[math] \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow} \newcommand{\ddxof}[1]{\frac{d #1}{d x}} \newcommand{\ddx}{\frac{d}{dx}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\dydx}{\ddxof y} \newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}} \newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}} \newcommand{\dist}{\mathrm{distance}} \newcommand{\arccot}{\mathrm{arccot\:}} \newcommand{\arccsc}{\mathrm{arccsc\:}} \newcommand{\arcsec}{\mathrm{arcsec\:}} \newcommand{\arctanh}{\mathrm{arctanh\:}} \newcommand{\arcsinh}{\mathrm{arcsinh\:}} \newcommand{\arccosh}{\mathrm{arccosh\:}} \newcommand{\sech}{\mathrm{sech\:}} \newcommand{\csch}{\mathrm{csch\:}} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\mathds}{\mathbb} [/math]

Let [math]P:[a,b] \goesto \R^2[/math] and [math]Q:[c,d]\goesto R^2[/math] be two parametrizations of the same curve [math]C[/math] such that all four coordinate functions are continuously differentiable. (A function is continuously differentiable if its derivative exists and is continuous at every number in its domain.) Then [math]P[/math] and [math]Q[/math] are called equivalent parametrizations of [math]C[/math] if there exists a continuously differentiable function [math]f[/math] with domain [math][a,b][/math] and range [math][c,d][/math] which has a continuously differentiable inverse function, and in addition satisfies (i) [math]f(a) = c[/math] and [math]f(b) = d[/math], (ii) [math]P(t) = Q(f(t)),[/math] for every [math]t[/math] in [math][a,b][/math].

• Using the Chain Rule and the Change of Variable Theorem for Definite Integrals (for the latter, see Theorem \ref{thm 4.6.6}), prove that equivalent parametrizations assign the same arc length to [math]C[/math].
• Show that
  [[math]] P(t) = (\cos t, \sin t), \quad 0 \leq t \leq \frac{\pi}2 , [[/math]]
  [[math]] Q(s) = \left( \frac{1-s^2}{1+s^2}, \frac{2s}{1+s^2}\right), \quad 0 \leq s \leq 1 , [[/math]]
  are equivalent parametrizations of the same curve [math]C[/math], and identify the curve.
• Show that
  [[math]] P(t) = (\cos t, \sin t), \quad 0 \leq t \leq 2\pi , [[/math]]
  and
  [[math]] Q(s) = (\cos 5t, \sin 5t), \quad 0 \leq t \leq 2\pi , [[/math]]
  are nonequivalent parametrizations of the circle.