PX275 - A2c- maxwell's thermodynamic relation

\begin{aligned} (1) & d U & = T d S - p d V \\ (2) & d f & = \frac{\partial f}{\partial x} d x + \frac{\partial f}{\partial y} d y \end{aligned}

\begin{matrix} T = \frac{\partial U}{\partial S} \\ ⟹ \frac{\partial T}{\partial V} = \frac{\partial^{2} U}{\partial V \partial S} \\ and, - p = \frac{\partial U}{\partial V} \\ ⟹ - \frac{\partial p}{\partial S} = \frac{\partial^{2} U}{\partial S \partial V} \\ ∴ \frac{\partial T}{\partial V} = - \frac{\partial p}{\partial S} \end{matrix}