Path dependent integrals

As indicated below, the extent of work performance of type i can always be either measured or calculated this permits us to construct a conceptual framework for the thermodynamic properties of a system of interest. In general, / itself depends on X and is not necessarily coUinear with it. Then the element of work is specified by the vectorial dot product as dW = /(jc,) dxi, where the d symbol again indicates that this increment generally involves a path-dependent integral, Jf xi) dxi. Where more than one type of work is performed, the contributions must be summed. 

Although the difference in final strength f, integrated through both the actual shock wave and the computational shock wave, will be mitigated by dynamic recovery (saturation) processes, this is still a substantial effect, and one that should not be left to chance. These are very important practical considerations in dealing with path-dependent, micromechanical constitutive models of all kinds. 

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. 

Since /(r) is conservative, the value of the line integral in Eq. (179) does not depend on the path of integration chosen. Note that Eq. (179) has been written such that y(co) = 0 a standard choice of gauge for the potential. Eq. (179) may be used for the analysis, mentioned in the previous subsection, of approximate solutions, to be performed now in terms of the potential itself, rather than its gradient. 

It should be noted that the surface integral on the r.h.s. of Eqs. (34) and (35) depends on the curvature F as well as on the connection A entering the covariant derivatives, which is reminiscent of the path dependence of the curvature 3F in the operator approach. 

The important point is that the final value of the integral depends only on the two endpoints, i.e., the value of the function z at (x, yi) and (x2, y2), but not the chosen path of integration (as illustrated in Sidebar 1.4). Moreover, in the special case of a cyclic integral (denoted ), where initial and final limits coincide, the integral (1.15) necessarily vanishes for an exact differential, independent of how the cyclic path is chosen. We can therefore state the following integral criterion for exactness ... 

The first integral I is just the area under the curve y = y(x), as shown by the shaded region in panel (b). Similarly, the second integral I2 is the area to the left of this curve, as shown by the shaded region in panel (c). Clearly, the values of both Ii and I2 are dependent on the chosen path of integration, confirming that dz and dz2 are inexact. However, the sum of these differentials, dz = dz + dz2 = ydx- - xdy, is evidently exact [cf. part (a) of Sidebar 1.5]. By inspection, its integral... 

The distinction between reversible and irreversible work is one of the most important in thermodynamics. We shall first illustrate this distinction by means of a specific numerical example, in which a specified system undergoes a certain change of state by three distinct paths approaching the idealized reversible limit. Later, we introduce a formal definition for reversible work that summarizes and generalizes what has been learned from the path dependence in the three cases. In each case, we shall evaluate the integrated work w 2 from the basic path integral,... 

If the initial velocity field is a potential one, v = dS/dx, then it will remain potential subsequently as well, even when it becomes multivalued. In mathematics, multivalued potential fields are called Lagrange manifolds. More precisely, a submanifold of an intermediate dimension in phase space is called Lagrange if J pdx on it depends only on the end points, and not on the path of integration. 

In the non-Abelian Stokes theorem (482), on the other hand, the boundary conditions are defined because the phase factor is path-dependent, that is, depends on the covariant derivative [50]. On the U(l) level [50], the original Stokes theorem is a mathematical relation between a vector field and its curl. In 0(3) or SU(2) invariant electromagnetism, the non-Abelian Stokes theorem gives the phase change due to a rotation in the internal space. This phase change appears as the integrals... 

Path-dependent properties are a function of the sequencing of the transition steps from an initial to a final state, and it is only when each event is enacted in an identical way that the integrals of path-dependent functions are themselves identical. 

The third pathway is internal to the process. Here heat flows back and forth between different unit operations. The magnitude of this energy path depends upon the heating and cooling needs and the amount of heat integration implemented. Whenever the internal path is missing, and there is a heating requirement, the heat has to be supplied from utilities. The same amount of heat must eventually be rejected to the environment elsewhere in the process. 

It will be shown later that although qrev = Jdqrcv is not independent of path, the integral of the heat, dqrcv, divided by the temperature at which it is transferred—J(d qrev/T)—is independent of the path and depends only on the initial and final states of the system. We define entropy S, so that... 

While for an ideal gas, Cp will have a constant value, Cp for superheated steam will depend on both temperature and pressure. We will assume that it is possible to fix an average value of specific heat for steam at each pressure that takes into account the sometimes substantial variations with temperature over the path of integration. Hence equation (16.59) becomes... 

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