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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1 Answer

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 might check, however, if the author gives a technical definition of the notion of "equivalence", hinted at in the use of the term "equivalent functions" (or perhaps a technical definition of the term "change of parameter" in the discussion of the change of variable formula) that incorporates a hypothesis like this. It is quite possible, as it certainly removes annoying technicalities from the discussion of the behavior of definite integrals under a "change of parameter".

Of course it is also quite possible that this is just a goof. This is the kind of stuff that lots of books slip up on, or at least gloss over without much discussion.

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).

I would see this as nonessential detail, however. It was almost certainly intentional. The task of finding (or even just stating) the most general assumptions, or even not-obviously-too-restrictive assumptions, under which the tools of calculus like the fundamental theorem of calculus, the change of variable formula, integration by parts, etc apply, quickly takes one out of the subject matter of calculus. The Riemann integral is not good enough, and simple combinations of hypotheses like "continuous" and "differentiable" are not good enough. And the functions found in calculus books that meet these more general hypotheses are always either continuous or piecewise continuous.

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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