Let $\pi:E\to M$ be a smooth real vector bundle of rank $r$ over a smooth manifold $M$, and let $\operatorname{Fr}(E)$ denote the principal right $GL(r,\mathbb{R})$-bundle whose fibre over $x\in M$ consists of linear isomorphisms $u:\mathbb{R}^r\to E_x$. Then principal connections on $\operatorname{Fr}(E)$ are in natural bijection with covariant derivatives on $E$, that is, with $\mathbb{R}$-linear maps $\nabla:\Gamma(E)\to \Omega^1(M;E)$ satisfying the Leibniz rule

\begin{align*} \nabla(fs)=df\otimes s+f\nabla s \end{align*}

for every $f\in C^\infty(M)$ and every $s\in\Gamma(E)$.