Laplace operator

This article is about the mathematical operator. For the Laplace probablity distribution, see Laplace distribution.

In mathematics and physics, the Laplace operator or Laplacian, denoted by $\Delta\,$ or $\nabla^2$ and named after Pierre-Simon de Laplace, is a differential operator, specifically an important case of an elliptic operator, with many applications. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator In Mathematics, an elliptic operator is one of the major types of Differential operator. In physics, it is used in modeling of wave propagation and heat flow, forming the Helmholtz equation. Note The term model has a different meaning in Model theory, a branch of Mathematical logic. The wave equation is an important second-order linear Partial differential equation that describes the propagation of a variety of Waves such as Sound waves The heat equation is an important Partial differential equation which describes the distribution of Heat (or variation in temperature in a given region over time The Helmholtz equation, named for Hermann von Helmholtz, is the Elliptic partial differential equation (\nabla^2 + k^2 A = 0 It is central in electrostatics and fluid mechanics, anchoring in Laplace's equation and Poisson's equation. Electrostatics is the branch of Science that deals with the Phenomena arising from what seems to be stationary Electric charges Since Classical Fluid mechanics is the study of how Fluids move and the Forces on them In Mathematics, Laplace's equation is a Partial differential equation named after Pierre-Simon Laplace who first studied its properties In Mathematics, Poisson's equation is a Partial differential equation with broad utility in Electrostatics, Mechanical engineering and Theoretical In quantum mechanics, it represents the kinetic energy term of the Schrödinger equation. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The kinetic energy of an object is the extra Energy which it possesses due to its motion In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system 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. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, Mathematical physics and the theory of Stochastic processes a harmonic function is a twice continuously differentiable function In Mathematics, Hodge theory is one aspect of the study of the Algebraic topology of a Smooth manifold M. In Mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to Algebraic topology and to Differential topology, capable

1 Definition
2 Motivation
- 2.1 Diffusion
- 2.2 Energy minimization
3 Coordinate expressions
4 Identities
5 Spectral theory
6 Generalizations
- 6.1 Laplace-Beltrami operator
7 See also
8 References
9 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 Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar Thus if f is a twice-differentiable real-valued function, then the Laplacian of f is defined by

$\Delta f = \nabla^2 f = \nabla \cdot \nabla f,$ (1)

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

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

As a second-order differential operator, the Laplace operator maps C^k-functions to C^k-2-functions for k ≥ 2. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function 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 In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability The expression (1) (or equivalently (2)) defines an operator Δ : C^k(Rⁿ) → C^k-2(Rⁿ), or more generally an operator Δ : C^k(Ω) → C^k-2(Ω) for any open set Ω. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in

Motivation

Diffusion

In the physical theory of diffusion, the Laplace operator (via Laplace's equation) arises naturally in the mathematical description of equilibrium. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Diffusion is the net movement of particles (typically molecules from an area of high concentration to an area of low concentration by uncoordinated random movement In Mathematics, Laplace's equation is a Partial differential equation named after Pierre-Simon Laplace who first studied its properties ^[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:

$\int_{\partial V} \nabla u \cdot \mathbf{n}\, dS = 0,$

where n is the unit normal to the boundary of V. In the various subfields of Physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length By the divergence theorem,

$\int_V \mathrm{div} \nabla u\, dx = \int_{\partial V} \nabla u\cdot\mathbf{n}\, dS = 0.$

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

$\mathrm{div} \nabla u = \Delta u = 0.$

The right-hand side of this equation is the Laplace operator. In Vector calculus, the divergence theorem, also known as Gauss&rsquos theorem ( Carl Friedrich Gauss) Ostrogradsky&rsquos theorem ( Mikhail

Energy minimization

Another motivation for the Laplacian appearing in physics is that solutions to $Δ f = 0$ in a region U are functions that make the energy functional

$E(f) = \frac{1}{2} \int_U \Vert \nabla f \Vert^2 \mathrm{d}x$

stationary. In Mathematics, a functional is traditionally a map from a Vector space to the field underlying the vector space which is usually the Real In Mathematics, particularly in Calculus, a stationary point is an input to a function where the Derivative is zero (equivalently the To see this, suppose $f\colon U\to \mathbb{R}$ is a function, and $u\colon U\to \mathbb{R}$ is a function that vanishes on the boundary of U. Then

$\frac{d}{d\varepsilon}\Big|_{\varepsilon = 0} E(f+\varepsilon u) = \int_U \nabla f \cdot \nabla u \, \mathrm{d} x = -\int_U u \Delta f \mathrm{d} x$

where the last equality follows using Green's first identity. In Mathematics, Green's identities are a set of three identities in Vector calculus. This calculation shows that if $Δ f = 0$ , then E is stationary around f. Conversely, if E is stationary around f, then $Δ f = 0$ by the fundamental lemma of calculus of variations. In Mathematics, specifically in the Calculus of variations, the fundamental lemma in the calculus of variations is a lemma that is typically used to transform

Coordinate expressions

Two dimensions

The Laplace operator in two dimensions is given by

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

where x and y are the standard Cartesian coordinates of the xy-plane. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane

Three dimensions

In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems.

In Cartesian coordinates,

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

In cylindrical coordinates,

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

In spherical coordinates:

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

(here $\theta \$ represents the polar angle and $φ$ the azimuthal angle). In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane The cylindrical coordinate system is a three-dimensional Coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial The term ${1 \over r^2} {\partial \over \partial r} \left( r^2 {\partial f \over \partial r} \right)$ can be replaced by its equivalent ${1 \over r} {\partial^2 \over \partial r^2} \left( r f \right)$ as well. See also the article Del in cylindrical and spherical coordinates. This is a list of some Vector calculus formulae of general use in working with various Coordinate systems Table

N dimensions

In spherical coordinates in $N$ dimensions, with the parametrization $x=r\theta \in {\mathbb R}^N$ with $r \in [0,+\infty)$ and $\theta \in S^{N-1}$ ,

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

where $\Delta_{S^{N-1}}$ is the Laplace-Beltrami operator on the $N - 1$ dimensional sphere, or spherical Laplacian. One can also write the term ${\partial^2 f \over \partial r^2} + \frac{N-1}{r} \frac{\partial f}{\partial r}$ equivalently as $\frac{1}{r^{N-1}} \frac{\partial}{\partial r} \Bigl(r^{N-1} \frac{\partial f}{\partial r} \Bigr)$ .

As a consequence, the spherical Laplacian of a function defined on $S^{N-1}\subset{\mathbb R}^N$ can be computed as the ordinary Laplacian of the function extended to ${\mathbb R}^N \setminus\{0\}$ so that it is constant along rays.

Identities

The Laplacian of a function is the trace of the function's Hessian. In Linear algebra, the trace of an n -by- n Square matrix A is defined to be the sum of the elements on the Main diagonal In Mathematics, the Hessian matrix is the Square matrix of second-order Partial derivatives of a function.

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.

$\Delta Y_{\ell m}(\theta,\phi) = -\frac{\ell(\ell+1)}{r^2} Y_{\ell m}(\theta,\phi).$

Therefore,

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

Spectral theory

See also: Hearing the shape of a drum

Generalizations

The Laplacian can be generalized in certain ways to non-Euclidean spaces, where it may be elliptic, hyperbolic, or ultrahyperbolic. To hear the shape of a drum is to infer information about the shape of the Drumhead from the sound it makes i In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry In Mathematics, an elliptic operator is one of the major types of Differential operator. In Mathematics, a hyperbolic partial differential equation is usually a second-order Partial differential equation (PDE of the form A u_{xx} In the mathematical field of Partial differential equations the ultrahyperbolic wave equation is a partial differential equation for an unknown scalar function

In the Minkowski space the Laplacian becomes the d'Alembert operator or d'Alembertian

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

The D'Alembert operator is often used to express the Klein-Gordon equation and the four-dimensional wave equation. In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity In Special relativity, Electromagnetism and wave theory, the d'Alembert operator \Box also called the d'Alembertian or the The Klein–Gordon equation ( Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic version of the Schrödinger The wave equation is an important second-order linear Partial differential equation that describes the propagation of a variety of Waves such as Sound waves The sign in front of the fourth term is negative, while it would have been positive in the Euclidean space. The signature of a Metric tensor (or more generally a nondegenerate Symmetric bilinear form, thought of as Quadratic form) is the number of positive 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. In Physics, natural units are Physical units of Measurement defined in terms of universal Physical constants, such that some chosen physical

Laplace-Beltrami operator

Main article: Laplace-Beltrami operator

The Laplacian can also be generalized to an elliptic operator called the Laplace-Beltrami operator defined on a Riemannian manifold. In Differential geometry, the Laplace operator can be generalized to operate on functions defined on Surfaces or more generally on Riemannian and Pseudo-Riemannian In Differential geometry, the Laplace operator can be generalized to operate on functions defined on Surfaces or more generally on Riemannian and Pseudo-Riemannian In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M The d'Alembert operator generalizes to a hyperbolic operator on pseudo-Riemannian manifolds. In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. The Laplace-Beltrami operator can also be generalized to an operator (also called the Laplace-Beltrami operator) which operates on tensor fields. In Mathematics, Physics and Engineering, a tensor field is a very general concept of variable geometric quantity

Another way to generalize the Laplace operator to pseudo-Riemannian manifolds is via the Laplace-de Rham operator which operates on differential forms. In Differential geometry, the Laplace operator can be generalized to operate on functions defined on Surfaces or more generally on Riemannian and Pseudo-Riemannian In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is This is then related to the Laplace-Beltrami operator by the Weitzenböck identity. In Mathematics, in particular in Differential geometry, Mathematical physics, and Representation theory a Weitzenbock identity expresses a relationship

References

^ See Evans, Section 2. In Mathematics, Weyl's lemma is a result that provides a "very weak" form of the Laplace equation. 2.

Evans, L (1998). Partial Differential Equations. American Mathematical Society. ISBN 978-0821807729.
Feynman, R, Leighton, R, and Sands, M (1970). "Chapter 12: Electrostatic Analogs", The Feynman Lectures on Physics Volume 2. The Feynman Lectures on Physics by Richard Feynman, Robert Leighton, and Matthew Sands is perhaps Feynman's most accessible technical work Addison-Wesley-Longman.
Gilbarg, D and Trudinger, N (2001). Elliptic partial differential equations of second order. Springer. ISBN 978-3540411604.
Schey, H. M. (1996). Div, grad, curl, and all that. W W Norton & Company. ISBN 978-0393969979.

External links

M. A. Shubin (2001), “Laplace operator”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
MathWorld: Laplacian
Derivation of the Laplacian in Spherical coordinates by Swapnil Sunil Jain

The Encyclopaedia of Mathematics is a large reference work in Mathematics.

Dictionary

-noun

(mathematics, physics) An elliptic operator, Δ, used in the modelling of wave propagation, heat flow and many other applications.

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