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Laplacian (Definition)

Let $ (x_1, \ldots, x_n)$ be Cartesian coordinates for some open set $ \Omega$ in $ \mathbb{R}^n$. Then the Laplacian differential operator $ \Delta$ is defined as

$\displaystyle \Delta = \frac{\partial^2 }{\partial x_1^2} + \cdots + \frac{\partial^2 }{\partial x_n^2}. $
In other words, if $ f$ is a twice differentiable function $ f:\Omega\to \mathbb{C}$, then
$\displaystyle \Delta f = \frac{\partial^2 f}{\partial x_1^2} + \cdots + \frac{\partial^2 f}{\partial x_n^2}. $
A coordinate independent definition of the Laplacian is $ \Delta = \nabla \cdot \nabla$, i.e., $ \Delta$ is the composition of gradient and divergence.

A harmonic function is one for which the Laplacian vanishes.

Notes

An older symbol for the Laplacian is $ \nabla^2$ - conceptually the scalar product of $ \nabla$ with itself. This form is more favoured by physicists.

Derivation

Click here to see an article that derives the Laplacian in spherical coordinates.



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"Laplacian" is owned by matte. [ full author list (4) | owner history (1) ]
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See Also: D'Alembertian

Also defines:  Laplace operator
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Cross-references: spherical coordinates, PlanetMath, scalar product, vanishes, harmonic function, divergence, gradient, composition, independent, coordinate, differentiable function, differential operator, open set, Cartesian coordinates
There are 19 references to this entry.

This is version 14 of Laplacian, born on 2002-06-04, modified 2008-04-04.
Object id is 3030, canonical name is Laplacian.
Accessed 12546 times total.

Classification:
AMS MSC31B05 (Potential theory :: Higher-dimensional theory :: Harmonic, subharmonic, superharmonic functions)
 31B15 (Potential theory :: Higher-dimensional theory :: Potentials and capacities, extremal length)

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