Derivative calculating

The case where the point B a,b) is not surrounded by the path F (see Fig. 17a). In this case, both Xp and x, are analytic functions of the coordinates in the region enclosed by F, and therefore the integrands of the two integrals can be replaced by the corresponding derivatives calculated at the respective intermediate points, namely,... 

DERIV. Calculates the inverse of the Jacobian matrix used in isoparametric transformations. 

Figure 17.4 then is a typical Hartree-Fock analytical derivative calculation on fluoromethane. 

Interpretation of the images is still not straightforward even when there seems to be a simple one-to-one correspondence between black (or white) dots in the image and atom positions. Especially when quantitative data on interatomic distances is to be derived, detailed calculations based on many-beam dynamical theory ( ) must be applied to derive calculated images for comparison with experiment. For this purpose the experimental parameters describing the imaging conditions and the specimen thickness and orientation must be known with high accuracy. 

Close examination of Figure 54-2b will reveal a decrease in the slope of the difference curve at the point it crosses the X-axis, even though we are not using the denominator term of the derivative calculation. 

Now we look at the second derivative similarly. Some of this has been presented previously in the literature [9, 10], although in less detail than we do here. Figures 54-4a to 54-4c present second derivatives calculated using the same spacing as for the... 

We continue in our next chapter by examining the behavior of the derivative calculation when the division of the Ay term is divided by the AX term, to form an approximation to the true derivative. 

Figure 55-7 First derivatives calculated using different spacings for finite difference approximation to the true derivative. The underlying curve is the 20 run bandwidth absorbance band in Figure 54-1, with data points every nm. Figure 55-7a Difference spacings = 1-5 nm Figure 55-7b Spacings = 5 10 run Figure 55-7c Spacings = 40-90 nm. (see Color Plate 21)...

But things are not so simple. In this examination we have so far looked only at a derivative calculated from adjacent data points. What happens when we calculate a two-point derivative based on non-adjacent data points In fact we have already considered this question qualitatively in our previous chapter [3], when we noted that using the optimum spacing will result in an improved S/N ratio for the derivative. Of course, improved in this case is in comparison to the derivative computed using adjacent data points, it must be determined on a case-by-case basis whether the improvement is sufficient to exceed that of the actual direct absorbance signal. 

In principle, this method is more accurate than the direct derivative discretization method described in the previous subsection, because the derivative calculations are made several times for the same surface point. Nevertheless, the method suffers essentially the same problems related to the finite derivative discretization on the lattice as in the previous case. 

The purely electronic derivatives, calculated for the rigid molecular geometry, determine the system electronic chemical potential... 

Table 3.10. Second derivatives, first left- and right-derivatives calculated for each data point listed in Table 3.9 through equations (3.1.43) to (3.1.46).

Figure 18.3 shows a curve of the derivative calculated from the quartic best fit of V as a function of which are points from a numeric differentiation, and chords... 

Figure 3.15. Results after applying three different derivative preprocessing tools to a sample vector, (a) A sample vector with noise and an offset of one unit. ) The derivative calculated by simple difference, (c) The derivative calculated using a running mean difference with a window width of 15. d) The derivative calculated using the Gorry method with a window width of 15.

Tliis results in a smoothed derivative calculation (sec Section 3-1-3 for discu.s-sion of smoothing). Figure 3-13e.shows the result of applying a running me.an difference derivative with a window width t)f 13 to the data in Figure 3-13 /. The signal-to-noi.sc of this derivative is much better than the simple difference derivative. (The apparent shift of the derivative is due to the end effects.)... 

Partial differential operators will play a large role in the examples of Lie algebra representations that concern us. Hence it behooves us to consider partial derivative calculations carefully. Consider a simple example ... 

Section 13.06.4.3. A similar analysis of a C-3,N 4-pyrrolo-fused derivative calculated an interconversion energy of 12.2 kcal mol-1 <20050L5305>. 

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