Line integral, defined

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]. 

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... 

The circulation of a vector field along a closed curve c is defined by the line integral... 

The interpretation of Eq. (584) is that the potential A(3) is defined along the integration path of the line integral. The field Ba> is defined as being perpendicular to the plane or surface enclosed by the line integral. Neither A(3) nor B<3) exists in a U(l) invariant electrodynamics. Effects attributed to the topological... 

The line integral is defined over the circular path, exactly as in the 0(3) invariant explanation of the Sagnac effect discussed earlier in this review and in Vol. 114, part 2. The key difference between the 0(3) and U(l) invariant explanations of the Aharonov-Bohm effect is that, in the former, there is a magnetic field B(3) present at the point of contact with the electrons. Agreement with the empirical data is obtained because... 

Absorption lines are defined relative to the continuum. In the case of a resolved line, one may describe the line in terms of its depth relative to the local continuum across the line, say R(AA) = F[(A )/FC where AA is the wavelength measured from line centre, Fc is the flux in the continuum, and F) is the flux at a wavelength within the line. The total absorption by the line obtained by integrating R(AA) over the line profile is known as the equivalent width W. When a line profile is not resolved yet unaffected by blending from neighbouring lines, the equivalent width is independent of the resolution even though the line profile is set by the instrumental profile and not by the intrinsic stellar profile. 

To transform the line integral in (A.27) to a surface integral, the version of (A.23) that is defined inserting the dot product sign is relevant. Accordingly, introducing the dot product and a tensor field into (A.23), a specific version of the surface divergence theorem can be derived ... 

As a result of the foregoing considerations, the wave-mechanical calculation of the intensities of spectral lines and the determination of selection rules are reduced to the consideration of the electric-moment integrals defined in Equation 40-11. We shall discuss the results for special problems in the following sections. 

The projection p(s, 6) is the line integral of n x, y) along the LOS, defined by s and 9, times the absorption cross-section cr at the laser source line freqnency n T are the limits of integration. It can be related to the initial (1°) and transmitted (7 ) laser-beam intensities as shown. 

The concept of work is one that is familiar in the discipline of mechanics. We shall, of course, utilize the same concept here but shall consider some aspects not emphasized in mechanics. Work is defined as the line integral of a generalized force over a generalized path. Consequently, the differential element of work is... 

In thermodynamics, the quantity fV is defined as the work done by the system on its surroundings in a specified change of state along a specified path. The path must be specified, since dW is an inexact differential form. Physically, the line integral depends upon the path because of the inclusion of dissipative forces. Rigorously, the definition of W demands that the initial and the final states of the system be equilibrium states. The work done by the system when the only external force is a uniform normal pressure is given by... 

In this appendix we present a discussion of a few mathematical techniques frequently utilized in thermodynamics. We treat several topics in the analysis of real functions of several real variables. We assume that the functions considered have the continuity properties necessary for the operations performed upon them to be meaningful. In Sec. A-1, we discuss some of the properties of partial derivatives. In Sec. A-2 we define homogeneous functions and derive a useful relation. In Sec. A-3, we treat linear differential forms. Line integrals are discussed in Sec. A-4. 

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