Differential Expressions

If a Pfaff differential expression DF = Xdx + Tdy+Zdz has the property that every arbitrary neighbourhood of a point P(x, y, z) contains points that are inaccessible along a path corresponding to a solution of the equation DF = 0, then an integrating denominator exists. Physically this means that there are two mutually exclusive possibilities either a) a hierarchy of non-intersecting surfaces (x,y, z) = C, each with a different value of the constant C, represents the solutions DF = 0, in which case a point on one surface is inaccessible... 

It should be noted that the differential expressions on the right-hand side of equation (A2.1.33). equation (A2.1.34), equation (A2.1.35), equation (A2.1.36), equation (A2.1.37), equation (A2.1.38), equation (A2.1.39) and equation (A2.1.40) express for each fiinction the appropriate independent variables for that fiinction, i.e. the variables—read constraints—drat are kept constant during a spontaneous process. 

Because these are exact differential expressions. Maxwell equations can be written by inspection. The two most useful ones are derived from equations 67 and 68 ... 

GP Yang, DT Ross, WW Kuang, PO Brown, RJ Weigel. Combining SSH and cDNA microar-rays for rapid identification of differentially expressed genes. Nucleic Acids Res 27 1517-1523, 1999. 

Differential display is a method for identifying differentially expressed genes, using anchored oligo-dT, random oligonucleotide primers and polymerase chain reaction on reverse-transcribed RNA from different cell populations. The amplified complementary DNAs are displayed and comparisons are drawn between the different cell populations. 

PARs are coupled to multiple G-proteins and mediate a number of well-defined cellular responses via classical second messenger and kinase pathways. PARs are differentially expressed in cells of the vasculature as well in the brain, lung, gastrointestinal tract, skin as well as other highly vascularised tissues and evidence suggests distinct physiological functions and roles in disease states [2]. 

Example 1.1 Test the following differential expressions to determine which is an exact differential... 

Pfaffian Differential Expressions With Two Variables Before we undertake the arguments generalized for three or more variables, we digress to consider some examples involving only two variables. These do not provide the generality we must have to treat thermodynamic systems of three or more variables, but will provide concrete illustrations of the general behavior we will invoke in the development. 

In this discussion, we will limit our writing of the Pfaffian differential expression bq, for the differential element of heat flow in thermodynamic systems, to reversible processes. It is not possible, generally, to write an expression for bq for an irreversible process in terms of state variables. The irreversible process may involve passage through conditions that are not true states" of the system. For example, in an irreversible expansion of a gas, the values of p. V, and T may not correspond to those dictated by the equation of state of the gas. 

Because the two derivatives are not equal, the differential expression is inexact. 

It can be shown mathematically that a two-dimensional Pfaffian equation (1.27) is either exact, or, if it is inexact, an integrating denominator can always be found to convert it into a new, exact, differential. (Such Pfaffians are said to be integrable.) When three or more independent variables are involved, however, a third possibility can occur the Pfaff differential can be inexact, but possesses no integrating denominator.x Caratheodory showed that expressions for SqKV appropriate to thermodynamic systems fall into the class of inexact but integrable differential expressions. That is, an integrating denominator exists that can convert the inexact differential into an exact differential. 

Pfaffian Differential Expressions with Three or More Variables and the Conditions for the Existence of an Integrating Denominator We extend the expression for the Pfaffian differentials in three or more variables, by writing it as... 

As we have seen before, exact differentials correspond to the total differential of a state function, while inexact differentials are associated with quantities that are not state functions, but are path-dependent. Caratheodory proved a purely mathematical theorem, with no reference to physical systems, that establishes the condition for the existence of an integrating denominator for differential expressions of the form of equation (2.44). Called the Caratheodory theorem, it asserts that an integrating denominator exists for Pfaffian differentials, Sq, when there exist final states specified by ( V, ... x )j that are inaccessible from some initial state (.vj,.... v )in by a path for which Sq = 0. Such paths are called solution curves of the differential expression The connection from the purely mathematical realm to thermodynamic systems is established by recognizing that we can express the differential expressions for heat transfer during a reversible thermodynamic process, 6qrey as Pfaffian differentials of the form given by equation (2.44). Then, solution curves (for which Sqrev = 0) correspond to reversible adiabatic processes in which no heat is absorbed or released. 

We have previously shown that the Pfaff differential <5pressure-volume work equation (2.43) is an inexact differential. It is easy to show that division of equation (2.43) by the absolute temperature T yields an exact differential expression. The division gives... 

The Caratheodory treatment is grounded in the mathematical behavior of Pfaffian differential expressions (equation (2.44), and the observation that a... 

The importance of these four equations cannot be overemphasized. They are total differentials for U as f(S, V), H as /(S./ ), A as f V,T), and G as j p,T). Although they were derived assuming a reversible process, as total differentials they apply to both reversible and irreversible processes. They are the starting points for the derivation of general differential expressions in which we express U, H, A and Casa function of p, V, T, Cp and Ci. a These are the relationships that we will now derive. 

We then substitute our differential expressions from Table 3.1 into equations (3.52) or (3.53) and integrate to get AZ. 

Let us consider a function Z that depends on two variables, X and Y, and signify this with the notation Z = f(X, Y). In addition to designating Z as a function, we may also refer to Z as the dependent variable, and X and Y as the independent variables. We can write a differential expression dZ that tells us the change in the dependent variable Z arising from small changes in the independent variables, dX and d Y. The result is... 

When the differential expression is for a change in a state function, dZ, equation (A 1.12) follows almost trivially. It gains importance, however, in that... 

In order to develop this concept, it is convenient to distinguish between a differential expression, dZ, associated with a state function, Z, and a general differential expression whose possible connection with a state function is yet to be established. Let us introduce the notation 6Q to represent this latter differential expression. 

Suppose then, we encounter a general differential expression and want to know whether it is associated with a state function. The behavior of this differential expression integrated over a closed path provides a means to answer this question. Two possibilities need be considered. 

We have already established that an integral of a differential expression associated with a state function is zero over a closed path. Now, we must consider whether the converse of that statement is true. That is, if the integral of a general differential expression over a closed path is found to be zero, this expression is the differential expression of some state function. To answer the question, let us reconsider the example described in Figure (A 1.1) and equations (A 1.13) and (A 1.14) and assume that equation (A 1.17) is true for all closed (cyclic) paths. Then, for a path 1 and a path 3 that connect the same two states, 1 and 2,... 

Thus, the assumption that an integration of a differential expression over a closed path is zero leads to a conclusion that an integration between two different, but fixed, states is independent of path. But, this property coincides with those we have ascribed to state functions. Thus, we have shown that a differential for which equation (A 1.17) is true must correspond to the differential of some state function. 

By similar reasoning, one can show that differential expressions for which equation (Al.18) is true must yield integrals between two fixed states whose values depend upon the path. Such differential expressions cannot be associated with state functions because of the dependence upon path. Therefore, equations (Al.17) and (Al.18) distinguish between differentials that can ultimately be associated with state functions and that cannot. Expressions for which equation (Al.17) is true are called exact differentials while those for which equation (Al.18) is true are called inexact differentials. 

It is not convenient to test for exactness by showing that equation (Al.17) is true for all closed paths, but an easier test can be developed. Consider a general differential expression, 6Q, for a quantity Q that is associated with the variables X and Y ... 

Equation (A 1.25) is known as the Maxwell relation. If this relationship is found to hold for M and A in a differential expression of the form of equation (A 1.22), then 6Q — dQ is exact, and some state function exists for which dQ is the total differential. We will consider a more general form of the Maxwell relationship for differentials in three dimensions later. 

As we have seen above, a path must be specified to integrate an inexact differential expression between two states, because different paths give different integration results. Since the value of the integral... 

We started our discussion of differentials in Section A l. 4 and return to it now to develop some additional concepts. We start with differential expressions that contain three variables, because the results are more general than in the simpler two-dimensional case.e... 

In two dimensions, the Pfaffian differential expression reduces to the form we saw earlier... 

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