# Laplacian

Let $({x}_{1},\mathrm{\dots},{x}_{n})$ be Cartesian coordinates^{} for some open set $\mathrm{\Omega}$
in ${\mathbb{R}}^{n}$.
Then the *Laplacian* differential operator $\mathrm{\Delta}$ is defined as

$$\mathrm{\Delta}=\frac{{\partial}^{2}}{\partial {x}_{1}^{2}}+\mathrm{\cdots}+\frac{{\partial}^{2}}{\partial {x}_{n}^{2}}.$$ |

In other words, if $f$ is a twice differentiable function $f:\mathrm{\Omega}\to \u2102$, then

$$\mathrm{\Delta}f=\frac{{\partial}^{2}f}{\partial {x}_{1}^{2}}+\mathrm{\cdots}+\frac{{\partial}^{2}f}{\partial {x}_{n}^{2}}.$$ |

A coordinate^{} independent definition of the Laplacian
is $\mathrm{\Delta}=\nabla \cdot \nabla $, i.e., $\mathrm{\Delta}$ is the composition^{} of
gradient^{} and codifferential.

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

\htmladdnormallinkClick here¡http://planetmath.org/?method=l2h&from=collab&id=76&op=getobj”¿ to see an article that derives the Laplacian in spherical coordinates^{}.

Title | Laplacian |
---|---|

Canonical name | Laplacian |

Date of creation | 2013-03-22 12:43:48 |

Last modified on | 2013-03-22 12:43:48 |

Owner | matte (1858) |

Last modified by | matte (1858) |

Numerical id | 18 |

Author | matte (1858) |

Entry type | Definition |

Classification | msc 31B05 |

Classification | msc 31B15 |

Related topic | DAlembertian |

Related topic | Codifferential |

Defines | Laplace operator |