Problem:
Let's consider the collection of $C^1$ functions, where $k=1,2,\ldots,(n-1)$: $g_k:\mathbb{R}^k\rightarrow \mathbb{R},$ where: $ g_k=g_k(x_1,x_2,\ldots,x_k)$
Then a new map $f$ is defined as follows: $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ such that: $f(x_{1},x_{2},\ldots,x_n)=(x_1, g_1(x_1)+2x_2, g_2(x_1,x_2) + 3x_3, \ldots, g_{n-1}(x_1,x_2,\ldots,x_{n-1})+nx_n)$
How can find the volume of $f((0,1)^n)$ where $(0,1)^n$ is an open unit cube $(0,1)^n$?