A vector Laplacian can be defined for a vector by
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
In spherical coordinates, the vector Laplacian
, Tensor Laplacian
, Vector Poisson Equation
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ReferencesMoon, 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.
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Cite this as:
Weisstein, Eric W. "Vector Laplacian."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VectorLaplacian.html