Laplace Operator

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Laplace operator

[lə′pläs ‚äp·ə‚rād·ər]
(mathematics)
The linear operator defined on differentiable functions which gives for each function the sum of all its nonmixed second partial derivatives. Also known as Laplacian.

Laplace Operator

 

(also Laplacian), a linear differential operator, which associates to the function Φ(x1, x2, . . ., xn) of η variables x1, x2, . . ., xn the function

In particular, if Φ = Φ (x, y) is a function of two variables x, y, then the Laplace operator has the form

and if Φ= Φ (x) is a function of one variable, then the Laplacian of Φ coincides with the second derivative, that is,

The Laplace operator is encountered in those problems of mathematical physics where the properties of an isotropic homogeneous medium (for example, the propagation of light, heat flow, the motion of an ideal incompressible fluid) are studied.

The equation ΔΦ = 0 is usually called the Laplace equation and hence the name Laplace operator.

References in periodicals archive ?
This effect is easily justified in terms of the p-Laplacian operator which, for p = 2, turns into the classical Laplace-Beltrami operator [[DELTA].sub.2].
The Laplace-Beltrami operator is treated only summarily, there is no spectral theory, and the structure theory of Lie algebras is not discussed.
Part 2 is about applications, with chapters on implementation (discrete Laplace-Beltrami operator, generalized eigenvalue problem and matrix power), shape representation, geometry processing, feature definition and detection, and shape matching, registration, and retrieval.
The first part of the book is of interest in itself, as it considers the boundary value problems for the Laplace-Beltrami operator. Author information is not given, and there is no subject index.
Note that both are roots of the Laplace-Beltrami operator [nabla]* of the unit sphere in the sense that