Is the differential a sensible mathematical object?
When doing differential geometry, physicists often use
for many different things. For instance, they define the holonomic basis relative to a coordinate system by imposing
and they compute the quadratic form of the metric as .
Computing the differential of a vector field ( i, in this case) feels strange, as in differential geometry differentials are usually considered to be alternating k-forms, so it would only make sense to talk about the differential of a scalar field (aka its exterior derivative).
Not only that, the "true" definitions of holonomic bases and ds2 don't use this at all.
EDIT: in fact, taking the derivative of , or any other vector field, is something we are not allowed to do in a general differentiable manifold without a connection, so we obviously wouldn't define a holonomic basis like that. A holonomic basis would basically be the basis formed by the tangent vectors .
After thinking about it, I thought the differential of a vector field might just be
so maybe i means ? How is rigorously defined, otherwise?