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 \( E\subseteq \mathbb {R} \) have a limit point at \( t\in \mathbb {R} \). Let \( \gamma : E \to M \) be a path. Then the metric derivative of \( \gamma \) at t t, denoted \( | \gamma' | (t) \), is defined by

\( | \gamma' | (t) := \lim_{s \to 0} \frac{d (\gamma(t + s), \gamma (t))}{| s |}, \)

if this limit exists.

Properties

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

\( d \left( \gamma(s), \gamma(t) \right) \leq \int_{s}^{t} m(\tau) \, \mathrm{d} \tau \mbox{ for all } [s, t] \subseteq I \)

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 m ∈ Lp(I; R) such that the above inequality holds.

If Euclidean space R n {\displaystyle \mathbb {R} ^{n}} \mathbb {R} ^{n} is equipped with its usual Euclidean norm \( \| - \| \), and \( \dot{\gamma} : E \to V^{*} \) is the usual Fréchet derivative with respect to time, then

\( | \gamma' | (t) = \| \dot{\gamma} (t) \|, \)

where \( d(x, y) := \| x - y \| \) 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. p. 24. ISBN 3-7643-2428-7.

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