Appendix A. Differential Calculus

Exact Differentials vs. State Variables

There’s a subtle distinction between an exact differential and a state variable that’s worth considering. Formally, we say that “a state variable has an exact differential.” Alternatively, “the differential of a state variable is exact.” For instance, if we compress a balloon infinitesimally (\(dV_m\)) while slowly moving it toward a heat source (\(dT\)), the pressure \(P\) would change by:

\[dP = \left( \frac{\partial P}{\partial V_m} \right)_T dV_m + \left( \frac{\partial P}{\partial T} \right)_{V_m} dT\]

This illustrates how the value of a state variable \(P\) shifts in response to tiny changes in its independent variables. In short, while it’s correct to say “\(dP\) is exact (not \(P\)) because \(P\) is a state variable,” it’s more logically precise to note that “\(P\) is a state variable because \(dP\) is exact.”

Exact vs. Inexact Differentials

In thermodynamics, we encounter both exact and inexact differentials:

• Exact differentials: These have equal mixed partial derivatives and integrate to a unique function (for example, \(dU\) for the internal energy \(U\)).

• Inexact differentials: These have unequal mixed partial derivatives and do not integrate to a single function (examples include the differentials for heat and work).

The idea of seeking a method to convert an inexact differential into an exact one is a key step in the derivation of entropy. In fact, by applying an integrating factor—in this case, \(1/T\)—we can transform the inexact differential of heat \(\delta q\) into the exact differential \(dS = \delta q/T\). This approach was central to the original formulation of entropy.