TOPICS

# Vector Laplacian

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

 (2) (3) (4)

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)

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

## Explore with Wolfram|Alpha

More things to try:

## 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.

Vector Laplacian

## Cite this as:

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