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Vector Laplacian

A vector Laplacian can be defined for a vector A by

del ^2A=del (del ·A)-del x(del xA),
(1)

where the notation ✡ is sometimes used to distinguish the vector Laplacian from the scalar Laplacian del ^2 (Moon and Spencer 1988, p. 3). In tensor notation, A is written A_mu, and the identity becomes

del ^2A_mu=A_(mu;lambda)^(;lambda)
(2)
=(g^(lambdakappa)A_(mu;lambda))_(;kappa)
(3)
=g^lambdakappa_(;kappa)A_(mu;lambda)+g^(lambdakappa)A_(mu;lambdakappa).
(4)

A tensor Laplacian may be similarly defined.

In cylindrical coordinates, the vector Laplacian is given by

del ^2v=[(partial^2v_r)/(partialr^2)+1/(r^2)(partial^2v_r)/(partialphi^2)+(partial^2v_r)/(partialz^2)+1/r(partialv_r)/(partialr)-2/(r^2)(partialv_phi)/(partialphi)-(v_r)/(r^2); (partial^2v_phi)/(partialr^2)+1/(r^2)(partial^2v_phi)/(partialphi^2)+(partial^2v_phi)/(partialz^2)+1/r(partialv_phi)/(partialr)+2/(r^2)(partialv_r)/(partialphi)-(v_phi)/(r^2); (partial^2v_z)/(partialr^2)+1/(r^2)(partial^2v_z)/(partialphi^2)+(partial^2v_z)/(partialz^2)+1/r(partialv_z)/(partialr)].
(5)

In spherical coordinates, the vector Laplacian is

del ^2v=[1/r(partial^2(rv_r))/(partialr^2)+1/(r^2)(partial^2v_r)/(partialtheta^2)+1/(r^2sin^2theta)(partial^2v_r)/(partialphi^2)+(cottheta)/(r^2)(partialv_r)/(partialtheta)-2/(r^2)(partialv_theta)/(partialtheta)-2/(r^2sintheta)(partialv_phi)/(partialphi)-(2v_r)/(r^2)-(2cottheta)/(r^2)v_theta ; 1/r(partial^2(rv_theta))/(partialr^2)+1/(r^2)(partial^2v_theta)/(partialtheta^2)+1/(r^2sin^2theta)(partial^2v_theta)/(partialphi^2)+(cottheta)/(r^2)(partialv_theta)/(partialtheta)-2/(r^2)(cottheta)/(sintheta)(partialv_phi)/(partialphi)+2/(r^2)(partialv_r)/(partialtheta)-(v_theta)/(r^2sin^2theta) ; 1/r(partial^2(rv_phi))/(partialr^2)+1/(r^2)(partial^2v_phi)/(partialtheta^2)+1/(r^2sin^2theta)(partial^2v_phi)/(partialphi^2)+(cottheta)/(r^2)(partialv_phi)/(partialtheta)+2/(r^2sintheta)(partialv_r)/(partialphi)+(2cottheta)/(r^2sintheta)(partialv_theta)/(partialphi)-(v_phi)/(r^2sin^2theta) ].
(6)

See also

Derivative, Laplacian, Tensor Laplacian, Vector Derivative, Vector Poisson Equation

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References

Moon, P. and Spencer, D. E. "The Meaning of the Vector Laplacian." J. Franklin Inst. 256, 551-558, 1953.Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, 1988.

Referenced on Wolfram|Alpha

Vector Laplacian

Cite this as:

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

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