Mixed second partial derivative

In this case, y is held fixed in both differentiations. In addition, there are mixed second partial derivatives, such as the derivative with respect to x and then with respect to y ... 

The Euler reciprocity relation is an identity relating the mixed second partial derivatives. If z = z(x, y) is a differentiable function, then the two different mixed second partial derivatives must equal each other ... 

These coefficients are themselves, in general, functions of the independent variables and may be differentiated to give mixed second partial derivatives for example ... 

The following is called a mixed second partial derivative ... 

Euler s reciprocity relation states that the two mixed second partial derivatives of a differentiable function must be equal to each other ... 

We refer to the second partial derivatives in Eqs. (B-12a) and (B-12b) as mixed second partial derivatives. The Euler reciprocity relation is a theorem of mathematics If / is differentiable, then the two mixed second partial derivatives in Eq. (B-12a) and (B-12b) are the same function ... 

For a function of three variables, there are nine second partial derivatives, six of which are mixed derivatives. The mixed second partial derivatives obey relations exactly analogous to Eq. (B-13). For example. 

Last but not least, suppose / (xj,..., x ) exists such that / and its first and second partial derivatives are continuous in the neighborhood of a point r = ( Ti,..., Xk) G K Under this presupposition, the theorem of Schwarz holds, which states that for the mixed second-order partial derivatives the order of differentiation is irrelevant that is. 

The second partial derivative d //dydx, for instance, is the partial derivative with respect to y of the partial derivative of / with respect to x. It is a theorem of calculus that if a function / is single valued and has continuous derivatives, the order of differentiation in a mixed derivative is immaterial. Therefore the mixed derivatives jdydx and d f /dxdy,... 

For a quaternary Ai xBxCyDi y alloy, the stability criterion is determined by the second partial derivative of the free energy change, AG, for mixing ... 

These relations are often called equations of state because they relate different state properties. Since the variables T, P, and [nj] play this special role of yielding the other thermodynamic properties, they are referred to as the natural variables of G. Further information on natural variables is given in the Appendix of this chapter. In writing partial derivatives, subscripts are omitted to simplify the notation. The second type of interrelations are Maxwell equations (mixed partial derivatives). Ignoring the VdP term, equation 3.1-1 has two types of Maxwell relations ... 

Since is an exact differential, second mixed partial derivatives of are independent of the order in which differentiation is performed. This leads to a Maxwell relation between the temperature dependence of (Pa and the mass fraction dependence of specific entropy s. Hence,... 

Also, the second derivatives of the fundamental equations give useful relationships. According to the theorem of Schwarz, the mixed partial derivatives of continuous functions are independent of the order of differentiation. For example, Eq. (2.14) yields... 

The thermodynamic potentials, being a system s state functions of the corresponding (natural) parameters, arc of special importance in the system state description, their partial derivatives being the parameters of the system as well. The equalities between th( second mixed derivatives are a property of the state functions and lead to relation-ship.s between the system parameters (the Gibbs-Helmholtz equations). Hence, once any thermodynamic potential (usually, the Gibbs or the Helmholtz one) has been evaluated, by means of either simulation or experiment, this means the complete characterization of the thermodynamic properties of the system. 

The time rate of change of shear strain, y (or dyidt, the dot is Newton s notation for the time derivative), is the so-called shear rate. Since the order in which the mixed second derivative is taken is immaterial (note that by sticking to two dimensions, we can write total rather than partial derivatives), then... 

Curves of binodal and spinodal points can be drawn as a function of the temperature up to the critical temperature (Tc) T2 in Figure 4.3) where these two curves meet. Beyond T, the system forms only one phase. At this critical temperature, the partial first- and second-order derivatives of the chemical potential (Afti) are equal to zero the chemical potential is the derivative of the free energy of mixing (AGmix) relative to the number of moles (Ni) ... 

Derive (14.11a). Hints Start with (14.6), with dV dy — 0. Make use of the definition of Helmholtz free energy A = U — TS. Remember that second mixed partial derivatives of thermodynamic state functions are independent of the order in which you take them. 

We now express first, second, and mixed partial derivatives in terms of finite differences. We show the development of these approximations using central differences, and in addition we summarize in tabular form the formulas obtained from using forward and backward differences. 

The partial structure factors for binary (Bhatia and Thorton, 1970) and multicomponent (Bhatia and Ratti, 1977) liquids have been expressed in terms of fluctuation correlation factors, which at zero wave number are related to the thermodynamic properties. An associated solution model in the limits of nearly complete association or nearly complete dissociation has been used to illustrate the composition dependence of the composition-fluctuation factor at zero wave number, Scc(0). For a binary liquid this is inversely proportional to the second derivative of the Gibbs energy of mixing with respect to atom fraction. 

An analysis of the cosolvent concentration dependence of the osmotic second virial coefficient (OSVC) in water—protein—cosolvent mixtures is developed. The Kirkwood—Buff fluctuation theory for ternary mixtures is used as the main theoretical tool. On its basis, the OSVC is expressed in terms of the thermodynamic properties of infinitely dilute (with respect to the protein) water—protein—cosolvent mixtures. These properties can be divided into two groups (1) those of infinitely dilute protein solutions (such as the partial molar volume of a protein at infinite dilution and the derivatives of the protein activity coefficient with respect to the protein and water molar fractions) and (2) those of the protein-free water—cosolvent mixture (such as its concentrations, the isothermal compressibility, the partial molar volumes, and the derivative of the water activity coefficient with respect to the water molar fraction). Expressions are derived for the OSVC of ideal mixtures and for a mixture in which only the binary mixed solvent is ideal. The latter expression contains three contributions (1) one due to the protein—solvent interactions which is connected to the preferential binding parameter, (2) another one due to protein/protein interactions (B p ), and (3) a third one representing an ideal mixture contribution The cosolvent composition dependencies of these three contributions... 

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