While reading a book, the author goes from writing that $$I=\int_0^1 \dots \int_0^1(u_1\dots u_s)^{-1+1/k}\frac{\sin 2\pi \lambda(u_1+\dots+u_s-1)}{\pi(u_1+\dots u_s-1)}\mathrm{d}u_1\dots \mathrm{d}u_s$$ to $$I=\int_0^s \phi(u)\frac{\sin 2\pi \lambda (u-1)}{\pi(u-1)} \mathrm{d}u \dots(\star)$$ where $$\phi(u)=\int_0^1 \dots \int_0^1 \Bigg(u_1\dots u_{s-1}(u-u_1-\dots-u_{s-1})\Bigg)^{-1+1/k} \mathrm{d}u_1\dots \mathrm{d}u_{s-1}$$ and the region of integration is over $(u_1,\dots, u_s)$ such that $u-1<u_1+\dots u_{s-1}<u$.
I don't really understand what theorem is being used to do this change of variables. Clearly, they are taking $u=u_1+\dots u_s$. Then $u$ can be between $0$ and $(1+\dots + 1)=s$. We have the following change of coordinates function $(u_1,\dots,u_{s-1}, u_s)\mapsto (u_1,\dots, u_{s-1}, u-u_1-\dots - u_{s-1})$. But then our Jacobian matrix is given by
\begin{pmatrix} 1 & 0 &\dots 0 & 0\\ 0 & 1 & \dots 0 & 0\\ 0 & \ddots & 1 & 0\\ -1 & -1 & \dots -1 & 0 \end{pmatrix}
After that, I am not sure what's going on here and how the author gets to $(\star)$. Also, it seems so weird that now this is a integral of one variable (sort of) and I am quite confused. Thanks, any help is appreciated!