# Dictionary:Laplace’s equation

(la plas’) A differential equation that describes field behavior in free space. The Laplacian ${\displaystyle {\nabla }^{2}U}$ of a potential function U vanishes in space that contains neither sources nor sinks. (${\displaystyle \nabla }$ is the operator ‘‘del’’.) In rectangular coordinates,
${\displaystyle {\nabla }^{2}U={\frac {{\partial }^{2}{U}}{\partial {x}^{2}}}+{\frac {{\partial }^{2}{U}}{\partial {y}^{2}}}+{\frac {{\partial }^{2}{U}}{\partial {z}^{2}}}=0}$