Problem Set 1   -   Questions

Let     be the position vector of a particle at time   t   as it moves along a curve   C, where the functions   x(t), y(t) and z(t)   are all differentiable for all values of   t.   Show that if     then the particle must always move in the same direction.

Let     be the position vector of a particle.   You may assume that     and     are positive constants.

1. 
2. Does the curve lie in one plane?
3. What type of curve is it?

There are three formulæ connecting arc length and the unit tangent, unit principal normal and unit binormal vectors, known as the Frenet-Serret formulæ.

1. Show that the first Frenet-Serret formula is ,
   where   s   is arc length and     is the curvature.

2. 
3. 

4. Hence prove the second Frenet-Serret formula   , where the torsion,   , is some scalar function of   s.   [The torsion is a measure of how much the curve is twisting out of the plane of   and .]

5. Use the fact that   , ,   form a mutually orthogonal right-handed triad of unit vectors, (from which it necessarily follows that   ),   to establish the third Frenet-Serret formula