Partial derivatives transformations

Since partial derivatives transform in just the same manner, we have... 

For partial derivatives an indication of which variable is being transformed avoids confusion. Thus, if... 

Although transformations to various curvilinear coordinates can be carried out relatively easily with the use of the vector relations introduced in Section 5.15, it is often of interest to make the substitutions directly. Furthermore, it is a very good exercise in the manipulation of partial derivatives. 

The electronic-nuclear coupling in molecules is also detected in the other partial Legendre-transformed representation H(/r, Q), which defines the combined Hessian G of Equation 30.27. Its first diagonal derivative,... 

Once the transformations have been made, a solution space has been defined. It is only within this framework that the concept of steepest ascents takes meaning. The proper direction in which to proceed can be determined by n + 1 cases if n is the number of controllable variables. In each of the n cases one variable is changed slightly from its value in the base case, while all other variables are held constant. This permits approximating the n partial derivatives of response with respect to each variable. The direction of steepest ascent is given by the vector which is the gradient of the response, R ... 

The evaluation of fi2 follows the same recipe as that given above through the transformation to new variables x and y, the various partial derivatives d2gl/dx2 etc., and then to gl i. This manipulation requires patience or a computer algebra package (preferably both), but really we are only interested in the resulting expression which is... 

To transform the governing equations from the x, y) to the (x, -) coordinates the partial derivatives with respect to x and y need to be re-evaluated in terms of x and as follows ... 

Canonical analysis achieves this geometric interpretation of the response surface by transforming the estimated polynomial model into a simpler form. The origin of the factor space is first translated to the stationary point of the estimated response surface, the point at which the partial derivatives of the response with respect to all of the factors are simultaneously equal to zero (see Section 10.5). The new factor... 

Shi and Geva [15] have also derived the QCLE in the adiabatic basis starting from the full path integral expression for the quantum mechanical problem. In this representation the derivation starts with the partial Wigner transform of the environmental degrees of freedom in contrast to what is done... 

The subscripts on the partial derivatives have been omitted because they are complicated, as indicated by the fundamental equation. The change in binding of coenzymes in a reaction can be studied at constant concentrations of coenzymes, just as the change in binding of hydrogen ions in a reaction can be studied at constant pH. The further transformed enthalpy H" of the system can be calculated by use of the Gibbs-Helmholtz equation or from G" = II" — TS". 

The three partial derivatives describe the kinetics of the reaction and (dE/dl) is the charge transfer resistance, Rct. It can be shown, using Laplace transformation, that... 

But notice now that H is the Legendre transform of E and hence, subject to Eq. (1.26.8). Thus, the sign of the second partial derivative of H with respect to P is opposite that of E with respect to V. Therefore, contrary to naive preconceptions,... 

The development described above transforms system (4.69) into the following system of n equations (k= 1, n) with partial derivatives ... 

Thus measurements of Af G, ° and Af //, ° at a single temperature yield Af 5, ° at that temperature. In the next chapter we will see that if Af //, ° is known, the standard transformed Gibbs energy of formation can be expressed as a function of temperature, and then all the other thermodynamic properties can be calculated by taking partial derivatives of this function. Note that in equations 3.4-5 to 3.4-9, the only Maxwell relation that does not involve a partial derivative with respect to the temperature is the one that yields Fh (0-... 

One of the Maxwell relations in equation 3.4-14 shows that the average binding of hydrogen ions by a reactant can be calculated by taking the partial derivative of the standard transformed Gibbs energy of formation of a reactant with respect to pH. 

Calculation of Other Standard Transformed Thermodynamic Properties of a Reactant by Taking Partial Derivatives... 

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