In mathematics, the metric derivative is a notion of derivative appropriate to parametrizedpaths in metric spaces. 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 spaces).
Definitionedit
Let be a metric space. Let have a limit point at . Let be a path. Then the metric derivative of at , denoted , is defined by
Recall that ACp(I; X) is the space of curves γ : I → X such that
for some m in the Lp spaceLp(I; R). For γ ∈ ACp(I; X), the metric derivative of γ exists for Lebesgue-almost all times in I, and the metric derivative is the smallest m ∈ Lp(I; R) such that the above inequality holds.
Ambrosio, L., Gigli, N. & Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures. ETH Zürich, Birkhäuser Verlag, Basel. p. 24. ISBN3-7643-2428-7.{{cite book}}: CS1 maint: multiple names: authors list (link)
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metric, derivative, mathematics, metric, derivative, notion, derivative, appropriate, parametrized, paths, metric, spaces, generalizes, notion, speed, absolute, velocity, spaces, which, have, notion, distance, metric, spaces, direction, such, vector, spaces, d. In mathematics the metric derivative is a notion of derivative appropriate to parametrized paths in metric spaces 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 spaces Definition editLet M d displaystyle M d nbsp be a metric space Let E R displaystyle E subseteq mathbb R nbsp have a limit point at t R displaystyle t in mathbb R nbsp Let g E M displaystyle gamma E to M nbsp be a path Then the metric derivative of g displaystyle gamma nbsp at t displaystyle t nbsp denoted g t displaystyle gamma t nbsp is defined by g t lims 0d g t s g t s displaystyle gamma t lim s to 0 frac d gamma t s gamma t s nbsp if this limit exists Properties editRecall that ACp I X is the space of curves g I X such that d g s g t stm t dt for all s t I displaystyle d left gamma s gamma t right leq int s t m tau mathrm d tau mbox for all s t subseteq I nbsp for some m in the Lp space Lp I R For g ACp I X the metric derivative of g exists for Lebesgue almost all times in I and the metric derivative is the smallest m Lp I R such that the above inequality holds If Euclidean space Rn displaystyle mathbb R n nbsp is equipped with its usual Euclidean norm displaystyle nbsp and g E V displaystyle dot gamma E to V nbsp is the usual Frechet derivative with respect to time then g t g t displaystyle gamma t dot gamma t nbsp where d x y x y displaystyle d x y x y nbsp is the Euclidean metric References editAmbrosio L Gigli N amp Savare G 2005 Gradient Flows in Metric Spaces and in the Space of Probability Measures ETH Zurich Birkhauser Verlag Basel p 24 ISBN 3 7643 2428 7 a href Template Cite book html title Template Cite book cite book a CS1 maint multiple names authors list link nbsp This metric geometry related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Metric derivative amp oldid 1169902467, wikipedia, wiki, book, books, library,