= Connection one-form
{title2=$A$}
= Connection matrix
{synonym}
Write a <local frame> of a rank-$r$ <vector bundle> as the row $e=(e_1,\ldots,e_r)$. The <connection one-form> of a <connection on a vector bundle> is the <matrix> of <one-forms> $A$ determined by
$$
\nabla e=eA,\qquad \nabla(eu)=e(du+Au).
$$
The placement of $A$ matters: its entry $A^i{}_j$ is the coefficient of $e_i$ in $\nabla e_j$. If $e'=eg$, substitution in the <Leibniz rule> gives
$$
A'=g^{-1}Ag+g^{-1}dg.
$$
The <derivative> term means that $A$ is not itself a global endomorphism-valued <one-form>. The <curvature form> $F=dA+A\wedge A$ instead changes by $F'=g^{-1}Fg$, and the difference of two <connections on a vector bundle> has the same homogeneous change law. These formulas fix the row-frame convention consistently.
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