Total derivative

The Cpg is expressed as the derivative of the enthalpy with respect to temperature at constant pressure. For an ideal gas it is a total derivative ... 

Flavor Formulas. Tables 7 and 8 give examples of modem flavor formulas. In Table 7 formula A is composed of fmit juice concentrate and essence distilled or extracted from the fmit juice. It is all natural and all from the named fmit, and is therefore termed a "natural flavor." It has a characterizing natural flavor. In Formula B the flavor is all natural, but is not all from the named fmit, ie, the fortifier is all natural but is not totally derived from the named fmit. Since the fortifier simulates, resembles, or reinforces the named flavor, eg, apple or pineapple, the flavor must be called "flavor with other natural flavors." It has a natural flavor with characterizing naturals added. Formula C is composed of both natural and artificial components with the natural usage outweighing the artificial. Therefore, it is a "flavor natural and artificial." It has a characterizing natural and artificial flavor. 

The total derivative is appropriate here because property changes of reaction are functions of temperature only. In combination with Eq. (4-343) this gives... 

Conservation of energy. Assuming that U and H do not depend explicitly on time or velocity (so that dH/dt = 0), it is easy to show from Eq. (8) that the total derivative dFUdt is zero i.e., the Hamiltonian is a constant of motion for Newton s equation. In other words, there is conservation of total energy under Newton s equation of motion. 

A number of risk importance measures have been defined for the interpretation of PSAs and for use in prioritization of operational and safety improvements. Some of these measures are similar to sensitivity defined as the total derivative (equation 2.8-1). 

Substituting the partial derivative for the total derivative in equation (2.25) gives... 

Next, various quantitative techniques were used to estimate releases by type of use. For use of benzene as an intermediate, we relied on the "emission factor" technique, which estimates the ratio of benzene release to total derivative production and then applies this ratio to the production rate at specific locations. Emissions factors were estimated from crude engineering assessments of the chemical processes entailed (such as open versus closed systems, continuous versus batch, and so on). 

Let represent some property of the flow, for example the velocity, temperature or density of the fluid. In general is a function of the time t and the spatial coordinates x, y, z. Then the total derivative of with respect to r is given by... 

Recall that P and T are intensive properties that are independent of the size of mass of the system, whereas E, El, G, and S (as well as V and n) are extensive properties that increase in proportion to mass or size. By writing the general relation for the total derivative of G with respect to the variables in Eq. (1.16), one obtains... 

Conversion Formulas. Often no convenient experimental method exists for evaluating a derivative needed for the numerical solution of a problem. In this case we must convert the partial derivative to relate it to other quantities that ate readily available. The key to obtaining an expression for a particular partial derivative is to start with the total derivative for the dependent variable and to realize that a derivative can be obtained as the ratio of two differentials [8]. For example, let us convert the derivatives of the volume function discussed in the preceding section. 

Consider a fluid element of constant mass pAxAyAz moving along with the local fluid velocity v. The x component of momentum of this fluid element is pvxAxAyAz. The momentum of the fluid element as it moves along with the local fluid velocity is a function of both space and time. The total derivative of the momentum of the fluid element with respect to time is then pAxAyAz Dvx/Dt). According to Newton s second law this quantity is to be equated to the forces acting on the element of mass the net force in the x direction due to the difference in pressure on faces a and b, which is [p x)AyAz — p(x + Ax)AyAz], the net force in the x direction due to the difference in the viscous stresses,2 which is... 

Several hydrocarbons occur in milk in trace amounts. Of these, carotenoids are the most significant. In quantitative terms, carotenes occur at only trace levels in milk (typically 200/rgl-1) but they contribute 10-50% of the vitamin A activity in milk (Table 3.5) and are responsible for the yellow colour of milk fat. The carotenoid content of milk varies with breed (milk from Channel Island breeds contains 2-3 times as much -carotene as milk from other breeds) and very markedly with season (Figure 3.4). The latter reflects differences in the carotenoid content of the diet (since they are totally derived from the diet) fresh pasture, especially if it is rich in clover and alfalfa, is much richer in carotenoids than hay or silage (due to oxidation on conservation) or cereal-based concentrates. The higher the carotenoid content of the diet, the more yellow will be the colour of milk and milk fat, e.g. butter from cows on pasture is yellower than that... 

An excellent recapitulation of the total derivation is given by M. J. Buerger, Elementary Crystallography, John Wiley Sons, New York, 1956. 

We have written a total derivative sign here to indicate that the dependence of Y on the ptJ is accounted for implicitly. 

To derive the equation that follows, you will need to make use of the relationship that a total derivative with a specified variable held constant is equal to the partial derivative. For example,... 

Biochemists often refer to enthalpies obtained in this manner as van t Hoff enthalpies and attach the subscript i/H to the AH to distinguish it from an enthalpy obtained directly from calorimetric measurements. In practice, one need not actually obtain the K values to extract the enthalpy change. Rather, there is a relationship between da/d T and d nK/dT. For the equilibrium of equation (16.15) (at constant pressure so that the partial derivative can be replaced by the total derivative),... 

The notation in Eqs. (6a) and (6b) corresponds to the usual notation of perturbation theory [18] in which it is understood that the derivatives of the Hamiltonian as well as all variational and nonvariational parameters (i.e., including the dependence of the wavefunction on the perturbations) are taken. Some authors prefer to write Eqs. (6a) and (6b) as total derivatives of E in order to indicate that all dependencies of E on the perturbation parameters must be considered [19,20], Ultimately one needs to consider an explicit energy expression in which no hidden dependencies on any of the perturbations are left. This also includes a possible dependence of the basis set on the perturbation. 

To extend Eq. (2) to functions of more than two variables, the equation for the total derivative must include a term for each variable, with the partial derivative for that variable holding all other variables constant. For h(x, y, z),... 

For a three-dimensional, unsteady flow field, with velocity components u,v and w in x, y and z directions, respectively, the rate of change of a property 0 in the flow field is provided by the substantial or total derivative D0/Dt defined as... 

The rate of increase of the energy contained in the volume element is given by the total derivative of the quantity in Eq. (6.32),... 

This pressure gradient is imposed on the outer edge of the liquid film. However, the liquid film on the surface is assumed to remain thin and the velocity component normal to the surface in the film is therefore assumed to be very small. As a result, as mentioned above, pressure changes across the liquid film are neglected. This is why a total derivative was used for the pressure in Eq. (11.1). Eq. (11.3) therefore gives the pressure gradient everywhere in the condensed film. 

We could equally welt have started with the given equation for H. Since dHfdxx is a total derivative, x2 cannot be treated as a constant. In fact, x2 = 1 — X, and dx2/dx] = — 1. Differentiation of the given equation for H therefore gives ... 

Total derivatives are appropriate here because the properties in the standard state are functions of temperature only. Multiplication of both sides of this equation by Vi and summation over all species gives... 

Strictly speaking, the concentration terms may represent amounts of particular groupings within the paper, such as amorphous cellulose, as well as discrete chemical species. The total derivatives on the right of Equation 11 are the kinetic expressions for the chemical (and perhaps morphological ) changes occurring in the paper. In Equation 12, ki is a chemical... 

---