Relations between Differentials

The Frechet differential of a functional is also the Gateaux differential. In turn, the Gateaux differential of a functional is also the variation. Thus, for a functional, the existence of the Frechet differential implies the existence of the Gateaux differential. In turn, the existence of the Gateaux differential implies the existence of the variation. However, there is no guarantee that the reverse relations hold. For example, a functional may have the variation but not the Gateaux differential. Using conditional statements, these relations are 

In the previous example, does the Gateaux differential of / exist at yo = 0 The Gateaux differential of I does not exist at yo = 0 since 

---

Figure 13.13 Compensation relation between differential heat and differential entropy of adsorption.

Fig. 3. The relation between differential G-values of crosslinking in polystyrene resist films and stopping powers. The G-values for gamma-ray irradiation ( ) are from Ref. 34. The typical error is shown for 1.0 MeV H+ beams. From Ref. 3...

Figure 2.3.a-2 Relation between differential and integral methods of kinetic analysis and differential and integral reactors. 

In general, for a function of state /, that is completely determined by variables X and y, df = A dx + B dy. Cross-differentiation in df gives (dA/dy = (dBtdx)y, known as a Maxwell relation. Similarly, cross-differentiation in dU, dH, dF, and dG yields a wide variety of Maxwell relations between differential quotients. For instance, by cross-differentiation in dG we find... 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

All of the described differential viscoelastic constitutive equations are implicit relations between the extra stress and the rate of deformation tensors. Therefore, unlike the generalized Newtonian flows, these equations cannot be used to eliminate the extra stress in the equation of motion and should be solved simultaneously with the governing flow equations. 

The natural laws in any scientific or technological field are not regarded as precise and definitive until they have been expressed in mathematical form. Such a form, often an equation, is a relation between the quantity of interest, say, product yield, and independent variables such as time and temperature upon which yield depends. When it happens that this equation involves, besides the function itself, one or more of its derivatives it is called a differential equation. 

A relation between the variables, involving no derivatives, is called a solution of the differential equation if this relation, when substituted in the equation, satisfies the equation. A solution of an ordinaiy differential equation which includes the maximum possible number of arbitrary constants is called the general solution. The maximum number of arbitrai y constants is exactly equal to the order of the dif-... 

A solution of a difference equation is a relation between the variables which satisfies the equation. If the difference equation is of order n, the general solution involves n arbitraty constants. The techniques for solving difference equations resemble techniques used for differential equations. 

Differentiating (5.24) with the use of the ehain rule and using (5.21) provides a relation between the normals to the elastie limit surfaee in strain space and the elastie limit surfaee in stress spaee, respeetively,... 

It is also possible to find a relation between the stretching tensor d and the rate of spatial strain e. Differentiating (A. 19)... 

Of the five magnitudes p, v, , s, u, any two may be chosen for the independent variables x and y, and for each pair we shall, by mean s of (11), obtain a relation between the differential coefficients of two other magnitudes with respect to the chosen variables. These are deduced as follows ... 

It must repeatedly have been remarked, however, that these equations are not in themselves sufficient to lead to a complete solution of the problems to which they have been applied. This arises from the fact that they are differential equations, in the solution of which there always appear arbitrary constants of integration (H. M., 73,101, 121). Thus, the relation between the pressure of a saturated vapour and the temperature is expressed by the differential equation of Clausius ( 80) ... 

The relation between the spatial distribution of the electrostatic potential /(jc) and the spatial distribution of charge density Qy(x) can be stated, generally, in terms of Poisson s differential equation. 

Differential equations are usually classified as ordinary or partial . In the former case only one independent variable is involved and its differential is exact. Thus there is a relation between the dependent variable, say y(x), its various derivatives, as well as functions of the independent variable x. Partial differential equations contain several independent variables, and hence partial derivatives. 

Differential methods based on differentiation of experimental concentration versus time data in order to obtain the actual rate of reaction. In these approaches one analyzes the data by postulating various functional relations between the rate of reaction and the concentrations of the various species in the reaction mixture and tests these hypotheses using appropriate plots. 

Data obtained in continuous stirred tank reactors have the merits of isothermicity and of an algebraic relation between the variables rather than a differential one. At steady state In a CSTR the material balance on a reactant A is... 

---