx^i} \wedge ... \wedge dx^k)((I^k_{(j,\alpha)}(x^1,...,x^{k-1}))(DI^k_{(j,\alpha)}(...))$$ but the matrix representing $DI^k_{(j,\alpha)}$ will have diagonal 1's everywhere except the jth row will be a 0 row. I can't see how this could possibly lead to the result stated
any help please

On the cube $I_{(j, \alpha)}^{k}$, the $1$-form $dx^{j}$ vanishes because $x^{j} \equiv \alpha$ is constant. If $i  \neq j$, the wedge product with $dx^{i}$ omitted contains $dx^{j}$.
