= Conservative vector field
{wiki}
A <vector field> $X$ is conservative when $X=\nabla\Phi$ for a <potential of a conservative vector field> $\Phi$. On a <simply connected> open subset of $\mathbb R^3$, every continuously differentiable <curl-free vector field> is conservative. Locally this follows from the <Poincaré lemma> applied to the <differential one-form> $X_i\,dx^i$; globally, simple connectivity makes its <line integral> independent of the path. The <fundamental theorem for line integrals> expresses that path independence as the difference of the potential at the endpoints.
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