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##### From Applied Electromagnetics Using QuickField and MATLAB
 In This ChapterSeparation of Variables Conformal MappingFinite Difference MethodsMonte Carlo MethodsFinite Element Method
 Cartesian CoordinatesIn Cartesian coordinates, Laplace’s equation S(x, y, z) = 0 is written aswhere −∞ < x < ∞, −∞ < y < ∞, −∞ < z < ∞ . Applying separation of variables we assume a product solutionSubstituting this form of S into Laplace's equation and dividing by S givesEach term in the above equation must be equal to a constant if the sum is zero for all x, y, and z, since these variables may vary independentlywhere α2 + β2 = γ2 . The solutions to these differential equations may be written as linear combinations of sine, cosine, and exponential functions so that we may construct a product solutionwith constants determined by the boundary conditions.As an example of the application of separation of variables, consider a unit square region with boundary condition S = 0 on three sides and S = 1 on the other side. Since there is no z dependence α2 + β2 equation (2.5) becomeApplying the boundary conditions S(0,y) =...

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This appendix is based on symbols.tex version 3.2 by David Carlisle, available from the CTAN Archives. Table B.1: Accents. \'{o} \~{o} ? \v{o}...

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