Triple Product Rule

$f(x,y,z)=0\to \frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=-1$

• Mathematical version of triple product rule

$\left(\frac{\partial P}{\partial V}\right)\left(\frac{\partial V}{\partial T}\right)\left(\frac{\partial T}{\partial P}\right)=-1$

${\left(\frac{\partial \sigma }{\partial V}\right)}_U{\left(\frac{\partial V}{\partial U}\right)}_{\sigma }{\left(\frac{\partial U}{\partial \sigma }\right)}_V=-1$

• ${\left(\frac{\partial \sigma }{\partial V}\right)}_U$ for example means the partial derivative of $\sigma$ with respect to $V$ at a constant $U$