Abstract
The partial differential equation which is identified with the name of Pierre Simon Marquis de Laplace (1749–1827) is one of the most important equations in mathematics which has wide applications to a number of topics relevant to mathematical physics and engineering. In the area of mathematical physics, Laplace’s equation can be used to formulate the general class of problems associated with the theory of gravitation, electrostatics, dielectrics and magnetostatics. In all these applications, the associated fields can be expressed as a gradient of a potential. For example, in the theory of gravitation, the force of attraction associated with matter is expressed as the gradient of a gravitational potential; in electrostatics, the electric vector is expressed as the gradient of an electrostatic potential; in magnetostatics, the magnetic vector is expressed in terms of a magnetostatic potential; in electricity, the conduction current vector is derived from the gradient of a potential function. This ability to express fundamental vector fields of interest to particular areas is an underlying theme in all applications which culminate in the development of Laplace’s equation.
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