# Relation between torsion of a curve and the curl of a vector field

The torsion of a curve in $\mathbb{R}^3$ indicates how much it twists around. The curl of a vector field indicates how much the vector field twists around. Is there a relation between the curl of a vector field and the torsion of a curve through that vector field at a given point?

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Does curve "through" a vector field mean a "flow curve" of that vector field? –  a.r. Sep 22 '10 at 10:42

Going the other way and recovering any information about the vector field from its tangent curves is not necessarily possible. As a simple counterexample, consider that all vector fields of the form $\mathbf{v}(r,\theta,z) = f(r)\mathbf{e}_\theta$ in cylindrical coordinates have exactly the same tangent curves (circles around the $z$-axis) but completely different curls.