Chapter 12.5, Problem 58ES

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 $\text{B}=\text{T}\times \text{N}\text{.}$ )

Show that $\frac{d\text{T}}{ds}=\kappa \left(s\right)\text{N}\left(s\right)$ and use this result to obtain the formulas in (10).