# Arc Length under change of parameter

This is from Apostol's Calculus Vol. I, Section 14.13 #21:

Let $C$ be a curve described by two equivalent functions $X$ and $Y$, where $Y(t)=X[u(t)]$ for $c\le t\le d$. If the function $u$ which defines a change of parameter has a continuous derivative in $[c,d]$ prove that

$$\int_{u(c)}^{u(d)} \! ||X'(u)||\,\mathrm du=\int_c^d \! ||Y'(t)||\, \mathrm d t$$ and deduce that the arc length of $C$ is invariant under such a change of parameter.

I believe that the condition placed on the derivative of $u$ should not have been continuity but rather non-negativity. First a counter-example: $$Y(t)=t\boldsymbol i\,,\quad X(t)=-t\boldsymbol i \, , \quad u(t)=-t\,.$$ Then $Y(t)=X[u(t)]$ over, say, $0\le t\le 1$ and $u'(t)=-1$ is certainly continuous. Now $||X'(t)||=1$ and $||Y'(t)||=1$ but $$\int_{u(0)}^{u(1)} \! ||X'(u)||\,\mathrm du=\int_0^{-1}\!\mathrm d u=-1$$ and $$\int_0^1 \! ||Y'(t)||\,\mathrm dt=\int_0^1\!\mathrm d t=1\,.$$

On the other hand, if we require $u'(t)\ge0$ (and I don't think we even need continuity, do we?) then we can write

\begin{align}Y'(t)&=X'[u(t)]u'(t)\\ ||Y'(t)||&=||X'[u(t)]u'(t)||\\ &=||X'[u(t)]||\cdot |u'(t)|\\ &=||X'[u(t)]||u'(t)\\ \implies\int_c^d\!||Y'(t)||\,\mathrm d t&=\int_c^d \!||X'[u(t)]||u'(t)\,\mathrm d t\\ &=\int_{u(c)}^{u(d)}\!||X'(u)||\,\mathrm d u \end{align}

Is this correct? Should the condition on $u'$ be non-negativity, rather than continuity, or do I need non-negativity in addition to continuity? I don't see anything in my proof at the end that requires continuity, but maybe I'm glossing over it.

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You are right; without the assumption that $u'(t) \geq 0$ the formula as stated won't work (although, as you probably are aware, if $u'(t) < 0$ for all $t$, the worst that can happen is that left and right hand sides have opposite sign).
You are also right that the assumption of continuity of $u'$ is stronger than needed. Its only purpose is to ensure that the "change of variable" formula used in computing the definite integral is justified. Since this formula holds more generally, the result could be stated more generally (e.g. obviously piecewise continuity would also be enough).
Thanks for the verification and suggestions. I checked, and the book doesn't seem to use "change of parameter" elsewhere in a way that would imply $u'(t)\ge 0$, in fact some questions preceding this one specifically addressed cases for when $u'(t)\ge 0$ and when $u'(t)\le 0$ separately. I would accept your answer now, but it's been suggested to me before to wait at least a few hours before accepting an answer, so I will likely do so soon. –  process91 Apr 26 '12 at 22:35