where Q, A, B, and C are functions of x, y, and z, is an exact differential if the following relations hold:

$\left( \frac{\partial A}{\partial y} \right)_{x,z} \!\!\!= \left( \frac{\partial B}{\partial x} \right)_{y,z}$ ; $\left( \frac{\partial A}{\partial z} \right)_{x,y} \!\!\!= \left( \frac{\partial C}{\partial x} \right)_{y,z}$ ; $\left( \frac{\partial B}{\partial z} \right)_{x,y} \!\!\!= \left( \frac{\partial C}{\partial y} \right)_{x,z}$

In other words, the three component functions of dQ must be differentially symmetric with respect to each other.

Four variable complete differential
The conditions above, for two-variable and three-variable differentials, are easy to generalize, in that they arise from the rule of independence of the order of differentiation in the calculation of second derivatives. Subsequently, for a differential of four variables to be exact, there are six conditions to satisfy. [3]

Entropy
It seems that Clausius gave so much attention to the use and understanding of the completed differential because the differential function of heat dQ is not a complete differential. In particular, according to American mechanical engineer William Durand, it is well known that the integral of the derivative of heat dQ:

$\int dQ$

involved in any reversible change in a given substance or system is not independent of the path followed, or in other words of the intermediate conditions passed through. It follows that dQ is not a complete differential and that the integral of dQ cannot be expressed as a function of the initial and terminal conditions. [9]

This may possibly, however, have been the mode of logic that French physicist Sadi Carnot was using in 1824 with his re-establishment of equilibrium of the caloric; whereby heat was considered as a indestructable fluid particle (caloric) that could from an initial to an final point in a body unchanged, independent of path. In other words, if heat was a caloric particle, the following relation would hold:

$\int_i^f dQ=Q(f)-Q(i)$

being independent of the path followed. This, however, is not generally true in that due to the principle of the mechanical equivalence of heat, part of the heat transforms into mechanical work inside the body in its path motion, and hence does not simply depend on the initial and final values. It is well known, however, says Durand, that it is possible by means of an integrating factor to reduce dQ to a complete differential, and hence to express its integral thus transformed, as a new function of the initial and end conditions. [9] In other words, a differential dQ that is not exact is said to be integrable when there is a function 1/τ such that the new differential dQ/τ is exact. The function 1/τ is called the integrating factor, τ being the integrating denominator. [10] The factor that Clausius used for this purpose is the reciprocal of the absolute temperature T, such that a new function of heat can be said to exist:

$\frac{dQ}{T} = dS$

In which dS is a complete differential. The integral of dS is then termed the entropy, symbol S, and convenient set of initial conditions being taken from which to measure its value.

It is also well known, says Durand, that where one integrating factor, as 1/T, exists, there will also exist an indefinite number of other factors, leading to a corresponding indefinite series of values of the complete integral. Hence, there may very well exist various resultant forms of the entropy function S. [9]