- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.13 Exercises
- Let $C$ be a curve described by two equivalent functions $X$ and $Y,$ where $Y(t) = X[u(t)]$ for $c \leq t \leq d.$ If the function $u$ which defines the change of parameter has a continuous derivative in $[c,d]$ prove that $$\int_{u(c)}^{u(d)} \|X'(u)\|\,du = \int_c^d \|Y'(t)\|\,dt$$ and deduce that the arc length of $C$ is invariant under such a change of parameter.
- Taking the derivative of $Y$ with respect to $t$ gives us: $$ \begin{align*} Y'(t) &= \frac{d}{dt}X[u(t)] \\ &= X'[u(t)]\frac{du}{dt} \end{align*} $$ Integrating the norm of $Y'(t)$ from $t = c$ to $t = d,$ noting that $c \leq t \leq d:$ $$ \begin{align*} \int_c^d \|Y'(t)\|\,dt &= \int_c^d \|X'[u(t)]\left|\frac{du}{dt}\right|\,dt \\ &= \int_{u(c)}^{u(d)} \|X'[u(t)]\|\,\left|du\right| \end{align*} $$ and since $u'(t)$ is continuous (and presumed to be positive) on $[c, d],$ $|du| = du,$ hence: $$ \begin{align*} \int_c^d \|Y'(t)\|\,dt &= \int_{u(c)}^{u(d)} \|X'[u(t)]\|\,du \end{align*} $$ Thus demonstrating that the arc-length of a function is invariant under a change of parameter. $\blacksquare$