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 Incoming links: Ordinary covector derivative under nonlinear coordinate changes

Antisymmetrized covector derivative is tensorial 2026-10-20

For a smooth one-form

A

, the components

\partial_{a} A_{b} - \partial_{b} A_{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_{a} A_{b} - \nabla_{b} A_{a}

. This is the coordinate expression of

d A

, without choosing a metric.

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