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.
There are three formulæ connecting arc length and the unit tangent, unit principal normal and unit binormal vectors, known as the Frenet-Serret formulæ.
Show that the first Frenet-Serret formula is
,
where s is arc length and
is the curvature.
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 .]
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
[To the Index of Problem Sets.] | [Solutions to this problem set] | ||
[Return to your previous page] |