Process function

Infinitesimal changes in a process function X are often indicated by $\delta X$ to distinguish them from infinitesimal changes in a state function Y which is written $dY$ . The quantity $dY$ is an exact differential, while $\delta X$ is not, it is an inexact differential. Infinitesimal changes in a process function may be integrated, but the integral between two states depends on the particular path taken between the two states, whereas the integral of a state function is simply the difference of the state functions at the two points, independent of the path taken.
In general, a process function X may be either holonomic or non-holonomic. For a holonomic process function, an auxiliary state function (or integrating factor) $\lambda$ may be defined such that $Y=\lambda X$ is a state function. For a non-holonomic process function, no such function may be defined. In other words, for a holonomic process function, $\lambda$ may be defined such that $dY=\lambda \delta X$ is an exact differential. For example, thermodynamic work is a holonomic process function since the integrating factor $\lambda =1/p$ (where p is pressure) will yield exact differential of the volume state function $dV=\delta W/p$ . The second law of thermodynamics as stated by Carathéodory essentially amounts to the statement that heat is a holonomic process function since the integrating factor $\lambda =1/T$ (where T is temperature) will yield the exact differential of an entropy state function $dS=\delta Q/T$ .