# codifferential

The codifferential $\delta $ of a $k$-form on an $n$-dimensional
Riemannian manifold^{} is given by:

$${(-1)}^{n(k+1)+1}\ast d\ast $$ |

where $\ast $ is the Hodge star operator and $d$ is the exterior
derivative^{}.

Let $g$ denote the matrix locally representing the metric with respect to co-ordinates ${x}_{1},\mathrm{\cdots},{x}_{n}$. Then for a 1-form $w$ we have:

$$\delta w=\frac{-1}{\sqrt{(\mathrm{Det}g)}}\frac{\partial}{\partial {x}_{i}}[\surd (\mathrm{Det}g){\{{g}^{-1}\}}_{ij}{w}_{j}]$$ |

Title | codifferential |
---|---|

Canonical name | Codifferential |

Date of creation | 2013-03-22 18:37:11 |

Last modified on | 2013-03-22 18:37:11 |

Owner | whm22 (2009) |

Last modified by | whm22 (2009) |

Numerical id | 5 |

Author | whm22 (2009) |

Entry type | Definition |

Classification | msc 53B21 |

Related topic | DifferentialForms |

Related topic | Laplacian |