Useful Partial Derivative Identities

The advantage of this method is that it avoids both the evaluation of partial derivatives and the inversion of the Jacobian. To start the iterations, an initial estimate H0 is also required an identity matrix is frequently used for this purpose. 

Only four of these partial derivatives are not identically zero. For these we may use the following relationships, which apply at the Hopf bifurcation point, to simplify... 

The use of the conditions expressed by Equations (5.97>-(5.102) and the fact that each second partial derivative in Equation (5.105) is identical to the same derivative for the double-prime part shows that 82E = d2E", and therefore Equation (5.108) can be written as... 

Next, we derive some relations which will prove useful in later applications of the theory. All of the following relations are obtainable by the application of simple identities between partial derivatives, such as (see also Appendix A)... 

There are some useful identities allowing manipulations of expressions containing partial derivatives. 

Equation (7.20) is an example of a set of useful equations. If each symbol is consistently replaced by another S3mibol, we will have a useful equation for other variables besides the thermodynamic energy. We regard this as our first identity for partial derivatives, which we call the variable-change identity. 

It is fairly common in thermodynamics to have measured values for some partial derivative such as (dH/dT)p n, which is equal to the heat capacity at constant pressure. However, some other partial derivatives are difficult or impossible to measure. It is convenient to be able to express such partial derivatives in terms of measurable quantities. We now obtain some identities that can be used for this purpose. 

Since the partial derivative of x with respect to x is equal to unity, this equation becomes identical with Eq. (7.26) when the reciprocal identity is used. Our derivation is indefensible, but the result is correct. 

Use Table 2.1 to derive an expression for the total differential of the enthalpy in terms of C° and a(= (1 /V)(dV/dT)p). In other words, start with dH = (dH/dT)pdT + (dH/dP)TdP and find expressions for the two partial derivative terms from Table 2.1. You will find that if in this resulting equation you let dH = 0, you get an expression for (dT/dP)H identical to that in equation 8.1 (Chapter 8). This derivation is used by Ramberg (1971) in his elegant discussion of the Joule-Thompson effect in a gravitational field. 

Similarly, the entropy of ionization, AS°, the standard partial molar heat capacity of ionization, AjC , and the standard partial molar volume of ionization, AV°, can be derived from AG using standard thermodynamic identities (Mesmer et al., 1988) ... 

Reciprocity relation. The reciprocity relation or the cross-differentiation identity is very useful in some thermodynamic derivations. Let us represent the partial derivatives in Eq. (1.77) by... 

A mathematical identity is an equation that is valid for all values of the variables contained in the equation. There are several useful identities involving partial derivatives. Some of these are stated in Appendix B. An important identity is the cycle rule, which involves three variables such that each can be expressed as a differentiable function of the other two ... 

We have here first used the fact that the total derivatives of the SCF energy and of the SCF Lagrangian are identical for the optimized wavefunction. Next, we have invoked the Hellmann-Feynman theorem for the fully variational Lagrangian, thereby reexpressing the SCF molecular gradient as the partial derivatives of the Lagrangian. Finally, we have inserted the expression for the Lagrangian (equation 34). 

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