Let $M$ be a smooth manifold, let $G$ be a Lie group, let $\omega\in\Omega^1(P;\mathfrak g)$ be a principal connection on the smooth principal $G$-bundle $\pi:P\to M$, and let $E=P\times_G V$ be the associated vector bundle associated to a smooth representation $\rho:G\to GL(V)$. There is a unique covariant derivative