Let $\pi:P\to M$ be a smooth principal $G$-bundle with right action, let $\rho:G\to GL(V)$ be a smooth finite-dimensional representation, and let $E=P\times_G V$ be the associated vector bundle with [equivalence relation](/page/Equivalence%20Relation) $(p,v)\sim(pg,\rho(g^{-1})v)$. Let $\omega\in\Omega^1(P;\mathfrak g)$ be a principal connection form, and let $\nabla^E$ be the covariant derivative on $E$ induced by $\omega$ and $\rho$.

Let $U_i\subset M$ be an [open set](/page/Open%20Set) and let $\sigma_i:U_i\to P$ be a smooth local section. Define the local connection form $A_i:=\sigma_i^*\omega\in\Omega^1(U_i;\mathfrak g)$. If $s\in\Gamma(E)$ and $s_i:U_i\to V$ is the local representative determined by

\begin{align*} s(x)=[\sigma_i(x),s_i(x)] \end{align*}

for every $x\in U_i$, then on $U_i$ the $V$-valued local representative of $\nabla^E s$ is

\begin{align*} (\nabla^E s)_i=ds_i+\rho_*(A_i)s_i. \end{align*}

Equivalently, for every $x\in U_i$ and every $X\in T_xM$,

\begin{align*} (\nabla^E_Xs)_i(x)=d(s_i)_x(X)+\rho_*(A_i(x)(X))s_i(x). \end{align*}