I'm trying to show that conductivity of a metal in uniform Electric field is:
$$
\sigma=\int \frac{d\textbf{k}}{4\pi^3}\left (- \frac{\partial f}{\partial \epsilon} \right )\textbf{v(k)u(k)}
$$
where u(k) is a solution to the integral equation
$$
\textbf{v(k)}=\int...