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In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇^{2} or ∆. The Laplacian ∆f(p) of a function f at a point p, up to a constant depending on the dimension, is the rate at which the average value of f over spheres centered at p, deviates from f(p) as the radius of the sphere grows. In a Cartesian coordinate system, the Laplacian is given by 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.
The Laplace operator is named after the French mathematician PierreSimon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics, where the operator gives a constant multiple of the mass density when it is applied to a given gravitational potential. Solutions of the equation ∆f = 0, now called Laplace's equation, are the socalled harmonic functions, and represent the possible gravitational fields in free space.
The Laplacian occurs in differential equations that describe many physical phenomena, such as electric and gravitational potentials, the diffusion equation for heat and fluid flow, wave propagation, and quantum mechanics. The Laplacian represents the flux density of the gradient flow of a function. For instance, the net rate at which a chemical dissolved in a fluid moves toward or away from some point is proportional to the Laplacian of the chemical concentration at that point; expressed symbolically, the resulting equation is the diffusion equation. For these reasons, it is extensively used in the sciences for modelling all kinds of physical phenomena. The Laplacian is the simplest elliptic operator, and is at the core of Hodge theory as well as the results of de Rham cohomology. In image processing and computer vision, the Laplacian operator has been used for various tasks such as blob and edge detection.
Definition
The Laplace operator is a second order differential operator in the ndimensional Euclidean space, defined as the divergence (∇·) of the gradient (∇ƒ). Thus if ƒ is a twicedifferentiable realvalued function, then the Laplacian of ƒ is defined by

where the latter notations derive from formally writing $\backslash nabla\; =\; \backslash left\; (\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\_1\}\; ,\; \backslash dots\; ,\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\_n\}\; \backslash right\; ).$ Equivalently, the Laplacian of ƒ is the sum of all the unmixed second partial derivatives in the Cartesian coordinates $x\_i$ :

As a secondorder differential operator, the Laplace operator maps C^{k}functions to C^{k−2}functions for k ≥ 2. The expression Template:EqNote (or equivalently Template:EqNote) defines an operator ∆ : C^{k}(R^{n}) → C^{k−2}(R^{n}), or more generally an operator ∆ : C^{k}(Ω) → C^{k−2}(Ω) for any open set Ω.
Motivation
Diffusion
In the physical theory of diffusion, the Laplace operator (via Laplace's equation) arises naturally in the mathematical description of equilibrium.^{[1]} 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 of any smooth region V is zero, provided there is no source or sink within V:
 $\backslash int\_\{\backslash partial\; V\}\; \backslash nabla\; u\; \backslash cdot\; \backslash mathbf\{n\}\backslash ,\; dS\; =\; 0,$
where n is the outward unit normal to the boundary of V. By the divergence theorem,
 $\backslash int\_V\; \backslash operatorname\{div\}\; \backslash nabla\; u\backslash ,\; dV\; =\; \backslash int\_\{\backslash partial\; V\}\; \backslash nabla\; u\; \backslash cdot\; \backslash mathbf\{n\}\backslash ,\; dS\; =\; 0.$
Since this holds for all smooth regions V, it can be shown that this implies
 $\backslash operatorname\{div\}\; \backslash nabla\; u\; =\; \backslash Delta\; u\; =\; 0.$
The lefthand side of this equation is the Laplace operator. The Laplace operator itself has a physical interpretation for nonequilibrium 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.
Density associated to a potential
If φ denotes the electrostatic potential associated to a charge distribution q, then the charge distribution itself is given by the Laplacian of φ:

This is a consequence of Gauss's law. Indeed, if V is any smooth region, then by Gauss's law the flux of the electrostatic field E is equal to the charge enclosed (in appropriate units):
 $\backslash int\_\{\backslash partial\; V\}\; \backslash mathbf\{E\}\backslash cdot\; \backslash mathbf\{n\}\backslash ,\; dS\; =\; \backslash int\_\{\backslash partial\; V\}\; \backslash nabla\backslash varphi\backslash cdot\; \backslash mathbf\{n\}\backslash ,\; dS\; =\; \backslash int\_V\; q\backslash ,dV,$
where the first equality uses the fact that the electrostatic field is the gradient of the electrostatic potential. The divergence theorem now gives
 $\backslash int\_V\; \backslash Delta\backslash varphi\backslash ,dV\; =\; \backslash int\_V\; q\backslash ,\; dV,$
and since this holds for all regions V, (1) follows.
The same approach implies that the Laplacian of the gravitational potential is the mass distribution. Often the charge (or mass) 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 $\backslash Delta\; f\; =\; 0$ in a region U are functions that make the Dirichlet energy functional stationary:
 $E(f)\; =\; \backslash frac\{1\}\{2\}\; \backslash int\_U\; \backslash Vert\; \backslash nabla\; f\; \backslash Vert^2\; \backslash ,dx.$
To see this, suppose
$f\backslash colon\; U\backslash to\; \backslash mathbb\{R\}$ is a function, and
$u\backslash colon\; U\backslash to\; \backslash mathbb\{R\}$ is a function that vanishes on the
boundary of U. Then
 $$
\frac{d}{d\varepsilon}\Big_{\varepsilon = 0} E(f+\varepsilon u)
= \int_U \nabla f \cdot \nabla u \, dx
= \int_U u \Delta f\, dx
where the last equality follows using Green's first identity.
This calculation shows that if $\backslash Delta\; f\; =\; 0$, then
E is stationary around f. Conversely, if E is stationary
around f, then $\backslash Delta\; f=0$ by the fundamental lemma of calculus of variations.
Coordinate expressions
Two dimensions
The Laplace operator in two dimensions is given by
 $\backslash Delta\; f\; =\; \backslash frac\{\backslash partial^2f\}\{\backslash partial\; x^2\}\; +\; \backslash frac\{\backslash partial^2\; f\}\{\backslash partial\; y^2\}$
where x and y are the standard Cartesian coordinates of the xyplane.
In polar coordinates,
 $\backslash begin\{align\}$
\Delta f
&= {1 \over r} {\partial \over \partial r}
\left( r {\partial f \over \partial r} \right)
+ {1 \over r^2} {\partial^2 f \over \partial \theta^2}\\
&= {1 \over r} {\partial f \over \partial r}
+ {\partial^2 f \over \partial r^2}
+ {1 \over r^2} {\partial^2 f \over \partial \theta^2}
.
\end{align}
Three dimensions
In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems.
In Cartesian coordinates,
 $$
\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}.
In cylindrical coordinates,
 $\backslash Delta\; f$
= {1 \over \rho} {\partial \over \partial \rho}
\left(\rho {\partial f \over \partial \rho} \right)
+ {1 \over \rho^2} {\partial^2 f \over \partial \varphi^2}
+ {\partial^2 f \over \partial z^2 }.
In spherical coordinates:
 $\backslash Delta\; f$
= {1 \over r^2} {\partial \over \partial r}
\left(r^2 {\partial f \over \partial r} \right)
+ {1 \over r^2 \sin \theta} {\partial \over \partial \theta}
\left(\sin \theta {\partial f \over \partial \theta} \right)
+ {1 \over r^2 \sin^2 \theta} {\partial^2 f \over \partial \varphi^2}.
(here φ represents the azimuthal angle and θ the zenith angle or colatitude).
In general curvilinear coordinates ($\backslash xi^1,\; \backslash xi^2,\; \backslash xi^3$):
$\backslash nabla^2\; =\; \backslash nabla\; \backslash xi^m\; \backslash cdot\; \backslash nabla\; \backslash xi^n\; \{\backslash partial^2\; \backslash over\; \backslash partial\; \backslash xi^m\; \backslash partial\; \backslash xi^n\}\; +\; \backslash nabla^2\; \backslash xi^m\; \{\backslash partial\; \backslash over\; \backslash partial\; \backslash xi^m\; \},$
where summation over the repeated indices is implied.
N dimensions
In spherical coordinates in N dimensions, with the parametrization x = rθ ∈ R^{N} with r representing a positive real radius and θ an element of the unit sphere S^{N−1},
 $\backslash Delta\; f$
= \frac{\partial^2 f}{\partial r^2}
+ \frac{N1}{r} \frac{\partial f}{\partial r}
+ \frac{1}{r^2} \Delta_{S^{N1}} f
where $\backslash Delta\_\{S^\{N1\}\}$ is the Laplace–Beltrami operator on the (N−1)sphere, known as the spherical Laplacian. The two radial terms can be equivalently rewritten as
 $\backslash frac\{1\}\{r^\{N1\}\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; r\}\; \backslash Bigl(r^\{N1\}\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; r\}\; \backslash Bigr).$
As a consequence, the spherical Laplacian of a function defined on S^{N−1} ⊂ R^{N} can be computed as the ordinary Laplacian of the function extended to R^{N}\{0} so that it is constant along rays, i.e., homogeneous of degree zero.
Spectral theory
The spectrum of the Laplace operator consists of all eigenvalues λ for which there is a corresponding eigenfunction ƒ with
 $\backslash Delta\; f\; =\; \backslash lambda\; f.$
This is known as the Helmholtz equation.
If Ω is a bounded domain in R^{n} then the eigenfunctions of the Laplacian are an orthonormal basis for the Hilbert space L^{2}(Ω). This result essentially follows from the spectral theorem on compact selfadjoint operators, applied to the inverse of the Laplacian (which is compact, by the Poincaré inequality and Kondrakov embedding theorem).^{[2]} It can also be shown that the eigenfunctions are infinitely differentiable functions.^{[3]} 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 Ω is the nsphere, the eigenfunctions of the Laplacian are the wellknown spherical harmonics.
Generalizations
Laplace–Beltrami operator
The Laplacian also can be generalized to an elliptic operator called the Laplace–Beltrami operator defined on a Riemannian manifold. The d'Alembert operator generalizes to a hyperbolic operator on pseudoRiemannian manifolds. The Laplace–Beltrami operator, when applied to a function, is the trace of the function's Hessian:
 $\backslash Delta\; f\; =\; \backslash mathrm\{tr\}(H(f))\backslash ,\backslash !$
where the trace is taken with respect to the inverse of the metric tensor. The Laplace–Beltrami operator also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates on tensor fields, by a similar formula.
Another generalization of the Laplace operator that is available on pseudoRiemannian manifolds uses the exterior derivative, in terms of which the “geometer's Laplacian" is expressed as
 $\backslash Delta\; f\; =\; d^*\; d\; f\backslash ,$
Here d^{∗} is the codifferential, which can also be expressed using the Hodge dual. Note that this operator differs in sign from the "analyst's Laplacian" defined
above, a point which must always be kept in mind when reading papers in global analysis.
More generally, the "Hodge" Laplacian is defined on differential forms α by
 $\backslash Delta\; \backslash alpha\; =\; d^*\; d\backslash alpha\; +\; dd^*\backslash alpha.\backslash ,$
This is known as the Laplace–de Rham operator, which is related to the Laplace–Beltrami operator by the Weitzenböck identity.
D'Alembertian
The Laplacian can be generalized in certain ways to nonEuclidean spaces, where it may be elliptic, hyperbolic, or ultrahyperbolic.
In the Minkowski space the Laplace–Beltrami operator becomes the d'Alembert operator or d'Alembertian:
 $\backslash square$
=
\frac {1}{c^2}{\partial^2 \over \partial t^2 }

{\partial^2 \over \partial x^2 }

{\partial^2 \over \partial y^2 }

{\partial^2 \over \partial z^2 }.
It is the generalisation of the Laplace operator in the sense that it is the differential operator which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time independent functions. Note that the overall sign of the metric here is chosen such that the spatial parts of the operator admit a negative sign, which is the usual convention in high energy particle physics. The D'Alembert operator is also known as the wave operator, because it is the differential operator appearing in the wave equations and it is also part of the Klein–Gordon equation, which reduces to the wave equation in the massless case.
The additional factor of c in the metric is needed in physics if space and time are measured in different units; a similar factor would be required if, for example, the x direction were measured in meters while the y direction were measured in centimeters. Indeed, theoretical physicists usually work in units such that c=1 in order to simplify the equation.
See also
Notes
References
External links
 Template:Springer
 MathWorld.
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