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Laplace operator[lə′pläs ‚äp·ə‚rād·ər]
(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.