Line Integrals of Exact Differentials

The line integral of an exact differential depends only on the endpoints of the path, but the line integral of an inexact differential depends on the path. 

Figure 7.3 Diagram illustrating the line integral of an exact differential.

Theorem 1 If dF is an exact differential, then the line integral J dF depends only on the initial and final points and not on the choice of curve joining these points. Further, the line integral equals the value of the function F at the final point minus the value of the function at the beginning point. We say that the line integral of an exact differential is path-independent... 

The line integral of an exact differential form depends only on the initial and final points of integration and is independent of the path of integration. This can be easily seen by substituting Eq. (A-31) into Eq. (A-33). The result is... 

We have now seen that the requirement that the two integrals be exact differentials is exactly the requirement that the Cauchy-Riemann conditions be satisfied. This means of course that the line integral... 

Since the pressure P is a state function, dP is an exact differential. The line integral of dP in part b of Example 2.2 is equal to the value of P at the end of the process minus the value of P at the beginning of the process ... 

In order to represent an arbitrary cycle we construct reversible isothermal and adiabatic steps that are smaller and smaller in size, until the curve of the arbitrary cycle is more and more closely approximated by isothermal and adiabatic steps. In the limit that the sizes of the steps approach zero, any curve is exactly represented and the line integral of dq ey/T vanishes for any cycle. The differential dS = dq y/T is therefore exact and 5 is a state function. For a simple system containing one substance and one phase, S must be a function of three state variables. We can write... 

There is an important theorem of mathematics concerning the line integral of an exact differential If dz is an exact differential it is the differential of a function z. 

Equation (2.18) is another example of a line integral, demonstrating that 6q is not an exact differential. To calculate q, one must know the heat capacity as a function of temperature. If one graphs C against T as shown in Figure 2.8, the area under the curve is q. The dependence of C upon T is determined by the path followed. The calculation of q thus requires that we specify the path. Heat is often calculated for an isobaric or an isochoric process in which the heat capacity is represented as Cp or Cy, respectively. If molar quantities are involved, the heat capacities are C/)m or CY.m. Isobaric heat capacities are more... 

Although the expression (2-131) is a perfectly general statement of the force balance at an interface, it is not particularly useful in this form because it is an overall balance on a macroscopic element of the interface. To be used in conjunction with the differential Navier-Stokes equations, which apply pointwise in the two bulk fluids, we require a condition equivalent to (2-131) that applies at each point on the interface. For this purpose, it is necessary to convert the line integral on C to a surface integral on A. To do this, we use an exact integral transformation (Problem 2-26) that can be derived as a generalization of Stokes theorem ... 

We have written M and N as the partial derivatives which they must be equal to in order for i/F to be exact (see Section 7.4). If du is not an exact differential, there is no such things as a function u, and the line integral will depend not only on the beginning and ending points, but also on the curve of integration joining these points. 

In thermodynamics, the equilibrium state of a system is represented by a point in a space whose axes represent the variables specifying the state of the system. A line integral in such a space represents a reversible process. A cyclic process is one that begins and ends at the same state of a system. A line integral that begins and ends at the same point is denoted by the symbol f du. Since the beginning and final points are the same, such an integral must vanish if du is an exact differential ... 

A differential equation contains one or more derivatives of an unknown function, and solving a differential equation means finding what that function is. One important class of differential equations consists of classical equations of motion, which come from Newton s second law of motion. We will discuss the solution of several kinds of differential equations, including linear differential equations, in which the unknown function and its derivatives enter only to the first power, and exact differential equations, which can be solved by a line integration. We will also introduce partial differential equations, in which partial derivatives occur and in which there are two or more independent variables. We will also discuss the solution of differential equations by use of Laplace transformations. Some differential equations can be solved either symbolically or numerically using Mathematica. 

The properties of exact versus inexact differentials are well illustrated by integrating them along different paths. An integral whose path is specified is called a line integral. Consider the expression y dx, that we have just seen is inexact. Integration of y dx can be viewed as determining the area under some curve in the x-y plane (Figure 2.7), but the obvious question is what curve ... 

Of particular significance are line integrals around closed paths, in which the initial and final points coincide. For such cyclic paths, the integral sign is written as The closed curve is by convention traversed in the counterclockwise direction. If dq(x, y) is an exact differential, then... 

In Chapter 5 we introduced the concept of the line integral as representing the area under a curve (or path) taking some function/( c) from x to xi. Exact and inexact differentials are directly related to line integrals and occupy a significant place in... 

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