In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable.
Informally, the Laplacian Δf (p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f (p).
The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that density distribution.
Solutions of Laplace's equation Δf = 0 are called harmonic functions and represent the possible gravitational potentials in regions of vacuum.
The Laplacian occurs in many differential equations describing physical phenomena.
In image processing and computer vision, the Laplacian operator has been used for various tasks, such as blob and edge detection.
The Laplacian is the simplest elliptic operator and is at the core of Hodge theory as well as the results of de Rham cohomology.
is the real-valued function defined by: where the latter notations derive from formally writing:
Explicitly, the Laplacian of f is thus the sum of all the unmixed second partial derivatives in the Cartesian coordinates xi: As a second-order differential operator, the Laplace operator maps Ck functions to Ck−2 functions for k ≥ 2.
[1] In the physical theory of diffusion, the Laplace operator arises naturally in the mathematical description of equilibrium.
[2] Specifically, if u is the density at equilibrium of some quantity such as a chemical concentration, then the net flux of u through the boundary ∂V (also called S) of any smooth region V is zero, provided there is no source or sink within V:
Solutions of the Laplace equation, i.e. functions whose Laplacian is identically zero, thus represent possible equilibrium densities under diffusion.
The Laplace operator itself has a physical interpretation for non-equilibrium diffusion as the extent to which a point represents a source or sink of chemical concentration, in a sense made precise by the diffusion equation.
The same approach implies that the negative of the Laplacian of the gravitational potential is the mass distribution.
Finding the potential function subject to suitable boundary conditions is equivalent to solving Poisson's equation.
In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems.
represents the radial distance, φ the azimuth angle and z the height.
where summation over the repeated indices is implied, gmn is the inverse metric tensor and Γl mn are the Christoffel symbols for the selected coordinates.
In arbitrary curvilinear coordinates in N dimensions (ξ1, ..., ξN), we can write the Laplacian in terms of the inverse metric tensor,
In spherical coordinates in N dimensions, with the parametrization x = rθ ∈ RN with r representing a positive real radius and θ an element of the unit sphere SN−1,
The Laplacian is invariant under all Euclidean transformations: rotations and translations.
(More generally, this remains true when ρ is an orthogonal transformation such as a reflection.)
The spectrum of the Laplace operator consists of all eigenvalues λ for which there is a corresponding eigenfunction f with:
just on the left of each vector field component is the (scalar) Laplace operator.
This can be seen to be a special case of Lagrange's formula; see Vector triple product.
An example of the usage of the vector Laplacian is the Navier-Stokes equations for a Newtonian incompressible flow:
Another generalization of the Laplace operator that is available on pseudo-Riemannian manifolds uses the exterior derivative, in terms of which the "geometer's Laplacian" is expressed as
Here δ is the codifferential, which can also be expressed in terms of the Hodge star and the exterior derivative.
The Laplacian can be generalized in certain ways to non-Euclidean spaces, where it may be elliptic, hyperbolic, or ultrahyperbolic.
Indeed, theoretical physicists usually work in units such that c = 1 in order to simplify the equation.