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