Inexact differential equation

Some inexact differential equations can be converted to exact differential equations by multiplication by an integrating factor. 

Solution of Inexact Differential Equations by the Use of Integrating Factors... 

Exact and Inexact Differentials Equation (B-1) gives the differential of a function, which is called an exact differential. We can also write a general differential in terms of dx, dy, and dz. ... 

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. 

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... 

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. 

When this equation is substituted for dV in Equation (2.16), the inexact differential expression... 

Some such differential forms are exact, which means that they are differentials of functions. Other differentials are inexact, which means that they are not differentials of functions. If the differential is exact, the equation is called an exact differential equation. 

The reason many differential equations are so difficult to solve is due to the fact that they have been formed by the elimination of constants as well as by the elision of some common factor from the primitive. Such an equation, therefore, does not actually represent the complete or total differential of the original equation or primitive. The equation is then said to be inexact. On the other hand, an exact differential equation is one that has been obtained by the differentiation of a function of x and y and performing no other operation involving x and y. 

SE7 Mathematically inexact deconvolution. Numerical procedures such as numerical integration, numerical solution of differential equations, and some matrix-vector formulations of linear systems are numerical approximations and as such contain errors. This type of error is largely eliminated in the direct deconvolution method where the deconvolution is based on a mathematical exact deconvolution formula (see above). Similarly, the prescribed input function method ( deconvolution through convolution ) wiU largely eliminate this numerical type of error if the convolution can be done analytically so that numerical convolution is avoided. 

In the above equation dQ denotes the amount of heat entering the system (the symbol d represents an inexact differential), —PdV is the amount of mechanical work performed on the system (P is the hydrostatic pressure and V is the system volume), and fidN is the so-called chemical work performed on the system (/u., and Ni are respectively the chemical potential and the molecular count of the ith chemical species). 

Equation (2.2-1) does not indicate that C is a derivative of q with respect to T. We will see that dq is an inexact differential so that the heat capacity C depends on the way in which the temperature of the system is changed. If the temperature is changed at constant pressure, the heat capacity is denoted by Cp and is called the heat capacity at constant pressure. If the temperature is changed at constant volume, the heat capacity is denoted by Cy and is called the heat capacity at constant volume. These two heat capacities are not generally equal to each other. 

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