Jacobian determinant of a map? 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
 A: 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 $n$.
