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A partial differential equation that governs potential fields (in regions where there are no sources) and is equivalent, in three dimensions, to the inverse square law of gravitational or electrical attraction. In Cartesian coordinates, the Laplace equation equates the sum of the second partial (spatial) derivatives of the field to zero. (When a source is present, this sum is equal to the strength of the source and the resulting equation is called Poisson's equation). The differential equation is named for French mathematician Pierre-Simon de Laplace (1749 to 1827), and applies to electrical, gravity and magnetic fields.
∇2u = ∂2u/∂x2 + ∂2u/∂y2 + ∂2u/∂z2 = 0,
where u(x,y,z) is a potential function.
See related terms: potential field, spherical harmonic
