Integrals, line

A line integral is an integral with an implicit single integration variable that constraints the integration to a path. 

The most frequently-seen line integral in this book, /p dV, will serve as an example. The integral can be evaluated in three different ways  

The integrand p can be expressed as a function of the integration variable V, so that there is only one variable. Lor example, if p equals c/V where c is a constant, the line integral is given by/pdF = c/ (l/F)dF = cln(F2/Fi). 

If p and F can be written as functions of another variable, such as time, that coordinates their values so that they follow the desired path, this new variable becomes the integration variable. 

The desired path can be drawn as a curve on a plot of p versus F then /pdV is equal in value to the area under the curve. 

The extension of the concept of integration considered in this section involves continuous summation of a differential expression along a specified path C. For the case of two independent variables, the line integral can be defined as follows  

The line integral (10.66) reduces to a Riemann integral when the path of integration is parallel to either coordinate axis. For example, along the linear path y = JO = const, we obtain 

More generally, when the curve C can be represeted by a functional relation y — six), y can be eliminated from Eq. (10.66) to give 

FIGURE10.9 Line integral as limit of summation at points (j ,, ) ) along path C between xa, ya) and (x/, yt)- The value of the integral along path C wUl, in general, be different. 

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 

If we are integrating from A to C there are an infinite number of paths to follow, giving an infinite number of areas. For example, the path ABC results in the area V x2 — x ), while ADC results in y2 x2 — xi), and intermediate paths such as the one shown give intermediate results. This is clearly also true of the expression x dy. However, the exact expression 

Thus in summary, exact differentials have coefficients that satisfy the reciprocity relations and have definite integrals that are independent of the path followed during integration. Exact differentials are obtained by differentiating some function. Inexact differentials have coefficients that do not satisfy the reciprocity relations, and have 

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 

In this text, we will not need to consider line integrals of the form L= MdX + NdY 

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 

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Under ideal conditions (e.g., point sources producing spherical waves and no multiple reflections) a rectified backscattered signal represents line integrals of the ultrasonic reflectivity over concentric arcs centered at the transducer position. To reconstruct the reflection tomo-... 

BE-1924 Monitoring On-line integrated technologies for operational reliability Mr. P. Hodgkinson British Aerospace Ltd... 

V is the derivative with respect to R.) We stress that in this formalism, I and J denote the complete adiabatic electronic state, and not a component thereof. Both /) and y) contain the nuclear coordinates, designated by R, as parameters. The above line integral was used and elaborated in calculations of nuclear dynamics on potential surfaces by several authors [273,283,288-301]. (For an extended discussion of this and related matters the reviews of Sidis [48] and Pacher et al. [49] are especially infonnative.)... 

Since to date summaiies about the practical implementation of the line integral have been given recently (in [108,282] as also in the chapter by Baer in the present volume), and the method was applied also to a pair of ci s [282], we do not elaborate here on the fonn of the phase associated with one or more ci s, as obtained through this method. 

An alternative method that can be used to characterize the topology of PES is the line integral technique developed by Baer [53,54], which uses properties of the non-adiabatic coupling between states to identify and locate different types of intersections. The method has been applied to study the complex PES topologies in a number of small molecules such as H3 [55,56] and C2H [57]. 

Assuming that the diabatic space can be truncated to the same size as the adiabatic space, Eqs. (64) and (65) clearly define the relationship between the two representations, and methods have been developed to obtain the tians-formation matrices directly. These include the line integral method of Baer [53,54] and the block diagonalization method of Pacher et al. [179]. Failure of the truncation assumption, however, leads to possibly important nonremovable derivative couplings remaining in the diabatic basis [55,182]. 

IV. The Adiabatic-to-Diabatic Transformation Matrix and the Line Integral Approach... 

IV. THE ADIABATIC-TO-DIABATIC TRANSFORMATION MATRIX AND THE LINE INTEGRAL APPROACH... 

The 3 X 3 A maUix has nine elements of which we are interested in only four, namely, an, an, 021, and 022- However, these four elements are coupled to 031 and 032 and, therefore, we consider the following six line integrals [see Eq. (27)] ... 

Eq. (46) we find that the line integral to solve A is perturbed to the second order, namely. 

The answer to this question can be given following a careful study of these effects employing the line integral approach presented in terms of Eq. (27). For this purpose, we analyze what happens along a certain line F that surrounds... 

In Section IX, we intend to present a geometrical analysis that permits some insight with respect to the phenomenon of sign flips in an M-state system (M > 2). This can be done without the support of a parallel mathematical study [9]. In this section, we intend to supply the mathematical foundation (and justification) for this analysis [10,12], Thus employing the line integral approach, we intend to prove the following statement ... 

Recently, Xu et al. [11] studied in detail the H3 molecule as well as its two isotopic analogues, namely, H2D and D2H, mainly with the aim of testing the ability of the line integral approach to distinguish between the situations when the contour surrounds or does not surround the conical intersection point. Some time later Mebel and co-workers [12,72-74,116] employed ab initio non-adiabatic coupling teiins and the line-integral approach to study some features related to the C2H molecule. 

These results as well as others presented in [11] are important because on various occasions it was implied that the line integral approach is suitable only... 

In this series of results, we encounter a somewhat unexpected result, namely, when the circle surrounds two conical intersections the value of the line integral is zero. This does not contradict any statements made regarding the general theory (which asserts that in such a case the value of the line integral is either a multiple of 2tu or zero) but it is still somewhat unexpected, because it implies that the two conical intersections behave like vectors and that they arrange themselves in such a way as to reduce the effect of the non-adiabatic coupling terms. This result has important consequences regarding the cases where a pair of electronic states are coupled by more than one conical intersection. 

Reference [73] presents the first line-integral study between two excited states, namely, between the second and the third states in this series of states. Here, like before, the calculations are done for a fixed value of ri (results are reported for ri = 1.251 A) but in contrast to the previous study the origin of the system of coordinates is located at the point of this particulai conical intersection, that is, the (2,3) conical intersection. Accordingly, the two polar coordinates (adiabatic coupling term i.e. X(p (— C,2 c>(,2/ )) again employing chain rules for the transformation... 

In some applications it may be necessary to prescribe a pressure datum at a node at the domain boundary. Although pressure has been eliminated from the working equations in the penalty scheme it can be reintroduced through the penalty terms appearing in the boundary line integrals. 

The last teim in the right-hand side of Equation (4.143) represents boundary line integrals. These result from the application of Green s theorem to... 

Let S be an open, two-sided surface bounded by a curve C, then the line integral of vector A (ciiiwe C is traversed in the positive direction) is expressed as... 

Developments. A variety of process modifications aimed at improving surface finish or weld line integrity have been described. They include gas assisted, co-injection, fusible core, multiple Hve feed, and push—pull injection mol ding (46,47). An important development includes computer-aided design (CAD) methods, wherein a proposed mold design is simulated by a computer and the melt flow through it is analy2ed (48). 

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