Let (x1,,xn) be Cartesian coordinatesMathworldPlanetmath for some open set Ω in n. Then the Laplacian differential operator Δ is defined as


In other words, if f is a twice differentiable function f:Ω, then


A coordinateMathworldPlanetmathPlanetmath independent definition of the Laplacian is Δ=, i.e., Δ is the compositionMathworldPlanetmath of gradientMathworldPlanetmath and codifferential.

A harmonic function is one for which the Laplacian vanishes.


An older symbol for the Laplacian is 2 – conceptually the scalar productMathworldPlanetmath of with itself. This form is more favoured by physicists.



Click here¡”¿ to see an article that derives the Laplacian in spherical coordinatesMathworldPlanetmath.

Title Laplacian
Canonical name Laplacian
Date of creation 2013-03-22 12:43:48
Last modified on 2013-03-22 12:43:48
Owner matte (1858)
Last modified by matte (1858)
Numerical id 18
Author matte (1858)
Entry type Definition
Classification msc 31B05
Classification msc 31B15
Related topic DAlembertian
Related topic Codifferential
Defines Laplace operator