Laplace operator

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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. In physics, it is used in modeling of wave propagation and heat flow, forming the Helmholtz equation. It is central in electrostatics and fluid mechanics, 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.

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

[edit] Definition

The Laplace operator is a second order differential operator in the n-dimensional Euclidean space, defined as the divergence of the gradient. 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. 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.

[edit] Motivation

[edit] 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:

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

where n is the unit normal to the boundary of V. 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.

[edit] 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. 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. 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.

[edit] Coordinate expressions

[edit] 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.

[edit] 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). 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.

[edit] 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.

[edit] Identities

The Laplacian of a function is the trace of the function's Hessian.

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,

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

[edit] Spectral theory

See also: Hearing the shape of a drum

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Further information might be found on the talk page or at requests for expansion.

[edit] Generalizations

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 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. 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.

[edit] 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. The d'Alembert operator generalizes to a hyperbolic operator on pseudo-Riemannian manifolds. The Laplace-Beltrami operator can also be generalized to an operator (also called the Laplace-Beltrami operator) which operates on tensor fields.

Another way to generalize the Laplace operator to pseudo-Riemannian manifolds is via the Laplace-de Rham operator which operates on differential forms. This is then related to the Laplace-Beltrami operator by the Weitzenböck identity.

[edit] See also

The vector Laplacian operator, a generalization of the Laplacian to vector fields
The discrete Laplace operator is an 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 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)

[edit] References

^ See Evans, Section 2.2.

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. 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.

[edit] 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

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