= Irrotational vector field
{title2=$\nabla\times X=0$}
= Curl-free vector field
{synonym}
= Irrotational
{synonym}
A continuously differentiable <vector field> on an open subset of three-dimensional <Euclidean space> is irrotational when its <curl> is zero. Under the Euclidean identification $\alpha=X_i\,dx^i$, this says that the <differential one-form> $\alpha$ is a <closed differential form>. The <Poincaré lemma> gives a <potential of a conservative vector field> locally. A global potential additionally requires zero <line integrals> around every closed curve; <simply connected> domains ensures this. On the punctured plane extended in the third direction, $X=(-y,x,0)/(x^2+y^2)$ has zero curl but integral $2\pi$ around a circle, so it is locally conservative and has no single-valued global potential.
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