Let $(M,g)$ be a semi-Riemannian manifold and $X$ a vector field. the divergence of $X$ is defined as
$$\text{div}X= \text{trace}(s \mapsto \nabla_sX)$$
Is it correct, that, in local coordinates, it holds that
$$\text{div}X=dx^i( \nabla_{\partial_i} X)$$?