Chain rule relation

The chain rule relates the partial derivatives of g to those of / ... 

Finally we give the following chain-rule relation among the derivatives, which is used later ... 

Partial derivatives of and may be related to each other by differentiating (5.187) using the chain rule... 

Expressing Eqs. (32) and (33) with the use of the chain rule leads to three relations which must be simultaneously satisfied, viz. 

Most such predictive methods are at best 50-70 per cent accurate. The relatively large inaccuracy stems from the fact that the folded (tertiary) structure imposes constraints upon the nature/extent of secondary structure within some regions of the polypeptide chain. Any generalized rules relating secondary structure to amino acid sequence data, by nature, will not take such issues into consideration. 

The frontier orbitals responses (or bare Fukui functions) f (r) and the Kohn-Sham Fukui functions (or screened Fukui functions)/, (r) are related by Dyson equations obtained by using the PRF and its inverse [32]. Indeed, by using Equation 24.57 and the chain rule for functional derivatives in Equation 24.36, one obtains... 

Obviously we should not allow multiple iterations of the same rule to increase the value of the consequent. If this were to be allowed then one could obtain any final value by simply re-iterating the same rule sufficient times. But redundancies in rules arise in subtle ways, e.g, B => A and C => A where B <-> C, i.e, B is another name for C. Finally, it can be shown that even if the chain of relation between B and C contains logical connectives other than <->, then allowing two successive inferences to increase the value of the consequence above that inferred by the strongest alone can lead to problems. 

The approximation that the MgCl2 mean ionic activity coefficient remains constant is introduced into the linkage relation by first expanding Eq. (21.26) by the standard chain rule ... 

Relation 4 The derivative of complicated functions can be reduced to the derivatives of simpler function by the chain rule ... 

Clearly, the d N and d Q vectors have different dimensions. Suppose that one is interested in relating the (m — 1) populational degrees of freedom for constant N, (6N)n to the equidimensional subset, Qs of Q, e.g., Qb = Rb, in noncyclic systems. In order to fix bond angles one requires an additional constraint transformation, ts = 8Q/dQs, which is available from geometrical considerations [23]. For example, the d/ b - (d/V)w mapping is given by the product (chain rule) transformation ... 

Using the chain rule of calculus, the tensor E can be related to the velocity gradient tensor Vv, as follows ... 

The chain rule provides a relation between the partial derivative of / with respect to the individual particle velocity c and the partial derivative of fc with respect to the peculiar velocity C. To understand the forthcoming transformation it might be informative to specify explicitly the meaning of the partial derivatives. 

By chain rule differentiation, the last term on the RHS of this relation can be... 

To convert the derivatives of scalars with respect to x,y,z into derivatives with respect to r,9,z, the chain rule of partial differentiation is used. The derivative operators are thus related as ... 

Given an some orbital- and eigenvalue-dependent xc-functional the main task is to derive a relation with allows the calculation of die xc potential (65). Two equivalent approaches to this task are possible. One can either minimize the total energy with respect to the total RKS potential under the subsidiary condition that the orbitals satisfy the RKS equations [48, 49,55,36] or one can use the chain rule for functional differentiation to replace the derivative with respect to 7 by derivatives with respect to and... 

The explicit formulae for all terms in the multipole expansion up to R 5, which includes the quadrupole-quadrupole, octupole-dipole and hexadecapole-charge terms, have been published (Price et al., 1984 Stone, 1991). The chain-rule type formalism for the associated forces, torques and second derivatives has also been established, along with the derivatives with respect to the strain matrix which defines the unit cell shape, by Willock et al. (1995), with related analyses by Popelier and Stone (1994). 

The local softness s(r) is related to the Fukui functions/(r) through a chain rule... 

Using the chain rule again we can determine how J relates to j ... 

Since the solid and fluid are in equilibrium, q is related to c and T through the isotherm After assuming that solid (e, Sp, K, and pj properties are constant, applying the chain rule and sinplifying, Eq. (18z65a) becomes... 

Models seldom express the conservation relations in a Lagrangian framework. The chain rule of calculus is used to convert to an Eulerian framework. 

The chain rule tells us how the partial derivatives of g(x,y, z) are related to those of In fact. 

The rheology of blends of linear and branched PLA architectures has also been comprehensively investigated [42, 44]. For linear architectures, the Cox-Merz rule relating complex viscosity to shear viscosity is valid for a large range of shear rates and frequencies. The branched architecture deviates from the Cox-Merz equality and blends show intermediate behavior. Both the zero shear viscosity and the elasticity (as measured by the recoverable shear compliance) increase with increasing branched content. For the linear chain, the compliance is independent of temperature, but this behavior is apparently lost for the branched and blended materials. These authors use the Carreau-Ya-suda model. Equation 10.29, to describe the viscosity shear rate dependence of both linear and branched PLAs and their blends ... 

Differentiating Eq. (4) with respect to the state time derivatives, states, inputs, outputs and disturbances using the chain rule the following relation is derived ... 

Differentiating the dispersion relation with respect to k and using the chain rule, we have = 0 and as long as 0, we have the necessary but not sufficient condition for absolute instability... 

An important property of first-order homogeneous functions is their relation to their partial derivatives. Differentiate both sides of Equation (7.12) with respect to A and use the chain rule (see page 77) to get... 

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