Cross-derivatives

The factor in wavy brackets is obviously an exact differential because the coefficient of d9 is a fiinction only of 9 and the coefficient of dVis a fiinction only of V. (The cross-derivatives vanish.) Manifestly then... 

Galego and Arroyo [14] described a simultaneous spectrophotometric determination of OTC, hydrocortisone, and nystatin in the pharmaceutical preparations by using ratio spectrum-zero crossing derivate method. The calculation was performed by using multivariate methods such as partial least squares (PLS)-l, PLS-2, and principal component regression (PCR). This method can be used to resolve accurately overlapped absorption spectra of those mixtures. 

As is shown below for the solute, the linear term can be included in the quadratic term by redefining the fluctuation operator. G(m s) is a rectangular matrix obtained from the second cross-derivatives coupling the solute to the solvent l(s m) is a vector obtained by similar procedures from the (dGm(X)/dRs)(n) linear coupling. How to actually perfonn such operation is not directly important now. We assume that, in principle, such operations are feasible. 

A ratio-spectra, zero-crossing, derivative spectrophotometric method was described for the analysis of spironolactone in presence of hydrochlorothiazide [16]. After extracting the drugs from their tablets with 1 1 0.1 N HC1 - methanol, the first derivative of the ratio of their absorption spectra to that of standard solution is computed at 270.7 and 269.9 nm. 

The latter two derivatives are known as cross-derivatives, and for any well-behaved function of x and y, they are equal1 ... 

This useful equation is known as the cross-derivative rule. There are nine second partial derivatives of a function h(x, y, z) of three variables. Calculating derivatives for a few of these functions can convince the reader that the cross-derivative rule also holds for such functions. Thus, using the notation of Eq. (14),... 

V, we obtain 829/8xdy - 829/8xdy - 0 i.e., the second cross derivatives of the potential are equal when taken in either order - a very comforting result. 

On finally setting F = —taking Fx = —d0/dx, Fy = —d0/dy, F = —d0 dz, and invoking Eq. (1.5.17) we find that d 0jdxdy — d 0/dydx = 0, and similarly for the other components. Thus, the second order cross derivatives of the potential function taken in either order are the same—a comforting result. 

Since is an exact differential the cross derivatives taken in either order must match ... 

The positive right-hand side furnishes both an upper and a negative lower bound for the cross derivative E y. If desired the quantity dP/dV)s may be rewritten via Eq. (1.3.8) in terms of partial derivatives that involve the entropy. 

As can be deduced, for m > 2, expression (2.67) leads to cross derivatives by x and y, whose evaluation is rather cumbersome. To alleviate this difficulty, only one fictitious point can be considered at each side of the interface and hence only the zero- and first-order jump conditions are implemented. While this notion gives reliable solutions, an alternative quasi-fourth-order strategy has been presented in [28] for the consideration of higher order conditions and crossderivative computation. A fairly interesting feature of the derivative matching method is that it encompasses various schemes with different orders that permit its hybridization with other high-accuracy time-domain approaches. 

An alternative and faster way to proceed has more recently been developed (Wertz, 1974). In the method described above, one was in essence calculating three of the second derivatives analytically, and neglecting the other three second derivatives. In the current method, all six second derivatives are analytically evaluated, in addition to the three first derivatives, all at the same point. This gives us, including the cross derivatives, a total of nine equations and nine unknowns, so this system may be solved directly. In practice, the second derivative scheme runs approximately three times faster than the first derivative scheme. 

Figure 2. Cross-derivational inter-relations between the VB-hierarchy of models and the orthogonal-orbital (MO-associated) PPP-Hubbard and Hiickel models.

The derivatives in Eq. (9.14) are sometimes called cross-derivatives because of their relation to the total differential, Eq. (9.12) ... 

Equation (9.15) is one of an important group of equations called the Maxwell relations its meaning will be discussed later along with that of the other members of the group. The equality of the cross-derivatives is used frequently in later arguments. 

With the cross-derivative rule and the cyclic relation at our disposal, we are ready to manipulate the equations of thermodynamics into useful forms. 

Since each of the expressions on the right-hand side of these equations is an exact differential expression, it follows that the cross-derivatives are equal. From this we immediately obtain the four Maxwell relations ... 

For reversiole processes, coefficients are related by cross derivatives hence. 

The thermodynamic consistency of the expressions used for electron and hole activity coefficients can be evaluated by application of the second cross derivative of the Gibbs function,58 i.e.,... 

Product price increases revenue at a constant rate while safety equipment, capital, and labor each increase revenue at decreasing rates. All cross derivatives among safety equipment, capital, and labor are positive. 

Then, cross derivation of the above equation gives ... 

In order to close these expressions for particulate pressures, we also need equations for the variance of total particle volume concentration in an assemblage of particles belonging to the two different types. For an arbitrary polydisperse particulate pseudo-gas, variances of partial volume concentrations for different particles can be evaluated on the basis of the thermodynamical theory of fluctuations. According to this theory, these variances are expressible in terms of the minors of a matrix that consists of the cross derivatives of the chemical potentials for particles of different species over the partial number concentrations of such particles [39]. For a binary pseudo-gas, these chemical potentials can be expressed as functions of number concentrations using the statistical theory of binary hard sphere mixtures developed in reference [77]. However, such a procedure leads to a very cumbersome and inconvenient final equation for the desired variance. To simplify the matter, it has been suggested in reference [76] to ignore a slight difference between this variance and the similar quantity for a monodisperse system of spherical particles of the same volume concentration. This means that the variance under question may be approximately described by Equation 7.4 even in the case of binary mixtures. 

FIGURE 12 Four-corner detection of Fig. 10a illustrating cross-derivative results (a) intensity plot and (b)3-D plot. 

In the absence of specific adsorption and at a sufficiently high negative charge, the adsorption of anions can be ignored as compared to the adsorption of cations Ta Ycl [In Eq. (101), account is taken only of the supporting electrolyte at concentration C, while the effect of a low concentration, Cf, of the reagent is disregarded.] Then, the last term in Eq. (101) can be omitted. From the properties of perfect differentials, there follows the equality of cross derivatives... 

Fourth, based on the Euler relation, set the cross-derivatives equal. Set the derivative of S (from the first term on the righthand side of Equation (9.8)) with respect to p (from the second term) equal to the derivative of V (from the second term) with respect to T (from the first term). -S in the first term on the right hand side of Equation (9.8) is -S = (BG/dT). Take its derivative with respect to p, to get... 

Since the energy is not a quadratic function of the coefficients, several Newton steps are needed to locate a minimum. In addition, the calculation and inversion of the full Hessian matrix can be extremely time consuming for large active spaces and/or basis sets. Often, the elements of the Hessian coupling orbital and configurational coefficients (cross derivatives) are neglected and the optimization of the two sets of coefficients are decoupled. 

For functions of many variables, there is a second derivative corresponding to every pair of variables d U/dTdV, d U/dNdV, d U/dT etc. For the cross derivatives such as d U/dT dV that are derivatives with respect to two different variables, the order of differentiation does not matter. That is... 

In addition, for any function of many variables, the cross derivatives must be equal, i.e. we must have equalities of the type... 

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