Mathematical identities

This is exact—see Problem 11-8. Notice that Eq. 11-14 is exactly what one would write, assuming the meniscus to be hanging from the wall of the capillary and its weight to be supported by the vertical component of the surface tension, 7 cos 6, multiplied by the circumference of the capillary cross section, 2ar. Thus, once again, the mathematical identity of the concepts of surface tension and surface free energy is observed. 

These equations are mathematically identical to longer forms such as... 

An alternative form of the fundamental property relation given by equation 60 is provided by the mathematical identity of equation 166 ... 

In Schrodinger s wave mechanics (which has been shown4 to be mathematically identical with Heisenberg s quantum mechanics), a conservative Newtonian dynamical system is represented by a wave function or amplitude function [/, which satisfies the partial differential equation... 

In this one-dimensional flat case the Laplace operator is simpler than in the case with spherical symmetry arising when deriving the Debye-Huckel limiting law. Therefore, the differential equation (B.5) can be solved without the simplification (of replacing the exponential factors by two terms of their series expansion) that would reduce its accuracy. We shall employ the mathematical identity... 

To solve this equation we employ the mathematical identity... 

If this is an explicit equation with respect to a the estimation of the vector k is mathematically identical to a differential analysis. The only difference is that values of ki are searched, for which the concentrations calculated from the above equation are as close as possible to the measured concentrations. Below, a simple example illustrating both techniques is given. 

For constant fluid density the design equations for plug flow and batch reactors are mathematically identical in form with the space time and the holding time playing comparable roles (see Chapter 8). Consequently it is necessary to consider only the batch reactor case. The pertinent rate equations were solved previously in Section 5.3.1.1 to give the following results. 

By rearranging equation 9-1, we can also express it as follows, wherein the fact that it is a mathematical identity becomes apparent ... 

We can also obtain these expressions from the energy-balance equation for the steady-state PFTR by simply transforming dzju dt with A,/ V replacing Pw/At. The solutions of these equations for the batch reactor are mathematically identical to those in the PFTR, although the physical interpretations are quite different. 

This minimization problem is mathematically identical to the solution of another equation system... 

If the osmotic pressure be a function of the hydrostatic pressure of the solution p and of the numerical concentration we have the mathematical identity... 

Mathematically, studies of diffusion often require solving a diffusion equation, which is a partial differential equation. The book of Crank (1975), The Mathematics of Diffusion, provides solutions to various diffusion problems. The book of Carslaw and Jaeger (1959), Conduction of Heat in Solids, provides solutions to various heat conduction problems. Because the heat conduction equation and the diffusion equation are mathematically identical, solutions to heat conduction problems can be adapted for diffusion problems. For even more complicated problems, including many geological problems, numerical solution using a computer is the only or best approach. The solutions are important and some will be discussed in detail, but the emphasis will be placed on the concepts, on how to transform a geological problem into a mathematical problem, how to study diffusion by experiments, and how to interpret experimental data. 

Equation (16) has appeared in the past in the literature [41, 42] and is mathematically identical to the Jarzynski equality [31]. We analyze this connection in Section III.C.l. 

This is the starting point for the four mathematical identities to be derived below. 

We now use the mathematical identity (1.13) to change the variable held constant from P to V ... 

Although these two equations are mathematically identical, eq. (48) should be used when 5 is dominated by the pseudo-contact contribution and eq. (49) should be used when S 313 is dominated by contact effects, thus maximizing the slopes of the resulting straight lines (Reuben and Elgavish, 1980). Plots of Sf /(Sz)j vs Cj/(Sz)j (eq. (48)) or 5 vs... 

Since equations (45a) and (45b) are mathematically identical, it follows that the solvent isotope-effect equation (50) applies to this case too. The implications of equation (50) are, however, different for the two mechanisms. The parameter i in equation (50) for the A-SE2 case can be obtained by an independent measurement. It corresponds to the fractionation factor of the proton (in the transition state) which becomes incorporated in the product. Provided that this hydrogen nucleus does not undergo ready isotope exchange after formation of the product, 1 can therefore be measured. It follows that the ratio of rates in H20 and D20 can be used to evaluate 2 since, according to equation (47b),... 

We first develop an alternative form of Eq. (10.2), just as was done in 6.2, where the fundamental property relation was restricted to phases of con composition. We make use of the same mathematical identity ... 

In algebra the equal sign stands between two algebraic expressions and indicates that two expressions are related by a reflexive, symmetric and transitive relation. The mathematical expressions on either side of the = sign are mathematically identical and interchangeable in equations. 

---