(a) Given a space curve , with a parameter (not necessarily arc-length), give mathematical expressions for the unit tangent, unit normal, and unit binormal vectors.
(b) Consider the closed curve given by
Show that the unit tangent vector may be written as
with each sign associated with a certain range of , which you should specify.
Calculate the unit normal and the unit binormal vectors, and hence deduce that the curve lies in a plane.
(c) A closed space curve lies in a plane with unit normal . Use Stokes' theorem to prove that the planar area enclosed by is the absolute value of the line integral
Hence show that the planar area enclosed by the curve given by is .