Let $P\to M$ be a principal $G$-bundle with connection $\omega$, let $\rho:G\to GL(V)$ be a representation, and let $E=P\times_\rho V$ be the associated vector bundle with induced covariant derivative $\nabla$. For vector fields $X,Y\in\mathfrak{X}(M)$ and a section $s\in\Gamma(E)$, the curvature operator