Editing Differentiable curve (section)

==Length and natural parametrization{{anchor|Length|Natural parametrization}}==
{{main|Arc length}}
{{see also|Curve#Length of a curve}}

The length {{mvar|l}} of a parametric {{math|''C''<sup>1</sup>}}-curve <math>\gamma : [a, b] \to \mathbb{R}^n</math> is defined as
<math display="block">l ~ \stackrel{\text{def}}{=} ~ \int_a^b \left\| \gamma'(t) \right\| \, \mathrm{d}{t}.</math>
The length of a parametric curve is invariant under reparametrization and is therefore a differential-geometric property of the parametric curve.

For each regular parametric {{math|''C''<sup>''r''</sup>}}-curve <math>\gamma : [a, b] \to \mathbb{R}^n</math>, where {{math|''r'' ≥ 1}}, the function is defined
<math display="block">\forall t \in [a,b]: \quad s(t) ~ \stackrel{\text{def}}{=} ~ \int_a^t \left\| \gamma'(x) \right\| \, \mathrm{d}{x}.</math>
Writing {{math|''{{overline|γ}}''(s) {{=}} ''γ''(''t''(''s''))}}, where {{math|''t''(''s'')}} is the inverse function of {{math|''s''(''t'')}}. This is a re-parametrization {{math|''{{overline|γ}}''}} of {{mvar|γ}} that is called an ''{{vanchor|arc-length parametrization}}'',  ''natural parametrization'', ''unit-speed parametrization''. The parameter {{math|''s''(''t'')}} is called the {{em|natural parameter}} of {{mvar|γ}}.

This parametrization is preferred because the natural parameter {{math|''s''(''t'')}} traverses the image of {{mvar|γ}} at unit speed, so that
<math display="block">\forall t \in I: \quad \left\| \overline{\gamma}'\bigl(s(t)\bigr) \right\| = 1.</math>
In practice, it is often very difficult to calculate the natural parametrization of a parametric curve, but it is useful for theoretical arguments.

For a given parametric curve {{mvar|γ}}, the natural parametrization is unique up to a shift of parameter.

The quantity
<math display="block">E(\gamma) ~ \stackrel{\text{def}}{=} ~ \frac{1}{2} \int_a^b \left\| \gamma'(t) \right\|^2 ~ \mathrm{d}{t}</math>
is sometimes called the {{em|energy}} or [[action (physics)|action]] of the curve; this name is justified because the [[geodesic]] equations are the [[Euler–Lagrange equation]]s of motion for this action.