Proof of Equations

The bulk modulus is appropriate for situations where the material is subjected to hydrostatic stresses. TTie proof of equations (2.15) and (2.16) is given by Benham et al. 

When derivations or proofs of equations are called for, start from fundamental definitions and principles. 

Part of the proof of equation 13 acknowledges this is not so for a Hartree Product. To remedy this, first consider a two-electron system, such as helium. Two equivalent Hartree Product wavefunctions for this system are... 

We will illustrate the proof of equation (13.5) as an example of the application of conventional thermodynamics to this problem. Let C be the heat capacity of a magnetic system at constant field strength, M be the magnetization, and Xt = (dM/dW)T be the magnetic susceptibility. From Table 13.2 we can see that just below the critical temperature ... 

This is not seen easily in considering the defining Equations 6.22 and 6.23, so a mathematical proof of Equation 6.29 is given in Appendix A. Hence the modified Thiele modulus 0m now becomes... 

In view of these uncertainties, it is fortunate that an independent proof of Equation (5.28) exists in solid-state physics.Equation (5.29), the Fermi distribution function, is used to count the number of electrons in the valence band and in the conduction band. The result is that... 

Figure 4 Geometric proof of equation (39) for the rotation about the axis a, a = 1... 

This section deals with the proof of equation (11.32). It is based on the identity... 

Let us scetch the proof of Equation (99) in the case of a smooth electrostatic potential. 

The proof of Equations (5.49) and (5.50) can be found in the references [54,56]. Note, incidentally, that other techniques for measuring the MMD, such as MALDI (matrix-assisted laser desorption ionization mass spectrometry) will have a different relation between the signal and the MMD. Note also that the assumption that the GPC signal is proportitHial to the mass of polymer depends on what sort of detection is used for example, this will not hold if the detector is sensitive just to the end-group of a polymer chain. 

The proof of equation (8 1) is easily given on the basis of considerations similar to those in section 5. Taking the incident X-rays again as propagated in the 2-direction and as Hnearly polarised in the o -direction, and letting the nucleus of the first atom coincide with the origin A (see fig. 7), w e may replace this atom in its coherent scattering effect, as we have seen there, by... 

Upon this identity, which relates the material derivative of the mean value (10.209) to the mean value of the material derivatives, all our proofs of equations of balance in a heterogeneous medium are founded. 

---