Chain rule, for

Reference [73] presents the first line-integral study between two excited states, namely, between the second and the third states in this series of states. Here, like before, the calculations are done for a fixed value of ri (results are reported for ri = 1.251 A) but in contrast to the previous study the origin of the system of coordinates is located at the point of this particulai conical intersection, that is, the (2,3) conical intersection. Accordingly, the two polar coordinates (adiabatic coupling term i.e. X(p (— C,2 c>(,2/ )) again employing chain rules for the transformation... 

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... 

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... 

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

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

Hint this is an exercise in careful, correct application of the chain rule for functions of several variables.)... 

The axial Poiseuille flow occurs when 6 = n/2, with x taking the role of z, and y taking the role of De/2 — r. The metric coefficient reduces to h = De/2 — y = r. The expected axisymmetric momentum equations for axial Poiseuille flow can be recovered by substituting f — ur and carrying out the independent variable transformation to exchange r for v. The chain rule for the independent variable transformation provides that... 

Now (S 10.4-6) is apparently the chain rule for a function having the special property (for some constant c0)... 

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. 

Defining f = Vip into Gauss theorem and using the chain rule for the divergence of the vector, the so-called Green s first identity is obtained (Theorem (10.1.2)). This identity is also valid when we use it for a vector g = when we substract the fist identity for g to the first identity of f we obtain the Green s second identity (Theorem (10.1.3)). 

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

Using the chain rule for partial derivatives [Eq. (7) of Appendix A] express (dP/dT)r in terms of the thermal expansion coefficient a and the isothermal compressibility k. 

The last equation looks just like the chain rule for ordinary derivatives, except for the minus sign, which is often omitted by students. However, the need for the minus sign is fairly obvious. If z increases with x at constant y and with y at constant x, for z to remain constant, as we increase x, we must decrease y. We will call Eq. (7) the chain rule for partial derivatives. 

For further justification of the chain rule for partial derivatives, consider the following example Let z equal the money in your checking account, x he the number of times you go shopping for clothes in a month, and y the number of times you eat out in a month. What... 

When the variational energy is a functional of the reference state well-defined for nonintegral occupation numbers. This implies two chain rules, for m 0 ... 

---