codifferential

The codifferential $\delta$ of a $k$-form on an $n$-dimensionalRiemannian manifold is given by:

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

where $\ast$ is the Hodge star operator and $d$ is the exteriorderivative.

Let $g$ denote the matrix locally representing the metric withrespect to co-ordinates $x_1,\mathrm{\cdots },x_n$. Then for a 1-form $w$ wehave:

$$\delta w=\frac{-1}{\sqrt{(\mathrm{Det}g)}}\frac{\partial }{\partial x_i}[\surd (\mathrm{Det}g){\{g^{-1}\}}_{ij}{w}_j]$$