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:

$$ \delta w = \frac{-1}{\surd{({\rm Det } g)}} \frac{\partial}{\partial x_i}\left[\surd{({\rm Det } g)} \{g^{-1}\}_{ij} w_j \right] $$