Euler relations

By expanding the Helmholtz free energy F at constant T in an arithmetic series in terms of ujk, we see that the linear terms vanish in view of the equilibrium condition (homogeneous functions of second order, F is given as... 

Scalar counting relations for sets of structural components can seen as expressions for characters under the identity operation of more general relations between representations of those sets. For example, the Euler relation in topology can be generalised to connect not only the numbers of edges, vertices and faces of a polyhedron, but also various symmetries associated with the structural features. The well-known Euler theorem... 

Equation (1.107) is more useful if it is integrated with the Pfaffian form however, this is not a straightforward step, since intensive properties are functions of all the independent variables of the system. The Euler relation for... 

Each phase is a simple system, and we may write the appropriate Euler relations... 

Graphite is of course the classical example here, though there are many other possibilities. The Euler relation of the preceding subsection continues to apply for 2D... 

X = x(K) per site and demand that it be strictly negative (even for the infinite graph limit). The Euler relation then devolves ° to... 

The last identity in Eq. (5.22) arises from the Euler relation of homogeneous functions of first order U = TS - pV + iJ,n and the definition of the enthalpy as... 

We note that using Euler relation (3.17) we can write the mass balance (3.63) as... 

It is clear that analogues of (3.12), denoted Ja, are valid for each constituent as well as that of Euler relation (3.17) and Reynolds theorem (3.24) with a instead of the dot in the original equations, cf. (d) in Rem. 3, but material volume (containing the same particles of constituent a) is not as important as for the pure substance. 

In the derivation of the relations between invariants of polyhexes the Euler relation is frequently invoked (se, e.g. HaU 1988). In our notation it reads... 

Substituting these definitions into the Euler relation gives dp ... 

Fourth, based on the Euler relation, set the cross-derivatives equal. Set the derivative of S (from the first term on the righthand side of Equation (9.8)) with respect to p (from the second term) equal to the derivative of V (from the second term) with respect to T (from the first term). -S in the first term on the right hand side of Equation (9.8) is -S = (BG/dT). Take its derivative with respect to p, to get... 

According to the Euler relation, the two second derivatives must be equal, so... 

By evaluating the last two functional derivatives under the integrals of (3.413) with the aid of the total spin energy and by using the Euler relations (3. 412), one can re-rewrite the last total differential as ... 

Using the Euler relation e = cos 0 -F sin 0, we obtain the complex form of the Fourier series... 

While the evaluation of pi can be simphfied using Euler relations for certain classes of monomer structures, these Euler relations [50] do not apply, for example, to the three polyolefin chains depicted in Fig. lb because they have short side chains. (Of the monomer structures in Fig. la, the Euler relations for pi are valid only for PEE and PHI.) Therefore, the geometrical index pi = /Mi = nPVsi is evaluated by directly enumerating all sets of three sequential bonds (nP ) that traverse a monomer of species i. [46] siunmarizes the details of these calculations and tabulates values of p,- for several monomer structures, so we pass now to a consideration of how thermodynamic properties depend on n and p, in the high molecular weight, incompressible system limit of the LCT. 

The absence of a Euler relation for for arbitrary monomer structures renders the calculation of y K and y2 much more involved. The coefficients... 

The final calculation uses the recurrent Gamma-Euler relations, see Eq. (2.183a) ... 

Wells, A.F. The Geometrical Basis of Crystal Chemistry Part 7, Wells, A.F. Sharpe, R.R. ActaCryst, 963,16, 857. This paper reports on a modification of the Euler relation for application to 3-dimensional polyhedra. Such polyhedra are infinite in extent and are represented by the Schlaefli symbols (3, p) where p is greater than 6, and other Schlaefli symbols, some of which are identical to those of the ordinary 3-dimensional nets. 

Bucknum [1] in work first described in 1997, outlined a general scheme for the systematic classification and mapping of the polyhedra, 2-dimen-sional tessellations and 3-dimensional networics in a self-consistent topological space for these structures. This general scheme begins with a consideration of the Euler relation [2] for the polyhedra, shown as Eq. (1), which was first proposed in 1758 to the Russian Academy by Euler, and was, in fact, the point of departure for Euler into a new area of mathematics thereafter known explicitly as topology. 

---