Let $(M,g)$ be a Riemannian manifold and let $e_1,...,e_n$ be locally frame field defined on $U\subset M$ and moreover, let $\omega _1 ,...,\omega _n$ be its dual. If we represent the connection $1$-forms of $e_1,...,e_n$ by $\omega_{ij}$ then I have the following questions:
What is the definition of $\mathrm{d} e_k$ and how can we calculate it in terms of $\omega_i \,, \omega _{ij}$?
For example, let $F:TU \to TU$ given by $F(u,v)=\sum _{i=1}^nf_i(u,v)e_i(u)$ be a mapping for some functions $f_i:TU \to \mathbb{R}$ where $(u,v)\in TM$ is a vector. Then how can I calculate the differential of $F$?
The usual meaning associated to an expression like $de_k$ where $e_k$ is a section of a bundle with connection (in this case the tangent bundle with Levi-Civita connection) is the covariant exterior derivative, defined on a section $s$ by $ds(v) = \nabla_v s$, where $\nabla$ is a covariant derivative. In terms of local frame $e_k$ we have $\nabla_v e_k= \omega_{ik}(v)e_i$ since the $e_k$ span. This defines the connection one-form (which may not be an actual globally defined one-form).
So we compute $de_k$ simply as $\omega_{ik}e_i$.