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We find solutions of Laplace's equation with special boundary conditions, using a general curvilinear system of coordinates. We call this purely geometrical solutions Basic Harmonic Functions (BHF's). From them we obtain more general solutions with arbitrary constant values on the boundaries. Further, the BHF's are used to obtain the capacitance of many electrostatic configurations of conductors....
We prove that the matrix of capacitance in electrostatics is a positive-singular matrix with a non-degenerate null eigenvalue. We explore the physical implications of this fact, and study the physical meaning of the eigenvalue problem for such a matrix. Many properties are easily visualized by constructing a “potential space” isomorphic to the euclidean space. The problem of minimizing the internal...
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