Integral for the configurational part of the canonical partition function (classical monatomic gas) S. Salinas, Introduction to Statistical Physics (p. 111) computes the following integral for the configurational part of the canonical partition function of a classical monatomic gas (volume $V$, temperature $T$), here $f(r)=e^{-{\cal V}(r)/(k_BT)}-1$, $k_B$ is the Boltzmann constant, ${\cal V}(r)$ is the intermolecular potential:
$$\iint f(|{\bf r_1}-{\bf r_2}|)\text{d}^3{\bf r_1}\text{d}^3{\bf r_2}=4\pi V\int f(r)r^2 \text{d}r \qquad $$
Could someone please explain how such vectorial integrals are computed, or supply a reference which works out the above ? Thanks.
 A: The vector $\mathbf{r_i}$ denotes the 3-dimensional Cartesian coordinates of $i$th particle, so we treat it as the usual Cartesian variables. The transition from the left side equation to the right side equation is a change of variable. To accomplish this, imagine fixing $\mathbf{r_1}$ and varying $\mathbf{r_2}$ from $-\infty$ to $+\infty$. Under a change of variable to spherical coordinate, it is equivalent to varying $r$ from $0$ to $\infty$, $\theta$ from $0$ to $\pi$ as well as $\phi$ from $0$ to $2\pi$ from the fixed point $\mathbf{r_1}$. Since the potential only depends on the radial coordinate from $\mathbf{r_1}$, we get the $4\pi$ from the full solid angle and only the radial coordinate integration is left. Then we complete the integration of $\mathbf{r_1}$ to get $V$. Under the change of variables, following the notation of Salinas, $f_{12}$ is transformed into $f(r)$.
Note: In your question, the integrand in the left-hand side equation should be written as $f(|\mathbf{r_1}-\mathbf{r_2}|)$ to avoid any ambiguity.
