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In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties.This is often written as = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.
Laplace obtained these equations by simplifying the fluid dynamic equations. But they can also be derived from energy integrals via Lagrange's equation . For a fluid sheet of average thickness D , the vertical tidal elevation ζ , as well as the horizontal velocity components u and v (in the latitude φ and longitude λ directions, respectively ...
In mathematics and mathematical physics, potential theory is the study of harmonic functions.. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the gravitational potential and electrostatic potential, both of which ...
The slight changes that result from the terms in the equations – which themselves may have been simplified yet again – are used as corrections to the original solution. Because simplifications are made at every step, the corrections are never perfect, but even one cycle of corrections often provides a remarkably better approximate solution ...
Laplace obtained these equations by simplifying the fluid dynamics equations, but they can also be derived from energy integrals via Lagrange's equation. For a fluid sheet of average thickness D , the vertical tidal elevation ζ , as well as the horizontal velocity components u and v (in the latitude φ and longitude λ directions, respectively ...
In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions .There are two kinds: the regular solid harmonics (), which are well-defined at the origin and the irregular solid harmonics (), which are singular at the origin.
As is the case with spherical coordinates, Laplace's equation may be solved by the method of separation of variables to yield solutions in the form of prolate spheroidal harmonics, which are convenient to use when boundary conditions are defined on a surface with a constant prolate spheroidal coordinate (See Smythe, 1968).
If one were asked to name the two most important works in the progress of mathematics and physics, the answer would undoubtedly be, the Principia of Newton and the Mécanique Céleste of Laplace. In their historical and philosophical aspects these works easily outrank all others, and furnish thus the standard by which all others must be measured.