Metric derivative

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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[edit]

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

if this limit exists.

Properties[edit]

Recall that ACp(I; X) is the space of curves γ : IX such that

for some m in the Lp space Lp(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 mLp(I; R) such that the above inequality holds.

If Euclidean space is equipped with its usual Euclidean norm , and is the usual Fréchet derivative with respect to time, then

where is the Euclidean metric.

References[edit]

  • 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. ISBN 3-7643-2428-7.