Chapter 12.5, Problem 59ES

Assume that s is an arc length parameter for a smooth vector-valued function r(s) in 3-space and that $d\text{T}/ds\text{and }d\text{N}/ds$ exist at each point on the curve. (This implies that $d\text{B/}ds$ exists as well, since $\text{B}=\text{T}\times \text{N}\text{.}$ )

(a) Show that $d\text{B/}ds$ is perpendicular to $\text{B}\left(s\right).$

(b) Show that $d\text{B/}ds$ is perpendicular to $\text{T}\left(s\right).$

(c) Use the results in parts (a) and (b) to show that $d\text{B/}ds$ is a scalar multiple of $\text{N}\left(s\right).$ The negative of this scalar is called the torsion of $\text{r}\left(s\right)$ and is denoted by $\tau \left(s\right).$ Thus,

$\frac{d\text{B}}{ds}=-\tau \left(s\right)\text{N}\left(s\right)$

(d) Show that $\tau \left(s\right)=0$ for all s if the graph of r(s) lies in a plane. [Note: For reasons that we cannot discuss here, the torsion is related to the “twisting� properties of the curve, and $\tau \left(s\right)$ is regarded as a numerical measure of the tendency for the curve to twist out of the osculating plane.]