Euler equations theorem

The set K in Theorem 1.11 may coincide with the space V. For a differentiable functional J it guarantees the solvability of the Euler equation... 

By either a direct integration in which Z is held constant, or by using Euler s theorem, we have accomplished the integration of equation (5.16), and are now prepared to understand the physical significance of the partial molar property. For a one-component system, Z = nZ, , where Zm is the molar property. Thus, Zm is the contribution to Z for a mole of substance, and the total Z is the molar Zm multiplied by the number of moles. For a two-component system, equation (5.17) gives... 

Most branches of theoretical science can be expounded at various levels of abstraction. The most elegant and formal approach to thermodynamics, that of Caratheodory [1], depends on a familiarity with a special type of differential equation (Pfaff equation) with which the usual student of chemistry is unacquainted. However, an introductory presentation of thermodynamics follows best along historical lines of development, for which only the elementary principles of calculus are necessary. We follow this approach here. Nevertheless, we also discuss exact differentials and Euler s theorem, because many concepts and derivations can be presented in a more satisfying and precise manner with their use. 

Extensive thermodynamic properties at constant temperature and pressure are homogeneous functions of degree 1 of the mole numbers. From Euler s theorem [Equation (2.33)] for a homogeneous function of degree n... 

The virtue of this theorem is that it reduces the dual problem to the question of solving the Euler equation PQ = 0, a second-order algebraic equation for the... 

Proof. By property R5 listed at the end of Section II, the elements of the Pauli subspace S are traceless, from which we infer by Theorems 12 and 13 that the energy problem and the spectral optimization problem have optimal solutions. By Theorem 10 these solutions are characterized by the Euler equation PQ = 0. ... 

In an infinite solid this set of critical points obeys a number of theorems, the chief being the Euler equation (eqn (14.1)) ... 

Equation (6.27) merely says that if the independent extensive arguments of U are multiplied by A [cf. (6.25b-d)], then U itself must be multiplied by the same factor [cf. (6.25a)]. [Mathematically, the property (6.27) identifies the internal energy function (6.26) as a homogeneous function of first order, and the consequence to be derived is merely a special case of what is called Euler s theorem for homogeneous functions in your college algebra textbook.]... 

The last member of Equation (2) shows that n, is the partial molar quantity associated with the Gibbs free energy, G. Euler s theorem gives then... 

Alternatively, equation 2.2-14 can be regarded as a result of Euler s theorem. A function f(xux2,...,xN) is said to be homogeneous of degree n if... 

These expressions are formally exact and the first equality in Eq. (123) comes from Euler s theorem stating that the AT potential u3(rn, r23) is a homogeneous function of order -9 of the variables r12, r13, and r23. Note that Eq. (123) is very convenient to realize the thermodynamic consistency of the integral equation, which is based on the equality between both expressions of the isothermal compressibility stemmed, respectively, from the virial pressure, It = 2 (dp/dE).,., and from the long-wavelength limit S 0) of the structure factor, %T = p[.S (0)/p]. The integral in Eq. (123) explicitly contains the tripledipole interaction and the triplet correlation function g (r12, r13, r23) that is unknown and, according to Kirkwood [86], has to be approximated by the superposition approximation, with the result... 

This helps indicate why groups of multiplicative type are important. But it should be said that solvability is definitely a necessary hypothesis. Let S for example be the group of all rotations of real 3-space. For g in S we have gtf — 1, so all complex eigenvalues of g have absolute value 1. The characteristic equation of g has odd degree and hence has at least one real root. Since det( ) = 1, it is easy to see that 1 is an eigenvalue. In other words, each rotation leaves a line fixed, and thus it is simply a rotation in the plane perpendicular to that axis (Euler s theorem). Each such rotation is clearly separable. But obviously the group is not commutative (and not solvable). Finally, since U is nilpotent, we have the following result. 

Plateau borders. Euler s theorem then relates the numbers of polygons (P), sides (S), and vertices (V) by P — 5 -f V = 2. This equation, combined with the observation that three sides meet at every vertex, and every side connects two vertices, so that 3V = 25, leads to the conclusion that for an infinite two-dimensional foam the average number of sides per polygon is six. 

This is Euler s theorem. Furthermore it follows from the theory of partial differential equations that conversely any function/(a , y, z. ..) which satisfies (1.12) is homogeneous of the mth degree in x, y, z. ... ... 

The TEC model developed by Teo also has been successfully applied to rationalize the geometries of a large number of cluster compounds. The TEC model combines Lauher s rule with Euler s theorem and adds an adjustable parameter This parameter X is equal to the number of electron pairs present in excess of that predicted by the 18-electron rule. " X has also been interpreted in terms of the number of missing antibonding orbitals. Given a value for X, determined by the shape of the cluster, an equation predicts the electron count for a cluster. Theoretical justification of the parameter X is based largely upon the classical molecular orbital calculations performed by Hoffmann and Lipscomb via the extended Hiickel method on the corresponding polyhedral boron hydride clusters The values... 

Euler s theorem states that the number of vertices v, edges e, and faces / of a simple convex polyhedron are related by the equation v + f = e + Z... 

Using Euler s theorem on homograeous functions, Watson [36] showed that the sum in this equation is equal to s/2 and, therefore,... 

This equation tells us that the interfacial free energy can be described (under our conditions of constant temperature and pressure) by the variables A and n, all of which are extensive. This feature allows us to invoke Euler s theorem, which yields... 

Prove this result by actual differentiation. It of course follows directly from Euler s theorem, since the equation is homogeneous and of the third degree. jc3 + x y + y3 "du... 

Euler s theorem for homogeneous equations yields G = in-,n whereby... 

Since Zi is homogeneous in the zeroth degree in ni, n2,..., etc., Euler s theorem (Chapter 2) gives the Gibbs-Duhem equation ... 

Equation (14.17) is also the result of Euler s Theorem applied to G as a function ... 

Equation (A-29) holds for all values of A. For A = 1, Eq. (A-29) reduces to Eq. (A-23) and we have completed the proof of Euler s theorem. Equation (A-23) is of use in relating extensive thermodynamic properties to the corresponding partial molal properties. 

Equation fQ.iol is a special case of Euler s theorem. If a mathematical function satisfies the condition fax, ay) = afx, y),... 

Equation (48) results from Euler s theorem, since Zp j is a homogeneous function of degree 1 it entails the relations... 

But we don t have to use Euler s theorem. We can simply expand our definition of G, which so far is restricted to closed (constant composition) systems. If we exclude chemical work, which means we deal only with systems at complete stable equilibrium, we know from Equation (4.65)... 

---