Editing Laplace operator

{{short description|Differential operator}}
{{about|the mathematical operator|the integral transform|Laplace transform|other uses|List of things named after Pierre-Simon Laplace}}
{{Calculus |Vector}}

In [[mathematics]], the '''Laplace operator''' or '''Laplacian''' is a [[differential operator]] given by the [[divergence]] of the [[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]]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. 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 [[celestial mechanics]]: the Laplacian of the [[gravitational potential]] due to a given mass density distribution is a constant multiple of that density distribution. Solutions of [[Laplace's equation]] {{math|1=Δ''f'' = 0}} are called [[harmonic function]]s and represent the possible [[gravitational potential]]s in regions of [[vacuum]].

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

==Definition==
The Laplace operator is a [[Second-order differential equation|second-order differential operator]] in the ''n''-dimensional [[Euclidean space]], defined as the [[divergence]] (<math>\nabla \cdot</math>) of the [[gradient]] (<math>\nabla f</math>). Thus if <math>f</math> is a [[derivative|twice-differentiable]] [[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]]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|[[Continuously differentiable|''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]] {{math|Ω ⊆ '''R'''<sup>''n''</sup>}}.

Alternatively, the Laplace operator can be defined as:

<math display="block">\nabla^2 f(\vec{x}) = \lim_{R \rightarrow 0} \frac{2n}{R^2} (f_{shell_R} - f(\vec{x})) = \lim_{R \rightarrow 0} \frac{2n}{A_{n-1} R^{1+n}} \int_{shell_R} f(\vec{r}) - f(\vec{x}) d r^{n-1} </math>

where <math>n</math> is the dimension of the space, <math>f_{shell_R}   </math> is the average value of <math>f</math> on the surface of an [[n-sphere]] of radius <math>R</math>, <math>\int_{shell_R} f(\vec{r}) d r^{n-1}</math> is the surface integral over an [[n-sphere]] of radius <math>R</math>, and <math>A_{n-1}</math> is the [[Unit sphere#Volume and area|hypervolume of the boundary of a unit n-sphere]].<ref>{{Cite journal |last=Styer |first=Daniel F. |date=2015-12-01 |title=The geometrical significance of the Laplacian |url=https://pubs.aip.org/aapt/ajp/article-abstract/83/12/992/1057202/The-geometrical-significance-of-the-Laplacian?redirectedFrom=fulltext |journal=American Journal of Physics |volume=83 |issue=12 |pages=992–997 |doi=10.1119/1.4935133 |bibcode=2015AmJPh..83..992S |issn=0002-9505 |archive-url=https://www2.oberlin.edu/physics/dstyer/Electrodynamics/Laplacian.pdf |archive-date=20 November 2015}}</ref>

== Analytic and geometric Laplacians ==

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

* The "analytic" Laplacian, which could be characterized in <math>\R^n</math> as
<math display="block">\Delta=\nabla^2=\sum_{j=1}^n\Big(\frac{\partial}{\partial x_j}\Big)^2,</math>
which is [[Definite_quadratic_form|negative-definite]] in the sense that 
<math display="block">\int_{\R^n}\overline{\varphi(x)}\Delta\varphi(x)\,dx=-\int_{\R^n}|\nabla\varphi(x)|^2\,dx<0</math>
for any [[Smoothness|smooth]] [[Support_(mathematics)#Compact_support|compactly supported]] function <math>\varphi\in C^\infty_c(\R^n)</math> which is not identically zero);
* The "geometric", [[Definite_quadratic_form|positive-definite]] Laplacian defined by 

<math display="block">\Delta=-\nabla^2=-\sum_{j=1}^n\Big(\frac{\partial}{\partial x_j}\Big)^2.</math>

== Motivation ==

=== Diffusion ===
In the [[physics|physical]] theory of [[diffusion]], the Laplace operator 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 {{math|∂''V''}} (also called {{math|''S''}}) of any smooth region {{math|''V''}} is zero, provided there is no source or sink within {{math|''V''}}:
<math display="block">\int_{S} \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 display="block">\int_V \operatorname{div} \nabla u\, dV = \int_{S} \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]]. 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>, 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 [[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]]. 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 [[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|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]].

== 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===
{{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 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 and second 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} \left(\cos \theta \frac{\partial f}{\partial \theta} + \sin \theta \frac{\partial^2 f}{\partial \theta^2} \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 particular, the above is equivalent to

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

where <math>\Delta_{S^2}f</math> is the [[Laplace–Beltrami operator|Laplace-Beltrami operator]] on the unit sphere. 
<!---**************************************************************-->

In general '''[[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 [[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.

=== {{mvar|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 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]-[[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=[[YouTube]] | date=16 April 2014 | 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]] {{math|[[N sphere|''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 transformation]]s: [[rotation]]s and [[Translation (geometry)|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]] 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 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]] 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|{{mvar|n}}-sphere]], the eigenfunctions of the Laplacian are the [[spherical harmonics]].

==Vector Laplacian==
The '''vector Laplace operator''', also denoted by <math>\nabla^2</math>, is a [[differential operator]] defined over a [[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]] and returns a scalar quantity, the vector Laplacian applies to a [[vector field]], returning a vector quantity. When computed in [[orthonormal]] [[Cartesian coordinates]], the returned vector field is equal to the vector field of the scalar Laplacian applied to each vector component.

The '''vector Laplacian''' of a [[vector field]] <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>
This definition can be seen as the [[Helmholtz decomposition]] of the vector Laplacian.

In [[Cartesian coordinate]]s, this reduces to the much simpler expression
<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]] <math>\mathbf{T}</math> ("tensor" includes scalar and vector) is defined as the [[divergence]] of the [[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]] 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]], 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 [[Newtonian fluid|Newtonian]] [[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 [[velocity]] field <math>\mu\left(\nabla ^2 \mathbf{v}\right)</math> represents the [[viscosity|viscous]] [[Stress (physics)|stress]]es in the fluid.

Another example is the wave equation for the electric field that can be derived from [[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 [[Klein–Gordon equation]].

==Some properties==
First of all, we say that a smooth function <math>u \colon \Omega \subset \mathbb R^N \to \mathbb R</math> is superharmonic whenever <math>-\Delta u \geq 0</math>.

Let <math>u \colon \Omega \to \mathbb R</math> be a smooth function, and let <math>K \subset \Omega</math> be a connected compact set. If <math>u</math> is superharmonic, then, for every <math>x \in K</math>, we have
<math display="block"> u(x) \geq \inf_\Omega u + c\lVert u \rVert_{L^1(K)} \;, </math>
for some constant <math>c > 0</math> depending on <math>\Omega</math> and <math>K</math>. <ref>{{Cite book |last=Ponce |first=Augusto C. |date=2016-10-14 |title=Elliptic PDEs, Measures and Capacities |url=https://ems.press/books/etm/141 |access-date=2024-11-26 |series=EMS Tracts in Mathematics |volume=23 |publisher=EMS Press |doi=10.4171/140 |isbn=978-3-03719-140-8 |language=en}}</ref>

== 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 [[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 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]]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 operator|hyperbolic]], or [[ultrahyperbolic operator|ultrahyperbolic]].

In [[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]] 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]]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 {{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]]s.

==See also==
*[[Laplace–Beltrami operator]], generalization to submanifolds in Euclidean space and Riemannian and pseudo-Riemannian manifold.
*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}}
*[https://feynmanlectures.caltech.edu/II_12.html The Feynman Lectures on Physics Vol. II Ch. 12: Electrostatic Analogs]
*{{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}}.

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