Exercise 13

In this exercise we verify the curvature formula $$\begin{gathered} \kappa = \frac{\vecmag{\vr'(t) \times \vr''(t)}}{\vecmag{\vr'(t) }^3}. \end{gathered}$$

1. Explain why $$\begin{gathered} \vecmag{\vr'(t)} = \frac{ds}{dt}. \end{gathered}$$

2. Use the fact that $\vT(t) = \frac{\vr'(t)}{\vecmag{\vr'(t)}}$ and $\vecmag{\vr'(t)} = \frac{ds}{dt}$ to explain why $$\begin{gathered} \vr'(t) = \frac{ds}{dt} \vT(t). \end{gathered}$$

3. The Product Rule shows that $$\begin{gathered} \vr''(t) = \frac{d^2s}{dt^2} \vT(t) + \frac{ds}{dt} \vT'(t). \end{gathered}$$ Explain why $$\begin{gathered} \vr'(t) \times \vr''(t) = \left(\frac{ds}{dt}\right)^2 (\vT(t) \times \vT'(t)). \end{gathered}$$

4. In Key Idea on $\vT$ and $\vN$ we showed that $\vecmag{\vT(t)} = 1$ implies that $\vT(t)$ is orthogonal to $\vT'(t)$ for every value of $t$. Explain what this tells us about $\vecmag{\vT(t) \times \vT'(t)}$ and conclude that $$\begin{gathered} \vecmag{\vr'(t) \times \vr''(t)} = \left(\frac{ds}{dt}\right)^2 \vecmag{\vT'(t)}. \end{gathered}$$

5. Finally, use the fact that $\kappa = \frac{\vecmag{\vT'(t)} }{\vecmag{\vr'(t)}}$ to verify that $$\begin{gathered} \kappa = \frac{\vecmag{vr'(t) \times \vr''(t) }}{\vecmag{\vr'(t)}^3}. \end{gathered}$$