Order of derivatives

Compute the desired order of derivative of that polynomial... 

What determines the number of rows and columns The number of rows is determined by the number of coefficients that are to be calculated. In this example, therefore, we will compute a set (sets, actually, as we will see) of seven coefficients. The number of columns is determined by the degree of the polynomial that will be used as the fitting function. The number of columns also determines the maximum order of derivative that can be computed. In our example we will use a third-power fitting function and we can produce up to a third derivative. As we shall see, coefficients for lower-order derivatives are also computed simultaneously. 

Exchanging the order of derivation and integration on the left-hand side integral and using the divergence theorem on the right-hand side, we get... 

By comparing Eq. (2.153) to Eq. (2.78) for the desired diffusion equation, while taking due care with the order of derivatives, we hnd that the two forms of the diffusion equation become equivalent if and only if... 

The derivative spectrophotometry methods provide higher selectivity and higher sensitivity than do the methods based on normal (zero-order) absorption spectra. The increase in selectivity (with reduction or elimination of the effect of the spectrum of one substance on the spectrum of another one) results from reducing the band-width in the derivative spectra. An appropriate order of derivative spectrum may give complete separation of the spectra owing to the corresponding components of the system). 

Fcr clarity, we use a superscript containing a counting number in parentheses to denote a particular order of derivative,... 

Due to this property broad zero-order spiectra are quenched with generation of higher orders of derivatives while narrow undergo amplification. If the zero-order sp>ectrum possess two bands A and B which differ from their half- heights width (Lb>La), after a generation of n-order derivative a ratio of derivatives intensity can be expressed as ... 

The shape of derivative spectrum is more complicated than its parent one (Fig. 1). New maxima and minima appeared as results of derivatisation. The generation of n-th order derivative spectrum produces (n+1) new signals an intense main signal and weaker bands, so called satellite or wings signals. Position of maxima or minima depend on order of derivative. The main extreme of derivative spectra of even order is situated at the same wavelength as maximum in zero-order spectrum, but for 2, 6 and 10-th order it becomes minimum in... 

The order of derivative is stored in the data block use the Show Parameter command to display the content of the block. 

The derivatives that appear more frequently in the equations of reactive flows are of the first and second orders. Conventional approaches to these derivatives are presented in the following. It is important not to confuse order of derivative with order of its approximation. 

Figure 2-11. Influence of the order of derivatives on FWHM. a) Relative peak half-width (for a fundamental peak and its second derivative), with reaching 66.6% (schematic drawing) b) FWHM as a function of derivative order n, where P varies with the mode of the distribution curve.

It is important to acknowledge symmetry properties of the susceptibility tensors. Let us consider first non-dispersive media, in which the frequency dependence of susceptibility components can be omitted. Basic theorems of differential calculus tell us that for partial derivatives of order higher than one, the order of derivation doesn t change the derivative, so, for instance, = X - This property, which is called intrinsic permutation [3], has the effect, as an example, that of the 27 components of x only 18 are really independent. There is another relevant symmetry property of the tensor, known as Kleinman symmetry [3], that allows permutation of all three indices. In this case the independent components drop to 10. 

As these partial derivatives do not depend on the order of derivation, we deduce the following theorem ... 

Mechanical work at every particle of a continuum results from acting force and respective displacement or local stress and strain, correspondingly. So the above axiom of Remark 3.1 actually comprises two principles involving either virtual loads or virtual displacements. A brief derivation of both will be given in the following subsections. Similarly, the electric work can be treated, but we will present only one of the variants. The different formulations of the principle of virtual work are independent of a constitutive law and may be denoted as the weak forms of equilibrium, as only the equilibrium conditions have to be fulfilled in the integral mean. Weaker requirements with regard to differentiability of the involved functions have to be fulfilled, since the order of derivatives is reduced in comparison to the equilibrium formulation of Eqs. (3.f4) and (3.34). 

G is a function of state, which means that the order of derivation is unimportant ... 

Equation 7.64 is obtained, like many others, by taking advantage of the fact that free energy is a state function and therefore second derivatives are insensitive to the order of derivation, i.e. (9 G/9x 9y) = (d G/dy dx). Equations 7.65 and 7.66 are useful in... 

The weak form is characterized by the feet that the order of derivative in the field variable and the weighting function is the same. This statement is normally used to derive finite element formulations. Integration by parts of the first integral in the weak form according to O Eq. 26.37 gives... 

In each case there are four boundary conditions that must be satisfied in combination with the differential equations. In all such boundary value problems, the number of boundary value equations must match the total order of derivatives in the coupled differential equation set - two second order derivatives in one case and four first order derivatives in the other case. 

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