Taylor’s series expansion

Suppose we have a function F x,y), and we carry out a Taylor s series expansion about the point (jCo,yo), thus... 

This weighting procedure for the linearized Arrhenius equation depends upon the validity of Eq. (6-7) for estimating the variance of y = In k. It will be recalled that this equation is an approximation, achieved by truncating a Taylor s series expansion at the linear term. With poor precision in the data this approximation may not be acceptable. A better estimate may be obtained by truncating after the quadratic term the result is... 

They point out that it is invalid to retain a higher-order (in T) term while omitting a lower-order term, for this is inconsistent with the Taylor s series expansion. Least-... 

An equation is said to show analytic behavior if a Taylor s series expansion about a point in the solution set of the equation converges in the neighborhood of the point. 

The calculation of AH° and AS° values from the pK-temperature data in each solvent mixture was performed by the nonempirical method of Clarke and Glew (26) as simplified by Bolton (27). In this method the thermodynamic parameters are considered to be continuous, well-behaved functions of temperature, and their values are expressed as perturbations of their values at some reference temperature 0 by a Taylor s series expansion. The basic equation is ... 

Further simplification can be achieved by linearising the water gas-shift reaction rate, and using Taylor s series expansion the flowing expression for the shift reaction can be obtained... 

Several methods have been devised to solve this problem. - All involve approximations. The method of Denting is based on linearization of the nonlinear function by a truncated Taylor s series expansion. The treatment will not be given here very detailed descriptions are available. - However, it will be... 

Of course, this corresponds to an adiabatic potential-energy surface with two potential-energy minima separated by a well-defined barrier. Note that y A contains anharmonicities induced into Xr by D/A interactions as well as energy corrections that originate from D/A coupling. To some extent (i.e., as in first-order perturbation theory or in a Taylor s series expansion around the diabatic values of... 

A Taylor s series expansion to first-order in the infinitesimals yields... 

Use a Taylor s series expansion in terms of potential to obtain a relationship for the charge-transfer resistance in the Tafel regime. 

Nowadays refinement of the trial structure is nearly always carried out by a least squares procedure based on intensity curves13 in the forms defined by Eq. (5), (6a), or (6b). The geometry of the molecular model is usually expressed in some suitable set of internal coordinates-such as bond distances, bond angles, and torsion angles-and the dynamics of the model by a set of It is apparent from the forms of the intensity functions that the least squares procedure is nonlinear in these parameters. The usual linearizing methods based on Taylor s series expansions are used so that the parameters actually adjusted are the shifts in the internal coordinates and in the ,/. The successful conclusion of the re-... 

The force Fj is assumed to be derivable from a potential such as that given by Eq. 5 and has only indirect dependence on the time. The time dependence of the integrand of Eq. 22 is obtained by Taylor s series expansions which will here be carried out in a somewhat more elementary manner than in the original presentation of Kirkwood. However, the assumptions introduced and the final results are the same. 

Unlike the traditional Taylor s series expansion method, the Galerkin approach utilizes basis functions, such as linear piecewise polynomials, to approximate the true solution. For example, the Galerkin approximation to the sample problem Equation 23.1 would require evaluating Equation 23.13 for the specific grid formation and specific choice of basis function ... 

First, consider the approximation of the potential field O (x, y) by a 2D Taylor s series expansion about a point (x,y) ... 

A common procedure for solving this overdetermined system is the method of variation of parameters (also referred to in the mathematical literature as Gauss-Newton non-linear least squares algorithm) (Vanicek and Krakiwsky 1982), and this procedure is described in the following. As approximate values of coordinates x° are known a priori, by Taylor s series expansion of the function / about point x°. 

Like all other meshless methods, the first step in GFD is to scatter nodal points in the computational domain and along the boimdary. To each node (point), a collection of neighboring nodes are associated which is called star. The number and the position of nodes in each star are decisive factors affecting the finite difference approximation. Particular node patterns can lead to ill-conditioned situations and ultimately degenerated solutions. Using the Taylor s series expansion, the value of any sufficiently differentiable smooth function u at the central node of star, uq, can be expressed in terms of the value of the same function at the rest of nodes, with i = 1,. .N where N is the total number of neighboring nodes in the star and is one less than the total number of nodes in it. In two dimensions, a second-order accurate Taylor series expansion can be written as... 

Kiikwood and Buff (1951) included expressions for Y2 in single solvents from a Taylor s series expansion in solute mole fraction. The coefficients were collections of infinite-dilution (pure component 2) pair KBIs at the first order and pair and triplet KBIs at the second order. The form used for applications is... 

Ray [31] details the continuous variable approximation by considering equations (3.13)-(3.16), and using the equal reactivity assumption. If the chain length j is taken to be a continuous variable then the Taylor s series expansion for Pj j about Pj results in... 

---