 Metric derivative

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
Let (M,d) be a metric space. Let have a limit point at . Let be a path. Then the metric derivative of γ at t, denoted  γ'  (t), is defined by
if this limit exists.
Properties
Recall that AC^{p}(I; X) is the space of curves γ : I → X such that
for some m in the L^{p} space L^{p}(I; R). For γ ∈ AC^{p}(I; X), the metric derivative of γ exists for Lebesguealmost all times in I, and the metric derivative is the smallest m ∈ L^{p}(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
 Ambrosio, L., Gigli, N. & Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures. ETH Zürich, Birkhäuser Verlag, Basel. ISBN 3764324287.
Categories: Differential calculus
 Metric geometry
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