The Laplacian for a scalar function is a scalar differential operator defined by

(1)

where the
are the scale factors of the coordinate system (Weinberg
1972, p. 109; Arfken 1985, p. 92).

Note that the operator
is commonly written as
by mathematicians (Krantz 1999, p. 16).

The Laplacian is extremely important in mechanics, electromagnetics, wave theory, and quantum mechanics, and appears in Laplace's
equation

(2)

the Helmholtz differential equation

(3)

the wave equation

(4)

and the Schrödinger equation

(5)

The analogous operator obtained by generalizing from three dimensions to four-dimensional spacetime is denoted
and is known as the d'Alembertian . A version of
the Laplacian that operates on vector functions is known as the vector
Laplacian , and a tensor Laplacian can be
similarly defined. The square of the Laplacian is known as the biharmonic
operator .

A vector Laplacian can also be defined, as can
its generalization to a tensor Laplacian .

The following table gives the form of the Laplacian in several common coordinate systems.

The finite difference form is

(6)

For a pure radial function ,

Using the vector derivative identity

(10)

so

Therefore, for a radial power law,

An identity satisfied by the Laplacian is

(17)

where is the Hilbert-Schmidt
norm ,
is a row vector , and is the transpose of .

To compute the Laplacian of the inverse distance function , where , and integrate the Laplacian over a volume,

(18)

This is equal to

where the integration is over a small sphere of radius . Now, for and , the integral becomes 0. Similarly, for and , the integral becomes . Therefore,

(24)

where is the delta
function .

See also Laplacian Matrix ,

Vector
Laplacian ,

Tensor Laplacian
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References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985. Moon,
P. and Spencer, D. E. Field
Theory Handbook, Including Coordinate Systems, Differential Equations, and Their
Solutions, 2nd ed. New York: Springer-Verlag, 1988.
SeeAlso

Antilaplacian , Biharmonic Operator , d'Alembertian , Helmholtz
Differential Equation , Laplace's Equation ,
Schrödinger Equation , Tensor
Laplacian , Vector Laplacian , Wave
Equation

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985. Krantz,
S. G. Handbook
of Complex Variables. Boston, MA: Birkhäuser, p. 16, 1999. Moon,
P. and Spencer, D. E. Field
Theory Handbook, Including Coordinate Systems, Differential Equations, and Their
Solutions, 2nd ed. New York: Springer-Verlag, 1988. Weinberg,
S. Gravitation
and Cosmology: Principles and Applications of the General Theory of Relativity.
New York: Wiley, 1972. Referenced on Wolfram|Alpha Laplacian
Cite this as:
Weisstein, Eric W. "Laplacian." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/Laplacian.html

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