How to do a change of variable here 
How to do a change of variable here $\displaystyle\int_{0}^{2\pi}\int_{0}^{2\pi}2\left|\sin\frac{(\theta-\theta')}{2}\right|\ d\theta\ d\theta'$

If I replace $\theta-\theta'$ by $u$, double integral will be reduced to a single integral, but what happens to the limits? 
Thanks for your help.
 A: The Jacobian is the usual way to change the variables. In order to give an intuitive understanding of what happens, the figure below shows that the scales in the $(x,y)$ system and in the $(u,v)$ system are not equal : length ratio $=\sqrt(2)$. This explain why the ratio between $(dx dy)$ and $(du dv)$ is $=(\sqrt(2) )^2 = 2$
One understand that if only one integral (variable $v$) is taken into account, the length of a segment is obtained. In order to scan the whole square, another variable $u$ is necessary. This explaines why the change of the two initial variables requires not only one, but two new variables.
Finally, the result of the double integral is $16\pi$  
Of course, in more complicated cases of change of variables, a graphical explanation and the related calculus would become too ardous. Then, thanks to the Jacobian.

A: You can reinterpret the integral as follows:
$$\int_0^{2\pi} \int_0^{2\pi} 2 \left|\sin\frac{(\theta-\theta')}{2}\right| \mathrm{d}\theta \mathrm{d}\theta' = \int_0^{2\pi} \int_{-y}^{-y+2\pi} 2 \left| \sin\frac{x}{2} \right | \mathrm{d}x \mathrm{d}y$$
which sort of corresponds to the change of variables you wish to perform. I'd think of this as a change of variables
$$\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 &-1\\0 & 1\end{pmatrix} \begin{pmatrix} \theta \\ \theta' \end{pmatrix},$$
where the determinant of the Jacobian is just unity.
Here you can of course abuse the symmetry to actually evaluate the integral:
$$\int_0^{2\pi} \int_{-y}^{-y+2\pi} 2 \left| \sin\frac{x}{2} \right | \mathrm{d}x \mathrm{d}y = 2\int_0^{2\pi} \int_{0}^{-y+2\pi} 2 \sin\frac{x}{2} \mathrm{d}x \mathrm{d}y.$$
