(a) Let be a smooth curve parametrised by arc length . Explain the meaning of the terms in the equation
where is the curvature of the curve.
Now let . Show that there is a scalar (the torsion) such that
and derive an expression involving and for .
(b) Given a (nowhere zero) vector field , the field lines, or integral curves, of are the curves parallel to at each point . Show that the curvature of the field lines of satisfies
(c) Use to find an expression for the curvature at the point of the field lines of .