A vector Laplacian can be defined for a vector by

(1)

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

A tensor Laplacian may be similarly defined.

In cylindrical coordinates , the vector
Laplacian is given by

(5)

In spherical coordinates , the vector Laplacian
is

(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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