How to describe $\mathrm{d}_\nabla$ in a chart? Let

be a section (everything is analogous for $\mathbb{C}^k$)

The first part may be symbolically expressed as

as long as we keep in mind that this is component-wise derivation, non-intrinsic but chart-dependent. In the second part we find terms $\mathrm{d}_\nabla e_i$, which are really defining the derivative. Let $\mathrm{d}_\nabla e_i=\omega_i^j\otimes e_j$, where $\omega_i^j$ are 1-forms. We then get

with $A$ 1-form-valued matrix. Summing up, we may simbolically write $\mathrm{d}_\nabla\sigma=(\mathrm{d}+A)\sigma$, the first term acting component-wise, the second acting like matrix product