Calculus with Vector Functions — Question 9

Question 9

Let $\vec r(t)=\left\langle\cos t,\sin t,t\right\rangle$ and reparameterize by $t=u^3-u$. Find $d\vec r/du$. Identify every $u$ where the reparameterized curve momentarily stops, and explain why the geometric helix itself remains smooth there.

Question 9 – Solution

Strategy Apply the vector chain rule and distinguish the image curve from the chosen parameter clock.

Chain rule With $t=g(u)=u^3-u$, $$\frac{d\vec r}{du}=\vec r'(g(u))g'(u)=(3u^2-1)\left\langle-\sin(u^3-u),\cos(u^3-u),1\right\rangle.$$ It vanishes when $3u^2-1=0$, so $\boxed{u=\pm 1/\sqrt 3}$.

Interpretation The original derivative $\left\langle-\sin t,\cos t,1\right\rangle$ never vanishes. The stops occur because $dt/du=0$; they reflect a nonregular reparameterization, not a cusp in the helix.