NationMaster - Encyclopedia: Laplace operator

In mathematics and physics, the Laplace operator or Laplacian, denoted by Δ, is a differential operator, specifically an important case of an elliptic operator, with many applications. In physics, it is used in modeling of wave propagation and heat flow, forming the Helmholtz equation. It is central in electrostatics, anchoring in Laplace's equation and Poisson's equation. In quantum mechanics, it represents the kinetic energy term of the Schrödinger equation. In mathematics, functions with vanishing Laplacian are called harmonic functions; the Laplacian is at the core of Hodge theory and the results of de Rham cohomology. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Physics (from the Greek, Ï†Ï…ÏƒÎ¹ÎºÏŒÏ‚ (physikos), natural, and Ï†ÏÏƒÎ¹Ï‚ (physis), nature) is the science of the natural world, dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... In mathematics, an elliptic operator is one of the major types of differential operator P. It can also be defined on spaces of complex-valued functions, or some more general function-like objects. ... A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. ... The wave equation is an important partial differential equation that describes a variety of waves, such as sound waves, light waves and water waves. ... The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... The Helmholtz equation, named for Hermann von Helmholtz, is the following elliptic partial differential equation: The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. ... Electrostatics is the branch of physics that deals with the forces exerted by a static (i. ... In mathematics, Laplaces equation is a partial differential equation named after its discoverer Pierre-Simon Laplace. ... In mathematics, Poissons equation is a partial differential equation with broad utility in electrostatics, mechanical engineering and theoretical physics. ... For a non-technical introduction to the topic, please see Introduction to Quantum mechanics. ... Kinetic jkljfkdffmdklcjenergy (SI unit: the [[klof its motion. ... In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, is the definition of energy of a quantum system. ... Partial plot of a function f. ... In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U â†’ R (where U is an open subset of Rn) which satisfies Laplaces equation, i. ... In mathematics, Hodge theory is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric... In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. ...

1 Definition
- 1.1 Coordinate expressions
- 1.2 Identities
2 Laplace-Beltrami operator
3 Laplace-de Rham operator
- 3.1 Properties
4 See also
5 References
6 External links

Definition

The Laplace operator is a second order differential operator in the n-dimensional Euclidean space, defined as the divergence of the gradient: In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ...

$Delta = nabla^2 = nabla cdot nabla.$

Equivalently, the Laplacian is the sum of all the unmixed second partial derivatives: In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ...

$Delta = sum_{i=1}^n frac {partial^2}{partial x^2_i}.$

Here, it is understood that the $x i$ are Cartesian coordinates on the space; the equation takes a different form in spherical coordinates and cylindrical coordinates, as shown below. Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ...

In the three-dimensional space the Laplacian is commonly written as

$Delta = frac{partial^2} {partial x^2} + frac{partial^2} {partial y^2} + frac{partial^2} {partial z^2}.$

As we shall see later, the Laplacian can be generalized to non-Euclidean spaces, where it may be elliptic or hyperbolic. For example, in the Minkowski space the Laplacian becomes the d'Alembert operator or d'Alembertian The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... In mathematics, an elliptic operator is one of the major types of differential operator P. It can also be defined on spaces of complex-valued functions, or some more general function-like objects. ... A hyperbolic partial differential equation is usually a second-order partial differential equation of the form with . ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In special relativity, electromagnetism and wave theory, the dAlembert operator , also called the dAlembertian or the Wave operator, is the Laplace operator of Minkowski space and other solutions of the Einstein equation. ...

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

The D'Alembert operator is often used to express the Klein-Gordon equation and the four-dimensional wave equation. The sign in front of the fourth term is negative, while it would have been positive in the Euclidean space. The additional factor of c is required because space and time are usually measured in different units; a similar factor would be required if, for example, the x direction were measured in inches, and the y direction were measured in centimeters. Indeed, physicists usually work in units such that c=1 in order to simplify the equation. The Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is the relativistic version of the SchrÃ¶dinger equation. ... The wave equation is an important partial differential equation that describes a variety of waves, such as sound waves, light waves and water waves. ... The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ...

Coordinate expressions

In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems. Given a function f, in cylindrical coordinates, one has: This article describes some of the common coordinate systems that appear in elementary mathematics. ...

$Delta f = {1 over r} {partial over partial r} left( r {partial f over partial r} right) + {1 over r^2} {partial^2 f over partial theta^2} + {partial^2 f over partial z^2 }.$

In spherical coordinates: This article describes some of the common coordinate systems that appear in elementary mathematics. ...

$Delta f = {1 over r^2} {partial over partial r} left( r^2 {partial f over partial r} right) + {1 over r^2 sin theta} {partial over partial theta} left( sin theta {partial f over partial theta} right) + {1 over r^2 sin^2 theta} {partial^2 f over partial phi^2}.$

The spherical coordinates Laplacian can also be written in this form:

$Delta f = {1 over r} {partial^2 over partial r^2} left( rf right) + {1 over r^2 sin theta} {partial over partial theta} left( sin theta {partial f over partial theta} right) + {1 over r^2 sin^2 theta} {partial^2 f over partial phi^2}.$

See also the article Nabla in cylindrical and spherical coordinates. This is a list of some vector calculus formulae of general use in working with standard coordinate systems. ...

Identities

If f and g are functions, then the Laplacian of the product is given by

$Delta(fg)=(Delta f)g+2(nabla f)cdot(nabla g)+f(Delta g).$

Note the special case where f is a radial function $f (r)$ and g is a spherical harmonic, $Y l m (θ,φ)$ . One encounters this special case in numerous physical models. The gradient of $f (r)$ is a radial vector and the gradient of an angular function is tangent to the radial vector, therefore

$2(nabla f(r))cdot(nabla Y_{lm}(theta,phi))=0.$

In addition, the spherical harmonics have the special property of being eigenfunctions of the angular part of the Laplacian in spherical coordinates.

$Delta Y_{lm}(theta,phi) = -frac{l(l+1)}{r^2} Y_{lm}(theta,phi)$

Therefore,

$Delta( f(r)Y_{lm}(theta,phi) ) = left(frac{d^2f(r)}{dr^2} + frac{2}{r} frac{df(r)}{dr} - frac{l(l+1)}{r^2} f(r)right)Y_{lm}(theta,phi)$

Laplace-Beltrami operator

The Laplacian can be extended to functions defined on surfaces, or more generally, on Riemannian and pseudo-Riemannian manifolds. This more general operator goes by the name Laplace-Beltrami operator. One defines it, just as the Laplacian, as the divergence of the gradient. To be able to find a formula for this operator, one will need to first write the divergence and the gradient on a manifold. An open surface with X-, Y-, and Z-contours shown. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ...

If $g$ denotes the (pseudo)-metric tensor on the manifold, one finds that the volume form in local coordinates is given by In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ... In mathematics, the volume form is a differential form that represents a unit volume of a Riemannian manifold or a pseudo-Riemannian manifold. ... Local coordinates are measurement indices into a local coordinate system or a local coordinate space. ...

$mathrm{vol}_n := sqrt{|g|} ;dx^1wedge ldots wedge dx^n$

where the $d x i$ are the 1-forms forming the dual basis to the basis vectors (Redirected from 1-form) A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, the linear algebra concept of dual basis can be applied in the context of a finite extension L/K, by using the field trace. ...

$partial_i := frac {partial}{partial x^i}$

for the local coordinate system, and $wedge$ is the wedge product. Here $| g | : = | det g |$ is the absolute value of the determinant of the metric tensor. The divergence of a vector field X on the manifold can then be defined as In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In algebra, a determinant is a function depending on n that associates a scalar det(A) to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

$mathcal{L}_X mathrm{vol}_n = (mbox{div} X) ; mathrm{vol}_n$

where $mathcal{L}_X$ is the Lie derivative along the vector field X. In local coordinates, one obtains In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of smooth functions over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

$mbox{div} X = frac{1}{sqrt{|g|}} partial_i sqrt {|g|} X^i$

Here (and below) we use the Einstein notation, so the above is actually a sum in i. In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae. ...

The gradient of a scalar function f may be defined through the inner product $langlecdot,cdotrangle$ on the manifold, as // Definition Inner Product of two vectors Given twoN-by-1 column vectors v and u, the inner product is defined as the scalar quantity Î± resulting from where or equivalently indicates the conjugate transpose operator applied to vector v. ...

$langle mbox{grad} f(x) , v_x rangle = df(x)(v_x)$

for all vectors $v x$ anchored at point x in the tangent bundle $T x M$ of the manifold at point x. Here, df is the exterior derivative of the function f; it is a 1-form taking argument $v x$ . In local coordinates, one has In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x âˆˆ M and v âˆˆ Tx(M), the tangent space... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ...

$left(mbox{grad} fright)^i = partial^i f = g^{ij} partial_j f$

Combining these, the formula for the Laplace-Beltrami operator applied to a scalar function f is, in local coordinates

$Delta f = mbox{div grad} ; f = frac{1}{sqrt {|g|}} partial_i sqrt{|g|} partial^i f$ .

Here, $g i j$ are the components of the inverse of the metric tensor $g$ , so that $g^{ij}g_{jk}=delta^i_k$ with $delta^i_k$ the Kronecker delta. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

Note that the above definition is, by construction, valid only for scalar functions $f:Mrightarrow mathbb{R}$ . One may want to extend the Laplacian even further, to differential forms; for this, one must turn to the Laplace-deRham operator, defined in the next section. A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

Under local parametrization $u 1, u 2$ , the Laplace-Beltrami operator can be expanded in terms of the metric tensor and Christoffel symbols as follows: In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829-1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ...

$Delta f = g^{ij}(frac{partial^2 f}{partial u^i partial u^j} - Gamma_{ij}^k frac{partial f}{partial u^k} )$

One may show that the Laplace-Beltrami operator reduces to the ordinary Laplacian in Euclidean space by noting that it can be re-written using the chain rule as In calculus, the chain rule is a formula for the derivative of the composite of two functions. ...

$Delta f = partial_i partial^i f + (partial^i f) partial_i ln sqrt{|g|}.$

When $| g | = 1$ , such as in the case of Euclidean space, one then easily obtains In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

$Delta f = partial_i partial^i f$

which is the ordinary Laplacian. Using the Minkowski metric with signature (+++-), one regains the D'Alembertian given previously. Note also that by using the metric tensor for spherical and cylindrical coordinates, one can similarly regain the expressions for the Laplacian in spherical and cylindrical coordinates. The Laplace-Beltrami operator is handy not just in curved space, but also in ordinary flat space endowed with a non-linear coordinate system. In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ... In special relativity, electromagnetism and wave theory, the dAlembert operator, also called dAlembertian, is the Laplace operator of Minkowski space. ...

Note that the exterior derivative d and -div are adjoint:

$int_M df(X) ;mathrm{vol}_n = - int_M f mbox{div} X ;mathrm{vol}_n$ (proof)

where the last equality is an application of Stokes theorem. Note also, the Laplace-Beltrami operator is symmetric: This mathematics article is devoted entirely to providing proofs and backup support for claims and statements made in the article Laplace operator. ... Stokes Theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...

$int_M fDelta h ;mathrm{vol}_n = int_M langle mbox{grad} f, mbox{grad} h rangle ;mathrm{vol}_n = int_M hDelta f ;mathrm{vol}_n$

for functions f and h.

Laplace-de Rham operator

In the general case of differential geometry, one defines the Laplace-de Rham operator as the generalization of the Laplacian. It is a differential operator on the exterior algebra of a differentiable manifold. On a Riemannian manifold it is an elliptic operator, while on a Lorentzian manifold it is hyperbolic. The Laplace-de Rham operator is defined by In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... In mathematics, the exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, an elliptic operator is one of the major types of differential operator P. It can also be defined on spaces of complex-valued functions, or some more general function-like objects. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... A hyperbolic partial differential equation is usually a second-order partial differential equation of the form with . ...

$Delta= mathrm{d}delta+deltamathrm{d} = (mathrm{d}+delta)^2,;$

where d is the exterior derivative or differential and δ is the codifferential. When acting on scalar functions, the codifferential may be defined as δ = −∗d∗, where ∗ is the Hodge star; more generally, the codifferential may include a sign that depends on the order of the k-form being acted on. In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... In mathematics, the Hodge star operator is a linear map on the exterior algebra of an oriented inner product space which establishes a correspondence between the space of k-vectors and the space of (n-k)-vectors. ... In mathematics, the Hodge star operator on an oriented inner product space V is a linear operator on the exterior algebra of V, interchanging the subspaces of k-vectors and n−k-vectors where n = dim V, for 0 ≤ k ≤ n. ...

One may prove that the Laplace-de Rham operator is equivalent to the previous definition of the Laplace-Beltrami operator when acting on a scalar function f; see the Laplace operator article proofs for details. Notice that the Laplace-de Rham operator is actually minus the Laplace-Beltrami operator; this minus sign follows from the conventional definition of the properties of the codifferential. Unfortunately, Δ is used to denote both; which can sometimes be a source of confusion. This mathematics article is devoted entirely to providing proofs and backup support for claims and statements made in the article Laplace operator. ... In mathematics, the Hodge star operator is a linear map on the exterior algebra of an oriented inner product space which establishes a correspondence between the space of k-vectors and the space of (n-k)-vectors. ...

Properties

Given scalar functions f and h, and a real number a, the Laplace-de Rham operator has the following properties:

$Delta(af + h) = aDelta f + Delta h!$
$Delta(fh) = f Delta h + 2 partial_i f partial^i h + h Delta f$ (proof)

This mathematics article is devoted entirely to providing proofs and backup support for claims and statements made in the article Laplace operator. ...

References

Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0. (Provides a basic review of differential geometry in the special case of four-dimensional space-time.)
Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-4267-2 . (Provides a general introduction to curved surfaces).

External links

MathWorld: Laplacian
Derivation of the Laplacian in Spherical coordinates

Categories: Multivariate calculus | Riemannian geometry

Results from FactBites:

Laplace operator - definition of Laplace operator in Encyclopedia (229 words)
	In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable.
	Since it can be calculated as Δφ = div(grad φ) the Laplace operator is also written as
	The operator occurs for example in Laplace's equation and Poisson's equation.

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