Chapter 12 - Vector-Valued Functions - 12.5 Curvature - Exercises Set 12.5 - Page 879: 21

False.

Step 1 False. Since \[ \kappa(s) = \left\| \mathbf{T'}(s) \right\| | \] and \[ \mathbf{T'}(s) = \frac{\mathbf{r'}(s)}{\left\| \mathbf{r'}(s) \right\|}, \quad \left\| \mathbf{r'}(s) \right\| = 1 \Rightarrow \mathbf{T''}(s) = \mathbf{r''}(s) \] Hence \[ \kappa(s) = \left\| \mathbf{T''}(s) \right\| = \left\| \mathbf{r''}(s) \right\| \] Result: False.