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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

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The paper "Curvature Measures of 3D Vector Fields and Their Applications" by Weinkauf et al. describes how to compute the curvature and torsion of the tangent curves in terms of the spatial derivatives of the underlying vector field.

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.

In general, the difficulty is that the properties of the tangent curve can only tell you about the derivatives of the vector field along the parallel direction; you get no information about how the vector field varies in the plane normal to the curve.

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