Directional derivative

The directional derivative of a function /(y) at y = yo along a unit vector v is defined as 

Vv/(yo) quantifies the rate of change in / from y = yo along v. 

---

The described direct derivation of shape functions by the formulation and solution of algebraic equations in terms of nodal coordinates and nodal degrees of freedom is tedious and becomes impractical for higher-order elements. Furthermore, the existence of a solution for these equations (i.e. existence of an inverse for the coefficients matrix in them) is only guaranteed if the elemental interpolations are based on complete polynomials. Important families of useful finite elements do not provide interpolation models that correspond to complete polynomial expansions. Therefore, in practice, indirect methods are employed to derive the shape functions associated with the elements that belong to these families. 

IV, Other Heterocycles Directly Derived from Selenaaole... 

Solving the rightmost eqiiahty for the directional derivative using the quadratic formula gives... 

The generic name of a drug is not directly derived from systematic nomenclature. Furthermore, different pharmaceutical companies will call the same drug by their own trade name, which is different from its generic name. Generic names are invented on request (for a fee) by the U.S. Adopted Names Council, a private organization founded by the American Medical Association, the American Pharmaceutical Association, and the U.S. Pharma-copeial Convention. 

In a different approach, Stalcup and co-workers [25] used sulfated (3-cyclodextrin for the enantioseparation of piperoxan in work directly derived from earlier CE and classical gel results. Their results were obtained using a continuous free flow apparatus developed by R S Technologies, Inc. Processing rates on the order of 4.5 mg h were reported. 

A crucial element in MTR is the profile of the localized state density as a function of eneigy, the so-called density of states (DOS). Unfortunately, a direct derivation of the DOS from the variation of the mobility is not straightforward. In two papers published in 1972 and 1976 [116, 117], Spear and Le Comber developed a method based on a simplified description of the accumulation layer, which was assumed to behave like a depletion (Schottky) layer, with a constant density of carrier up to a given thickness L This method has been more recently analyzed by Powell [118], who concluded that is was only able to give a rough estimate of the DOS. Nevertheless, we have used this method to estimate the DOS in 6T and DH6T [115] and found an exponential distribution of the form... 

On the other hand, as already discussed in Chapter 2, Eq. (2.21), AO is directly related to the coverage, 0Na> of the backspillover Na8 species and to its dipole moment Pna via the Helmholz equation (2.21). Thus one can directly derive the equations ... 

The same result as in the case po2 = Po2 is obtained if the electrolyte is not present but the two metals are brought in direct contact so that jlw = Pr, from which Eq. (5.36) is directly derived. 

On the basis of the above global promotional mles the following two very simple local (L) rules can be directly derived ... 

IV. Other Heterocvcles Directly Derived from Selenazole... 

The future evolution of ehemistiy will be more and more based on theoretieal eoneepts, and we have to ascribe to "in-depth computors" the merit of this evolution, even if quite probably the most significant progresses will not directly derive from very aeeurate calculations. 

Since the three steps are considered to be elementary processes, their rates can be directly derived from their reaction rate expressions ... 

Before beginning the direct derivation of equation (H.2), we first derive a useful relationship. Consider the integral... 

The above dependence, which has been known for a long time, can be directly derived from Eq. (3) and the electroneutrality conditions of Eq. (14) which for that case are in the form ... 

Eq. 6 is a general result which applies even when Eq. 4b does not it is directly derivable from Eq. 2 provided that the segment distribution is isotropic. 

What is the donor/acceptor ratio in a given cell Again, this ratio cannot be directly derived because it concerns two quantities that stem from fluorophores with different properties (absorption coefficient, quantum yield, spectra) and that emit into two channels differing in gain, filters, and excitation intensity. Thus, the (overlap corrected) intensity of acceptors in channel A will be a factor k times that of donors in D, at equimolar concentrations,3 or ... 

---