TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the Laplace operator or Laplacian is a
differential operator 300px, A harmonic function defined on an annulus. Harmonic functions are exactly those functions which lie in the kernel of the Laplace operator, an important differential operator. In mathematics, a differential operator is an Operator (mathe ...
given by the
divergence In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ...

of the
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

of a
scalar function In mathematics and physics, a scalar field or scalar-valued function (mathematics), function associates a Scalar (mathematics), scalar value to every point in a space (mathematics), space – possibly physical space. The scalar may either be a ...

on
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
. It is usually denoted by the symbols $\nabla\cdot\nabla$, $\nabla^2$ (where $\nabla$ is the
nabla operator Del, or nabla, is an operator used in mathematics (particularly in vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the proce ...

), or $\Delta$. In a
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...
, the Laplacian is given by the sum of second
partial derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s of the function with respect to each
independent variable Dependent and Independent variables are variables in mathematical modeling A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to f ...
. In other
coordinate systems In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

, such as
cylindrical A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism (geometry), prism with a circle as its base. This traditi ...

and
spherical coordinates File:3D Spherical 2.svg, 240px, Spherical coordinates as often used in ''mathematics'': radial distance , azimuthal angle , and polar angle . The meanings of and have been swapped compared to the physics convention. As in physics, (rho) is of ...

, 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 Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering Engineering is the use of scientific method, scientific principles ...

(1749–1827), who first applied the operator to the study of
celestial mechanics Celestial mechanics is the branch of astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and cel ...
: the Laplacian of the
gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical wo ...

due to a given mass density distribution is a constant multiple of that density distribution. Solutions of
Laplace's equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
are called
harmonic function 300px, A harmonic function defined on an annulus. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and cal ...
s and represent the possible gravitational potentials in regions of
vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Gree ...

. The Laplacian occurs in many
differential equations In mathematics, a differential equation is an equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...
describing physical phenomena.
Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
describes
electric Electricity is the set of physics, physical Phenomenon, phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnet ...

and
gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical wo ...

s; the
diffusion equation The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's law ...
describes
heat In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these ...
and
fluid flow In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and h ...
, the
wave equation The wave equation is a second-order linear for the description of s—as they occur in —such as (e.g. waves, and ) or waves. It arises in fields like , , and . Historically, the problem of a such as that of a was studied by , , , and ...
describes
wave propagation Wave propagation is any of the ways in which wave In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, ...
, and the
Schrödinger equation The Schrödinger equation is a linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a ma ...
in
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
. In
image processing Digital image processing is the use of a digital computer A computer is a machine A machine is a man-made device that uses power to apply forces and control movement to perform an action. Machines can be driven by animals and people ...
and
computer vision Computer vision is an interdisciplinary scientific field that deals with how computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern computers can perform ge ...
, the Laplacian operator has been used for various tasks, such as
blob Blob may refer to: Science Computing * Binary blob A proprietary device driver is a closed-source device driver published only in binary code. In the context of free and open-source software, a Proprietary software, closed-source device dr ...
and
edge detection Edge detection includes a variety of mathematical methods that aim at identifying points in a digital image at which the image brightness changes sharply or, more formally, has discontinuities. The points at which image brightness changes sharpl ...

. The Laplacian is the simplest
elliptic operator In the theory of partial differential equations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...
and is at the core of
Hodge theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
, defined as the
divergence In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ...

($\nabla \cdot$) of the
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

($\nabla f$). Thus if $f$ is a
real-valued function Mass measured in grams is a function from this collection of weight to positive number">positive Positive is a property of Positivity (disambiguation), positivity and may refer to: Mathematics and science * Converging lens or positive lens, i ...
, then the Laplacian of $f$ is the real-valued function defined by: where the latter notations derive from formally writing: $\nabla = \left ( \frac , \ldots , \frac \right ).$ Explicitly, the Laplacian of is thus the sum of all the ''unmixed'' second
partial derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s in the
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

: 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
.

# Motivation

## Diffusion

In the
physical Physical may refer to: *Physical examination, a regular overall check-up with a doctor *Physical (album), ''Physical'' (album), a 1981 album by Olivia Newton-John **Physical (Olivia Newton-John song), "Physical" (Olivia Newton-John song) *Physical ( ...

theory of
diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration In chemistry Chemistry is the study of the properties and behavior of . It is a that covers ...

, the Laplace operator arises naturally in the mathematical description of
equilibrium List of types of equilibrium, the condition of a system in which all competing influences are balanced, in a wide variety of contexts. Equilibrium may also refer to: Film and television * Equilibrium (film), ''Equilibrium'' (film), a 2002 scien ...
. 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 : $\int_ \nabla u \cdot \mathbf\, dS = 0,$ where is the outward
unit normal Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit ...
to the boundary of . By the
divergence theorem In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

, $\int_V \operatorname \nabla u\, dV = \int_ \nabla u \cdot \mathbf\, dS = 0.$ Since this holds for all smooth regions , one can show that it implies: $\operatorname \nabla u = \Delta u = 0.$ The left-hand side of this equation is the Laplace operator, and the entire equation is known as
Laplace's equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. 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 diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's law ...
. This interpretation of the Laplacian is also explained by the following fact about averages.

## Averages

Given a twice continuously differentiable function $f : \R^n \to \R$, a point $p\in\R^n$ and a real number $h > 0$, we let $\overline_B\left(p,h\right)$ be the average value of $f$ over the ball with radius $h$ centered at $p$, and $\overline_S\left(p,h\right)$ be the average value of $f$ over the sphere (the boundary of a ball) with radius $h$ centered at $p$. Then we have: $\overline_B(p,h)=f(p)+\frac h^2 +o(h^2) \quad\text\;\; h\to 0$ and $\overline_S(p,h)=f(p)+\frac h^2 +o(h^2) \quad\text\;\; h\to 0.$

## Density associated with a potential

If denotes the
electrostatic potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work Work may refer to: * Work (human activity), intentional activity people perform to support the ...
associated to a
charge distribution In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the Systeme Internati ...
, then the charge distribution itself is given by the negative of the Laplacian of : $q = -\varepsilon_0 \Delta\varphi,$ where is the
electric constant Vacuum permittivity, commonly denoted (pronounced as "epsilon nought" or "epsilon zero") is the value of the absolute dielectric permittivity of classical vacuum. Alternatively it may be referred to as the permittivity of free space, the elec ...
. This is a consequence of
Gauss's law In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
. 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: $\int_ \mathbf\cdot \mathbf\, dS = \int_V \operatorname\mathbf\,dV=\frac1\int_V q\,dV.$ where the first equality is due to the
divergence theorem In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

. Since the electrostatic field is the (negative) gradient of the potential, this gives: $-\int_V \operatorname(\operatorname\varphi)\,dV = \frac1 \int_V q\,dV.$ Since this holds for all regions , we must have $\operatorname(\operatorname\varphi) = -\frac 1 q$ The same approach implies that the negative of the Laplacian of the
gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical wo ...

is the
mass distribution{{Other uses, Weight distribution In physics and mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces appl ...
. 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 Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
.

## Energy minimization

Another motivation for the Laplacian appearing in physics is that solutions to in a region are functions that make the
Dirichlet energyIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) In architecture File:Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Gallica 2011 (adjusted).jpg, upright=1.45, alt=Pl ...
stationary In addition to its common meaning, stationary may have the following specialized scientific meanings: Mathematics * Stationary point In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity ...
: $E(f) = \frac \int_U \lVert \nabla f \rVert^2 \,dx.$ To see this, suppose is a function, and is a function that vanishes on the boundary of . Then: $\left. \frac\_ E(f+\varepsilon u) = \int_U \nabla f \cdot \nabla u \, dx = -\int_U u \, \Delta f\, dx$ where the last equality follows using Green's first identity. This calculation shows that if , then is stationary around . Conversely, if is stationary around , then by the
fundamental lemma of calculus of variations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
.

# Coordinate expressions

## Two dimensions

The Laplace operator in two dimensions is given by: In
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

, $\Delta f = \frac + \frac$ where and are the standard
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

of the -plane. In
polar coordinates In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, $\begin \Delta f &= \frac \frac \left( r \frac \right) + \frac \frac \\ &= \frac + \frac \frac + \frac \frac, \end$ 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 A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

, $\Delta f = \frac + \frac + \frac.$ In
cylindrical coordinates 240px, A cylindrical coordinate system with origin , polar axis , and longitudinal axis . The dot is the point with radial distance , angular coordinate , and height . A cylindrical coordinate system is a three-dimensional coordinate system that s ...

, $\Delta f = \frac \frac \left(\rho \frac \right) + \frac \frac + \frac,$ where $\rho$ represents the radial distance, the azimuth angle and the height. In
spherical coordinates File:3D Spherical 2.svg, 240px, Spherical coordinates as often used in ''mathematics'': radial distance , azimuthal angle , and polar angle . The meanings of and have been swapped compared to the physics convention. As in physics, (rho) is of ...

: $\Delta f = \frac \frac \left(r^2 \frac \right) + \frac \frac \left(\sin \theta \frac \right) + \frac \frac,$ or $\Delta f = \frac \frac (r f) + \frac \frac \left(\sin \theta \frac \right) + \frac \frac,$ where represents the
azimuthal angle An azimuth (; from Arabic اَلسُّمُوت ''as-sumūt'', 'the directions', the plural form of the Arabic noun السَّمْت ''as-samt'', meaning 'the direction') is an angular measurement in a spherical coordinate system. The vector fr ...
and the
zenith angle The zenith is an imaginary point directly "above" a particular location, on the imaginary celestial sphere In astronomy and navigation, the celestial sphere is an abstraction, abstract sphere that has an arbitrarily large radius and is concen ...
or co-latitude. In general
curvilinear coordinates In , curvilinear coordinates are a for in which the s may be curved. These coordinates may be derived from a set of s by using a transformation that is (a one-to-one map) at each point. This means that one can convert a point given in a C ...
(): $\Delta = \nabla \xi^m \cdot \nabla \xi^n \frac + \nabla^2 \xi^m \frac = g^ \left(\frac - \Gamma^_\frac \right),$ where summation over the repeated indices is implied, is the inverse
metric tensor In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
and are the
Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surface (topology), surfaces or other manifolds endowed with a metric ...
for the selected coordinates.

## dimensions

In arbitrary
curvilinear coordinates In , curvilinear coordinates are a for in which the s may be curved. These coordinates may be derived from a set of s by using a transformation that is (a one-to-one map) at each point. This means that one can convert a point given in a C ...
in dimensions (), we can write the Laplacian in terms of the inverse
metric tensor In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
, $g^$: $\Delta = \frac 1\frac \left( \sqrt g^ \frac\right) ,$ from th
Voss
Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a Germany, German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he ...

formula for the
divergence In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ...
. In spherical coordinates in dimensions, with the parametrization with representing a positive real radius and an element of the
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center (geometry), center. More generally, it is the Locus (mathematics), set of points of distance 1 from a fixed central point, where different norm (mathematics), norm ...
, $\Delta f = \frac + \frac \frac + \frac \Delta_ f$ where is the
Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian manifold, Riemannian and pseudo-Riemannian manifol ...
on the -sphere, known as the spherical Laplacian. The two radial derivative terms can be equivalently rewritten as: $\frac \frac \left(r^ \frac \right).$ 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 Homogeneity and heterogeneity are concepts often used in the sciences Science () is a systematic enterprise that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about th ...
of degree zero.

# Euclidean invariance

The Laplacian is invariant under all
Euclidean transformation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s:
rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...

s and
translations Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
. In two dimensions, for example, this means that: $\Delta ( f(x\cos\theta - y\sin\theta + a, x\sin\theta + y\cos\theta + b)) = (\Delta f)(x\cos\theta - y\sin\theta + a, x\sin\theta + y\cos\theta + b)$ for all ''θ'', ''a'', and ''b''. In arbitrary dimensions, $\Delta (f\circ\rho) =(\Delta f)\circ \rho$ whenever ''ρ'' is a rotation, and likewise: $\Delta (f\circ\tau) =(\Delta f)\circ \tau$ whenever ''τ'' is a translation. (More generally, this remains true when ''ρ'' is an
orthogonal transformation In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and t ...
such as a
reflectionReflection or reflexion may refer to: Philosophy * Self-reflection Science * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal r ...
.) 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 A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum Continuum may refer to: * Continuum (measurement) Continuum theories or models expla ...
of the Laplace operator consists of all
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a Linear map, linear transformation is a nonzero Vector space, vector that changes at most by a Scalar (mathematics), scalar factor when that linear transformation is applied to i ...

s for which there is a corresponding
eigenfunction In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
with: $-\Delta f = \lambda f.$ This is known as the
Helmholtz equation In mathematics, the eigenvalue In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their r ...
. If is a bounded domain in , then the eigenfunctions of the Laplacian are an
orthonormal basis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
for the
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. This result essentially follows from the spectral theorem on
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
self-adjoint operator In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s, applied to the inverse of the Laplacian (which is compact, by the
Poincaré inequalityIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
and the
Rellich–Kondrachov theoremIn mathematics, the Rellich–Kondrachov theorem is a compactly embedded, compact embedding theorem concerning Sobolev spaces. It is named after the Austrian-German mathematician Franz Rellich and the Russian mathematician Vladimir Iosifovich Kondras ...
). It can also be shown that the eigenfunctions are
infinitely differentiable is a smooth function with compact support. In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain. At the very minimum, a function could be considered " ...
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 In the theory of partial differential equations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...
with smooth coefficients on a bounded domain. When is the -sphere, the eigenfunctions of the Laplacian are the
spherical harmonics In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

.

# Vector Laplacian

The vector Laplace operator, also denoted by $\nabla^2$, is a
differential operator 300px, A harmonic function defined on an annulus. Harmonic functions are exactly those functions which lie in the kernel of the Laplace operator, an important differential operator. In mathematics, a differential operator is an Operator (mathe ...
defined over a
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

. The vector Laplacian is similar to the scalar Laplacian; whereas the scalar Laplacian applies to a
scalar field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

and returns a scalar quantity, the vector Laplacian applies to a
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

, returning a vector quantity. When computed in
orthonormalIn linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vector In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quanti ...
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

, 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 In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

$\mathbf$ is defined as $\nabla^2 \mathbf = \nabla(\nabla \cdot \mathbf) - \nabla \times (\nabla \times \mathbf).$ In
Cartesian coordinate A Cartesian coordinate system (, ) in a plane is a coordinate system In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the o ...

s, this reduces to the much simpler form as $\nabla^2 \mathbf = (\nabla^2 A_x, \nabla^2 A_y, \nabla^2 A_z),$ where $A_x$, $A_y$, and $A_z$ are the components of the vector field $\mathbf$, and $\nabla^2$ 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. For expressions of the vector Laplacian in other coordinate systems see
Del in cylindrical and spherical coordinatesThis is a list of some vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculu ...
.

## Generalization

The Laplacian of any
tensor field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
$\mathbf$ ("tensor" includes scalar and vector) is defined as the
divergence In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ...

of the
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

of the tensor: $\nabla ^2\mathbf = (\nabla \cdot \nabla) \mathbf.$ For the special case where $\mathbf$ is a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...
(a tensor of degree zero), the
Laplacian In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
takes on the familiar form. If $\mathbf$ is a vector (a tensor of first degree), the gradient is a
covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection (mathematics), connection on a manifold ...
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 In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Produ ...
shown below for the gradient of a vector: $\nabla \mathbf= (\nabla T_x, \nabla T_y, \nabla T_z) = \begin T_ & T_ & T_ \\ T_ & T_ & T_ \\ T_ & T_ & T_ \end , \text T_ \equiv \frac.$ And, in the same manner, a dot product, which evaluates to a vector, of a vector by the gradient of another vector (a tensor of 2nd degree) can be seen as a product of matrices: $\mathbf \cdot \nabla \mathbf = \begin A_x & A_y & A_z \end \nabla \mathbf = \begin \mathbf \cdot \nabla B_x & \mathbf \cdot \nabla B_y & \mathbf \cdot \nabla B_z \end.$ This identity is a coordinate dependent result, and is not general.

## Use in physics

An example of the usage of the vector Laplacian is the for a
NewtonianNewtonian refers to the work of Isaac Newton, in particular: * Newtonian mechanics, i.e. classical mechanics * Newtonian telescope, a type of reflecting telescope * Newtonian cosmology * Newtonian dynamics * Newtonianism, the philosophical principle ...
incompressible flow In fluid mechanics Fluid mechanics is the branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the o ...

: $\rho \left(\frac+ ( \mathbf \cdot \nabla ) \mathbf\right)=\rho \mathbf-\nabla p +\mu\left(\nabla ^2 \mathbf\right),$ where the term with the vector Laplacian of the
velocity The velocity of an object is the rate of change of its position with respect to a frame of reference In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical scie ...

field $\mu\left\left(\nabla ^2 \mathbf\right\right)$ represents the
viscous The viscosity of a fluid In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, ...

stresses in the fluid. Another example is the wave equation for the electric field that can be derived from
Maxwell's equations Maxwell's equations are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
in the absence of charges and currents: $\nabla^2 \mathbf - \mu_0 \epsilon_0 \frac = 0.$ This equation can also be written as: $\Box\, \mathbf = 0,$ where $\Box\equiv\frac \frac-\nabla^2,$ is the D'Alembertian, used in the
Klein–Gordon equation The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz covariance, ...
.

# Generalizations

A version of the Laplacian can be defined wherever the Dirichlet energy functional makes sense, which is the theory of Dirichlet forms. For spaces with additional structure, one can give more explicit descriptions of the Laplacian, as follows.

## Laplace–Beltrami operator

The Laplacian also can be generalized to an elliptic operator called the
Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian manifold, Riemannian and pseudo-Riemannian manifol ...
defined on a
Riemannian manifold In differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a c ...
. The Laplace–Beltrami operator, when applied to a function, is the
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band) Trace was a Netherlands, Dutch progressive rock trio founded by Rick van der Linden in 1974 after leavin ...
() of the function's
Hessian A Hessian is an inhabitant of the German state of Hesse. Hessian may also refer to: Named from the toponym *Hessian (soldier), eighteenth-century German regiments in service with the British Empire **Hessian (boot), a style of boot **Hessian fa ...
: $\Delta f = \operatorname\big(H(f)\big)$ where the trace is taken with respect to the inverse of the
metric tensor In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
. The Laplace–Beltrami operator also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates on
tensor field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s, by a similar formula. Another generalization of the Laplace operator that is available on pseudo-Riemannian manifolds uses the
exterior derivative On a differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-interse ...
, in terms of which the "geometer's Laplacian" is expressed as $\Delta f = \delta d f .$ Here is the codifferential, which can also be expressed in terms of the
Hodge star In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
and the exterior derivative. This operator differs in sign from the "analyst's Laplacian" defined above. More generally, the "Hodge" Laplacian is defined on
differential form In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
s by $\Delta \alpha = \delta d \alpha + d \delta \alpha .$ 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 non-Euclidean spaces, where it may be elliptic operator, elliptic, hyperbolic operator, hyperbolic, or ultrahyperbolic operator, ultrahyperbolic. In Minkowski space the
Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian manifold, Riemannian and pseudo-Riemannian manifol ...
becomes the D'Alembert operator $\Box$ or D'Alembertian: $\square = \frac\frac - \frac - \frac - \frac.$ 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. 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 equation The wave equation is a second-order linear for the description of s—as they occur in —such as (e.g. waves, and ) or waves. It arises in fields like , , and . Historically, the problem of a such as that of a was studied by , , , and ...
s, and it is also part of the
Klein–Gordon equation The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz covariance, ...
, which reduces to the wave equation in the massless case. The additional factor of 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 direction were measured in meters while the direction were measured in centimeters. Indeed, theoretical physicists usually work in units such that in order to simplify the equation. The d'Alembert operator generalizes to a hyperbolic operator on pseudo-Riemannian manifolds.

*
Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian manifold, Riemannian and pseudo-Riemannian manifol ...
, generalization to submanifolds in Euclidean space and Riemannian and pseudo-Riemannian manifold. *The vector Laplacian operator, a generalization of the Laplacian to
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

s. *The Laplace operators in differential geometry, Laplacian in differential geometry. *The discrete Laplace operator is a finite-difference analog of the continuous Laplacian, defined on graphs and grids. *The Laplacian is a common operator in
image processing Digital image processing is the use of a digital computer A computer is a machine A machine is a man-made device that uses power to apply forces and control movement to perform an action. Machines can be driven by animals and people ...
and
computer vision Computer vision is an interdisciplinary scientific field that deals with how computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern computers can perform ge ...
(see the Laplacian of Gaussian, blob detection, blob detector, and scale space). *The list of formulas in Riemannian geometry contains expressions for the Laplacian in terms of Christoffel symbols. *Weyl's lemma (Laplace equation). *Earnshaw's theorem which shows that stable static gravitational, electrostatic or magnetic suspension is impossible. *
Del in cylindrical and spherical coordinatesThis is a list of some vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculu ...
. *Other situations in which a Laplacian is defined are: analysis on fractals, time scale calculus and discrete exterior calculus.

* * *. *.