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

$$\sigma=\begin{pmatrix} \sigma^1 \\ \sigma^2 \\ \vdots \\ \sigma^k \end{pmatrix}=\sigma^i e_i:U_\alpha\longrightarrow\mathbb{R}^k$$

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

$$\mathrm{d}_\nabla(\sigma)=\mathrm{d}_\nabla(\sigma^i e_i)=\mathrm{d}(\sigma^i)\otimes e_i+\sigma^i\mathrm{d}_\nabla e_i$$

The first part may be symbolically expressed as

$$ \begin{pmatrix} \mathrm{d}\sigma^1 \\ \mathrm{d}\sigma^2 \\ \vdots \\ \mathrm{d}\sigma^k \end{pmatrix}=\mathrm{d}\begin{pmatrix} \sigma^1 \\ \sigma^2 \\ \vdots \\ \sigma^k \end{pmatrix}=\mathrm{d}(\sigma) $$

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

$$\sigma^i\mathrm{d}_\nabla e_i=\sigma^i\omega_i^j\otimes e_j=\begin{pmatrix} \sigma^i\omega_i^1 \\ \sigma^i\omega_i^2 \\ \vdots \\ \sigma^i\omega_i^k \end{pmatrix}= \begin{pmatrix} \omega_1^1 & \omega_2^1 & \cdots & \omega_k^1 \\ \omega_1^2 & \omega_2^2 & \cdots & \omega_k^2 \\ \vdots & \vdots & \ddots & \vdots \\ \omega_1^k & \omega_2^k & \cdots & \omega_k^k \\ \end{pmatrix}\begin{pmatrix} \sigma^1 \\ \sigma^2 \\ \vdots \\ \sigma^k \end{pmatrix}=A\sigma$$

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