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Laplacian

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

del ^2phi=1/(h_1h_2h_3)[partial/(partialu_1)((h_2h_3)/(h_1)partial/(partialu_1))+partial/(partialu_2)((h_1h_3)/(h_2)partial/(partialu_2))+partial/(partialu_3)((h_1h_2)/(h_3)partial/(partialu_3))]phi,
(1)

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

Note that the operator del ^2 is commonly written as Delta 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

del ^2phi=0,
(2)

the Helmholtz differential equation

del ^2psi+k^2psi=0,
(3)

the wave equation

del ^2psi=1/(v^2)(partial^2psi)/(partialt^2),
(4)

and the Schrödinger equation

ih(partialPsi(x,y,z,t))/(partialt)=[-(h^2)/(2m)del ^2+V(x)]Psi(x,y,z,t).
(5)

The analogous operator obtained by generalizing from three dimensions to four-dimensional spacetime is denoted square ^2 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 (del ^2)^2=del ^4 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.

coordinate systemdel ^2f
Cartesian coordinates(partial^2f)/(partialx^2)+(partial^2f)/(partialy^2)+(partial^2f)/(partialz^2)
cylindrical coordinates1/rpartial/(partialr)(r(partialf)/(partialr))+1/(r^2)(partial^2f)/(partialtheta^2)+(partial^2f)/(partialz^2)
parabolic coordinates1/(uv(u^2+v^2))[partial/(partialu)(uv(partialf)/(partialu))+partial/(partialv)(uv(partialf)/(partialv))]+1/(u^2v^2)(partial^2f)/(partialtheta^2)
parabolic cylindrical coordinates1/(u^2+v^2)((partial^2f)/(partialu^2)+(partial^2f)/(partialv^2))+(partial^2f)/(partialz^2)
spherical coordinates1/(r^2)partial/(partialr)(r^2(partialf)/(partialr))+1/(r^2sin^2phi)(partial^2f)/(partialtheta^2)+1/(r^2sinphi)partial/(partialphi)(sinphi(partialf)/(partialphi))

The finite difference form is

del ^2psi(x,y,z)=1/(h^2)[psi(x+h,y,z)+psi(x-h,y,z)+psi(x,y+h,z)+psi(x,y-h,z)+psi(x,y,z+h)+psi(x,y,z-h)-6psi(x,y,z)].
(6)

For a pure radial function g(r),

del ^2g(r)=del ·[del g(r)]
(7)
=del ·[(partialg(r))/(partialr)r^^+1/r(partialg(r))/(partialtheta)theta^^+1/(rsintheta)(partialg(r))/(partialphi)phi^^]
(8)
=del ·(r^^(dg)/(dr)).
(9)

Using the vector derivative identity

del ·(fA)=f(del ·A)+(del f)·A,
(10)

so

del ^2g(r)=del ·[del g(r)]
(11)
=(dg)/(dr)del ·r^^+del ((dg)/(dr))·r^^
(12)
=2/r(dg)/(dr)+(d^2g)/(dr^2).
(13)

Therefore, for a radial power law,

del ^2r^n=2/rnr^(n-1)+n(n-1)r^(n-2)
(14)
=[2n+n(n-1)]r^(n-2)
(15)
=n(n+1)r^(n-2).
(16)

An identity satisfied by the Laplacian is

del ^2||xA||=(||A||_(HS)^2-||(xA)A^(T)||^2)/(||xA||^3),
(17)

where ||A||_(HS) is the Hilbert-Schmidt norm, x is a row vector, and A^(T) is the transpose of A.

To compute the Laplacian of the inverse distance function 1/r, where r=|r-r^'|, and integrate the Laplacian over a volume,

int_Vdel ^2(1/(|r-r^'|))d^3r.
(18)

This is equal to

int_Vdel ^21/rd^3r=int_Vdel ·(del 1/r)d^3r
(19)
=int_S(del 1/r)·da
(20)
=int_Spartial/(partialr)(1/r)r^^·da
(21)
=int_S-1/(r^2)r^^·da
(22)
=-4pi(R^2)/(r^2),
(23)

where the integration is over a small sphere of radius R. Now, for r>0 and R->0, the integral becomes 0. Similarly, for r=R and R->0, the integral becomes -4pi. Therefore,

del ^2(1/(|r-r^'|))=-4pidelta^3(r-r^'),
(24)

where delta(x) 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.

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Laplacian

Cite this as:

Weisstein, Eric W. "Laplacian." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Laplacian.html

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