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:

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.