Euler s relationship

The connectivity h of a polyhedral fragment is one greater than the number of holes and satisfies the modified Euler s relationship v+t-e=3-h. 

Using Euler s relationship where j is the imaginary number ... 

There are exceptions to this simple equation that occur infrequentiy but nevertheless must be considered. A more complete relationship for the number of exchangers, E, in a network is obtained by applying Euler s network relation from graph theory (6) ... 

Equation (3-159) is the basic relationship of this method. Several techniques have been developed for the estimation of AylAx. The simplest of these, known as Euler s method, is to evaluate Ay/Ax at jcq. From Eq. (3-156), this gives (A v/Ajc)o = kyoi which, used in Eq. (3-159), yields... 

A relationship, known as Euler s formula, exists between a complex number [x + jy] (x is the real part, y is the imaginary part of the complex number (j = P )) and a sine and cosine function. Many authors and textbooks prefer the complex number notation for its compactness and convenience. By substituting the Euler equations cos(r) = d + e -")/2 and sin(r) = (d - e t )l2j in eq. (40.1), a compact complex number notation for the Fourier transform is obtained as follows ... 

In connection with the development of the thermodynamic concept of partial molar quantities, it is desirable to be familiar with a mathematical relationship known as Euler s theorem. As this theorem is stated with reference to homogeneous functions, we will consider briefly the namre of these functions. 

Let us now apply Euler s explicit and implicit algorithms with a constant step size h. They are defined by the relationships (in vector notation)... 

In analysing polyhedra and the relationships between them, a useful formula is Euler s relation... 

This relationship is known as Euler s Theorem for Homogeneous Functions of Degree One. However, in addition to the dependence on the x,- the function F may also display a dependence on parameters such as pressure P or temperature T that, of course, remains unaffected by the above manipulations. 

This is Euler s equation, ft contains, in the special case of one-dimensional steady-state flow, the relationship... 

Thus there is a unique relationship between E and E that depends only on po and Ap. The boundary condition associated with Equation 2.37 is E = 1 at E= 1. An analytical solntion is possible, bnt nnmerical integration of the ODE is easier. Euler s method works, but note that the independent variable, E, starts at 1.0 and is decreased in small steps until the desired final value is reached. This means that the integration step, AE, in Euler s method is negative. Once E has been determined, k f can be found from Equation 2.35. The following table shows values for the case of Poo = 1040 and po = 900 ... 

Thus, when the dQ /d/ -, are evaluated at the s assumed to make trial n, Euler s theorem22 gives the following relationship (see App. A). 

Here,/, v, and e are the numbers of the faces, vertices, and edges, respectively, and/ is equal to p+ h. Since each edge shares two polygons, we have the relationship 2e = 5p+ 6h. We also have the relationship 3v = 5p+ 6h because each vertex shares three polygons. Thus, we obtain the relationshipp = 6(/+ v — e), which givesp = 12, and V = 20+ 2h from Euler s rule. Namely, p must always be 12, and fullerenes must have even numbers of carbon atoms. If = 0, u is 20. This is the smallest fullerene... 

In the relationship above we have used Euler s identity (see Section 7.2) to express sin ((onT) ... 

Studies of aqueous solutes traditionally use molalities, which implies use of still another standard state. To develop this subject we will resort to using Euler s theorem, which makes the derivation less than completely intuitive, but basically we are just looking for the relationship between p and m, the solute molality, in the dilute solution region where Henry s Law is obeyed. 

This is an example for Euler s chain relation. Euler s chain relationship can be derived more formally with Jacobian determinants, as shown in Example 1.23. Rewriting Eq. (1.19) without arguments results in... 

Equation (1.80) is Euler s chain relationship. The relationship can be resolved into... 

This relationship is essentially Euler s chain relation [Funher information 2.2]. 

Fig. 2.2 Euler s method applied to the trimer. Left, the solution trajectories for different step-sizes right the maximum global errors in position projections of solutions computed using four different time-steps appear to show an asymptotic linear relationship to step-size, when plotted in log-log scale...

However, all microstructures satisfy a topological relationship represented by Euler s law which relates various dimensions in a geometrical figure. Let C (corner), E (edge), P (polygon) and B (body) be the numbers of... 

Naturally, it is not essential to understand or to use Euler s theorem in learning or using thermodynamics. But some people find these more mathematical relationships satisfying and illuminating, while others wonder why anyone would bother. 

It is worth deriving a last interesting relationship. From Equations 9.108 and 9.109 the Euler s formula can be directly deduced ... 

This relationship of the impedance as a complex function, shown in Equation 8.8, is often the reason that MXC researchers sometimes look at EIS as a complicated, mysterious technique. We believe that this may be a contributing factor to the sparse and incomplete application of EIS in MXC studies. The principles of EIS have foundations in basic mathematics of complex numbers. As we apply sinusoidal amplitude on voltage, both the voltage and the current with time have to be represented as a sine function, and it is through Euler s formula that these and the resulting impedance can be represented as complex functions. 

We now derive the Maxwell relations. These are relationships between partial derivatives that follow from Euler s reciprocal relation. Equation (5.39) (page 75). 

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