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{{short description|Differential operator}}
{{about|the [[mathematical operator]]|the Laplace probability distribution|Laplace distribution|graph theoretical notion|Laplacian matrix}}
{{redirect|Del Squared}}
{{Calculus |Vector}}

In [[mathematics]], the '''Laplace operator''' or '''Laplacian''' is a [[differential operator]] given by the [[divergence]] of the [[gradient]] of a [[function (mathematics)|function]] on [[Euclidean space]]. It is usually denoted by the symbols {{math|∇·∇}}, {{math|∇<sup>2</sup>}}. The Laplacian {{math|∇·∇''f''(''p'')}} of a function {{math|''f''}} at a point {{math|''p''}}, is (up to a factor) the rate at which the average value of {{math|''f''}} over spheres centered at {{math|''p''}} deviates from {{math|''f''(''p'')}} as the radius of the sphere shrinks towards 0. In a [[Cartesian coordinate system]], the Laplacian is given by the sum of second [[partial derivative]]s of the function with respect to each [[independent variable]]. In other [[coordinate systems]] such as [[cylindrical coordinates|cylindrical]] and [[spherical coordinates]], the Laplacian also has a useful form.

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]], where the operator gives a constant multiple of the mass density when it is applied to the  [[gravitational potential]] due to the mass distribution with that given density. Solutions of the equation {{math|∇·∇''f'' {{=}} 0}}, now called [[Laplace's equation]], are the so-called [[harmonic function]]s, and represent the possible [[gravitational fields]] in regions of [[vacuum]].

The Laplacian occurs in [[differential equations]] that describe many physical phenomena, such as [[electric potential|electric]] and [[gravitational potential]]s, the [[diffusion equation]] for [[heat equation|heat]] and [[fluid mechanics|fluid flow]], [[wave equation|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 detection|blob]] and [[edge detection]].

==Definition==
The Laplace operator is a second order differential operator in the ''n''-dimensional [[Euclidean space]], defined as the [[divergence]] ({{math|∇·}}) of the [[gradient]] ({{math|∇''f ''}}). Thus if {{math|''f''}} is a [[derivative|twice-differentiable]] [[real-valued function]], then the Laplacian of {{math|''f''}} is defined by

{{NumBlk|:|<math>\Delta f = \nabla^2 f = \nabla \cdot \nabla f </math>|{{EqRef|1}}}}

where the latter notations derive from formally writing
:<math>\nabla = \left ( \frac{\partial}{\partial x_1} , \ldots , \frac{\partial}{\partial x_n} \right ).</math>
Equivalently, the Laplacian of {{math|''f''}} is the sum of all the ''unmixed'' second [[partial derivative]]s in the [[Cartesian coordinates]] {{math|''x<sub>i</sub>''}}:

{{NumBlk|:|<math>\Delta f = \sum_{i=1}^n \frac {\partial^2 f}{\partial x^2_i}</math>|{{EqRef|2}}}}

As a second-order differential operator, the Laplace operator maps {{math|[[continuously differentiable|''C<sup>k</sup>'']]}} functions to {{math|''C''<sup>''k''−2</sup>}} functions for {{math|''k'' ≥ 2}}. The expression {{EqNote|1}} (or equivalently {{EqNote|2}}) defines an operator {{math|Δ : ''C''<sup>''k''</sup>(ℝ<sup>''n''</sup>) → ''C''<sup>''k''−2</sup>(ℝ<sup>''n''</sup>)}}, or more generally an operator {{math|Δ : ''C''<sup>''k''</sup>(Ω) → ''C''<sup>''k''−2</sup>(Ω)}} for any [[open set]] {{math|Ω}}.

== Motivation ==

=== Diffusion ===
In the [[physics|physical]] theory of [[diffusion]], the Laplace operator (via [[Laplace's equation]]) arises naturally in the mathematical description of [[Diffusion equilibrium|equilibrium]].<ref>{{harvnb|Evans|1998|loc=§2.2}}</ref> Specifically, if {{math|''u''}} is the density at equilibrium of some quantity such as a chemical concentration, then the [[net flux]] of {{math|''u''}} through the boundary of any smooth region {{math|''V''}} is zero, provided there is no source or sink within {{math|''V''}}:

:<math>\int_{\partial V} \nabla u \cdot \mathbf{n}\, dS = 0,</math>

where {{math|'''n'''}} is the outward [[unit normal]] to the boundary of {{math|''V''}}. By the [[divergence theorem]],

:<math>\int_V \operatorname{div} \nabla u\, dV = \int_{\partial V} \nabla u \cdot \mathbf{n}\, dS = 0.</math>

Since this holds for all smooth regions {{math|''V''}}, it can be shown that this implies
: <math>\operatorname{div} \nabla u = \Delta u = 0.</math>
The left-hand side of this equation is the Laplace operator. 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]].

=== Density associated with a potential ===
If {{math|''φ''}} denotes the [[electrostatic potential]] associated to a [[charge distribution]] {{math|''q''}}, then the charge distribution itself is given by the negative of the Laplacian of {{math|''φ''}}:

:<math>q = -\varepsilon_0 \Delta\varphi,</math>

where {{math|''ε''<sub>0</sub>}} is the [[electric constant]].

This is a consequence of [[Gauss's law]]. Indeed, if {{math|''V''}} is any smooth region, then by Gauss's law the flux of the electrostatic field {{math|'''E'''}} is proportional to the charge enclosed:

:<math>\int_{\partial V} \mathbf{E}\cdot \mathbf{n}\, dS = \int_V \operatorname{div}\mathbf{E}\,dV=\frac1{\varepsilon_0}\int_V q\,dV.</math>

where the first equality is due to the [[divergence theorem]]. Since the electrostatic field is the (negative) gradient of the potential, this now gives

:<math>-\int_V \operatorname{div}(\operatorname{grad}\varphi)\,dV = \frac1{\varepsilon_0} \int_V q\,dV.</math>

So, since this holds for all regions {{math|''V''}}, we must have

:<math>\operatorname{div}(\operatorname{grad}\varphi) = -\frac 1 {\varepsilon_0}q</math>

The same approach implies that the negative of 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 {{math|Δ''f'' {{=}} 0}} in a region {{math|''U''}} are functions that make the [[Dirichlet energy]] [[functional (mathematics)|functional]] [[stationary point|stationary]]:

: <math> E(f) = \frac{1}{2} \int_U \Vert \nabla f \Vert^2 \,dx.</math>

To see this, suppose {{math|''f'' : ''U'' → ℝ}} is a function, and {{math|''u'' : U → ℝ}} is a function that vanishes on the 
boundary of {{math|''U''}}. Then

: <math>\left. \frac{d}{d\varepsilon}\right|_{\varepsilon = 0} E(f+\varepsilon u) = \int_U \nabla f \cdot \nabla u \, dx = -\int_U u \, \Delta f\, dx </math>

where the last equality follows using [[Green's first identity]]. This calculation shows that if {{math|Δ''f'' {{=}} 0}}, then {{math|''E''}} is stationary around {{math|''f''}}. Conversely, if {{math|''E''}} is stationary around {{math|''f''}}, then {{math|Δ''f'' {{=}} 0}} by the [[fundamental lemma of calculus of variations]].

== Coordinate expressions ==

=== Two dimensions ===
The Laplace operator in two dimensions is given by:

In '''[[Cartesian coordinates]]''',

:<math>\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}</math>

where {{math|''x''}} and {{math|''y''}} are the standard [[Cartesian coordinates]] of the {{math|''xy''}}-plane.

In '''[[polar coordinates]]''',

:<math>\begin{align}
\Delta f &= \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial f}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2} \\
&= \frac{\partial^2 f}{\partial r^2} + \frac{1}{r} \frac{\partial f}{\partial r} + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2},
\end{align}</math>

where {{math|''r''}} represents the radial distance and {{math|''θ''}} the angle.

===Three dimensions===
{{See also|Del in cylindrical and spherical coordinates}}
In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems.

In '''[[Cartesian coordinates]]''',

:<math>\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}.</math>

In '''[[cylindrical coordinates]]''',

:<math>\Delta f = \frac{1}{r} \frac{\partial}{\partial r} \left(r \frac{\partial f}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \varphi^2} + \frac{\partial^2 f}{\partial z^2 },</math>

where {{math|''r''}} represents the radial distance, {{math|''φ''}} the azimuth angle and {{math|''z''}} the height.

In '''[[spherical coordinates]]''':

:<math>\begin{align}
\Delta f &= \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial f}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \varphi^2} \\
&= \frac{1}{r} \frac{\partial^2}{\partial r^2} (rf) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \varphi^2}
\end{align}</math>

<!---**********PLEASE SEE THE DISCUSSION PAGE BEFORE CHANGING THIS.**********-->
where {{math|''φ''}} represents the [[azimuthal angle]] and {{math|''θ''}} the [[zenith angle]] or [[colatitude|co-latitude]].
<!---**************************************************************-->

In general '''[[curvilinear coordinates]]''' ({{math|''ξ''<sup>1</sup>, ''ξ''<sup>2</sup>, ''ξ''<sup>3</sup>}}):

:<math>\nabla^2 = \nabla \xi^m \cdot \nabla \xi^n \frac{\partial^2}{\partial \xi^m \partial \xi^n} + \nabla^2 \xi^m \frac{\partial}{\partial \xi^m } = g^{mn} \left(\frac{\partial^2}{\partial\xi^m \partial\xi^n} - \Gamma^{l}_{mn}\frac{\partial}{\partial\xi^l} \right),</math>

where [[Einstein summation convention|summation over the repeated indices is implied]],
{{math|''g<sup>mn</sup>''}} is the inverse [[metric tensor]] and  {{math|Γ''<sup>l</sup> <sub>mn</sub>''}} are the [[Christoffel symbols]] for the selected coordinates.

=== {{math|''N''}} dimensions ===
In arbitrary [[curvilinear coordinates]] in {{math|''N''}} dimensions ({{math|''ξ''<sup>1</sup>,..., ''ξ<sup>N</sup>''}}), we can write the Laplacian in terms of the inverse [[metric tensor]], <math> g^{ij} </math>:

:<math>\nabla^2 = \frac1{\sqrt{\det g}}\frac{\partial}{\partial\xi^i} \left( \sqrt{\det g} g^{ij}  \frac{\partial}{\partial \xi^j}\right)</math>,

from the [https://www.genealogy.math.ndsu.nodak.edu/id.php?id=59087 Voss]-[[Hermann_Weyl| Weyl]] formula<ref>{{cite web|last1=Grinfeld| first1=Pavel| title=The Voss-Weyl Formula|url=https://www.youtube.com/watch?v=BD2AiFk651E&list=PLlXfTHzgMRULkodlIEqfgTS-H1AY_bNtq&index=23|accessdate=9 January 2018|language=English}}</ref> for the [[divergence#General coordinates | divergence]].

In '''spherical coordinates in {{math|''N''}} dimensions''', with the parametrization {{math|''x'' {{=}} ''rθ'' ∈ ℝ<sup>''N''</sup>}} with {{math|''r''}} representing a positive real radius and {{math|''θ''}} an element of the [[unit sphere]] {{math|[[N sphere|''S''<sup>''N'' − 1</sup>]]}},

:<math> \Delta f = \frac{\partial^2 f}{\partial r^2} + \frac{N-1}{r} \frac{\partial f}{\partial r} + \frac{1}{r^2} \Delta_{S^{N-1}} f</math>

where {{math|Δ<sub>''S''<sup>''N''−1</sup></sub>}} is the [[Laplace–Beltrami operator]] on the {{math|(''N'' − 1)}}-sphere, known as the spherical Laplacian. The two radial derivative terms can be equivalently rewritten as

:<math>\frac{1}{r^{N-1}} \frac{\partial}{\partial r} \left(r^{N-1} \frac{\partial f}{\partial r} \right).</math>

As a consequence, the spherical Laplacian of a function defined on {{math|''S''<sup>''N''−1</sup> ⊂ ℝ<sup>''N''</sup>}} can be computed as the ordinary Laplacian of the function extended to {{math|ℝ<sup>''N''</sup>\{0<nowiki>}</nowiki>}} so that it is constant along rays, i.e., [[homogeneous function|homogeneous]] of degree zero.

==Euclidean invariance==
The Laplacian is invariant under all [[Euclidean transformation]]s: [[rotation]]s and [[translation (mathematics)|translations]].  In two dimensions, for example, this means that

:<math>\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)</math>

for all &theta;, ''a'', and ''b''.  In arbitrary dimensions,
:<math>\Delta (f\circ\rho) =(\Delta f)\circ \rho</math>
whenever &rho; is a rotation, and likewise
:<math>\Delta (f\circ\tau) =(\Delta f)\circ \tau</math>
whenever &tau; is a translation.  (More generally, this remains true when &rho; is an [[orthogonal transformation]] such as a [[reflection (mathematics)|reflection]].)

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==
{{see also|Hearing the shape of a drum|Dirichlet eigenvalue}}
The [[spectral theory|spectrum]] of the Laplace operator consists of all [[eigenvalue]]s {{math|''λ''}} for which there is a corresponding [[eigenfunction]] {{math|''f''}} with

:<math>-\Delta f = \lambda f.</math>

This is known as the [[Helmholtz equation]].

If {{math|Ω}} is a bounded domain in {{math|ℝ<sup>''n''</sup>}} then the eigenfunctions of the Laplacian are an [[orthonormal basis]] for the [[Hilbert space]] {{math|[[Lp space|''L''<sup>2</sup>(Ω)]]}}. This result essentially follows from the [[spectral theorem]] on [[compact operator|compact]] [[self-adjoint operator]]s, applied to the inverse of the Laplacian (which is compact, by the [[Poincaré inequality]] and the [[Rellich–Kondrachov theorem]]).<ref>{{harvnb|Gilbarg|Trudinger|2001|loc=Theorem 8.6}}</ref> It can also be shown that the eigenfunctions are [[infinitely differentiable]] functions.<ref>{{harvnb|Gilbarg|Trudinger|2001|loc=Corollary 8.11}}</ref> 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 {{math|Ω}} is the [[N-sphere|{{math|''n''}}-sphere]], the eigenfunctions of the Laplacian are the well-known [[spherical harmonics]].

== Generalizations ==
A version of the Laplacian can be defined wherever the [[Dirichlet energy|Dirichlet energy functional]] makes sense, which is the theory of [[Dirichlet form]]s.  For spaces with additional structure, one can give more explicit descriptions of the Laplacian, as follows.  

=== Laplace–Beltrami operator ===
{{main article|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 [[pseudo-Riemannian manifold]]s. The Laplace–Beltrami operator, when applied to a function, is the [[trace (linear algebra)|trace]] ({{math|tr}}) of the function's [[Hessian matrix|Hessian]]:

:<math>\Delta f = \operatorname{tr}\big(H(f)\big)</math>

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 field]]s, by a similar formula.

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

:<math> \Delta f = \delta d f .</math>

Here {{math|''δ''}} 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 form]]s {{math|''α''}} by

:<math>\Delta \alpha = \delta d \alpha + d \delta \alpha .</math>

This is known as the '''[[Laplace–Beltrami operator|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 the [[Minkowski space]] the [[Laplace–Beltrami operator]] becomes the [[D'Alembert operator]] {{math|⧠}} or D'Alembertian:

:<math>\square = \frac1{c^2}\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}-\frac{\partial^2}{\partial z^2}.</math>

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 equation]]s and it is also part of the [[Klein–Gordon equation]], which reduces to the wave equation in the massless case.

The additional factor of {{math|''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 {{math|''x''}} direction were measured in meters while the {{math|''y''}} direction were measured in centimeters. Indeed, theoretical physicists usually work in units such that [[natural units|{{math|''c'' {{=}} 1}}]] in order to simplify the equation.

==See also==
*The [[vector Laplacian]] operator, a generalization of the Laplacian to [[vector field]]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]] and [[computer vision]] (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 coordinates]]
*Other situations in which a Laplacian is defined are: [[analysis on fractals]], [[time scale calculus]] and [[discrete exterior calculus]].

==Notes==
{{reflist|20em}}

==References==
*{{citation|last=Evans|first=L.|title=Partial Differential Equations|publisher=American Mathematical Society|year=1998|isbn=978-0-8218-0772-9}}
*{{citation|last1=Feynman|first1=R.|last2=Leighton|first2=R|last3=Sands|first3=M.|title=[[The Feynman Lectures on Physics]]|volume=2|chapter=Chapter 12: Electrostatic Analogs|publisher=Addison-Wesley-Longman|year=1970}}
*{{citation|author2-link=Neil Trudinger|first1=D.|last1=Gilbarg|first2=N.|last2=Trudinger|title=Elliptic Partial Differential Equations of Second Order|year=2001|publisher=Springer|isbn=978-3-540-41160-4}}.
*{{citation|last=Schey|first=H. M.|title=Div, Grad, Curl, and All That|publisher=W. W. Norton|year=1996|isbn=978-0-393-96997-9}}.

==External links==
*{{springer|title=Laplace operator|id=p/l057510}}
*{{MathWorld | urlname=Laplacian | title=Laplacian}}
*[http://blog.wolfram.com/2017/08/25/how-laplace-would-hide-a-goat-the-new-science-of-magic-windows/ How Laplace Would Hide a Goat: The New Science of Magic Windows]
*[http://ramanujan.math.trinity.edu/rdaileda/teach/s12/m3357/lectures/lecture_3_27_2_short.pdf Laplacian in polar coordinates derivation]

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[[Category:Elliptic partial differential equations]]
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[[Category:Pierre-Simon Laplace|Operator]]
[[Category:Harmonic functions]]
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