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Topics in Calculus

Fundamental theorem
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Vector calculus 

Gradient
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Laplacian
Gradient theorem
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Divergence theorem

In mathematics and physics, the Laplace operator or Laplacian, named after Pierre-Simon de Laplace, is a differential operator, specifically an important case of an elliptic operator, with many applications. It is denoted by the symbols Δ, ∇2, or ∇·∇. In physics, it is used in the modeling of wave propagation, heat flow, and fluid mechanics. It is central in electrostatics, where it represents the charge associated to a given potential. It is the main part of Laplace's equation for a conservative potential, Poisson's equation for the gravitational potential associated to a given mass, and the Helmholtz equation for the vibrations of a drum. 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. In image processing and computer vision, the Laplacian is used as a blob detector for tasks such as object recognition and texture analysis as well as in early works on edge detection.

Contents

Definition

The Laplace operator is a second order differential operator in the n-dimensional Euclidean space, defined as the divergence (\nabla\cdot) of the gradient (\nabla f). 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 xi:

\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 Ck-functions to Ck−2-functions for k ≥ 2. The expression (1) (or equivalently (2)) defines an operator Δ : Ck(Rn) → Ck−2(Rn), or more generally an operator Δ : Ck(Ω) → Ck−2(Ω) for any open set Ω.

The Laplacian of a function is also the trace of the function's Hessian:

\Delta f = \mathrm{tr}(H(f)).\,\!

Motivation

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

\int_V \mathrm{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

\mathrm{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.

Density associated to a potential

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

   

q = \Delta\phi.\,

 

 

 

 

(1)

   

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 equal to the charge enclosed (in appropriate units):

\int_{\partial V} \mathbf{E}\cdot \mathbf{n} = \int_{\partial V} \nabla\phi\cdot \mathbf{n} = \int_V q\,dV,

where the first equality uses the fact that the electrostatic field is the gradient of the electrostatic potential. The divergence theorem now gives

\int_V \Delta\phi\,dV = \int_V q\, dV,

and since this holds for all regions V, (1) follows.

The same approach implies that 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 Δf = 0 in a region U are functions that make the Dirichlet energy functional stationary:

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

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 Δf = 0, then E is stationary around f. Conversely, if E is stationary around f, then Δf = 0 by the fundamental lemma of calculus of variations.

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 polar coordinates,

 \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} .

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 \varphi^2}.

(here φ represents the azimuthal angle and θ the polar angle).

N dimensions

In spherical coordinates in N dimensions, with the parametrization x = rθ ∈ R N with r representing a positive real radius and θ an element of the unit sphere SN−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)-sphere, known as the spherical Laplacian. The two radial terms can be equivalently rewritten 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 SN−1 ⊂ R N 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.

Spectral theory

The spectrum of the Laplace operator consists of all eigenvalues λ for which there is a corresponding eigenfunction ƒ with

− Δf = λf.

If Ω is a bounded domain in Rn then the eigenfunctions of the Laplacian are an orthonormal basis in the Hilbert space L2(Ω). 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 Kondrakov embedding theorem).[2] It can also be shown that the eigenfunctions are infinitely differentiable functions.[3] 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 Ω is the n-sphere, the eigenfunctions of the Laplacian are the well-known spherical harmonics.

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

The D'Alembert operator is also known as the wave operator, because it is the differential operator appearing in the four-dimensional wave equation. It is also the leading part of the Klein–Gordon equation. The signs in front of the spatial derivatives are negative, while they would have been positive in the Euclidean space. The additional factor of c is needed 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, physicists usually work in units such that c=1 in order to simplify the equation.

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.

See also

Notes

  1. ^ Evans 1998, §2.2
  2. ^ Gilbarg & Trudinger 2001, Theorem 8.6
  3. ^ Gilbarg & Trudinger 2001, Corollary 8.11

References

External links


Wiktionary

Up to date as of January 15, 2010

Definition from Wiktionary, a free dictionary

Contents

English

Etymology

From Pierre-Simon Laplace.

Adjective

Laplacian (not comparable)

Positive
Laplacian

Comparative
not comparable

Superlative
none (absolute)

  1. Of, or relating to Laplace.

Translations

  • Swedish: Laplace-

Noun

Singular
Laplacian

Plural
uncountable

Laplacian (uncountable)

  1. (mathematics) The Laplace operator.

Translations


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