Chapter 12.5, Problem 84E

(a)

To determine

To prove: The relation $K=\frac{\Vert T^{\prime }\left(t\right)\Vert }{\Vert r^{\prime }\left(t\right)\Vert }$ for the curvature $k=\Vert T^{\prime }\left(s\right)\Vert =\Vert r^{″}\left(s\right)\Vert$.

(b)

To determine

To prove: The relation $K=\frac{\Vert r^{\prime }\left(t\right)\times {r^{\prime }}^{\prime }\left(t\right)\Vert }{{\Vert r^{\prime }\left(t\right)\Vert }^3}$ for the curvature $k=\Vert T^{\prime }\left(s\right)\Vert =\Vert r^{″}\left(s\right)\Vert$.

(c)

To determine

To prove: The relation $K=\frac{a\left(t\right)\cdot N\left(t\right)}{{\Vert v\left(t\right)\Vert }^2}$.