Change of frame of a vector-bundle connection ($A^{\prime }=g^{-1}Ag+g^{-1}dg$)

With local frame written as a row $e$ and $D(eu)=e(du+Au)$, a change $e^{\prime }=eg$ gives $A^{\prime }=g^{-1}Ag+g^{-1}dg$. The vector-bundle curvature transforms by conjugation. This convention makes the inhomogeneous derivative term and every subsequent vector-bundle curvature sign explicit.