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# Laplacian

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|                                      | The n-dimensional Laplacian operator in Cartesian coordinates is defined by                                                                                                             |                                                                                                      | The n-dimensional Laplacian operator in Cartesian coordinates is defined by                                                                                                                                                                                                                                                                                                                   |
|                                      |                                                                                                                                                                                         |                                                                                                      |                                                                                                                                                                                                                                                                                                                                                                                               |
| -                                    | <math>                                                                                                                                                                                  | +                                                                                                    | :<math>                                                                                                                                                                                                                                                                                                                                                                                       |
|                                      | \Delta u = \sum_{i=1}^n \frac{\partial^2 u}{\partial x_i^2}                                                                                                                             |                                                                                                      | \Delta u = \sum_{i=1}^n \frac{\partial^2 u}{\partial x_i^2}                                                                                                                                                                                                                                                                                                                                   |
|                                      | </math>                                                                                                                                                                                 |                                                                                                      | </math>                                                                                                                                                                                                                                                                                                                                                                                       |
|                                      |                                                                                                                                                                                         |                                                                                                      |                                                                                                                                                                                                                                                                                                                                                                                               |
| -                                    | It is an important differential operator which occurs in many equations of mathematical physics and is usually associated with dissipative effects. Some of the important equations are | +                                                                                                    | It is an important differential operator which occurs in many equations of mathematical physics and is usually associated with dissipative effects (except in the case of [[wave equation]]). Some of the important equations are                                                                                                                                                             |
|                                      |                                                                                                                                                                                         |                                                                                                      |                                                                                                                                                                                                                                                                                                                                                                                               |
| -                                    | * Laplace equation                                                                                                                                                                      | +                                                                                                    | * [[Laplace equation]]                                                                                                                                                                                                                                                                                                                                                                        |
|                                      |                                                                                                                                                                                         |                                                                                                      |                                                                                                                                                                                                                                                                                                                                                                                               |
| -                                    | <math>                                                                                                                                                                                  | +                                                                                                    | :<math>                                                                                                                                                                                                                                                                                                                                                                                       |
|                                      | \Delta u = 0                                                                                                                                                                            |                                                                                                      | \Delta u = 0                                                                                                                                                                                                                                                                                                                                                                                  |
|                                      | </math>                                                                                                                                                                                 |                                                                                                      | </math>                                                                                                                                                                                                                                                                                                                                                                                       |
|                                      |                                                                                                                                                                                         |                                                                                                      |                                                                                                                                                                                                                                                                                                                                                                                               |
| -                                    | * Poisson equation                                                                                                                                                                      | +                                                                                                    | * [[Poisson equation]]                                                                                                                                                                                                                                                                                                                                                                        |
|                                      |                                                                                                                                                                                         |                                                                                                      |                                                                                                                                                                                                                                                                                                                                                                                               |
| -                                    | <math>                                                                                                                                                                                  | +                                                                                                    | :<math>                                                                                                                                                                                                                                                                                                                                                                                       |
| -                                    | \Delta u = f                                                                                                                                                                            | +                                                                                                    | -\Delta u = f                                                                                                                                                                                                                                                                                                                                                                                 |
|                                      | </math>                                                                                                                                                                                 |                                                                                                      | </math>                                                                                                                                                                                                                                                                                                                                                                                       |
|                                      |                                                                                                                                                                                         |                                                                                                      |                                                                                                                                                                                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         | +                                                                                                    | * [[Heat equation]]                                                                                                                                                                                                                                                                                                                                                                           |
|                                      |                                                                                                                                                                                         | +                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         | +                                                                                                    | :<math>                                                                                                                                                                                                                                                                                                                                                                                       |
|                                      |                                                                                                                                                                                         | +                                                                                                    | \frac{\partial u}{\partial t} - \Delta u = 0                                                                                                                                                                                                                                                                                                                                                  |
|                                      |                                                                                                                                                                                         | +                                                                                                    | </math>                                                                                                                                                                                                                                                                                                                                                                                       |
|                                      |                                                                                                                                                                                         | +                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         | +                                                                                                    | * [[Wave equation]]                                                                                                                                                                                                                                                                                                                                                                           |
|                                      |                                                                                                                                                                                         | +                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         | +                                                                                                    | :<math>                                                                                                                                                                                                                                                                                                                                                                                       |
|                                      |                                                                                                                                                                                         | +                                                                                                    | \frac{\partial^2 u}{\partial t^2} - \Delta u = 0                                                                                                                                                                                                                                                                                                                                              |
|                                      |                                                                                                                                                                                         | +                                                                                                    | </math>                                                                                                                                                                                                                                                                                                                                                                                       |
|                                      | Solutions of these equations are very smooth and in most cases are infinitely differentiable (when the associated data of the problem are sufficiently smooth).                         |                                                                                                      | Solutions of these equations are very smooth and in most cases are infinitely differentiable (when the associated data of the problem are sufficiently smooth).                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         |                                                                                                      |                                                                                                                                                                                                                                                                                                                                                                                               |
| -                                    | The Laplacian operator is invariant under coordinate rotation.                                                                                                                          | +                                                                                                    | Folland (see reference below) explains the ubiquitous appearance of the Laplacian.                                                                                                                                                                                                                                                                                                            |
|                                      |                                                                                                                                                                                         | +                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         | +                                                                                                    | ''Why is it so ubiquitous ? The answer, which we shall prove, is that it commutes with translations and rotations and generates the ring of all differential operators with this property. Hence, the Laplacian is likely to turn up in the description of any physical process whose underlying physics is homogeneous (independent of position) and isotropic (independent of direction).'' |
|                                      |                                                                                                                                                                                         | +                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         | +                                                                                                    | Moreover it can shown that any linear operator which commutes with translations and rotations must be a polynomial in <math>\Delta</math>, i.e., it must be of the form <math>\sum_j a_j \Delta^j</math> where the <math>a_j</math> are constants (see Folland).                                                                                                                              |
|                                      |                                                                                                                                                                                         | +                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         | +                                                                                                    | The Laplacian operator is invariant under coordinate translation and rotation.                                                                                                                                                                                                                                                                                                                |
|                                      |                                                                                                                                                                                         | +                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         | +                                                                                                    | The Laplace operator is also denoted as <math>\nabla^2</math> since it is the divergence of the gradient operator                                                                                                                                                                                                                                                                             |
|                                      |                                                                                                                                                                                         | +                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         | +                                                                                                    | :<math>                                                                                                                                                                                                                                                                                                                                                                                       |
|                                      |                                                                                                                                                                                         | +                                                                                                    | \Delta = \nabla^2 = \nabla \cdot \nabla                                                                                                                                                                                                                                                                                                                                                       |
|                                      |                                                                                                                                                                                         | +                                                                                                    | </math>                                                                                                                                                                                                                                                                                                                                                                                       |
|                                      |                                                                                                                                                                                         | +                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         | +                                                                                                    | == Laplacian in cylindrical coordinates ==                                                                                                                                                                                                                                                                                                                                                    |
|                                      |                                                                                                                                                                                         | +                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         | +                                                                                                    | If <math>(x,r,\phi)</math> are cylindrical coordinates, then the Laplacian of a scalar field variable <math>u</math> is                                                                                                                                                                                                                                                                       |
|                                      |                                                                                                                                                                                         | +                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         | +                                                                                                    | :<math>                                                                                                                                                                                                                                                                                                                                                                                       |
|                                      |                                                                                                                                                                                         | +                                                                                                    | \Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial u}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 u}{\partial \phi^2}                                                                                                                                                                                                    |
|                                      |                                                                                                                                                                                         | +                                                                                                    | </math>                                                                                                                                                                                                                                                                                                                                                                                       |
|                                      |                                                                                                                                                                                         | +                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         | +                                                                                                    | == Laplacian in spherical coordinates ==                                                                                                                                                                                                                                                                                                                                                      |
|                                      |                                                                                                                                                                                         | +                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         | +                                                                                                    | If <math>(r,\theta,\phi)</math> are spherical coordinates, then the Laplacian of a scalar field variable <math>u</math> is                                                                                                                                                                                                                                                                    |
|                                      |                                                                                                                                                                                         | +                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         | +                                                                                                    | :<math>                                                                                                                                                                                                                                                                                                                                                                                       |
|                                      |                                                                                                                                                                                         | +                                                                                                    | \Delta u =                                                                                                                                                                                                                                                                                                                                                                                    |
|                                      |                                                                                                                                                                                         | +                                                                                                    | \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial u}{\partial r} \right) +                                                                                                                                                                                                                                                                                                  |
|                                      |                                                                                                                                                                                         | +                                                                                                    | \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial \theta} \left( \sin\theta \frac{\partial u}{\partial \theta} \right) +                                                                                                                                                                                                                                                                      |
|                                      |                                                                                                                                                                                         | +                                                                                                    | \frac{1}{r^2 \sin^2\theta} \frac{\partial^2 u}{\partial \phi^2}                                                                                                                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         | +                                                                                                    | </math>                                                                                                                                                                                                                                                                                                                                                                                       |
|                                      |                                                                                                                                                                                         | +                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         | +                                                                                                    | == References ==                                                                                                                                                                                                                                                                                                                                                                              |
|                                      |                                                                                                                                                                                         | +                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                               |
|                                      |                                                                                                                                                                                         | +                                                                                                    | * {{reference-book | author=Gerald B. Folland | year=1995 | title=Introduction to partial differential equations | rest=Princeton University Press}}                                                                                                                                                                                                                                          |

## Latest revision as of 08:19, 12 April 2007

The n-dimensional Laplacian operator in Cartesian coordinates is defined by

$\Delta u = \sum_{i=1}^n \frac{\partial^2 u}{\partial x_i^2}$

It is an important differential operator which occurs in many equations of mathematical physics and is usually associated with dissipative effects (except in the case of wave equation). Some of the important equations are

$\Delta u = 0$
$-\Delta u = f$
$\frac{\partial u}{\partial t} - \Delta u = 0$
$\frac{\partial^2 u}{\partial t^2} - \Delta u = 0$

Solutions of these equations are very smooth and in most cases are infinitely differentiable (when the associated data of the problem are sufficiently smooth).

Folland (see reference below) explains the ubiquitous appearance of the Laplacian.

Why is it so ubiquitous ? The answer, which we shall prove, is that it commutes with translations and rotations and generates the ring of all differential operators with this property. Hence, the Laplacian is likely to turn up in the description of any physical process whose underlying physics is homogeneous (independent of position) and isotropic (independent of direction).

Moreover it can shown that any linear operator which commutes with translations and rotations must be a polynomial in $\Delta$, i.e., it must be of the form $\sum_j a_j \Delta^j$ where the $a_j$ are constants (see Folland).

The Laplacian operator is invariant under coordinate translation and rotation.

The Laplace operator is also denoted as $\nabla^2$ since it is the divergence of the gradient operator

$\Delta = \nabla^2 = \nabla \cdot \nabla$

## Laplacian in cylindrical coordinates

If $(x,r,\phi)$ are cylindrical coordinates, then the Laplacian of a scalar field variable $u$ is

$\Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial u}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 u}{\partial \phi^2}$

## Laplacian in spherical coordinates

If $(r,\theta,\phi)$ are spherical coordinates, then the Laplacian of a scalar field variable $u$ is

$\Delta u = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial u}{\partial r} \right) + \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial \theta} \left( \sin\theta \frac{\partial u}{\partial \theta} \right) + \frac{1}{r^2 \sin^2\theta} \frac{\partial^2 u}{\partial \phi^2}$

## References

• Gerald B. Folland (1995), Introduction to partial differential equations, Princeton University Press.
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