Laplace operator
Laplacian

The left-hand side of this equation is the Laplace operator.

The main example for boundary value problems is the Laplace operator,

The Laplace operator in two dimensions is given by

The Laplace operator of particle i is :.

As a matrix representation of the negative discrete Laplace operator

It is also used in numerical analysis as a stand-in for the continuous Laplace operator.

The radial Laplace operator is split in two differential operators

It can be applied to the Laplace operator, the wave equation and the heat equation.

As a simple example, consider the Laplace operator Δ.

The Laplace operator is an example of an unbounded positive linear operator.

The simplest example of an operator on a metric graph is the Laplace operator.

It is a generalization of the Laplace operator, where is allowed to range over .

Δ = / r is the Laplace operator.

The total Laplace operator yields after applying the second differential operator

where is the Laplace operator (sometimes also written ).

The spectrum of the Laplace operator on a finite graph can be conveniently described

It can be obtained by letting the Laplace operator work on the indicator function of some domain D.

We also define the gradient , the Laplace operator and the divergence .

If is the Laplace operator in the x-y plane, the result is:

Let denote the usual Laplace operator.

where we can take the flat spacetime Laplace operator on the right.

The Laplace operator, a differential operator often denoted by the symbol .

The Laplace operator occurs frequently in electrostatics.

is the speed of light in the medium, and is the Laplace operator.

where m is a positive integer, and represents the Laplace operator.

Note that with this definition, the connection Laplacian has negative spectrum.

The Laplacian is given above in terms of spherical polar coordinates.

The Laplacian is more easily computed by noting that .

The same approach implies that the Laplacian of the gravitational potential is the mass distribution.

The energy can only have a minimum at points where the Laplacian of the energy is greater than zero.

All eigenvalues of the normalized Laplacian are real and non-negative.

This is a general fact about elliptic operators, of which the Laplacian is a major example.

The combinatorial Laplacian is not the only operator on graphs to select from.

Note that the standard Laplacian is just .

They include for example the Laplacian of potential theory and other major examples of mathematical physics.

Closely related to the discrete Laplacian is the averaging operator:

This is another self-adjoint operator and, moreover, has the same leading symbol as the spin Laplacian.

The Laplacian of a function is equal to the divergence of the gradient.

For a tensor field, , the laplacian is generally written as:

This definition of the Laplacian is commonly used in numerical analysis and in image processing.

This technique is fairly popular in a variety of fields and is known as the graph laplacian.

The expressions for the gradient, divergence, and Laplacian can be directly extended to n-dimensions.

For the special case where is a scalar (a tensor of rank zero), the Laplacian takes on the familiar form.

As remarked above, the Laplacian is diagonalized by the Fourier transform.

Krylov basis construction uses the actual transition matrix instead of random walk Laplacian.

It is proven here that the Laplacian of each individual component of a magnetic field is zero.

For operators that approximate the underlying continuous Laplacian the eigenvalues are a sequence of positive real numbers.

The Laplacian represents the flux density of the gradient flow of a function.

The Laplacian of the indicator has been used in fluid dyamics, e.g. to model the interfaces between different media.

Specifically, they allow for the direct calculation of the Green's function and the inverse graph Laplacian.