Example 2-3-3 - Maple Help



Chapter 2: Space Curves



Section 2.3: Tangent Vectors



Example 2.3.3



Let $\mathbf{R}\left(p\right)$ be the position-vector representation of the parametric curve , $p\ge 0$, and let $\mathbf{R}\left(s\right)=\mathbf{R}\left(p\left(s\right)\right)$ be the reparametrization obtained in Example 2.2.6. (Recall that $s$ is the arc length along the curve.)

 a) Obtain $\mathrm{ρ}=∥\mathbf{R}\prime \left(p\right)∥$.
 b) Obtain the unit tangent vector $\mathbf{T}\left(p\right)=\mathbf{R}\prime \left(p\right)/\mathrm{ρ}$.
 c) Show that $\mathbf{T}\left(p\left(s\right)\right)=\frac{d}{\mathrm{ds}}\mathbf{R}\left(s\right)$, thus verifying that $\mathbf{R}\prime \left(s\right)$ is automatically a unit tangent vector.