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| In [[mathematics]], the '''metric derivative''' is a notion of [[derivative]] appropriate to [[Parametric equation|parametrized]] [[path (topology)|paths]] in [[metric space]]s. It generalizes the notion of "speed" or "absolute velocity" to spaces which have a notion of distance (i.e. metric spaces) but not direction (such as [[vector space]]s).
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| ==Definition==
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| Let <math>(M, d)</math> be a metric space. Let <math>E \subseteq \mathbb{R}</math> have a [[limit point]] at <math>t \in \mathbb{R}</math>. Let <math>\gamma : E \to M</math> be a path. Then the '''metric derivative''' of <math>\gamma</math> at <math>t</math>, denoted <math>| \gamma' | (t)</math>, is defined by
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| :<math>| \gamma' | (t) := \lim_{s \to 0} \frac{d (\gamma(t + s), \gamma (t))}{| s |},</math>
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| if this [[Limit (mathematics)|limit]] exists.
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| ==Properties==
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| Recall that [[absolute continuity|AC<sup>''p''</sup>(''I''; ''X'')]] is the space of curves ''γ'' : ''I'' → ''X'' such that
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| :<math>d \left( \gamma(s), \gamma(t) \right) \leq \int_{s}^{t} m(\tau) \, \mathrm{d} \tau \mbox{ for all } [s, t] \subseteq I</math> | |
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| for some ''m'' in the [[Lp space|''L''<sup>''p''</sup> space]] ''L''<sup>''p''</sup>(''I''; '''R'''). For ''γ'' ∈ AC<sup>''p''</sup>(''I''; ''X''), the metric derivative of ''γ'' exists for [[Lebesgue measure|Lebesgue]]-[[almost all]] times in ''I'', and the metric derivative is the smallest ''m'' ∈ ''L''<sup>''p''</sup>(''I''; '''R''') such that the above inequality holds.
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| If [[Euclidean space]] <math>\mathbb{R}^{n}</math> is equipped with its usual Euclidean norm <math>\| - \|</math>, and <math>\dot{\gamma} : E \to V^{*}</math> is the usual [[Fréchet derivative]] with respect to time, then
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| :<math>| \gamma' | (t) = \| \dot{\gamma} (t) \|,</math>
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| where <math>d(x, y) := \| x - y \|</math> is the Euclidean metric.
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| ==References==
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| * {{cite book | author=Ambrosio, L., Gigli, N. & Savaré, G. | title=Gradient Flows in Metric Spaces and in the Space of Probability Measures | publisher=ETH Zürich, Birkhäuser Verlag, Basel | year=2005 | isbn=3-7643-2428-7 | page=24}}
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| [[Category:Differential calculus]]
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| [[Category:Metric geometry]]
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They contact me Emilia. Years ago we moved to North Dakota. To gather badges is what her family and her enjoy. For years he's been working as a meter reader and it's some thing he really enjoy.
Feel free to surf to my web site - karachicattleexpo.com