I am reading "Calculus on Manifolds" by Michael Spivak.
In the proof of Theorem 3-13 (Change of variable), the author wrote as follows:
Let $W\subset U$ be a rectangle of the form $D\times [a_n,b_n]$, where $D$ is a rectangle in $\mathbb{R}^{n-1}$. By Fubini's theorem $$\int_{h(W)} 1=\int_{[a_n,b_n]}\left(\int_{h(D\times\{x^n\})} 1 dx^1\cdots dx^{n-1}\right)dx^n.$$
Let $h_{x^n}:D\to\mathbb{R}^{n-1}$ be defined by $h_{x^n}(x^1,\dots,x^{n-1})=(g^1(x^1,\dots,x^n),\dots,g^{n-1}(x^1,\dots,x^n))$.
$h:U\to\mathbb{R}^n$ is defined by $h(x)=(g^1(x),\dots,g^{n-1}(x),x^n)$.
I think I understand what the author intend.
But $h(D\times\{x^n\})\subset\mathbb{R}^n$.
And the author wrote $\int_{h(D\times\{x^n\})} 1 dx^1\cdots dx^{n-1}$.
The function $1$ in $\int_{h(D\times\{x^n\})} 1 dx^1\cdots dx^{n-1}$ is defined on $\mathbb{R}^{n-1}$ not on $\mathbb{R}^n$.
Is this ok?
I think the following is perfect:
$$\int_{h(W)} 1=\int_{[a_n,b_n]}\left(\int_{h_{x^n}(D)} 1 dx^1\cdots dx^{n-1}\right)dx^n.$$
The author defined $h_{x^n}$ but the author didn't use $h_{x^n}$ in $$\int_{h(W)} 1=\int_{[a_n,b_n]}\left(\int_{h(D\times\{x^n\})} 1 dx^1\cdots dx^{n-1}\right)dx^n.$$
I wonder why.