For $m,n\in \mathbb N$, let $f$ is the map given by $$\begin{align} f: & \quad \mathbb R^m \times \mathbb R^n \longrightarrow \mathbb R^m \times \mathbb R^n \\ & (x,y)\mapsto f(x,y) = (x+x',y+y'+[x,x']); \quad \mbox{for fixed } \, (x',y')\in \mathbb R^m \times \mathbb R^n, \end{align}$$ where $[.,.]$ is a map $[.,.] : \mathbb R^m \times \mathbb R^m \longrightarrow \mathbb R^n.$
How to prove that the differential of $f$ is lower triangular, and thus the Jacobian determinant is $1$ (i.e., $|J_{f}|=1$) ?
Thank you in advance
For notational convenience, define the map \begin{equation} \begin{split} g:\mathbb{R}^m\times\mathbb{R}^m&\longmapsto\mathbb{R}^n \\ (u,v)&\longmapsto[u,v]. \end{split} \end{equation} The map $f$ is a composition of the translation map \begin{equation} \begin{split} T:\mathbb{R}^m\times\mathbb{R}^n&\longrightarrow\mathbb{R}^m\times\mathbb{R}^n \\ (x,y)&\longmapsto(x+x',y+y') \end{split} \end{equation} and the translation map \begin{equation} \begin{split} T':\mathbb{R}^m\times\mathbb{R}^n&\longrightarrow\mathbb{R}^m\times\mathbb{R}^n \\ (x,y)&\longmapsto(x,y+g(x,x')). \end{split} \end{equation} Now the Jacobian matrix of a translation by a constant element is easily seen to be the identity, so we need only consider the Jacobian of $T'$. Since the map is non-trivial only in the "$\mathbb{R}^n$-part" of the map $T'$, and the map $g$ only depends on the first $m$ factors, the only non-zero entries of the Jacobian matrix are the diagonal and entries below the diagonal, so that the Jacobian is lower-triangular. As a simple example, if $T':\mathbb{R}^2\times\mathbb{R}^3$ and $g:\mathbb{R}^2\times\mathbb{R}^2\to\mathbb{R}^3$, then \begin{equation} DT'=\begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \frac{\partial g_1}{\partial x_1} &\frac{\partial g_1}{\partial x_2} & 1 & 0 & 0 \\ \frac{\partial g_2}{\partial x_1} & \frac{\partial g_2}{\partial x_2} & 0 & 1 & 0 \\ \frac{\partial g_3}{\partial x_1} & \frac{\partial g_3}{\partial x_2} & 0 & 0 & 1 \end{pmatrix}, \end{equation} which is of course easily generalised for arbitrary $m$ and $n$.