# Why is the sum of two inexact differentials exact?

From the first law of thermodynamics, $$\mathrm{d}U = đQ + đW$$, where $$\mathrm{d}$$ represents an exact differential and $$đ$$ an inexact differential. Exact differentials correspond to state functions and are path-independent.

How is it possible, and what does it imply, when we say that the sum of two inexact differentials is an exact differential? Is the sum of two path-dependent functions always a path-independent function? Also, what is the difference between a partial derivative and an inexact derivative / differential, apart from mathematical implications?

• I saw essentially the question (somehow the answer is still unclear to me) posted in Physics SE. Perhaps you'll be able find it. It might be a case of notation abuse as per Wikipedia. I also think that an elegant answer might be given here too. When I say the answer is not clear to me, I mean it is my problem... I always remain with the same taste even if at first I think to understand... Feb 8, 2021 at 12:31
• It obviously is not valid generally, but particularly for the Q and W case. BTW, it is exact/inexact differential, not derivative. Feb 8, 2021 at 12:33
• @Alchimista Sir,could you provide me that quesion link?
– PV.
Feb 8, 2021 at 14:41
• @Poutnik Sir, how do you differentiate between a derivative and a differential?
– PV.
Feb 8, 2021 at 14:42
• I have a pie. Amy and Bill eat pieces of pie. From the fact that the pie is gone there is no way to show who ate how much pie... Feb 8, 2021 at 22:16

There's not much to do, as you nearly spotted the difference by yourself: $$U$$ is a state function, thus is path-independent. In other terms, one can estimate it at any moment only with an initial and a final value (and express it in terms of its variables only). But with $$Q$$ and $$W$$, it's different: these are path functions (or process functions) consequently depending on the path. Hence, their differential is not something one might express depending only of its variables. (these are not "real" functions for mathematicians...)

Is the sum of two path-dependent functions a path-independent function?

The sum of inexact differentials is an exact differential in that particular case. In other cases, it depends on the units involved: for instance, with entropy, we have $$\mathrm{d}S = \delta Q_\mathrm{rev} / T$$. So, no typical case here!

what is the difference between a partial derivative and an inexact derivative?

These are absolutely not the same! A partial derivative is a differentiation of a multi-variable function regarding one of its variables only, while what we have here is an inexact differential, thus not a derivative as 1) no variable is specified here, and 2) an inexact differential can be of a function depending on only 1 variable.

How is it possible, and what does it imply, when we say that the sum of two inexact differentials is an exact differential?

In the example at hand it means that we divide the ways we can change the energy of the system into exactly two bins. The first we call heat, and all the other ones we call work.

Heat is an energy transfer that occurs when two entities of different temperature are in contact. The difference in average kinetic energy of the particles causes an energy transfer during collisions which overall increases the temperature of the cooler one and decreases the temperature of the hotter one.

Work is everything else, from compression of a gas to stirring to powering a heating coil inside the system with an electrical power source outside of the system.

We can have different amount of heat and work depending how we run a process. If the starting and final state match for two processes, the change of the system's energy will be the same, no matter what path we choose (i.e. what the individual contributions are).

Is the sum of two path-dependent functions always a path-independent function?

We could have divided up the work into pV-work and non-pV-work. Both would be inexact differentials. If we add them up, we would still have an inexact differential.

A partial derivative is when you form the derivative of a multi-variable function with respect to a single variable. So if my function T(r) describes the temperature distribution in 3D space, and I form the derivative with respect to the height $$z$$, that is a partial derivative. This is different from the total derivative, which would tell me the gradient of the temperature at a given point (i.e. in which direction it changes most). So a partial derivative and an inexact differential are quite distinct.