Double integral

The shape of the electrocapillary curve is easily calculated if it is assumed that the double layer acts as a condenser of constant capacity C. In this case, double integration of Eq. V-50 gives... 

Mciny of the theories used in molecular modelling involve multiple integrals. Examples include tire two-electron integrals formd in Hartree-Fock theory, and the integral over the piriitii >ns and momenta used to define the partition function, Q. In fact, most of the multiple integrals that have to be evaluated are double integrals. 

A Iraditional or one-dimensional integral corresponds to the area under the curve between Ihc imposed limit, as illustrated in Figure 1.11. Multiple integrals are simply extensions of llu vc ideas to more dimensions. We shall illustrate the principles using a frmction of two vai ialiles,/(r. yj. The double integral... 

Fig. l-ll Single and double integrals. (Figure adapted in part from Boas M L, 1983, Mathematical Methods in the Physical Sciences. 2nd Edition. New York, Wiley.)... 

Integration It is sometimes useful to generate a double integral to solve a problem. By this approach, the fundamental theorem indicated by Eq. (3-66) can be used. 

The double integral of Eq. (7.79) across the outlet area lA x IB results in the following equation for a heat flux through a given point of a jet supplied through a rectangular outlet ... 

The double integral of a function with two independent variables is of the form... 

Although Eqs. (14.2) and (14.4) can be solved analytically (as will be shown in the next section), here it is sufficient to use the renowned depletion, or Schott-ky approximation, in which it is assumed that the effective density of chaige is constant and equal to qNd in the depletion region, and zero outside this region. In this case, a double integration of Eq. (14.2) directly gives... 

In this procedure, the value of the integrand can be determined numerically for every pair of elements and the double integral, approximately the sum of these values, then becomes ... 

The double integral in Equation (8.4) is a fairly general definition of the mixing-cup average. It is applicable to arbitrary velocity profiles and noncircular cross sections but does assume straight streamlines of equal length. Treatment of curved streamlines requires a precise and possibly artificial definition of the system boundaries. See Nauman and Buffham. ... 

ESR characterization was performed in situ in order to avoid any contact of the pretreated solids with air. Spectra, recorded as the first derivative of the absorption, were obtained at room temperature or 77K using a Varian E9 spectrometer working in the X band. The g values were measured relative to a DPPH reference (g = 2.0036). The sample tubes were filled with the solid to a height greater than the depth of the resonant cavity and the number of paramagnetic species was calculated by double integration of the recorded spectra normalized to that of Varian Strong Pitch sample (g = 2.0028, 3. lO spins, cm" ). 

Since the spherical harmonics are normalized, the value of the double integral is unity. 

Double integration with respect to EA yields the surface excess rB+ however, the calculation requires that the value of this excess be known, along with the value of the first differential 3TB+/3EA for a definite potential. This value can be found, for example, by measuring the interfacial tension, especially at the potential of the electrocapillary maximum. The surface excess is often found for solutions of the alkali metals on the basis of the assumption that, at potentials sufficiently more negative than the zero-charge potential, the electrode double layer has a diffuse character without specific adsorption of any component of the electrolyte. The theory of diffuse electrical double layer is then used to determine TB+ and dTB+/3EA (see Section 4.3.1). 

To get the potential energy of two overlapping charge clouds, consider the interaction of small elements of each cloud and then form a double integral over each of them. Calling the clouds 1 and 2, and the volume elements within them dvi and dv2, each of which carries a charge equal to the charge density of each cloud times the element s volume, the potential is ... 

Perhaps the greatest source of error is introduced by the double integration of the experimental derivative curve. The exact location of the baseline is critical since the outer regions or wings of the spectrum are weighted more heavily than the central portion. One necessary requirement is that the areas enclosed by the curve above and below the baseline must be equal. After the baseline and the initial and terminal points on the spectrum have been determined, the integration can be carried out rather easily by numerical techniques. 

Although the error in the absolute spin concentration will vary from one type of study to another, a reasonable value is about 20%. The error in relative concentrations between samples having identical line shapes is probably closer to 2%. In the latter case the relative amplitude of the derivative spectrum may be used hence, the troublesome double integration may be avoided. 

Thirdly, when we separated equation 43-51 into two terms, we only worked with the first term. The second term, which we presented in equation 43-52B, was neglected. Is it possible that the nonlinear effects observed for equation 43-52A will also operate on equation 43-52B The answer is yes, it will, but... And the but..is this AEs is a random variable, just as AEr is. Furthermore, it is uncorrelated with AEx. Therefore, in order to evaluate the integral representing the variation of both AEs and AEr, it would be necessary to perform a double integration over both variables. Now, for each value of AEs, the nonlinearity caused by the presence of AEr in the denominator would apply. However, AEs is symmetrically distributed around zero, therefore for every positive value of AEs there is an equal but negative value that is subject to exactly the same nonlinear effect. The net result is that these pairs always form equal and opposite contributions to the integral, which therefore cancel, leaving no effect due to AEs. 

We are now ready to evaluate the expressions in equation 53-59 and substitute then into equation 53-5. We will use the same value of k for both sample and reference beams. By having k the same, the results will be independent of the transmittance of the sample, as discussed previously. It also eases our task, since we will not have to compute a family of curves, but only one curve representing the change in computed transmittance as k varies. Evaluating it this way also eliminates the need to perform a double integration we can simply keep the sample transmittance constant at unity, and plot the variation in computed transmittance. 

For a small noise intensity, the double integral may be evaluated analytically and finally we get the following expression for the escape time (inverse of the eigenvalue yj) of the considered bistable potential ... 

If the steadystate error is to go to zero, the term l/s(l + Gm, <,)) must go to zero as s goes to zero. This requires that B(,)Gj f(,) must contain a 1/s term. Double integration is needed to drive the steadystate error to zero for a ramp input (to make the output track the changing setpoint). 

Comparing this controller with that designed for a step input [Eq. (20.21)], we can see that the ramp setpoint design yields a controller that contains a double integrator. 

The double integral represents the nonzero terms of the dissipation rate tensor as adapted by Middleman [61] and Bernhardt and McKelvey for adiabatic extrusion [62]. The nontensorial approach was adopted by Tadmor and Klein in their classical text on extrusion [9]. In essence these are the nonzero terms of the dissipation rate tensor when it is applied to the boundary of the fluid at the solid-fluid interface. In the following development this historic analysis was adopted for energy dissipation for a rotating screw. In this case the velocities Ui are evaluated at the screw surface s and calculated in relation to screw rotation theory. The work in the flight clearance was previously described in the literature [9]. The shear... 

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