Chapter 12.5, Problem 62ES

(a) Use the chain rule and the first two Frenet-Serret formulas in Exercise 61 to show that

$\text{<msup> T <mo> ′ </mo> </msup>}=\kappa s^{\prime }\text{N}\text{and}\text{<msup> N <mo> ′ </mo> </msup>}=-\kappa s^{\prime }\text{T}+\tau s^{\prime }\text{B}$ where primes denote differentiation with respect to t.

(b) Show that Formulas (4) and (6) can be written in the form

$\text{<msup> r <mo> ′ </mo> </msup>}\left(t\right)=s^{\prime }\text{T}\text{and}\text{<msup> r <mo> ″ </mo> </msup>}\left(t\right)=s^{″}\text{T}+\kappa {\left(s^{\prime }\right)}^{\text{2}}\text{N}$

(c) Use the results in parts (a) and (b) to show that

$\text{r}\text{'}\text{'}\text{'}\left(t\right)=[s^{‴}-{\kappa }^2{\left(s^{\prime }\right)}^{\text{3}}\text{]T}\text{+}\text{[3}\kappa \text{<msup> s <mo> ′ </mo> </msup>}{s}^{″}+{\kappa }^{\prime }{\left(s^{\prime }\right)}^2]\text{N}+\kappa \tau {\left(s^{\prime }\right)}^{\text{3}}\text{B}$

(d) Use the results in parts (b) and (c) to show that

$\tau \left(t\right)=\frac{[\text{<msup> r <mo> ′ </mo> </msup>}\left(t\right)\times \text{<msup> r <mo> ″ </mo> </msup>}\left(t\right)]\cdot \text{<msup> r <mo> ‴ </mo> </msup>}\left(t\right)}{{‖\text{<msup> r <mo> ′ </mo> </msup>}\left(t\right)\times \text{<msup> r <mo> ″ </mo> </msup>}\left(t\right)‖}^2}$