Laplace Operator

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.

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Laplace Operator 

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