Chain rule, for differentiation

In conjunction with the use of isoparametric elements it is necessary to express the derivatives of nodal functions in terms of local coordinates. This is a straightforward procedure for elements with C continuity and can be described as follows Using the chain rule for differentiation of functions of multiple variables, the derivative of a function in terms of local variables ij) can be expressed as... 

By using the chain rule for differentiation and the orthogonality of the basis vectors, we then obtain... 

The second integration technique, known as the substitution method, derives from the inversion of the chain rule for differentiation described in Chapter 4. The objective here, once again, is to transform the integrand into a simpler or, preferably, a standard form. However, just like the integration by parts method, there is usually a choice of substitutions and although, in some cases, different substitutions yield different answers, these answers must only differ by a constant (remember that, for an indefinite integral, the answer is determined by inclusion of a constant). The substitution method is best illustrated using a worked problem. 

The chain rule for differentiating the volume integral of eqn. (10.69) and the following identities [50]... 

Finally, we mention the chain rule for differentiation for instance, if we have the derivative of pA with respect to the density pA at some set of constant variables C, we now want the derivative of pA with respect to, say, the mole fraction xA, at the same set of constant variables C. We have... 

Since we know how to calculate the functional derivative with respect to v the functional derivative with respect to n seems rather straightforward. By the chain rule for differentiation we obtain... 

We finally make some comments on the calculation of functional derivatives in this section. We stress this point since careless use of the chain rule for differentiation has led to wrong results in the literature [48], As an example we... 

First the Laplacian. Using the chain rule for differentiation... 

Equation 2-13 can be simplified by expanding the accumulation term using the chain rule for differentiation of a product ... 

As [N] is expressed in terms of natural coordinates the chain rule for differentiation must be used. First, the Jacobian is calculated ... 

A fifth formula, for use in situations in which a new variable X(P,T) is to be introduced, is an example of the chain rule of differential calculus. The formula is... 

Treating as an independent variable, the chain rule for functional differentiation gives... 

To prove (5.30), let us begin by writing the known fundamental equation for entropy in the more general form S = S(U, Xh X2,...) to focus on its dependence on U, where 2Q are the remaining extensive arguments (e.g., V, N, N2,..., Nc for a system with c independent chemical components). By the chain rule, the differential variations dS can be written as... 

The proof can be done by mathematical induction. For convenience, denote the /th derivative by f. The fust derivative appears in Equation (9-34). Just by plugging in i=l, it is clear that f satisfies the relationship. Now, use the chain rule to differentiate f,... 

The chain rule of differentiation will give us the Green s first identity for vector... 

We give a brief overlook to the derivation of the vibration-rotation Hamiltonian with the coordinate chain rules of differentiation (e.g., Refs. 22-28). For example, in Ref. 23, is written as... 

The chain rule for the differentiation of products has the matrix generalization... 

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... 

We define a function U(S, V, n). Then, using the chain rule for a function with multiple variables, the differential with respect to a general variable X is... 

The substitution rule is the reverse of the chain rule of differential calculus. The method looks more complicated than it actually is. Let us consider an example We are looking for an antiderivative of... 

In each iteration of the Newton-Raphson method, when the guesses are close to the true values, the length of the error vector, y, is the square of its length after the previous iteration that is, when the length of the initial error vector is 0.1, the subsequent error vectors are reduced to 0.01, 10 , 10", . However, this rapid rate of convergence requires that n" partial derivatives be evaluated at x. Since most recycle loops involve many process units, each involving many equations, the chain rule for partial differentiation cannot be implemented easily. Consequently, the partial derivatives are evaluated by numerical perturbation that is, each guess, Xj, i = 1,..., , is perturbed, one at a time. For each... 

Written in terms of this hypothetical pressure, eq. (10.5-5) does not have direct application as we do not know the hypothetical pressure directly but rather we have to solve for them from eq. (10.5-4). It is desirable, however, that we express the flux equation in terms of the adsorbed concentration as this is known from the solution of mass balance equations in a crystal. Now that the partial pressure Pi is a function of the adsorbed concentrations of all species (eq. 10.5-4), we apply the chain rule of differentiation to get ... 

Certain interval methods make use of interval gradients which require derivatives. Automatic Differentiation (AD) is a method for simultaneously computing partial derivatives using a multicomponent object, called gradient, whose algebraic properties incorporate the chain rule of differentiation. The rules, together with the gradient type, form an extended type that we call AD. This type can be used with intervals. 

Substitution of Equations (5.50) into Equation (5.49) produces the chain rule for partial differentiation ... 

The simplest way to derive the OPM equation is the transformation of the functional derivative (2.4) into derivatives with respect to and Cfc, using the chain rule for functional differentiation [18],... 

Show that for constant r and V = 0, equation 11.56 becomes equation 11.46. Hint you will have to apply the chain rule of differentiation to the derivatives in the second term of equation 11.56.)... 

Similar expressions can be derived for second spatial derivatives. The final form of the equations that result after a generalized coordinate transformation depends on the degree of differentiation by using the chain rule, i.e. on the treatment of the metrics x, x, and y. For more details we refer to the... 

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