Differentiation definition

This is the differential definition of the absolute intensity. The total absolute intensity can be deduced by integration from Eq. (7.19) and Eq. (7.20) for any normal transmission geometry. Geometries are discriminated by the shape and size of the irradiated volume, the image of the primary beam in the registration plane17 of the detector, and the dimensions of the detector elements18. 

Of course, a differential definition can also be given, viz. dine,-... 

Evaluation of some thermodynamic derivatives may require us to differentiate definite integrals. The general prescription for so doing was given by Leibniz. The problem is to find the general expression for dF/dx, when F x) is given by... 

This is the Leibniz rule for differentiating definite integrals. In those special cases in which one or both limits (a and b) are constants, independent of x, then (A.11.7) simplifies accordingly. 

The differential definition more appropriate to atomic system is on the basis that for a system of N electrons with ground state energy E [N,v],... 

With this expression the number of the mode of vibrations are calculated (based on differential definition above) with the expression... 

When a system is spatially homogeneous, none of its features depends on the localization inside the system or on its orientation in space. In that simple case, peculiar operators are not required for knowing a state variable in each geometrical subdivision of the system—curve, surface, or volume. A simple division by a length, an area, or a volume provides the pertinent quantity, a density or a concentration. On the contrary, in spatially heterogeneous systems, localized variables are dependent on their localization. In that case, a differential definition must be used around each point of the system (in a continuous space). For the first localization step, a line or a curve, the definition of the 1-localized quantity, is given by the differential equation ... 

The differential definition of the exponential function can be recognized in Equation 9.146, leading to the simple expression... 

This equation is the differential definition of the exponential function as already seen, thus modeling the wave function of the spatiotemporal oscillator with exactly the same expression as for the temporal and spatial oscillators... 

The classical definition of pressnre as the ratio of a force on an area is not always of easy application. For instance, in a gas, in a cloud of vapor, or in plasma, it is difficult to find a well-defined area unless one uses a differential definition, which amounts to the local pressure. Second, if one needs to have a surface systematically available to be able to define a pressure inside a volume, one comes up sometimes against serious difficulties, for instance, in the case of finely divided surfaces (fractal medium) or randomly varying ones (fluctuations, chaos). 

As one has regularly done in case of differential definition of a variable, one has associated to it an integral variable defined by simple proportionality ... 

In opposition, the federal Department of Canada has proposed, via Health Canada, differentiated definitions for nutraceuticals and functional foods. A nutraceutical is a product isolated or purified from foods that is generally sold in medicinal forms not usually associated with food. It is demonstrated to have a physiological benefit or provide protection against chronic disease. A functional food is similar in appearance to, or may be, a conventional food, is consumed as part of a usual diet, and is demmistrated to have physiological benefits and/or reduce the risk of chronic disease beymid basic nutritional functions. In this case, the difference of formulatiOTi between both kinds of products is well established [7]. 

Cartesian coordinates, 4 definition of, 4 spherical polar coordinates, 6 Differentials exact, definition of, 84 significance of, 86 inexact, definition of, 84 partial, 38 of a cylinder, 39 total. 37-39 Differentiation definition of, 31... 

Let us consider the consequence of mechanics for the ensemble density. As in subsection A2.2.2.1. let D/Dt represent differentiation along the trajectory in F space. By definition,... 

Whenever an economic evaluation is undertaken, a corresponding problem definition should be provided as the basis on which the evaluation is made. This definition, sometimes called an economic scope, should clearly differentiate between specifications that have actually been selected and features that have been assumed for the evaluation. In a comparison of alternatives, all of the assumptions, data, and conditions must be consistent, reaUstic, and devoid of bias. 

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. 

The natural laws in any scientific or technological field are not regarded as precise and definitive until they have been expressed in mathematical form. Such a form, often an equation, is a relation between the quantity of interest, say, product yield, and independent variables such as time and temperature upon which yield depends. When it happens that this equation involves, besides the function itself, one or more of its derivatives it is called a differential equation. 

Together with Eos. (4-6) and (4-7), this definition allows ehmination of all the partial differential coefficients from the preceding equation ... 

Additional thermodynamic properties are related to these and arise by arbitrary definition. Multiplication ofEq. (4-11) hy n and differentiation yields the general expression ... 

The Berlin City electrical engineer M. Kallmann reported in 1899 on a system for controlling stray currents of electric railways [64]. As early as 1894, the Board of Trade in London issued a safety regulation for the British electric railways which specified a potential differential of not more than 1.5 V where the pipeline was positive to the rails, but 4.5 V with the rails positive. Extensive research was undertaken on reducing the risk of stray current in the soil by metallic connections from pipes to rails. However, as one writer noted, a procedure on these lines should definitely be discouraged as it carries the seed of its own destruction [64]. 

By substimting the definition of H [Eq. (1)] into Eq. (8), we regain Eq. (6). The first first-order differential equation in Eq. (8) becomes the standard definition of momentum, i.e.. Pi = miFi = niiVi, while the second turns into Eq. (6). A set of two first-order differential equations is often easier to solve than a single second-order differential equation. 

What is the best choice of differential cost function A variety of definitions of the cost function have been proposed. One stems form the highly original work of Fiber and Karplus [38] and Czerminski and Fiber [39], where... 

Using a differential cost function such as that of Fiber and Karplus, the potential energy is averaged over the path by including a factor of 1/L. In other definitions, such as the one employed in the MaxFlux method, there is no such nonnalization. Therefore, if the potential is set to zero, the MaxFlux method will find that the best path is the straight line path connecting reactants and products. However, methods where the differential cost is proportional to 1 /L will find that all paths are equally good. 

Solubility parameters are generally tabulated, together with the corresponding liquid molar volumes, only at 25°C. Although solubility parameters are themselves temperature-dependent, the combination of quantities in Eq. 70 is not. Differentiating Eq. 70 with respect to temperature gives — the excess entropy, a quantity which has been assumed to be zero in accord with the definition of a regular solution. Thus only data at 25°C are needed. Solubility parameters may be... 

The definition is intended to differentiate these adhesives from merely sticky materials like flypaper or materials that may have only substrate specific adhesion. 

Perhaps the most definitive result to come from the early nickel-aluminia synthesis work was the thermal analysis investigation of Hammetter [88HO 88W01], which showed explicit data on substantial changes in the shockec-but-unreacted mixtures. Differential thermal analysis was carried out on th -starting powder compacts of both the mechanically mixed and composite powders. Shocked and unreacted powders were compared to provide direc evidence for substantial changes introduced by the shock process. 

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