Metric derivative

Short description: Mathematical concept

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

[math]\displaystyle{ | \gamma' | (t) := \lim_{s \to 0} \frac{d (\gamma(t + s), \gamma (t))}{| s |}, }[/math]

if this limit exists.

Properties

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

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

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 Lebesgue-almost 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 [math]\displaystyle{ \mathbb{R}^{n} }[/math] is equipped with its usual Euclidean norm [math]\displaystyle{ \| - \| }[/math], and [math]\displaystyle{ \dot{\gamma} : E \to V^{*} }[/math] is the usual Fréchet derivative with respect to time, then

[math]\displaystyle{ | \gamma' | (t) = \| \dot{\gamma} (t) \|, }[/math]

where [math]\displaystyle{ d(x, y) := \| x - y \| }[/math] 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.