Let $G$ be a Lie group with Lie algebra $\mathfrak g$, let $\pi:P\to M$ be a smooth principal right $G$-bundle, and let $\omega\in \Omega^1(P;\mathfrak g)$ be a connection form. Let $\Omega\in \Omega^2(P;\mathfrak g)$ be its curvature form,
Let $U_i,U_j\subset M$ be open sets with $U_i\cap U_j\neq \varnothing$, and let $s_i:U_i\to P$ and $s_j:U_j\to P$ be smooth local sections. Define the local connection forms and local curvature forms by