Dictionary:Laplace’s equation

{{#category_index:L|Laplace’s equation}} (la plas’) A differential equation that describes field behavior in free space. The Laplacian ${\nabla }^{2}U$ of a potential function U vanishes in space that contains neither sources nor sinks. ($\nabla$ is the operator ‘‘del’’.) In rectangular coordinates,

${\nabla }^{2}U={\frac {{\partial }^{2}{U}}{\partial {x}^{2}}}+{\frac {{\partial }^{2}{U}}{\partial {y}^{2}}}+{\frac {{\partial }^{2}{U}}{\partial {z}^{2}}}=0$

Gravity, magnetic, electrical, electromagnetic fields obey Laplace's equation in free space (where there are no sources). See Figure C-14 for the Laplacian in cylindrical and spherical coordinates. Compare Poisson’s equation. Named for Pierre Simon Laplace (1749–1827), French mathematician.