Overview

The idea of a “complete differential” plays an important part in the theory of differential equations. An expression X

In this case, then, under certain restrictions, the value of ∫ (X

dW = Xdx + Ydy

where supposing that

W = F(x,y) + c

where

If then we supposeF(xf,yf)–F(xo,yo)

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'

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

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]

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

dz = M dx + N dy

where

is the "condition for an exact differential", which states that the partial of the function

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

; ;

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

dQ = A dx + B dy + C dz

where

; ;

In other words, the three component functions of

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

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

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:

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

In which

It is also well known, says Durand, that where one integrating factor, as

References

1. Clausius, Rudolf. (1875).

2. Zemansky, Mark W. (1943).

3. Perrot, Pierre. (1998).

4. Cengel, Yunus A. and Boles, Michael A. (2002).

5. Neumann, Carl. (1875).

6. Laider, Keith, J. (1993).

7. Challis, J. (1841). “A New Method of Investigating the Resistance of the Air to an Oscillating Spring”,

8. Clausius, Rudolf. (1858). “On the Treatment of Differential Equations which are not Directly Integrable.” Dingler’s

9. (a) Durand, William F. (1897). “Note on Different Forms of the Entropy Function”.

(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),

Further reading

● Forsyth, Andrew R. (1890).

● Zill, Dennis G. (1993).