= Antisymmetrized covector derivative is tensorial
{title2=$(dA)_{ab}=\partial_aA_b-\partial_bA_a$}
For a smooth <one-form> $A$, the components $\partial_aA_b-\partial_bA_a$ transform as a covariant rank-two <tensor>. The non-tensorial second derivatives in the <ordinary covector derivative under nonlinear coordinate changes> are symmetric in $a,b$ and cancel. For a <torsion-free connection> the same components are $\nabla_aA_b-\nabla_bA_a$. This is the coordinate expression of $dA$, without choosing a metric.
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