Laplace operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols, , or. 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. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian of a function at a point measures by how much the average value of over small spheres or balls centered at deviates from.
The Laplace operator is named after the French mathematician Pierre-Simon de Laplace, 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 are called harmonic functions and represent the possible gravitational potentials in regions of vacuum.
The Laplacian occurs in many differential equations describing physical phenomena. Poisson's equation describes electric and gravitational potentials; the diffusion equation describes heat and fluid flow; the wave equation describes wave propagation; and the Schrödinger equation describes the wave function in quantum mechanics. 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.

Definition

The Laplace operator is a second-order differential operator in the n-dimensional Euclidean space, defined as the divergence of the gradient. Thus if is a twice-differentiable real-valued function, then the Laplacian of is the real-valued function defined by:
where the latter notations derive from formally writing:
Explicitly, the Laplacian of is thus the sum of all the unmixed second partial derivatives in the Cartesian coordinates :
As a second-order differential operator, the Laplace operator maps functions to functions for. It is a linear operator, or more generally, an operator for any open set.
Alternatively, the Laplace operator can be defined as:
where is the dimension of the space, is the average value of on the surface of an n-sphere of radius, is the surface integral over an -sphere of radius, and is the hypervolume of the boundary of a unit -sphere.

Analytic and geometric Laplacians

There are two conflicting conventions as to how the Laplace operator is defined:

The "analytic" Laplacian, which could be characterized in as which is negative-definite in the sense that for any smooth compactly supported function that is not identically zero);
The "geometric", positive-definite Laplacian defined by
Motivation

Diffusion

In the physical theory of diffusion, the Laplace operator arises naturally in the mathematical description of equilibrium. Specifically, if is the density at equilibrium of some quantity such as a chemical concentration, then the net flux of through the boundary of any smooth region is zero, provided there is no source or sink within :
where is the outward unit normal to the boundary of. By the divergence theorem,
Since this holds for all smooth regions, one can show that it implies:
The left-hand side of this equation is the Laplace operator, and the entire equation is known as Laplace's equation. 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. This interpretation of the Laplacian is also explained by the following fact about averages.

Averages

Given a twice continuously differentiable function and a point, the average value of over the ball with radius centered at is:
Similarly, the average value of over the sphere with radius centered at is:

Density associated with a potential

If denotes the electrostatic potential associated to a charge distribution, then the charge distribution itself is given by the negative of the Laplacian of :
where is the electric constant.
This is a consequence of Gauss's law. Indeed, if is any smooth region with boundary, then by Gauss's law the flux of the electrostatic field across the boundary is proportional to the charge enclosed:
where the first equality is due to the divergence theorem. Since the electrostatic field is the gradient of the potential, this gives:
Since this holds for all regions, we must have
The same approach implies that the negative of the Laplacian of the gravitational potential is the mass distribution. Often the charge distribution are given, and the associated potential is unknown. Finding the potential function subject to suitable boundary conditions is equivalent to solving Poisson's equation.

Energy minimization

Another motivation for the Laplacian appearing in physics is that solutions to in a region are functions that make the Dirichlet energy functional stationary:
To see this, suppose is a function, and is a function that vanishes on the boundary of. Then:
where the last equality follows using Green's first identity. This calculation shows that if, then is stationary around. Conversely, if is stationary around, then by the fundamental lemma of calculus of variations.

Coordinate expressions

Two dimensions

The Laplace operator in two dimensions is given by:
In Cartesian coordinates,
where and are the standard Cartesian coordinates of the -plane.
In polar coordinates,
where represents the radial distance and the angle.

Three dimensions

In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems.
In Cartesian coordinates,
In cylindrical coordinates,
where represents the radial distance, the azimuth angle and the height.
In spherical coordinates:
or
by expanding the first and second term, these expressions read
where represents the azimuthal angle and the zenith angle or co-latitude. In particular, the above is equivalent to
where is the Laplace-Beltrami operator on the unit sphere.
In general curvilinear coordinates :
where summation over the repeated indices is implied,
is the inverse metric tensor and are the Christoffel symbols for the selected coordinates.

''N'' dimensions

In arbitrary curvilinear coordinates in dimensions, we can write the Laplacian in terms of the inverse metric tensor, :
from the -Weyl formula for the divergence.
In spherical coordinates in dimensions, with the parametrization with representing a positive real radius and an element of the unit sphere,
where is the Laplace–Beltrami operator on the -sphere, known as the spherical Laplacian. The two radial derivative terms can be equivalently rewritten as:
As a consequence, the spherical Laplacian of a function defined on can be computed as the ordinary Laplacian of the function extended to so that it is constant along rays, i.e., homogeneous of degree zero.

Euclidean invariance

The Laplacian is invariant under all Euclidean transformations: rotations and translations. In two dimensions, for example, this means that:
for all,, and. In arbitrary dimensions,
whenever is a rotation, and likewise:
whenever is a translation.
In fact, the algebra of all scalar linear differential operators, with constant coefficients, that commute with all Euclidean transformations, is the polynomial algebra generated by the Laplace operator.

Spectral theory

The spectrum of the Laplace operator consists of all eigenvalues for which there is a corresponding eigenfunction with:
This is known as the Helmholtz equation.
If is a bounded domain in, then the eigenfunctions of the Laplacian are an orthonormal basis for the Hilbert space. This result essentially follows from the spectral theorem on compact self-adjoint operators, applied to the inverse of the Laplacian. It can also be shown that the eigenfunctions are infinitely differentiable functions. More generally, these results hold for the Laplace–Beltrami operator on any compact Riemannian manifold with boundary, or indeed for the Dirichlet eigenvalue problem of any elliptic operator with smooth coefficients on a bounded domain. When is the -sphere, the eigenfunctions of the Laplacian are the spherical harmonics.

Vector Laplacian

The vector Laplace operator, also denoted by, is a differential operator defined over a vector field. The vector Laplacian is similar to the scalar Laplacian; whereas the scalar Laplacian applies to a scalar field and returns a scalar quantity, the vector Laplacian applies to a vector field, returning a vector quantity. When computed in orthonormal Cartesian coordinates, the returned vector field is equal to the vector field of the scalar Laplacian applied to each vector component.
The vector Laplacian of a vector field is defined as
This definition can be seen as the Helmholtz decomposition of the vector Laplacian.
In Cartesian coordinates, this reduces to the much simpler expression
where,, and are the components of the vector field, and just on the left of each vector field component is the Laplace operator. This can be seen to be a special case of Lagrange's formula; see Vector triple product.
For expressions of the vector Laplacian in other coordinate systems see Del in cylindrical and spherical coordinates.

Generalization

The Laplacian of any tensor field is defined as the divergence of the gradient of the tensor:
For the special case where is a scalar, the Laplacian takes on the familiar form.
If is a vector, the gradient is a covariant derivative which results in a tensor of second degree, and the divergence of this is again a vector. The formula for the vector Laplacian above may be used to avoid tensor math and may be shown to be equivalent to the divergence of the Jacobian matrix shown below for the gradient of a vector:
And, in the same manner, a dot product, which evaluates to a vector, of a vector by the gradient of another vector can be seen as a product of matrices:
This identity is a coordinate dependent result, and is not general.