Let $s:U\to P$ be a local section, let $A=s^*\omega\in\Omega^1(U;\mathfrak g)$, and let a gauge transformation $\Phi:P\to P$ be a smooth principal bundle automorphism covering the identity on $U$. Suppose $\Phi$ is represented in this section by a smooth map $u:U\to G$, meaning $\Phi(s(x))=s(x)u(x)$. For the gauge action $\Phi\cdot\omega=\Phi^*\omega$, the local representative in the original section $s$ is

\begin{align*} A'=s^*(\Phi^*\omega)=\operatorname{Ad}_{u^{-1}}A+u^{-1}du. \end{align*}

Equivalently, $A'$ is the representative of the original connection $\omega$ in the changed section $s'=su$.

For a matrix Lie group this is written

\begin{align*} A'=u^{-1}Au+u^{-1}du. \end{align*}