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Revised distributional forms of the Laplacian and Poisson's equation, their implications, and all related generalizations
The theory of distributions of L. Schwartz is a very useful and convenient way for the analysis of physical problems since physical distributions, especially charge distributions yielding the discontinuity of the potential and boundary conditions, can be correctly described in terms of mathematical distributions. To obtain the charge distributions, the distributional form of the Laplacian is applied to the Poisson's equation; therefore, for the correct representations and interpretations, the distributional forms and their proper applications are very important. In this article, it is shown that the distributional form of the Laplacian has been presented by Schwartz and also others with a missing term, leading to confusing and wrong results mathematically, and as a result electromagnetically; and the revised, correct, and complete distributional representations of the Laplace operator, the Poisson equation, and double layers, defined as the dipole layer and equidensity layer, are obtained and presented with detailed discussions and explanations including boundary conditions. By using the revised form of the Laplacian, Green's theorem is obtained explicitly with special emphases about important points and differences with previous works. The generalized forms of the Laplacian, Poisson's equation, charge densities, boundary conditions, and Greens theorem are also presented when there is a multi-layer on the surface of discontinuity.