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Laplacian (Definition)

The Laplacian is a vector differential operator. Like all vector operators, it is given in different forms in different coordinate systems. In general it is given by:

$\displaystyle \nabla^2 f = \Delta f = \sum_i \frac{\partial f_i}{\partial x^2_i} $
where the subscript $i$ refers to the different coordinate components of the vector $f$.

Laplacian in Cartesian coordinates

As usual with vector operators, the Cartesian form is the easiest to remember and apply.

$\displaystyle \nabla^2 = {\partial \over \partial x^2} + {\partial \over \partial y^2} + {\partial \over \partial z^2} $

Laplacian in spherical coordinates

$\displaystyle \nabla _{sph}^{2} = \frac{1}{r^2} \frac{\partial}{\partial r}\lef... ...\theta}\right) + \frac{1}{r^2 sin^2 \theta} \frac{\partial^2}{\partial \phi^2} $

Laplacian in cylindrical coordinates

$\displaystyle \nabla ^2 = \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{... ...1}{r^2} \frac{\partial^2}{\partial \theta^2} + \frac{\partial^2}{\partial z^2} $



"Laplacian" is owned by invisiblerhino.

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See Also: Laplacian in Spherical Coordinates, Laplacian in Cylindrical Coordinates, Laplacian in Cartesian Coordinates

Other names:  Laplace operator

Cross-references: systems, operators, operator, vector
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This is version 2 of Laplacian, born on 2008-03-25, modified 2008-03-25.
Object id is 276, canonical name is Laplacian.
Accessed 1955 times total.

Classification:
Physics Classification: 02.40.Dr (Euclidean and projective geometries)
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