Conformal maps finding

In some cases, analytic solutions exist. Powerful techniques to find them are conformal mapping and separation of the variables. These solutions are also very important to check numerical methods. 

To find primary current distributions, a lot of classical analytical tools are available (superposition, conformal mapping, method of images,. .) [ 15]. Once the geometry is too complex or when non-linear overpotentials are considered, approximate methods must be used. 

This is Laplace s equation in rectangular coordinates. If suitable boundary conditions exist or are known, Eq. (3.9-9) can be solved to give (x, y). Then the velocity at any point can be obtained using Eq. (3.9-5). Techniques for solving this equation include using numerical analysis, conformal mapping, and functions of a complex variable and are given elsewhere (B2, S3). Euler s equations can then be used to find the pressure distribution. 

Steric information is most easily summarised in a conformational energy map. Such calculations have been made by a number of workers but they require semi-empirical potentials or the effective van der Waals radii which are subject to error and dispute. We have carried out calculations of this sort for polyethylene units using different potentials and find that although the relative energies of the minima vary considerably, their locations are not particularly sensitive to the different potentials. Thus the conformations for which the scattering will be calculated may be restricted to those of low energy without loss of generality. 

The concept of conformational Probability Density Map often is improperly associated with that of potential energy map. The potential energy map indeed gives a relatively good image of the probability density map of finding a molecule in a given conformation [53,54]. To determine the conformational chemical physic properties of a molecule, as a function of temperature, statistical methods have to be used. 

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