i do not understand the application of the Weizenböck identity, $$ \Delta = \nabla^*\nabla + \text{Ric}, $$
where $\Delta = \text{d}\delta+\delta\text{d}$. It is applied like this
$$ \eqalign{ (\text{d}\delta\text{d} u)\cdot \text{d}v &= ((\text{d}\delta+\delta\text{d})\text{d}u)\cdot(\text{d}v) \\ &=(\nabla^*\nabla\text{d}u)\cdot(\text{d}v)+\text{Ric}(\text{d}u,\text{d}v) } $$
and i do not understand why the second argument in $\text{Ric}$ is $\text{d}v$ and not $\text{d}u$ since the operator that is exchanged operates on $\text{d}u$. Please correct me :).