Chapter 13.3, Problem 63E

Use ihe 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 + κ(s^r)²N

(b) r^r × rⁿ = κ(s^r)³B

(c) r^m = [s^m − κ²(s^r)³]T + [3κs′s″ + κ^′(s′)²]N + κτ(s′)³B

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