Arc Length with Vector Functions — Question 8

Question 8

Suppose $\vec q(u)=\vec r(\phi(u))$ traces the same regular curve, where $\phi$ is continuously differentiable and strictly monotone. Prove that the arc length is unchanged by this reparameterization, including the case in which orientation is reversed.

Question 8 – Solution

Strategy Apply the chain rule to speed, then use substitution while carefully retaining the absolute value.

Transformed speed $$\vec q'(u)=\vec r'(\phi(u))\phi'(u),\qquad \|\vec q'(u)\|=\|\vec r'(\phi(u))\|\,|\phi'(u)|.$$

Increasing case If $\phi'\ge 0$ and $\phi(\alpha)=a,\phi(\beta)=b$, substitution gives $$\int_\alpha^\beta\|\vec q'(u)\|\,du=\int_a^b\|\vec r'(t)\|\,dt.$$

Decreasing case If $\phi'\le 0$, then $\phi(\alpha)=b$ and $\phi(\beta)=a$. Since $|\phi'|=-\phi'$, reversing the substituted limits again produces $\int_a^b\|\vec r'(t)\|dt$. Thus length depends on the traced curve, not its orientation or timing.