Laplacian

The Laplacian for a scalar function is a scalar differential operator. In an orthogonal coordinate system it is defined by

(1)

where the are the scale factors of the coordinate system (Weinberg 1972, p. 109; Arfken 1985, p. 92).

Note that the operator is commonly written as by mathematicians (Krantz 1999, p. 16).

The operator is often used in spectral problems. On a nonempty bounded region with smooth boundary, this sign convention gives positive eigenvalues for Dirichlet boundary conditions and nonnegative eigenvalues for Neumann boundary conditions. The latter include , corresponding to constant eigenfunctions. Weyl's law describes the asymptotic growth of these eigenvalues, while the heat trace encodes geometric information such as volume and boundary measure (Vassilevich 2003, Ivrii 2016).

The Laplacian is extremely important in mechanics, electromagnetics, wave theory, and quantum mechanics, and appears in Laplace's equation

(2)

the Helmholtz differential equation

(3)

the wave equation

(4)

and the Schrödinger equation

(5)

The analogous operator obtained by generalizing from three dimensions to four-dimensional spacetime is denoted and is known as the d'Alembertian. A version of the Laplacian that operates on vector functions is known as the vector Laplacian, and a tensor Laplacian can be similarly defined. The square of the Laplacian is known as the biharmonic operator.

The following table gives the form of the Laplacian in several common coordinate systems.

coordinate system
Cartesian coordinates
cylindrical coordinates
parabolic coordinates
parabolic cylindrical coordinates
spherical coordinates

For a C⁴ function , the finite difference approximation for small is

(6)

For a pure radial function ,

			(7)
			(8)
			(9)

Using the vector derivative identity

(10)

so

			(11)
			(12)
			(13)

Therefore, for a radial power law,

			(14)
			(15)
			(16)

For a real matrix and a row vector with , an identity satisfied by the Laplacian with respect to is

(17)

where is the Hilbert-Schmidt norm and is the transpose of . The vector norms are Euclidean norms.

For the inverse distance function in , where , the classical Laplacian vanishes away from . On a sphere of radius centered at , the outward flux is

(18)

The singularity at prevents applying the classical divergence theorem across that point. In the sense of generalized functions,

(19)

where is the three-dimensional delta function (Hunter 2014, pp. 33-34).

On a bounded domain, the eigenvalues of with zero boundary values are its Dirichlet eigenvalues. Their multiplicities depend on the geometry of the domain, with sharp restrictions for convex domains and arbitrary finite multiplicity patterns on smooth star-shaped domains in dimensions at least three (Chen et al. 2026).