Laplace Operator
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Laplace Operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ?·?, ?2 (where ? is the nabla operator) or ?. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian ?f(p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f(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, 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 ?f = 0, now called Laplace's equation, are the so-called harmonic functions and represent the possible gravitational fields in regions of vacuum.

The Laplacian occurs in differential equations that describe many physical phenomena, such as electric and gravitational potentials, the diffusion equation for heat and fluid flow, wave propagation, and quantum mechanics. The Laplacian represents the flux density of the gradient flow of a function. For instance, the net rate at which a chemical dissolved in a fluid moves toward or away from some point is proportional to the Laplacian of the chemical concentration at that point; expressed symbolically, the resulting equation is the diffusion equation. For these reasons, it is extensively used in the sciences for modelling a variety of physical phenomena. The Laplacian is the simplest elliptic operator and is at the core of Hodge theory as well as the results of de Rham cohomology. In image processing and computer vision, the Laplacian operator has been used for various tasks, such as blob and edge detection.

## Definition

The Laplace operator is a second-order differential operator in the n-dimensional Euclidean space, defined as the divergence () of the gradient (?f ). Thus if f is a twice-differentiable real-valued function, then the Laplacian of f is defined by:

where the latter notations derive from formally writing:

${\displaystyle \nabla =\left({\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}\right).}$

Equivalently, the Laplacian of f is the sum of all the unmixed second partial derivatives in the Cartesian coordinates xi:

As a second-order differential operator, the Laplace operator maps Ck functions to Ck-2 functions for k >= 2. The expression (1) (or equivalently (2)) defines an operator ? : Ck(Rn) -> Ck-2(Rn), or more generally, an operator ? : Ck(?) -> Ck-2(?) for any open set ?.

## Motivation

### Diffusion

In the physical theory of diffusion, the Laplace operator (via Laplace's equation) arises naturally in the mathematical description of equilibrium.[1] Specifically, if u is the density at equilibrium of some quantity such as a chemical concentration, then the net flux of u through the boundary of any smooth region V is zero, provided there is no source or sink within V:

${\displaystyle \int _{\partial V}\nabla u\cdot \mathbf {n} \,dS=0,}$

where n is the outward unit normal to the boundary of V. By the divergence theorem,

${\displaystyle \int _{V}\operatorname {div} \nabla u\,dV=\int _{\partial V}\nabla u\cdot \mathbf {n} \,dS=0.}$

Since this holds for all smooth regions V, it can be shown that this implies:

${\displaystyle \operatorname {div} \nabla u=\Delta u=0.}$

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.

### Averages

Given a twice continuously differentiable function ${\displaystyle f:{\mathbb {R}}^{n}\to {\mathbb {R}}}$, a point ${\displaystyle p\in {\mathbb {R}}^{n}}$ and a real number ${\displaystyle h>0}$, we let ${\displaystyle {\overline {f}}_{B}(p,h)}$ be the average value of ${\displaystyle f}$ over the ball with radius ${\displaystyle h}$ centered at ${\displaystyle p}$, and ${\displaystyle {\overline {f}}_{S}(p,h)}$ be the average value of ${\displaystyle f}$ over the sphere with radius ${\displaystyle h}$ centered at ${\displaystyle p}$. Then we have:[2]

${\displaystyle {f}_{B}(p,h)=f(p)+{\frac {\Delta f(p)}{2(n+2)}}h^{2}+o(h^{2})\;\;\;\;{\mbox{for}}\;\;h\to 0}$

and

${\displaystyle {f}_{S}(p,h)=f(p)+{\frac {\Delta f(p)}{2n}}h^{2}+o(h^{2})\;\;\;\;{\mbox{for}}\;\;h\to 0.}$

### Density associated with a potential

If ? denotes the electrostatic potential associated to a charge distribution q, then the charge distribution itself is given by the negative of the Laplacian of ?:

${\displaystyle q=-\varepsilon _{0}\Delta \varphi ,}$

where ?0 is the electric constant.

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

${\displaystyle \int _{\partial V}\mathbf {E} \cdot \mathbf {n} \,dS=\int _{V}\operatorname {div} \mathbf {E} \,dV={\frac {1}{\varepsilon _{0}}}\int _{V}q\,dV.}$

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

${\displaystyle -\int _{V}\operatorname {div} (\operatorname {grad} \varphi )\,dV={\frac {1}{\varepsilon _{0}}}\int _{V}q\,dV.}$

So, since this holds for all regions V, we must have

${\displaystyle \operatorname {div} (\operatorname {grad} \varphi )=-{\frac {1}{\varepsilon _{0}}}q}$

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 ?f = 0 in a region U are functions that make the Dirichlet energy functional stationary:

${\displaystyle E(f)={\frac {1}{2}}\int _{U}\Vert \nabla f\Vert ^{2}\,dx.}$

To see this, suppose f : U -> R is a function, and u : U -> R is a function that vanishes on the boundary of U. Then:

${\displaystyle \left.{\frac {d}{d\varepsilon }}\right|_{\varepsilon =0}E(f+\varepsilon u)=\int _{U}\nabla f\cdot \nabla u\,dx=-\int _{U}u\,\Delta f\,dx}$

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

## Coordinate expressions

### Two dimensions

The Laplace operator in two dimensions is given by:

${\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}}$

where x and y are the standard Cartesian coordinates of the xy-plane.

{\displaystyle {\begin{aligned}\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{aligned}}}

where r 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.

${\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}.}$
${\displaystyle \Delta f={\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}\left(\rho {\frac {\partial f}{\partial \rho }}\right)+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}},}$

where ${\displaystyle \rho }$ represents the radial distance, ? the azimuth angle and z the height.

{\displaystyle {\begin{aligned}\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{aligned}}}

where ? represents the azimuthal angle and ? the zenith angle or co-latitude.

In general curvilinear coordinates (?1, ?2, ?3):

${\displaystyle \Delta =\nabla \xi ^{m}\cdot \nabla \xi ^{n}{\frac {\partial ^{2}}{\partial \xi ^{m}\partial \xi ^{n}}}+\nabla ^{2}\xi ^{m}{\frac {\partial }{\partial \xi ^{m}}}=g^{mn}\left({\frac {\partial ^{2}}{\partial \xi ^{m}\partial \xi ^{n}}}-\Gamma _{mn}^{l}{\frac {\partial }{\partial \xi ^{l}}}\right),}$

where summation over the repeated indices is implied, gmn is the inverse metric tensor and ?lmn are the Christoffel symbols for the selected coordinates.

### N dimensions

In arbitrary curvilinear coordinates in N dimensions (?1, ..., ?N), we can write the Laplacian in terms of the inverse metric tensor, ${\displaystyle g^{ij}}$:

${\displaystyle \Delta ={\frac {1}{\sqrt {\det g}}}{\frac {\partial }{\partial \xi ^{i}}}\left({\sqrt {\det g}}g^{ij}{\frac {\partial }{\partial \xi ^{j}}}\right)}$,

from the Voss- Weyl formula[3] for the divergence.

In spherical coordinates in N dimensions, with the parametrization x = r? ? RN with r representing a positive real radius and ? an element of the unit sphere SN-1,

${\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {N-1}{r}}{\frac {\partial f}{\partial r}}+{\frac {1}{r^{2}}}\Delta _{S^{N-1}}f}$

where ?SN-1 is the Laplace-Beltrami operator on the (N - 1)-sphere, known as the spherical Laplacian. The two radial derivative terms can be equivalently rewritten as:

${\displaystyle {\frac {1}{r^{N-1}}}{\frac {\partial }{\partial r}}\left(r^{N-1}{\frac {\partial f}{\partial r}}\right).}$

As a consequence, the spherical Laplacian of a function defined on SN-1 ? RN can be computed as the ordinary Laplacian of the function extended to RN\{0} so that it is constant along rays, i.e., homogeneous of degree zero.

## Euclidean invariance

The Laplacian is invariant under all Euclidean transformations: rotations and translations. In two dimensions, for example, this means that:

${\displaystyle \Delta (f(x\cos \theta -y\sin \theta +a,x\sin \theta +y\cos \theta +b))=(\Delta f)(x\cos \theta -y\sin \theta +a,x\sin \theta +y\cos \theta +b)}$

for all θ, a, and b. In arbitrary dimensions,

${\displaystyle \Delta (f\circ \rho )=(\Delta f)\circ \rho }$

whenever ρ is a rotation, and likewise:

${\displaystyle \Delta (f\circ \tau )=(\Delta f)\circ \tau }$

whenever τ is a translation. (More generally, this remains true when ρ is an orthogonal transformation such as a 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 spectrum of the Laplace operator consists of all eigenvalues ? for which there is a corresponding eigenfunction f with:

${\displaystyle -\Delta f=\lambda f.}$

This is known as the Helmholtz equation.

If ? is a bounded domain in Rn, then the eigenfunctions of the Laplacian are an orthonormal basis for the Hilbert space L2(?). This result essentially follows from the spectral theorem on compact self-adjoint operators, applied to the inverse of the Laplacian (which is compact, by the Poincaré inequality and the Rellich-Kondrachov theorem).[4] It can also be shown that the eigenfunctions are infinitely differentiable functions.[5] 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 ? is the n-sphere, the eigenfunctions of the Laplacian are the spherical harmonics.

## Vector Laplacian

The vector Laplace operator, also denoted by ${\displaystyle \nabla ^{2}}$, is a differential operator defined over a vector field.[6] 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 ${\displaystyle \mathbf {A} }$ is defined as

${\displaystyle \nabla ^{2}\mathbf {A} =\nabla (\nabla \cdot \mathbf {A} )-\nabla \times (\nabla \times \mathbf {A} ).}$

In Cartesian coordinates, this reduces to the much simpler form:

${\displaystyle \nabla ^{2}\mathbf {A} =(\nabla ^{2}A_{x},\nabla ^{2}A_{y},\nabla ^{2}A_{z}),}$

where ${\displaystyle A_{x}}$, ${\displaystyle A_{y}}$, and ${\displaystyle A_{z}}$ are the components of ${\displaystyle \mathbf {A} }$. 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 ${\displaystyle \mathbf {T} }$ ("tensor" includes scalar and vector) is defined as the divergence of the gradient of the tensor:

${\displaystyle \nabla ^{2}\mathbf {T} =(\nabla \cdot \nabla )\mathbf {T} .}$

For the special case where ${\displaystyle \mathbf {T} }$ is a scalar (a tensor of degree zero), the Laplacian takes on the familiar form.

If ${\displaystyle \mathbf {T} }$ 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:

${\displaystyle \nabla \mathbf {T} =(\nabla T_{x},\nabla T_{y},\nabla T_{z})={\begin{bmatrix}T_{xx}&T_{xy}&T_{xz}\\T_{yx}&T_{yy}&T_{yz}\\T_{zx}&T_{zy}&T_{zz}\end{bmatrix}},{\text{ where }}T_{uv}\equiv {\frac {\partial T_{u}}{\partial v}}.}$

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:

${\displaystyle \mathbf {A} \cdot \nabla \mathbf {B} ={\begin{bmatrix}A_{x}&A_{y}&A_{z}\end{bmatrix}}\nabla \mathbf {B} ={\begin{bmatrix}\mathbf {A} \cdot \nabla B_{x}&\mathbf {A} \cdot \nabla B_{y}&\mathbf {A} \cdot \nabla B_{z}\end{bmatrix}}.}$

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 incompressible flow:

${\displaystyle \rho \left({\frac {\partial \mathbf {v} }{\partial t}}+(\mathbf {v} \cdot \nabla )\mathbf {v} \right)=\rho \mathbf {f} -\nabla p+\mu \left(\nabla ^{2}\mathbf {v} \right),}$

where the term with the vector Laplacian of the velocity field ${\displaystyle \mu \left(\nabla ^{2}\mathbf {v} \right)}$ represents the viscous stresses in the fluid.

Another example is the wave equation for the electric field that can be derived from the Maxwell's equations in the absence of charges and currents:

${\displaystyle \nabla ^{2}\mathbf {E} -\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}=0.}$

The previous equation can also be written as:

${\displaystyle \Box \,\mathbf {E} =0,}$

where

${\displaystyle \Box \equiv {\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2},}$

is the D'Alembertian, used in the Klein-Gordon equation.

## 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 defined on a Riemannian manifold. The d'Alembert operator generalizes to a hyperbolic operator on pseudo-Riemannian manifolds. The Laplace-Beltrami operator, when applied to a function, is the trace (tr) of the function's Hessian:

${\displaystyle \Delta f=\operatorname {tr} {\big (}H(f){\big )}}$

where the trace is taken with respect to the inverse of the metric tensor. The Laplace-Beltrami operator also can be generalized to an operator (also called the Laplace-Beltrami operator) which operates on tensor fields, by a similar formula.

Another generalization of the Laplace operator that is available on pseudo-Riemannian manifolds uses the exterior derivative, in terms of which the "geometer's Laplacian" is expressed as

${\displaystyle \Delta f=\delta df.}$

Here ? is the codifferential, which can also be expressed in terms of the 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 forms ? by

${\displaystyle \Delta \alpha =\delta d\alpha +d\delta \alpha .}$

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, hyperbolic, or ultrahyperbolic.

In the Minkowski space the Laplace-Beltrami operator becomes the D'Alembert operator ? or D'Alembertian:

${\displaystyle \square ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-{\frac {\partial ^{2}}{\partial x^{2}}}-{\frac {\partial ^{2}}{\partial y^{2}}}-{\frac {\partial ^{2}}{\partial z^{2}}}.}$

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 equations, and it is also part of the Klein-Gordon equation, which reduces to the wave equation in the massless case.

The additional factor of c in the metric is needed in physics if space and time are measured in different units; a similar factor would be required if, for example, the x direction were measured in meters while the y direction were measured in centimeters. Indeed, theoretical physicists usually work in units such that c = 1 in order to simplify the equation.

## Notes

1. ^ Evans 1998, §2.2
2. ^ Ovall, Jeffrey S. (2016-03-01). "The Laplacian and Mean and Extreme Values" (PDF). The American Mathematical Monthly. 123 (3): 287-291.
3. ^ Grinfeld, Pavel. "The Voss-Weyl Formula". Retrieved 2018.
4. ^ Gilbarg & Trudinger 2001, Theorem 8.6
5. ^ Gilbarg & Trudinger 2001, Corollary 8.11
6. ^ MathWorld. "Vector Laplacian".