Derivative defined

Thus, the first derivative defines the rate of change of the function U x), while the second derivative shows how its average value differs from the value of the function at the same point. For instance, if U" x) [/ (v) and there is a maximum. Next, suppose that within some interval of v the average value of the function coincides at each point with the value of this function ... 

Finally, the mixed second derivatives define the nuclear Fukui function (NFF) indices ... 

The four blocks of V can be alternatively expressed in terms of the principal geometric derivatives defining the generalized Hessian of Equation 30.8. This can be accomplished first by expressing AQ as function of AN and AF, using the second Equation 30.9, and then by inserting the result into the first Equation 30.9 ... 

The relative partial molar enthalpies of the species are obtained by using Eqs. (70) and (75) in Eq. (41). When the interaction coefficients linear functions of T as assumed here, these enthalpies can be written down directly from Eq. (70) since the partial derivatives defining them in Eq. (41) are all taken at constant values for the species mole fractions. Since the concept of excess quantities measures a quantity for a solution relative to its value in an ideal solution, all nonzero enthalpy quantities are excess. The total enthalpy of mixing is then the same as the excess enthalpy of mixing and a relative partial molar enthalpy is the same as the excess relative partial molar enthalpy. Therefore for brevity the adjective excess is not used here in connection with enthalpy quantities. By definition the relation between the relative partial molar entropy of species j, Sj, and the excess relative partial molar entropy sj is... 

Analogous quantities to die electric moments can be defined when the external perturbation takes the form of a magnetic field. In this instance die first derivative defines the permanent magnetic moment (always zero for non-degenerate electronic states), the second derivative the magnetizability or magnetic susceptibility, etc. 

For a dense system of hard, smooth, and elastic spherical particles, a transport theorem based on the analogy of the kinetic theory of dense gases [Reif, 1965] may be derived. Define an ensemble average of any property xjr of a particle as... 

Methanol can be used as an alternative fuel for gasoline, diesel and fuel oil. Tests are being made for these three alternative uses. The methanol fuel will substitute for oil derivatives defined by Brazil s energy policy. What has to be done now is to verify the technical advantages and disadvantages to each form of utilization. 

What are the sufficient conditions that must hold at a solution point of problem [2] These conditions are simply extensions to R" of the well-known first and second derivative conditions for univariate functions. Let us assume that f(x) is a smooth function with continuous first, second, and third derivatives defined for all x. We further suppose that the following conditions hold for the gradient and Hessian of / at x R ... 

A tentative catalytic mechanism of ascorbate oxidase has been proposed based on the refined X-ray structure and on spectroscopic and mechanistic studies of ascorbate oxidase and the related laccase. The results of these studies have been discussed in detail (74). The X-ray structure determinations of the fully reduced and peroxide derivatives define two important intermediate states during the catalytic cycle. A proposal for the catalytic mechanism incorporating this new information is given in Messerschmidt et al. (150) and presented in Fig. 14. This scheme should be valid in principle also for laccase due to the close similarities of both blue oxidases. 

The dot in denotes a time derivative. In flows that are not spatially homogeneous, this time derivative is the substantial time derivative defined by... 

At steady state, the upper-convected derivative, defined by Eq. (3-33), reduces to... 

If [x4] is a constant matrix, that is, [x4] = [x4(O], then its eigenvalues A- and modal matrix [ ] will also be independent of t. Thus, we may bring [ ] inside the derivative. Defining a column matrix of new variables (x) by... 

Introducing the substantial derivative, defined by the average velocity Vd, one obtains ... 

Note that we add one particle of species i (denoted 1 ) to obtain the derivative defining the chemical potential. 

The rate expression in Eq. (16.31) can be simplified under the following well-defined conditions the coupling between the R coordinate and the solvent coordinates is neglected, the surfaces are approximated to be harmonic along the R coordinate, and the R mode frequency is assumed to be the same for both the reactant and product surfaces. In this case, (SR ) = (SR) = 0 and the energy gap derivative defined in Eq. (16.31) becomes... 

Antithesis Orbital functional derivatives define linear operators 9... 

ANTITHESIS ORBITAL FUNCTIONAL DERIVATIVES DEFINE LINEAR OPERATORS... 

Eq is a sum of explicit orbital functionals, T + V + Eb + Ex, with explicit orbital functional derivatives. Defining t = — V2 for kinetic energy,... 

It should be observed that equation (52) directly follows from the following transformation of the derivative defining the softness kernel [equation (50)] ... 

The first derivatives define the (microscopic) magnetic moment vector /2(/tx, My, M2)... 

In terms of the material derivative defining a change in a quantity moving with a fluid element (Eq. 2.2.3), we have... 

---