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{{short description|Differential operator}}
{{Calculus |Vector}}

In [[Mathematics|mathematics]], the '''Laplace operator''' or '''Laplacian''' is a [[Differential operator|differential operator]] given by the [[Divergence|divergence]] of the [[Gradient|gradient]] of a [[Scalar field|scalar function]] on [[Euclidean space]]. It is usually denoted by the symbols <math>\nabla\cdot\nabla</math>, <math>\nabla^2</math> (where <math>\nabla</math> is the [[Del|nabla operator]]), or <math>\Delta</math>. In a [[Cartesian coordinate system]], the Laplacian is given by the sum of second [[Partial derivative|partial derivative]]s 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 {{math|Δ''f''&hairsp;(''p'')}} of a function {{math|''f''}} at a point {{math|''p''}} measures by how much the average value of  {{math|''f''}} over small spheres or balls centered at {{math|''p''}} deviates from {{math|''f''&hairsp;(''p'')}}.

The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of [[Physics:Celestial mechanics|celestial mechanics]]: the Laplacian of the [[Physics:Gravitational potential|gravitational potential]] due to a given mass density distribution is a constant multiple of that density distribution. Solutions of [[Laplace's equation]] {{math|1=Δ''f'' = 0}} are called [[Harmonic function|harmonic function]]s and represent the possible [[Physics:Gravitational potential|gravitational potential]]s in regions of [[Physics:Vacuum|vacuum]].

The Laplacian occurs in many differential equations describing physical phenomena. [[Physics:Poisson's equation|Poisson's equation]] describes  [[Physics:Electric potential|electric]] and [[Physics:Gravitational potential|gravitational potential]]s; the [[Diffusion equation|diffusion equation]] describes [[Heat equation|heat]] and [[Physics:Fluid mechanics|fluid flow]]; the [[Physics:Wave equation|wave equation]] describes [[Physics:Wave equation|wave propagation]]; and the [[Physics:Schrödinger equation|Schrödinger equation]] describes the [[Wave function|wave function]] in [[Physics:Quantum mechanics|quantum mechanics]]. In [[Image processing|image processing]] and [[Computer vision|computer vision]], the Laplacian operator has been used for various tasks, such as [[Blob detection|blob]] and [[Edge detection|edge detection]]. The Laplacian is the simplest [[Elliptic operator|elliptic operator]] and is at the core of Hodge theory as well as the results of [[De Rham cohomology|de Rham cohomology]].

==Definition==
The Laplace operator is a second-order differential operator in the ''n''-dimensional [[Euclidean space]], defined as the [[Divergence|divergence]] (<math>\nabla \cdot</math>) of the [[Gradient|gradient]] (<math>\nabla f</math>). Thus if <math>f</math> is a [[Derivative|twice-differentiable]] [[Real-valued function|real-valued function]], then the Laplacian of <math>f</math> is the real-valued function defined by:
{{NumBlk||<math display="block">\Delta f = \nabla^2 f = \nabla \cdot \nabla f </math>|{{EqRef|1}}}}
where the latter notations derive from formally writing:
<math display="block">\nabla  = \left ( \frac{\partial }{\partial x_1} , \ldots , \frac{\partial }{\partial x_n} \right ).</math>
Explicitly, the Laplacian of {{math|''f''}} is thus the sum of all the ''unmixed'' second [[Partial derivative|partial derivative]]s in the Cartesian coordinates {{math|''x<sub>i</sub>''}}:
{{NumBlk||<math display="block">\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|''C{{i sup|k}}''}} functions to {{math|''C''{{i sup|''k''−2}}}} functions for {{math|''k'' ≥ 2}}. It is a linear operator {{math|Δ : ''C''{{i sup|''k''}}('''R'''<sup>''n''</sup>) → ''C''{{i sup|''k''−2}}('''R'''<sup>''n''</sup>)}}, or more generally, an operator {{math|Δ : ''C''{{i sup|''k''}}(Ω) → ''C''{{i sup|''k''−2}}(Ω)}} for any [[Open set|open set]] {{math|Ω ⊆ '''R'''<sup>''n''</sup>}}.

== Motivation ==

=== Diffusion ===
In the [[Physics:Physics|physical]] theory of [[Physics:Diffusion|diffusion]], the Laplace operator arises naturally in the mathematical description of 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 {{math|∂''V''}} of any smooth region {{math|''V''}} is zero, provided there is no source or sink within {{math|''V''}}:
<math display="block">\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|divergence theorem]],
<math display="block">\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''}}, one can show that it implies:
<math display="block">\operatorname{div} \nabla u = \Delta u = 0.</math>
The left-hand side of this equation is the Laplace operator, and the entire equation {{math|1=Δ''u'' = 0}} 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|diffusion equation]]. This interpretation of the Laplacian is also explained by the following fact about averages.

=== Averages ===
Given a twice continuously differentiable function <math>f : \R^n \to \R </math> and a point <math>p\in\R^n</math>.
Then, the average value of <math>f </math> over the ball with radius <math>h</math> centered at <math>p</math> is:<ref>{{Cite journal | last=Ovall | first=Jeffrey S. | date=2016-03-01 | title=The Laplacian and Mean and Extreme Values|url=http://web.pdx.edu/~jovall/PDF/LaplaceMeanValue.pdf | journal=The American Mathematical Monthly | volume=123 | issue=3 | pages=287–291| doi=10.4169/amer.math.monthly.123.3.287 | s2cid=124943537 }}</ref>
<math display="block">\overline{f}_B(p,h)=f(p)+\frac{\Delta f(p)}{2(n+2)} h^2 +o(h^2) \quad\text{for}\;\; h\to 0</math>

Similarly, the average value of <math>f </math> over the sphere (the boundary of a ball) with radius <math>h</math> centered at <math>p</math> is:
<math display="block">\overline{f}_S(p,h)=f(p)+\frac{\Delta f(p)}{2n} h^2 +o(h^2) \quad\text{for}\;\; h\to 0.</math>

=== 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 display="block">q = -\varepsilon_0 \Delta\varphi,</math>
where {{math|''ε''<sub>0</sub>}} is the electric constant.

This is a consequence of [[Physics:Gauss's law|Gauss's law]]. Indeed, if {{math|''V''}} is any smooth region with boundary {{math|∂''V''}}, then by Gauss's law the flux of the electrostatic field {{math|'''E'''}} across the boundary is proportional to the charge enclosed:
<math display="block">\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|divergence theorem]]. Since the electrostatic field is the (negative) gradient of the potential, this gives:
<math display="block">-\int_V \operatorname{div}(\operatorname{grad}\varphi)\,dV = \frac1{\varepsilon_0} \int_V q\,dV.</math>

Since this holds for all regions {{mvar|V}}, we must have
<math display="block">\operatorname{div}(\operatorname{grad}\varphi) = -\frac 1 {\varepsilon_0}q</math>

The same approach implies that the negative of the Laplacian of the [[Physics:Gravitational potential|gravitational potential]] is the [[Physics:Mass distribution|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 [[Physics:Poisson's equation|Poisson's equation]].

=== Energy minimization ===
Another motivation for the Laplacian appearing in physics is that solutions to {{math|1=Δ''f'' = 0}} in a region {{math|''U''}} are functions that make the [[Dirichlet energy]] [[Functional (mathematics)|functional]] [[Stationary point|stationary]]:
<math display="block"> E(f) = \frac{1}{2} \int_U \lVert \nabla f \rVert^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 {{mvar|U}}. Then:
<math display="block">\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|1=Δ''f'' = 0}}, then {{math|''E''}} is stationary around {{math|''f''}}. Conversely, if {{math|''E''}} is stationary around {{math|''f''}}, then {{math|1=Δ''f'' = 0}} by the [[Fundamental lemma of calculus of variations|fundamental lemma of calculus of variations]].

== Coordinate expressions ==

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

In '''Cartesian coordinates''',
<math display="block">\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}</math>
where {{mvar|x}} and {{mvar|y}} are the standard Cartesian coordinates of the {{math|''xy''}}-plane.

In '''polar coordinates''',
<math display="block">\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 {{mvar|r}} represents the radial distance and {{mvar|θ}} 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''',
<math display="block">\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 display="block">\Delta f = \frac{1}{\rho} \frac{\partial}{\partial \rho} \left(\rho \frac{\partial f}{\partial \rho} \right) + \frac{1}{\rho^2} \frac{\partial^2 f}{\partial \varphi^2} + \frac{\partial^2 f}{\partial z^2 },</math>
where <math>\rho</math> represents the radial distance, {{math|''φ''}} the azimuth angle and {{math|''z''}} the height.

In '''spherical coordinates''':
<math display="block">\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},</math>
or 
<math display="block">\Delta f = \frac{1}{r} \frac{\partial^2}{\partial r^2} (r f) + \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},</math>
by expanding the first term, these expressions read
<math display="block">\Delta f = \frac{\partial^2 f}{\partial r^2} + \frac{2}{r}\frac{\partial f}{\partial r}+\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},</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|curvilinear coordinates]]''' ({{math|''ξ''<sup>1</sup>, ''ξ''<sup>2</sup>, ''ξ''<sup>3</sup>}}):
<math display="block">\Delta = \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 summation over the repeated indices is implied,
{{math|''g<sup>mn</sup>''}} is the inverse [[Physics:Metric tensor|metric tensor]] and  {{math|Γ''<sup>l</sup> <sub>mn</sub>''}} are the [[Christoffel symbols]] for the selected coordinates.

=== {{mvar|N}} dimensions ===
In arbitrary [[Curvilinear coordinates|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 [[Physics:Metric tensor|metric tensor]], <math> g^{ij} </math>:
<math display="block">\Delta = \frac 1{\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]-[[Biography:Hermann Weyl|Weyl]] formula<ref>Archived at [https://ghostarchive.org/varchive/youtube/20211211/BD2AiFk651E Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20190220065415/https://www.youtube.com/watch?v=BD2AiFk651E&gl=US&hl=en Wayback Machine]{{cbignore}}: {{cite web | last1=Grinfeld | first1=Pavel | title=The Voss-Weyl Formula | website=[[Company:YouTube|YouTube]] | url=https://www.youtube.com/watch?v=BD2AiFk651E&list=PLlXfTHzgMRULkodlIEqfgTS-H1AY_bNtq&index=23 | access-date=9 January 2018 | language=en}}{{cbignore}}</ref> for the [[Divergence#General coordinates|divergence]].

In '''spherical coordinates in {{mvar|N}} dimensions''', with the parametrization {{math|1=''x'' = ''rθ'' ∈ '''R'''<sup>''N''</sup>}} with {{mvar|r}} representing a positive real radius and {{mvar|θ}} an element of the [[Unit sphere|unit sphere]] {{math|''S''<sup>''N''−1</sup>}},
<math display="block"> \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 display="block">\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/>}} so that it is constant along rays, i.e., [[Homogeneous function|homogeneous]] of degree zero.

==Euclidean invariance==
The Laplacian is invariant under all Euclidean transformations: [[Physics:Rotation|rotation]]s and translations.  In two dimensions, for example, this means that:
<math display="block">\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 ''θ'', ''a'', and ''b''.  In arbitrary dimensions,
<math display="block">\Delta (f\circ\rho) =(\Delta f)\circ \rho</math>
whenever ''ρ'' is a rotation, and likewise:
<math display="block">\Delta (f\circ\tau) =(\Delta f)\circ \tau</math>
whenever ''τ'' is a translation. (More generally, this remains true when ''ρ'' is an [[Orthogonal transformation|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==
The [[Spectral theory|spectrum]] of the Laplace operator consists of all eigenvalues {{math|''λ''}} for which there is a corresponding [[Eigenfunction|eigenfunction]] {{math|''f''}} with:
<math display="block">-\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|orthonormal basis]] for the [[Hilbert space]] {{math|[[Lp space|''L''<sup>2</sup>(Ω)]]}}. This result essentially follows from the [[Spectral theorem|spectral theorem]] on [[Compact operator|compact]] [[Self-adjoint operator|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|elliptic operator]] with smooth coefficients on a bounded domain. When {{math|Ω}} is the [[N-sphere|{{mvar|n}}-sphere]], the eigenfunctions of the Laplacian are the [[Physics:Spherical harmonics|spherical harmonics]].

==Vector Laplacian==
The '''vector Laplace operator''', also denoted by <math>\nabla^2</math>, is a [[Differential operator|differential operator]] defined over a [[Vector field|vector field]].<ref>{{cite web | url = http://mathworld.wolfram.com/VectorLaplacian.html | title = Vector Laplacian 
| author = MathWorld}}</ref> The vector Laplacian is similar to the scalar Laplacian; whereas the scalar Laplacian applies to a [[Scalar field|scalar field]] and returns a scalar quantity, the vector Laplacian applies to a [[Vector field|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|vector field]] <math> \mathbf{A} </math> is defined as
<math display="block"> \nabla^2 \mathbf{A} = \nabla(\nabla \cdot \mathbf{A}) - \nabla \times (\nabla \times \mathbf{A}). </math>

In Cartesian coordinates, this reduces to the much simpler form as
<math display="block"> \nabla^2 \mathbf{A} = (\nabla^2 A_x, \nabla^2 A_y, \nabla^2 A_z), </math>
where <math>A_x</math>, <math>A_y</math>, and <math>A_z</math> are the components of the vector field <math>\mathbf{A}</math>, and <math> \nabla^2 </math> 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 coordinates]].

===Generalization===
The Laplacian of any [[Tensor field|tensor field]] <math>\mathbf{T}</math> ("tensor" includes scalar and vector) is defined as the [[Divergence|divergence]] of the [[Gradient|gradient]] of the tensor:
<math display="block">\nabla ^2\mathbf{T} = (\nabla \cdot \nabla) \mathbf{T}.</math>

For the special case where <math>\mathbf{T}</math> is a [[Scalar (mathematics)|scalar]] (a tensor of degree zero), the Laplacian takes on the familiar form.

If <math>\mathbf{T}</math> is a vector (a tensor of first degree), the gradient is a [[Covariant derivative|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:
<math display="block">\nabla \mathbf{T}= (\nabla T_x, \nabla T_y, \nabla T_z) = \begin{bmatrix}
T_{xx} & T_{xy} & T_{xz} \\
T_{yx} & T_{yy} & T_{yz} \\
T_{zx} & T_{zy} & T_{zz}
\end{bmatrix} ,
\text{ where } T_{uv} \equiv \frac{\partial T_u}{\partial v}.</math>

And, in the same manner, a [[Dot product|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:
<math display="block"> \mathbf{A} \cdot \nabla \mathbf{B}
= \begin{bmatrix} A_x & A_y & A_z \end{bmatrix} \nabla \mathbf{B}
= \begin{bmatrix} \mathbf{A} \cdot \nabla B_x & \mathbf{A} \cdot \nabla B_y & \mathbf{A} \cdot \nabla B_z \end{bmatrix}.</math>
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 Navier-Stokes equations for a [[Physics:Newtonian fluid|Newtonian]] [[Physics:Incompressible flow|incompressible flow]]:
<math display="block">\rho \left(\frac{\partial \mathbf{v}}{\partial t}+ ( \mathbf{v} \cdot \nabla ) \mathbf{v}\right)=\rho \mathbf{f}-\nabla p +\mu\left(\nabla ^2 \mathbf{v}\right),</math>
where the term with the vector Laplacian of the [[Physics:Velocity|velocity]] field <math>\mu\left(\nabla ^2 \mathbf{v}\right)</math> represents the [[Physics:Viscosity|viscous]] [[Physics:Stress|stress]]es in the fluid.

Another example is the wave equation for the electric field that can be derived from [[Physics:Maxwell's equations|Maxwell's equations]] in the absence of charges and currents:
<math display="block">\nabla^2 \mathbf{E} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0.</math>

This equation can also be written as:
<math display="block">\Box\, \mathbf{E} = 0,</math>
where <math display="block">\Box\equiv\frac{1}{c^2} \frac{\partial^2}{\partial t^2}-\nabla^2,</math> is the D'Alembertian, used in the [[Physics:Klein–Gordon equation|Klein–Gordon equation]].

== 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 Laplace–Beltrami operator, when applied to a function, is the [[Trace (linear algebra)|trace]] ({{math|tr}}) of the function's [[Hessian matrix|Hessian]]:
<math display="block">\Delta f = \operatorname{tr}\big(H(f)\big)</math>
where the trace is taken with respect to the inverse of the [[Physics:Metric tensor|metric tensor]]. The Laplace–Beltrami operator also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates on [[Tensor field|tensor field]]s, by a similar formula.

Another generalization of the Laplace operator that is available on pseudo-Riemannian manifolds uses the [[Exterior derivative|exterior derivative]], in terms of which the "geometer's Laplacian" is expressed as
<math display="block"> \Delta f = \delta d f .</math>

Here {{mvar|δ}} is the codifferential, which can also be expressed in terms of the [[Hodge star operator|Hodge star]] 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|differential form]]s {{mvar|α}} by
<math display="block">\Delta \alpha = \delta d \alpha + d \delta \alpha .</math>

This is known as the '''[[Laplace–Beltrami operator#Laplace–de Rham 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, or ultrahyperbolic.

In [[Physics:Minkowski space|Minkowski space]] the [[Laplace–Beltrami operator]] becomes the [[D'Alembert operator]] <math>\Box</math> or D'Alembertian:
<math display="block">\square = \frac{1}{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 generalization of the Laplace operator in the sense that it is the differential operator which is invariant under the [[Isometry group|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 [[Physics:Particle physics|particle physics]]. The D'Alembert operator is also known as the wave operator because it is the differential operator appearing in the [[Physics:Wave equation|wave equation]]s, and it is also part of the [[Physics:Klein–Gordon equation|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 {{mvar|x}} direction were measured in meters while the {{mvar|y}} direction were measured in centimeters. Indeed, theoretical physicists usually work in units such that {{math|1=[[Natural units|''c'' = 1]]}} in order to simplify the equation.

The d'Alembert operator generalizes to a hyperbolic operator on [[Pseudo-Riemannian manifold|pseudo-Riemannian manifold]]s.

==See also==
*[[Laplace–Beltrami operator]], generalization to submanifolds in Euclidean space and Riemannian and pseudo-Riemannian manifold.
*The [[Vector Laplacian|vector Laplacian]] operator, a generalization of the Laplacian to [[Vector field|vector field]]s.
*The [[Laplace operators in differential geometry|Laplacian in differential geometry]].
*The [[Discrete Laplace operator|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|image processing]] and [[Computer vision|computer vision]] (see the Laplacian of Gaussian, [[Blob detection|blob detector]], and [[Scale space|scale space]]).
*The [[List of formulas in Riemannian geometry|list of formulas in Riemannian geometry]] contains expressions for the Laplacian in terms of Christoffel symbols.
*[[Weyl's lemma (Laplace equation)]].
*[[Physics:Earnshaw's theorem|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|analysis on fractals]], time scale calculus and [[Discrete exterior calculus|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}}
*[https://feynmanlectures.caltech.edu/II_12.html The Feynman Lectures on Physics Vol. II Ch. 12: Electrostatic Analogs]
*{{citation|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}}.

==Further reading==
*[http://farside.ph.utexas.edu/teaching/em/lectures/node23.html The Laplacian - Richard Fitzpatrick 2006]

==External links==
*{{springer|title=Laplace operator|id=p/l057510}}
*{{MathWorld | urlname=Laplacian | title=Laplacian}}
*[http://ramanujan.math.trinity.edu/rdaileda/teach/s12/m3357/lectures/lecture_3_27_2_short.pdf Laplacian in polar coordinates derivation]
*[https://link.springer.com/article/10.1140/epjs/s11734-021-00317-4 Laplace equations on the fractal cubes and Casimir effect]
{{Calculus topics}}

[[Category:Differential operators]]
[[Category:Elliptic partial differential equations]]
[[Category:Fourier analysis]]
[[Category:Harmonic functions]]
[[Category:Linear operators in calculus]]
[[Category:Multivariable calculus]]

{{Sourceattribution|Laplace operator}}