Is there a reference out there somewhere that gives a dictionary of mathematician's notation vs. physicist's notation in differential geometry?
For example,
$$\nabla_v ( f \mathbf{X} ) = (vf) \mathbf{X} + f \nabla_v \mathbf{X}, $$
as opposed to
$$v^\mu \nabla_\mu (f X^\nu ) = (v^\mu \partial_\mu f) X^\nu + f v^\mu \nabla_\mu X^\nu $$
for covariant derivatives / connections on smooth manifolds. Of course, assuming that we can use local coordinates.