Normalized partial derivative

An analysis is terms of the normalized partial derivative (saturation parameter) is invariant with respect to the specific rate equation. Note that not all choices are necessarily equally plausible. 

It is straightforward to verify that both cases result in a well-defined interval for the (normalized) partial derivatives, namely,... 

Assuming a functional form of F = 1 + a,S + jS, with a and ft as auxiliary parameters, the (normalized) partial derivative is confined to the unit interval. Analogously, we evaluate the dependency on a product concentration P and obtain... 

Importantly, the contribution from thermodynamics is not restricted to finite interval. With y c [0,1], the normalized partial derivative may attain any (absolute) value between zero and infinity. In particular, for reactions close to equilibrium y 1, we obtain... 

Referring to the earlier treatment of linear least-squares regression, we saw that the key step in obtaining the normal equations was to take the partial derivatives of the objective function with respect to each parameter, setting these equal to zero. The general form of this operation is... 

The partial derivative of R with respect to each parameter is then minimized. The normal equations are... 

Here e is a unit vector normal to the surface at the POI defined by e = pH xpv/ pH xpv, and subscripts of p denote the partial derivatives. Thus the mean, H, and Gaussian, K, curvatures are expressed as... 

The governing dimensionless partial derivative equations are similar to those derived for cyclic voltammetry in Section 6.2.2 for the various dimerization mechanisms and in Section 6.2.1 for the EC mechanism. They are summarized in Table 6.6. The definition of the dimensionless variables is different, however, the normalizing time now being the time tR at which the potential is reversed. Definitions of the new time and space variables and of the kinetic parameter are thus changed (see Table 6.6). The equation systems are then solved numerically according to a finite difference method after discretization of the time and space variables (see Section 2.2.8). Computation of the... 

Up to this point, we have only considered ordinary partial derivatives. However, within MCA, it is often preferred to consider normalized (scaled) partial derivatives instead. Apart from minor exceptions, all equations are invariant under the normalization provided all involved quantities are scaled appropriately [315],... 

Overparameterization and frequently its sources are revealed by an eigenvalue-eigenvector analysis. In the module 1445 the matrix JT( )WJ(jB) is investigated. We call it normalized cross product matrix, because the partial derivatives are computed with respect to the normalized parameters... 

To quantify this idea in mathematical terms, we can recognize that we are really talking about the partial derivative quantities d(da/dt)/da and d(db/dt)/db. Stability has been associated in some sense with these two quantities being negative (i.e. da/dt decreases as a increases, so d(da/dt)/da < 0), instability with these being positive. In most normal chemical systems, e.g. those with deceleratory kinetics, the two partial derivatives will be negative. It is a characteristic of autocatalysis, however, that at least one of these may become positive — at least over some ranges of composition and experimental conditions. 

The constraint mo = 1 does not affect this result because the partial derivative is taken with the values of all moments other than m, fixed anyway. For the same reason, the second derivatives of the ideal part of gm are (up to a factor T) given by the inverse of the matrix of second-order normalized moments my, by analogy with (43). (Note that the first row and column of the second-order moment matrix, corresponding to i — 0 or j = 0, need to be retained they can only be discarded after the inverse has been taken.)... 

Note. These last four equations involve partial derivatives. If an equation involves more than two variables, then normally the effect of varying one of the quantities on a second quantity will be studied while the third is held constant, e.g. 

The partial derivatives listed at the bottom o/Figure 10.4, which are equated to zero in order to minimize the sum of the squares, may be rearranged to give the m normal equations for each xj, where j = 1 to m. 

A free surface is determined by fixing the normal component of the velocity at the surface to zero, and the partial derivative normal to the free surface of all other scalar quantities are set to zero. 

An outlet boundary can be determined assuming that the flow is fully developed, thus the partial derivatives of all scalar variables are set to zero normal to the outlet surface plane. 

To carry out a computation, H is prescribed and the surface operator Vs, the surface normal n, and the area element data are expressed in terms of the solution vector and its first partial derivatives. The Jacobian of the parametrization is easily calculated to depend only on w ... 

Discussion CJACOB computes a partial derivative of a species or a compartment with respect to a c—value of any species. The species numbers are input as arguments. If the dependent variable s number I is negative, it is the number of a compartment. CJACOB is normally called by PART but may be used independently if desired. ARITH should be called before using CJACOB separately. It is part of the Jacobian package of subroutines. 

Where - 1, and the two partial derivatives are proportional to the direction cosines of the normal to the surface at any point. Again, it follows from (10) and (11) that... 

The first partial derivative is the definition of the heat capacity, Cp, Notice this heat capacity is the extensive heat capacity of the reactor contents. Normally we express this quantity as an intensive heat capacity times the amount of material in the reactor. We can express the intensive heat capacity on either a molar or mass basis. We choose to use the heat capacity on a mass basis, so the total heat capacity can be expressed as... 

Thus, with the assumption of normal noise f, the MML principle requires the search for a minimum in the product of the squared terms of (f(a) - f ) in Eq. (10). This is the basis for the widely known Least Square Method (LSM). The minimum of the quadratic form (a) corresponds to a point with a zero gradient (a), i.e. to a point where all partial derivatives of (a) are equal to zero ... 

This conforms to the initial-value problem posed by the nonisothermal PFR, and the solution is always unique if /i and fj have continuous first partial derivatives. Functions such as the reaction rate and heat-transfer terms appearing in equations (6-109) and (6-110) normally satisfy this requirement. 

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