I'm trying to understand this paper, at some point the author applies a multivariate change of variable and I can't really understand why the Jacobian should be triangular instead of diagonal, here are the important passages directly from the paper:
Joint probability density of exactly $n$ events times:
\begin{equation}\label{26} \color{black}{f\left(u_{1}, u_{2}, \ldots, u_{n} \right)} = \color{black}{\prod_{k=1}^{n} \lambda\left(u_{k} \mid H_{u_{k}}\right) \exp \left\{-\int_{u_{k-1}}^{u_{k}} \lambda\left(u \mid H_{u}\right) d u\right\}} \end{equation}
We define the transformation
\begin{equation}\label{28} \Lambda\left(u_{k}\right)=\int_{0}^{u_{k}} \lambda\left(u \mid H_{u}\right) d u \end{equation}
\begin{equation} \tau_{k}=\Lambda\left(u_{k}\right)-\Lambda\left(u_{k-1}\right) \end{equation}
for $k=1, \ldots, n$.
Then, by the multivariate change-of-variable formula (Port, 1994), we have that
\begin{equation}\label{212} f\left(\tau_{1}, \tau_{2}, \ldots, \tau_{n}\right)=|J| f\left(u_{1}, u_{2}, \ldots, u_{n}\right) \end{equation}
where $|J|$ is the determinant of the Jacobian matrix $J$ of the transformation between $u_{j}, j=1, \ldots, n$ and $\tau_{k}, k=1, \ldots, n$.
Because $\tau_{k}$ is a function of $u_{1}, \ldots, u_{k}$, $J$ is a lower triangular matrix, and its determinant is the product of its diagonal elements defined as $|J|=\left|\prod_{k=1}^{n} J_{k k}\right|.$
My question is: why should J be a lower triangular matrix?
J should be something like
\begin{equation*} J = \begin{bmatrix} \frac{\partial u_1}{\partial \tau_1} & \frac{\partial u_1}{\partial\tau_2} & \dots &\frac{\partial u_1}{\partial\tau_n} \\ \frac{\partial u_2}{\partial \tau_1} & \frac{\partial u_2}{\partial\tau_2} & \dots &\frac{\partial u_2}{\partial\tau_n} \\ \vdots & \vdots & \ddots &\vdots \\ \frac{\partial u_n}{\partial \tau_1} & \frac{\partial u_n}{\partial\tau_2} & \dots &\frac{\partial u_n}{\partial\tau_n} \end{bmatrix} \end{equation*}
We have: \begin{equation} \begin{aligned} \color{red}{ \frac{\partial u_k}{\partial \tau_{k-1}} = \left(\frac{\partial \tau_{k-1}}{\partial u_{k}}\right)^{-1} \\ \frac{\partial \tau_{k-1}}{\partial u_{k}} = \frac{\partial}{\partial u_{k}}\int_{u_{k-2}}^{u_{k-1}} \lambda\left(u \mid H_{u}\right) du = 0 } \end{aligned} \end{equation}
and:
\begin{equation} \begin{aligned} \color{red}{ \frac{\partial u_{k-1}}{\partial \tau_{k}} = \left(\frac{\partial \tau_{k}}{\partial u_{k-1}}\right)^{-1} \\ \frac{\partial \tau_{k}}{\partial u_{k-1}} = \frac{\partial}{\partial u_{k-1}}\int_{u_{k-1}}^{u_{k}} \lambda\left(u \mid H_{u}\right) du = 0 } \end{aligned} \end{equation}
So my guess is:
$$ J = \begin{bmatrix} \frac{\partial u_1}{\partial \tau_1} & \color{red}0 & \dots & \color{red}0 \\ \color{red}{0} & \frac{\partial u_2}{\partial\tau_2} & \dots & \color{red}0 \\ \vdots & \vdots & \ddots &\vdots \\ \color{red}{0} & \color{red}{0} & \dots &\frac{\partial u_n}{\partial\tau_n} \end{bmatrix} $$
What am I doing wrong?
Edit: I fixed the paper link.
Moreover, I'm not sure that $\frac{\partial u_{k-1}}{\partial \tau_{k}} = \left(\frac{\partial \tau_{k}}{\partial u_{k-1}}\right)^{-1}$ and $\frac{\partial u_k}{\partial \tau_{k-1}} = \left(\frac{\partial \tau_{k-1}}{\partial u_{k}}\right)^{-1}$ hold in this case.