# 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 $x_{1},\cdots,x_{n}$. Then for a 1-form $w$ we have:

 $\delta w=\frac{-1}{\surd{({\rm Det}g)}}\frac{\partial}{\partial x_{i}}\left[% \surd{({\rm Det}g)}\{g^{-1}\}_{ij}w_{j}\right]$
Title codifferential Codifferential 2013-03-22 18:37:11 2013-03-22 18:37:11 whm22 (2009) whm22 (2009) 5 whm22 (2009) Definition msc 53B21 DifferentialForms Laplacian