\[ \newcommand{\d}{\mathrm{d}} \newcommand{\e}{\mathrm{e}} \newcommand{\i}{\mathrm{i}} \]

Every exact covector field is closed , which follows from Schwarz Theorem applied on the potential function.

Remarks
  • the converse is not true in general [@lee2013smooth_manifolds, Example 11.48]. However, this depends on the shape of the underlying manifold (see Poincaré lemma for covector fields ).

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