integration by substitution of multiple variables

Question

I have an integral

\begin{equation} \int_{\mathbb{R}^n}f(\mathbf{B}\mathbf{x})\mathrm{d}\mathbf{x} \end{equation}

where $f: \mathbb{R}^m \rightarrow \mathbb{R}$ and $\mathbf{B}\in\mathbb{R}^{m\times n}$. I also know

\begin{equation} \int_{\mathbb{R}^m}f(\mathbf{u})\mathrm{d}\mathbf{u}. \end{equation}

For $n=m$, we have

\begin{equation} \int_{\mathbb{R}^n}f(\mathbf{B}\mathbf{x})\mathrm{d}\mathbf{x} = \frac{1}{\det\mathbf{B}} \int_{\mathbb{R}^n}f(\mathbf{u})\mathrm{d}\mathbf{u}. \end{equation}

What do I do for $n\neq m$?

Answer 1

In short, you learn nothing about $I_1 := \int_{\mathbb{R}^n} f(Bx)dx$ by knowing $I_2 := \int_{\mathbb{R}^m} f(u)du$ when $n\neq m$. This is for geometric reasons:

• If $m>n$, then $I_1$ integrates just "one slice" of the integral $I_2$, which is a set of measure zero in $\mathbb{R}^m$ and so can be anything at all. For example, if $n=1$ and $m=2$ and $f$ is the characteristic function of a rectangle $[0,1]\times [0,2]$, then $I_2 = 2$, but $I_1$ can be anything from $1$ to $\sqrt{5}$ depending on $B$. If we made $f$ the characteristic function of the line $0\times [0,1]$, then $I_2 = 0$ and $I_1 = 1$ if $B = \left(1 \:\: 0 \right)^T$ but $I_1=0$ if $B$ has any non-zero second entry.
• On the other hand, $m < n$, then the integral $I_1$ generally won't converge at all, because there are infinitely many points taking on every non-zero value of the integrand. For example if $m=1$ and $n=2$ with $f$ being the characteristic function of the interval $[0,1]$, then for a $B$ like $\left(1\:\: 0\right)$ we see that $f(Bx)$ is the characteristic function of $[0,1]\times[-\infty,\infty]$, which has infinite area.

There is, however, some relationship between these integrals. In the case where $m>n$, we can consider comparing the integral $I_1$ with the surface integral in $\mathbb{R}^m$ over the slice spanned by $B$. Here there is a formula analogous to the one you mention, involving the "Gramian" of $B$. Here it is: $$ \int_{\mathbb{R}^n} f(Bx)dx = \frac{1}{\sqrt{\det{B^T B}}} \int_{R(B)} f(u)du$$

where $R(B)$ stands for the range of $B$, which is a subspace of $\mathbb{R}^m$. I recommend chapter 8 of this book for reference.