Let $G$ be a Lie group with Lie algebra $\mathfrak g$, let $\nu:P\to M$ be a smooth principal right $G$-bundle, and let $\omega\in\Omega^1(P;\mathfrak g)$ be a connection form. Let $U_i,U_j\subset M$ be open sets, and let $s_i:U_i\to P$ and $s_j:U_j\to P$ be smooth local sections. Write $U_{ij}:=U_i\cap U_j$, and suppose that there is a smooth transition map $g_{ij}:U_{ij}\to G$ such that

\begin{align*} s_j(x)=s_i(x)\cdot g_{ij}(x) \end{align*}

for every $x\in U_{ij}$.

Define the local connection forms $A_i:=s_i^*\omega\in\Omega^1(U_i;\mathfrak g)$ and $A_j:=s_j^*\omega\in\Omega^1(U_j;\mathfrak g)$. Let $\theta^L\in\Omega^1(G;\mathfrak g)$ denote the left Maurer-Cartan form, defined by $\theta^L_h(v)=(dL_{h^{-1}})_h(v)$ for $h\in G$ and $v\in T_hG$. Then on $U_{ij}$,

\begin{align*} A_j=\operatorname{Ad}(g_{ij}^{-1})A_i+g_{ij}^*\theta^L. \end{align*}

Equivalently, in matrix Lie group notation,

\begin{align*} A_j=\operatorname{Ad}(g_{ij}^{-1})A_i+g_{ij}^{-1}dg_{ij}. \end{align*}