Uniqueness theorem field equations

At the same time it is worth to notice that in modern numerical methods of a solution of boundary value problems, based on replacement of differential equations by finite difference, these steps are performed simultaneously. In accordance with the theorem of uniqueness, the field inside the volume V is defined by a distribution of masses inside this volume and boundary conditions, and correspondingly it is natural to derive an equation establishing this link. With this purpose in mind we will again proceed from Gauss s theorem,... 

Note that the solution of equation (10.62) and, correspondingly, equation (10.65), can be nonunique in a general case, since there may exist m (r) distributions that produce a zero external field, similar to the existence of the current distributions producing zero external field demonstrated in the beginning of this chapter. However, according to the uniqueness theorem by Gusarov (1981, see Chapter 1), the excess conductivity Aa (r) can be found uniquely in the class of piecewise-analytic functions. Therefore,... 

Ions are attracted to metal or other conductors that have no charge on them. The attraction arises from a process of induction. An ion in the neighborhood of a conducting surface induces a field. That field attracts the ion to the surface. Induction is readily modelled by recognizing an extraordinary mathematical property, the uniqueness of differential equations such as Poisson s equation. According to the uniqueness theorem, if you can find any solution to Poisson s equation that satisfies the boundary conditions for the problem of interest, you have found the only solution, even if you found it by wild guesses or clever tricks. 

The manifold M- a has locally stable and unstable manifolds that are close to the unperturbed locally stable and unstable manifolds and if these manifolds intersect transversely, then the Smale-Birkhoff theorem predicts the existence of horseshoes and their chaotic dynamics in the perturbed dissipative system. A 2D hyperbolic invariant torus Tja(Pi, P2) may be located on by averaging the perturbed dissipative vector field 7 > 0 and a > 0 restricted to M q, over the angular variables Qi and Q2- The averaged equations have a unique stable hyperbolic fixed point (Pi,P2) = (0,0) with two negative eigenvalues provided that the... 

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