The Taylor Series Expansion of

The double-bracket factor in the contnbution to the heat flux, given in the first line of Eq. (8.11), has already been evaluated in Eq. (12.21). Therefore, qf can be written as follows, provided that we neglect the last term in Eq. (12.21)  

As pointed out in connection with Eq. (5.9), the double-bracket quantities are always functions of the same variables as the associated configuration-space distribution function. It is for this reason that the T in Eq. (16.5) must be written as a function of the displaced coordinate (r — R ). However, according to Eq. (12.3), (r — R ,t) can replaced by 7 (r, t)(l — a RJ) and then T r,t) can be removed from the integral. In addition, terms quadratic m a are neglected. Then Eq. (16.5) becomes. 

Comparison of this equation with Eq. (6.10) shows that there is a simple relation between mass flux and the flux of kinetic energy by diffusion  

That IS, the contribution qf to the part of the heat-flux vector associated with species a is proportional to the mass flux of species a. Therefore, the Taylor-series results given in Eqs. (15.2-15.4) for the mass-flux vector can be taken over here. When Eq. (16.4) is used in these expressions, we get for the first three terms in the Taylor series for q ADEFPXC T V  

In obtaining these expressions we have used — R = -F R = iQ. This completes the development of the expressions for q for Hookean dumbbells.  

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Following the procedure described in Chatper 2, Section 2.5 the Taylor series expansion of the field unknowns at a time level equal to + oAt, where 0 < a < 1, are obtained as... 

The Taylor series expansion of the eleetronie energy is written as ... 

Recall that equations 9.86 and 9.100 have been both derived using only the first-order terms in the Taylor series expansion of our basic kinetic equation (equation 9.77). It is easy to show that if instead all terms through second-order in 6x and 6t are retained, the continuity equation ( 9.86) remains invariant but the momentum equation ( 9.100) requires correction terms [wolf86c]. The LHS of equation 9.100, to second order in (ia (5 << 1, is given by... 

SO that the series u and U2 converge for all finite values of To see what happens to u and M2 as 00, we consider the Taylor series expansion of e ... 

The derivatives of the Taylor series are all finite. It is not necessary to expand the series at xB = 0, but it is most common and convenient for dilute solutions. The Taylor series expansion of In yB may be expressed in a different notation as... 

Figure 3.9 An illustration of low-order terms in the Taylor series expansion of In y for dilute solutions using lnyT1 for the binary system Tl-Hg at 293 K as example. Here lny i =-2.069,fij11 =10.683 and/J1 =-14.4. Data are taken from reference [8],...

Successive linear programming (SLP) methods solve a sequence of linear programming approximations to a nonlinear programming problem. Recall that if g,(x) is a nonlinear function and x° is the initial value for x, then the first two terms in the Taylor series expansion of gt(x) around x° are... 

Hyper)polarizabilities are defined as the coefficients in the Taylor series expansion of the dipole moment - or the energy - in the presence of static and/or oscillating electric fields ... 

Stegun 1964) which has given the model its name. The three adjustable parameters are S, 13 and 14. The lowest three spectral moments may be obtained as functions of these parameters from the Taylor series expansion of the correlation function, according to... 

The Taylor series expansion of a function fix) around the point x0 is defined as a power series... 

The proof is given in Appendix A 5.1, where it will become clear that basically the theorem relates the Taylor series expansion of any given function f(x) to the expansion of In /(j ). As such it is a very general statement, of fundamental importance for the thermodynamic limit in any many-body theory. In the present context we note that % definition (Sect. 4.3)... 

Sometimes the Taylor series only converges for a specific domain of x values. For example, the Taylor series expansion of ln(l + x) (Problem 2-7) only converges for 1 1 < 1. There are even bizarre functions which do not converge at all to their Taylor series expansions. In practice, however, such pathological cases are almost never encountered in physics or chemistry problems, and Taylor series expansions are a very valuable tool. 

Comparison of Eqs. (337) and (340) shows that the term in the brackets in Eq. (340) is just the Taylor series expansion of 1 /Ship) around p — 0. Thus, the local equation converges for all values of and p. For this special case, we can also estimate the error in the solution when the local equation is truncated at... 

This is the descent condition and should be negative, rhe contribution of the first two terms in the Taylor series expansion of the energy. 

We can determine the velocity or flequency shift caused by such types of loading, which is in addition to the loading due to the liquid, by substituting into the denominator of the phase velocity expression of Equation 3.78 all the relevant masses per unit area that of die plate itself, M the equivalent loading of the liquid, pfSe, that of any selective biological or chemically soiptive layer, fnsorptive, und that due to the unknown itself. Am. If Am is much smaller than the sum of the other terms, as is usually the case when one is seeking the minimum detectable added mass, we can use the first term of the Taylor series expansion of the denominator of the square-root in the velocity expression, and write the approximate phase velocity as... 

The LTE is obtained expanding the terms y j and f j,j = 1(1)3 in (4) into Taylor series expansions and substituting the Taylor series expansions of the coefficients of the method. 

This approximate expression of the derivative in terms of differences is the finite difference form of the first derivative. The equation above can also be obtained by writing the Taylor series expansion of the function/about the point. t. 

Because we are working to linear order in iAL, we have used the fact that uUo,s) can contribute only higher powers of iL = ILq + iAL. Fligher powers of linear approximation. Effectively, it can be seen that the operator Ur(0, s) = e ° + 0(/AL), where O(iAL) indicates terms of order /AL and where we have used Eq. [99] and the time independence of /Lq. We also note that by definition, iLo/eq = 0. Thus only the first term in the Taylor series expansion of 17 (0, s) survives, yielding = /eq and we obtain Eq. [103]. 

Then, they proposed to examine the Taylor series expansion of the spherical average conditional pair density in the vicinity of the point s = 0 where one is measuring the short-range behaviour of the electron at point r2 approaching the reference point rl. The leading term of the Taylor series expansion is given by... 

To obtain these expressions, take the Taylor series expansions of the elements fIji in Eq. (125) and neglect higher than the first-order terms. If we can calculate the gradients of the Euler angles as... 

The truncation error of the linear approximation of the property gradient ( )e can be calculated in terms of two Taylor series expansions around V e-The Taylor series expansion of tjjp about V e yields [202] ... 

Dalgaard (1982) has a factor in front of the two terms on the right-hand side of this equation. He has included the factor coming from the Taylor series expansion of the response (see Eq. (33)) in the definition of the response functions. We follow the definitions used by Olsen and Jorgensen (1985). 

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