Editing Curve (section)

===Length of a curve===
{{main|Arc length}}
{{further|Differentiable curve#Length}}

If <math> X = \mathbb{R}^{n} </math> is the <math> n </math>-dimensional Euclidean space, and if <math> \gamma: [a,b] \to \mathbb{R}^{n} </math> is an injective and continuously differentiable function, then the length of <math> \gamma </math> is defined as the quantity
:<math>
\operatorname{Length}(\gamma) ~ \stackrel{\text{def}}{=} ~ \int_{a}^{b} |\gamma\,'(t)| ~ \mathrm{d}{t}.
</math>
The length of a curve is independent of the [[Parametrization (geometry)|parametrization]] <math> \gamma </math>.

In particular, the length <math> s </math> of the [[graph of a function|graph]] of a continuously differentiable function <math> y = f(x) </math> defined on a closed interval <math> [a,b] </math> is
:<math>
s = \int_{a}^{b} \sqrt{1 + [f'(x)]^{2}} ~ \mathrm{d}{x},
</math>
which can be thought of intuitively as using the [[Pythagorean theorem]] at the infinitesimal scale continuously over the full length of the curve.<ref>{{Cite book|url=https://books.google.com/books?id=OS4AAAAAYAAJ&dq=length+of+a+curve+formula+pythagorean&pg=RA2-PA108|title=The Calculus|last1=Davis|first1=Ellery W.|last2=Brenke|first2=William C.|date=1913|publisher=MacMillan Company|isbn=9781145891982|page=108|language=en}}</ref>

More generally, if <math> X </math> is a [[metric space]] with metric <math> d </math>, then we can define the length of a curve <math> \gamma: [a,b] \to X </math> by
:<math>
\operatorname{Length}(\gamma)
~ \stackrel{\text{def}}{=} ~
\sup \!
\left\{
\sum_{i = 1}^{n} d(\gamma(t_{i}),\gamma(t_{i - 1})) ~ \Bigg| ~ n \in \mathbb{N} ~ \text{and} ~ a = t_{0} < t_{1} < \ldots < t_{n} = b
\right\},
</math>
where the supremum is taken over all <math> n \in \mathbb{N} </math> and all partitions <math> t_{0} < t_{1} < \ldots < t_{n} </math> of <math> [a, b] </math>.

A rectifiable curve is a curve with [[wiktionary:finite|finite]] length. A curve <math> \gamma: [a,b] \to X </math> is called {{em|natural}} (or unit-speed or parametrized by arc length) if for any <math> t_{1},t_{2} \in [a,b] </math> such that <math> t_{1} \leq t_{2} </math>, we have
:<math>
\operatorname{Length} \! \left( \gamma|_{[t_{1},t_{2}]} \right) = t_{2} - t_{1}.
</math>

If <math> \gamma: [a,b] \to X </math> is a [[Lipschitz continuity|Lipschitz-continuous]] function, then it is automatically rectifiable. Moreover, in this case, one can define the speed (or [[metric derivative]]) of <math> \gamma </math> at <math> t \in [a,b] </math> as
:<math>
{\operatorname{Speed}_{\gamma}}(t) ~ \stackrel{\text{def}}{=} ~ \limsup_{s \to t} \frac{d(\gamma(s),\gamma(t))}{|s - t|}
</math>
and then show that
:<math>
\operatorname{Length}(\gamma) = \int_{a}^{b} {\operatorname{Speed}_{\gamma}}(t) ~ \mathrm{d}{t}.
</math>