Inexact differentials show up explicitly in the first law of thermodynamics, d U = δ δ Q − − δ δ W where U is the energy, δ δ Q is the differential change in heat and δ δ W is the differential change in work. Based on the constants of the thermodynamic system, we are able to parameterize the average energy in several different ways. E.g., in the first stage of the Carnot cycle a gas is heated by a reservoir, giving us an isothermal expansion of that gas. Some differential amount of heat δ δ Q = T d S enters the gas. During the second stage, the gas is allowed to freely expand, outputting some differential amount of work δ δ W = P d V . The third stage is similar to the first stage, except the heat is lost by contact with a cold reservoir, while the fourth cycle is like the second except work is done onto the system by the surroundings to compress the gas. Because the overall changes in heat and work are different over different parts of the cycle, there is a nonzero net change in the heat and work, indicating that the differentials δ δ Q and δ δ W must be inexact differentials.