Triple Product Rule

#Physics

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

• Mathematical version of triple product rule

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

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

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