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 mathematics and physics. In physics, it is used in modeling of wave propagation and heat flow; it occurs in the Helmholtz equation; it is central in electrostatics and represents the kinetic energy term of the Schroedinger 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.

Definition

The Laplace operator is a second order differential operator, defined as the divergence of the gradient:

<math>\Delta := \nabla^2 = \nabla \cdot \nabla <math>

In the n-dimensional Euclidean space, it is the sum of all the unmixed second partial derivatives:

<math>\Delta = \sum_{i=1}^n \frac {\partial^2}{\partial x^2_i}<math>.

Here, it is understood that the <math>x_i<math> are Cartesian coordinates on the space; the equation takes a different form in spherical coordinates and cylindrical coordinates, as shown below.

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

<math>\Delta =

\frac{\partial^2} {\partial x^2} + \frac{\partial^2} {\partial y^2} + \frac{\partial^2} {\partial z^2}. <math>

The Laplacian can also be defined in non-Euclidean spaces. For example, in the Minkowski spacetime the Laplacian becomes the d'Alembert operator or d'Alembertian

<math>\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 } <math>

This 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 was measured in inches, and the y direction was measured in centimeters. Indeed, physicists usually work in units such that c=1 in order to simplify the equation.

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:

<math> \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 \phi^2} + {\partial^2 f \over \partial z^2 }. <math>

In spherical coordinates:

<math> \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}. <math>

See also the article Nabla in cylindrical and spherical coordinates.

Identities

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

<math>\Delta(fg)=(\Delta f)g+2(\nabla f)\cdot(\nabla g)+f(\Delta g).<math>

Laplace-Beltrami operator

The Laplacian may be defined on curved surfaces, and specifically on Riemannian manifolds and pseudo-Riemannian manifolds. In this case, it is still defined as the divergence of the gradient; however, the definitions of the divergence and gradient are modified in order to take into account the curvature. There are several ways to define these. Given a metric tensor <math>g<math> on the manifold, one finds that the volume form in local coordinates is given by

<math>\mathrm{vol}_n := \sqrt{|g|} \;dx^1\wedge \ldots \wedge dx^n<math>

where the <math>dx^i<math> are the 1-forms forming the dual basis to the basis vectors <math>\partial_i := \frac {\partial}{\partial x^i}<math> for the local coordinate system, and <math>\wedge<math> is the wedge product. Here <math>|g|:=|\det g|<math> 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

<math>\mathcal{L}_X \mathrm{vol}_n = (\mbox{div} X) \; \mathrm{vol}_n<math>

where <math>\mathcal{L}_X<math> is the Lie derivative along the vector field X. In local coordinates, one obtains

<math>\mbox{div} X = \frac{1}{\sqrt{|g|}} \partial_i \sqrt {|g|} X^i

<math>

The gradient of a scalar function f may be defined through the inner product <math>\langle\cdot,\cdot\rangle<math> on the manifold, as

<math>\langle \mbox{grad} f(x) , v_x \rangle = df(x)(v_x)<math>

for all vectors <math>v_x<math> anchored at point x in the tangent bundle <math>T_xM<math> of the manifold at point x. Here, df is the exterior derivative of the function f; it is a 1-form taking argument <math>v_x<math>. In local coordinates, one has

<math> \left(\mbox{grad} f\right)^i =

\partial^i f = g^{ij} \partial_j f<math>

Combining these, one can express the Laplacian of a scalar function f in local coordinates as

<math>\Delta f = \mbox{div grad} \; f =

\frac{1}{\sqrt {|g|}} \partial_i \sqrt{|g|} \partial^i f<math>.

Keep in mind that <math>g^{ij}<math> are the components of the inverse of the metric tensor <math>g<math>, so that <math>g^{ij}g_{jk}=\delta^i_k<math> with <math>\delta^i_k<math> the Kronecker delta.

When defined in this way, the Laplacian is more commonly called the Laplace-Beltrami operator. Note that the above definition is, by construction, valid only for scalar functions <math>f:M\rightarrow \mathbb{R}<math>. One may want a more general definition of a Laplacian, valid for k-forms as well as scalar functions; for this, one must turn to the Laplace-deRham operator, defined in the next section.

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

<math>\Delta f = \partial_i \partial^i f + (\partial^i f) \partial_i \ln \sqrt{|g|}.<math>

When <math>|g| = 1<math>, such as in the case of Euclidean space, one then easily obtains

<math>\Delta f = \partial_i \partial^i f<math>

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 cylinderical coordinates, one can similarly regain the expressions for the Laplacian in spherical and cylinderical 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.

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

<math>\int_M df(X) \;\mathrm{vol}_n = – \int_M f \mbox{div} X \;\mathrm{vol}_n <math>     (proof)

where the last equality is an application of Stokes theorem. Note also, the Laplace-Beltrami operator is symmetric:

<math>\int_M f\Delta h \;\mathrm{vol}_n =

\int_M \langle \mbox{grad} f, \mbox{grad} h \rangle \;\mathrm{vol}_n = \int_M h\Delta f \;\mathrm{vol}_n<math>

for functions f and h.

Laplace-de Rham operator

In the genral 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, but on a pseudo-Riemannian manifold it is a hyperbolic operator. The Laplace-de Rham operator is defined by

<math>\Delta= \mathrm{d}\delta+\delta\mathrm{d} = (\mathrm{d}+\delta)^2,\;<math>

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.

One may prove that the Laplace-de Rahm operator is equivalent to the previous definition of the Laplace-Beltrami operator, when acting on a scalar function f. This proof reads as:

<math>\Delta f =

\mathrm{d}\delta f + \delta\mathrm{d}f = \delta \mathrm{d}f = \delta \partial_i f \mathrm{d}x^i = - *\mathrm{d}{*\partial_i f \mathrm{d}x^i} = - *\mathrm{d}(\varepsilon_{i J} \sqrt{|g|}\partial^i f \mathrm{d}x^J)<math>

<math> =

- *\varepsilon_{i J} \partial_j (\sqrt{|g|}\partial^i f) \mathrm{d} x^j\mathrm{d}x^J = - * \frac{1}{\sqrt{|g|}} \partial_i (\sqrt{|g|}\partial^i f) \mathrm{vol}_n = -\frac{1}{\sqrt{|g|}} \partial_i (\sqrt{|g|}\partial^i f),<math>

where ω is the volume form and ε is the completely antisymmetric Levi-Civita symbol. Note that in the above, the italic lower-case index i is a single index, whereas the upper-case roman J stands for all of the remaining (n-1) indecies. 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. Unfortuanatly, Δ is used to denote both; reader beware.

Properties

Given scalar functions f and h, and a real number a, the Laplacian has the following properties:

1. <math>\Delta(af + h) = a\Delta f + \Delta h<math>
2. <math>\Delta(fh) = f \Delta h + 2 \partial_i f \partial^i h + h \Delta f<math>

Proofs of properties

1. Clear from linearity of the exterior derivative.
2. <math>\Delta(fh) =

\delta\mathrm{d}fh = \delta(f\mathrm{d}h + h\mathrm{d}f) =

• \mathrm{d}(f{*\mathrm{d}h}) + *\mathrm{d}(h{*\mathrm{d}f})\;<math>

<math> = *(f\mathrm{d}*\mathrm{d}h +

\mathrm{d}f \wedge *\mathrm{d}h + \mathrm{d}h \wedge *\mathrm{d}f + h\mathrm{d}*\mathrm{d}f) = f*\mathrm{d}*\mathrm{d}h +

• (\mathrm{d}f \wedge *\mathrm{d}h +

\mathrm{d}h \wedge *\mathrm{d}f) + h*\mathrm{d}*\mathrm{d}f<math>

<math> = f \Delta h +

• (\partial_i f \mathrm{d}x^i \wedge

\varepsilon_{jJ} \sqrt{|g|} \partial^j h \mathrm{d}x^J + \partial_i h \mathrm{d}x^i \wedge \varepsilon_{jJ} \sqrt{|g|} \partial^j f \mathrm{d}x^J) + h \Delta f<math>

<math> = f \Delta h +

(\partial_i f \partial^i h + \partial_i h \partial^i f){*\mathrm{vol}_n} + g \Delta f = f \Delta h + 2 \partial_i f \partial^i h + h \Delta f<math>

where f and h are scalar functions.

See also

• Christoffel symbols
• The discrete Laplace operator is an analog of the continuous Laplacian, defined on graphs and grids.

External links

• MathWorld: Laplacian

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).