Chapter 13.3, Problem 63E

Use the Frenet-Serret formulas to prove each of the following. (Primes denote derivatives with respect to t. Start as in the proof of Theorem 10.)

(a) $r^{″}=s^{″}T+\kappa {\left(s^{\prime }\right)}^{2}N$

(b) $r^{\prime }\times r^{″}=\kappa {\left(s^{\prime }\right)}^{3}B$

(c) ${r^{\prime }}^{″}=\left[{s^{\prime }}^{″}-{\kappa }^2{\left(s^{\prime }\right)}^3\right]T+\left[3\kappa s^{\prime }{s}^{″}+{\kappa }^{\prime }{\left(s^{\prime }\right)}^2\right]N+\kappa \tau {\left(s^{\prime }\right)}^{3}B$

(d) $\tau =\frac{\left(r^{\prime }\times r^{″}\right)\cdot {r^{\prime }}^{″}}{{\left|r^{\prime }\times r^{″}\right|}^2}$