For a given function $f$, the functional derivative of the functional $\mathcal{F}[\rho]=\int f(x,\rho(x))\,dx$ is well-known to be $\frac{\delta}{\delta \rho(x)}\mathcal{F}[\rho]=\frac{\partial f}{\partial \rho}.$Suppose I instead have a functional of the form $$\mathcal{G}[\rho]=\int \int^x g(x,y,\rho(x),\rho(y))\,dy\,dx.$$ How does one compute the functional derivative in this case?
If it clarifies, in my case of interest I have $g((x,y,\rho(x),\rho(y))=\sin(x-y+\rho(x)-\rho(y))$ (with $\rho(x)$ some $\pi$-periodic function and outer integration over $[0,2\pi])$. But for the purpose of this problem I would prefer a generic integrand.