# Change of variable rule for multidimensional integral

Consider a real-valued function $g: \mathbb{R}^q\rightarrow \mathbb{R}$. I want to show that $\int g(x)dx=\frac{1}{h^q}\int g(\frac{x}{h})dx$. I guess I should use the change of variable rule but I don't see how to get $\frac{1}{h^q}$. Any hint would be really appreciated.

• What do you mean by $dx$? Should be $dx_1dx_2\cdots dx_n$ or $dA$. – Gregory Grant Jan 24 '16 at 12:18

Try rewriting that as $$\int g(u)~du~=~\frac{1}{h^q}\int g(\frac{x}{h})~dx.$$ Now the substitution $u = \frac{x}{h}$ (which really means $u_i = \frac{x_i}{h}$ for $i = 1, \ldots, q$), plus the change of variable theorem, does the trick.
• Ok, thanks, but what is $\frac{d(u_1,...,u_q)}{d(x_1,...,x_q)}$? – STF Jan 24 '16 at 12:24
• Since $u_i = x_i / k$, the matrix is diagonal, namely $\frac{1}{k} \mathbf{I}$. – John Hughes Jan 24 '16 at 13:37
$dx=dx_1\cdots dx_q$. Change each of the $x_i$ as $x_i =t_i/h$. Therefore $$dx=dx_1\cdots dx_q=\frac{dt_1}{h}\cdots\frac{dt_q}{h}=\frac{1}{h^q}dt_1\cdots dt_q\ .$$
• Ok, many thanks, when you write $dx=dx_1... dx_q$ are you using Fubini's Theorem? – STF Jan 24 '16 at 12:50
• The real question is: what do you mean by $\int g(x)dx$ [this being a multidimensional integral, according to your question]? The only way I can interpret your question is that your $\int g(x)dx$ is a bad notation for $\int d\mathbf{x}g(\mathbf{x})$, where $\mathbf{x}$ is a vector with $q$ components. Then I just use the change of variable theorem en.wikipedia.org/wiki/… whose Jacobian matrix in your case is diagonal (therefore its determinant is just the product of the diag.. entries). Fubini's theorem has no place here. – Pierpaolo Vivo Jan 24 '16 at 12:59
• Just to be more precise, the Jacobian matrix $J_{ij}=\frac{\partial x_i}{\partial t_j}=(1/h)\delta_{ij}$ so its determinant is $(1/h)^q$. – Pierpaolo Vivo Jan 24 '16 at 13:24
• Just to make sure I understood: $\int_{\mathbb{R}^q} dx g(x):=\int_{\mathbb{R}^q} d(x_1,...,x_q)g(x_1,...,x_q)$. Is this different from $\int_{\mathbb{R}}... \int_{\mathbb{R}} g(x_1,...,x_q)dx_1 dx_2... dx_q$ unless $\int_{\mathbb{R}^q} |g(x_1,...,x_q)|d(x_1,...,x_q)<\infty$ (which is Fubini's Theorem)? – STF Jan 24 '16 at 14:09