Editing Differentiable curve (section)

== Definitions ==
{{main|Curve}}
A ''parametric'' {{math|''C''<sup>''r''</sup>}}-''curve'' or a {{math|''C''<sup>''r''</sup>}}-''parametrization'' is a [[vector-valued function]] 
<math display="block">\gamma: I \to \mathbb{R}^{n}</math>
that is {{mvar|r}}-times [[continuously differentiable]] (that is, the component functions of {{mvar|γ}} are continuously differentiable), where <math>n \isin \mathbb{N}</math>, <math>r \isin \mathbb{N} \cup \{\infty\}</math>, and {{mvar|I}} is a non-empty [[Interval (mathematics)|interval]] of real numbers. The {{em|image}} of the parametric curve is <math>\gamma[I] \subseteq \mathbb{R}^n</math>. The parametric curve {{mvar|γ}} and its image {{math|''γ''[''I'']}} must be distinguished because a given subset of <math>\mathbb{R}^n</math> can be the image of many distinct parametric curves. The parameter {{mvar|t}} in {{math|''γ''(''t'')}} can be thought of as representing time, and {{mvar|γ}} the [[trajectory]] of a moving point in space. When {{mvar|I}} is a closed interval {{math|[''a'',''b'']}}, {{math|''γ''(''a'')}} is called the starting point and {{math|''γ''(''b'')}} is the endpoint of {{mvar|γ}}. If the starting and the end points coincide (that is, {{math|''γ''(''a'') {{=}} ''γ''(''b'')}}), then {{mvar|γ}} is a ''closed curve'' or a ''loop''. To be a {{math|''C''<sup>''r''</sup>}}-loop, the function {{mvar|γ}} must be {{mvar|r}}-times continuously differentiable and satisfy {{math|''γ''<sup>(''k'')</sup>(''a'') {{=}} ''γ''<sup>(''k'')</sup>(''b'')}} for {{math|0 ≤ ''k'' ≤ ''r''}}.

The parametric curve is {{em|simple}} if
<math display="block"> \gamma|_{(a,b)}: (a,b) \to \mathbb{R}^{n} </math>
is [[injective]]. It is {{em|analytic}} if each component function of {{mvar|γ}} is an [[analytic function]], that is, it is of class {{math|''C''<sup>''ω''</sup>}}.

The curve {{mvar|γ}} is ''regular of order'' {{mvar|m}} (where {{math|''m'' ≤ ''r''}}) if, for every {{math|''t'' ∈ ''I''}},
<math display="block">\left\{ \gamma'(t),\gamma''(t),\ldots,{\gamma^{(m)}}(t) \right\}</math>
is a [[linearly independent]] subset of <math>\mathbb{R}^n</math>. In particular, a parametric {{math|''C''<sup>1</sup>}}-curve {{mvar|γ}} is {{em|regular}} if and only if {{math|''γ''{{prime}}(''t'') ≠ '''0'''}} for any {{math|''t'' ∈ ''I''}}.