In thermodynamics, a complete differential (or exact differential) is a two or more variable differential equation that satisfies the condition for an exact differential (or the condition of immediate integrability).

Overview
The idea of a “complete differential” plays an important part in the theory of differential equations. An expression Xdx + Ydy is called a complete differential when X, Y are functions of the independent variables x, y, such that:

$\left(\frac{\partial Y}{\partial x}\right)_y = \left(\frac{\partial X}{\partial y}\right)_x$

In this case, then, under certain restrictions, the value of ∫ (Xdx + Ydy) depends only on the limiting values of the variables, and not on the intermediate ones by which these limits are connected, or, as generally expressed, on the path along which the integration is taken. [11] Hence, if a differential of the form

dW = Xdx + Ydy

where supposing that dW is the work done by force in the movement of a material point in the x-y plane, is determined to be complete differential, then it follows that:

W = F(x,y) + c

where c is a constant. In this case if we conceive the point to move from a given initial position (xi, yi) to any final position (xf, yf) the work done by the force during the motion will be resented by:

F(xf,yf) F(xo,yo)

If then we suppose F(x,y) to be such that it has only a single value for any one point in space, the work done will be completely determined by the orignal and final positions. [1]

Etymology
The use of the both terms “exact differential” and “complete differential”, to note, were in common use in 1841. [7] The explicit use of the exact different was brought into thermodynamics in 1858 by German physicist Rudolf Clausius. [8]

The term "complete differential" was used in the English translations of Clausius' The Mechanical Theory of Heat (1865, 1875), in the mathematical introduction, thus establishing its use to some extent. Into the 1940s, however, the terms "exact differential" (vs "inexact differential") were in common use by those as Joseph Keenan (1941) and Mark Zemansky (1943). A less used synonym is: perfect differential (vs imperfect differential). Sometimes, to note, the term total differential is used in this context, but this term has a unique and different meaning in calculus.

Notation
Point functions, i.e. functions that depend on the state of the body only, generally characterized by a point on the graph of the quantifying variables, and not on how the body reaches that state, have exact differentials. These types of differentials are signified by the symbol d. [4]

Path functions, i.e. functions in which their magnitudes depend on the path followed during a process as well as the end states, have inexact differentials. These types of differentials are signified by the symbols δ (Greek delta), often used in modern days, or đ (d-crossbar), used generally between 1875 and 1950.

This symbol differentiation seemed to have originated from the 1875 lectures on the mechanical theory of heat by German mathematician Carl Neumann. [5] Neumann, supposedly, began using the d crossbar symbol đ to signify that heat Q and work W functions, as used in the first law:

dU = đQ + đW

are not state functions, and that their values depend on how the processes are carried out. [6]

One-variable complete differential
A function of one variable, such as:

dQ = A(x) dx

is always exact. [3]

Two-variable complete differential
Given a function of the form:

dW = Xdx + Ydy

the expression on the right hand side is a complete differential if it satisfies the following condition of immediate integrability: [1]

$\frac{dX}{dy} = \frac{dY}{dx} \,$

To repeat, using Mark Zemansky's notation, differentials of the type: [2]

dz = M dx + N dy

where z, M, and N are all functions of x and y, the following equality:

$\left( \frac{\partial M}{\partial y} \right)_{x} = \left( \frac{\partial N}{\partial x} \right)_{y}$

is the "condition for an exact differential", which states that the partial of the function M with respect to y at constant x equals the partial of the function N with respect to x at constant y, signifies that the differential dz is exact. The thermodynamic potentials U, H, A, and G are actual functions and their differentials are exact. [2]

Three-variable complete differential
Given a function of the form:

dW = Xdx + Ydy + Zdz

the expression on the right hand side is the complete differential of a function of x, y, z, in which these may be treated as independent variables, if the following three conditions in integrability are satisfied:

$\frac{dX}{dy} = \frac{dY}{dx} \,$ ; $\frac{dY}{dz} = \frac{dZ}{dy} \,$ ; $\frac{dZ}{dx} = \frac{dX}{dz} \,$

To repeat using Pierre Perrot's notation, differentials of the type: [3]

dQ = A dx + B dy + C dz

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]

References
1. Clausius, Rudolf. (1875). The Mechanical Theory of Heat (section: Mathematical Introduction: on Mechanical Work, on Energy, and on the Treatment of Non-Integrable Differential Equations, pgs. 1-20). London: Macmillan & Co.
2. Zemansky, Mark W. (1943). Heat and Thermodynamics - an Intermediate Textbook for Students of Physics, Chemistry, and Engineering (exact differential, pg. 27, condition for, pg. 220). New York: McGraw-Hill Book Co., Inc.
3. Perrot, Pierre. (1998). A to Z of Thermodynamics (pg. 105-06). New York: Oxford University Press.
4. Cengel, Yunus A. and Boles, Michael A. (2002). Thermodynamics: an Engineering Approach (exact differential, pg. 127). New York: McGraw-Hill.
5. Neumann, Carl. (1875). Lectures on the Mechanical Theory of Heat (Vorlesungen über die Mechanische Theorie der Wärme). Germany.
6. Laider, Keith, J. (1993). The World of Physical Chemistry (pg. 98). Oxford University Press.
7. Challis, J. (1841). “A New Method of Investigating the Resistance of the Air to an Oscillating Spring”, Philosophical Magazine (pgs. 229-35). XXXII.
8. Clausius, Rudolf. (1858). “On the Treatment of Differential Equations which are not Directly Integrable.” Dingler’s Polytechnisches Journal, vol. cl. (pg. 29).
9. (a) Durand, William F. (1897). “Note on Different Forms of the Entropy Function”. Physical Review (pgs. 343-47). American Institute of Physics.
(b) William F. Durand – Wikipedia.
10. Integrating factor – Wikipedia.
11. Dixon, A.C. (1899). “On Simultaneous Partial Differential Equations” (sections 2-9: On bidifferentials: or the elements of double integrals, and on the conditions to be satisfied in order that a given bidifferential expression be a complete bidifferential, pgs. 152-59), Philosophical Transactions of the Royal Society of London (pgs. 151-91), Vol. 195. Great Britain: Royal Society.

Further reading
● Forsyth, Andrew R. (1890). Theory of Differential Equations (ch. 1: Single Exact Equation, pgs. 1-12). Cambridge University Press.
● Zill, Dennis G. (1993). A first Course in Differential Equations (section 2.4: Exact equations, pgs. 53-60). PWS-Kent Publishing Co.