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 $(M,d)$ 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 $t$, denoted $|{\gamma }^{\prime }|(t)$, is defined by

$|{\gamma }^{\prime }|(t):=\underset{s\to 0}{\lim }\frac{d(\gamma (t+s),\gamma (t))}{|s|},$

if this limit exists.

Properties

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

$d(\gamma (s),\gamma (t))\le \underset{s}{\overset{t}{\int }}m(\tau )\mathrm{d}\tau \text{ for all }[s,t]\subseteq I$

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 ${\mathbb{ℝ}}^n$ is equipped with its usual Euclidean norm $\Vert -\Vert$, and $\dot{\gamma }:E\to V^{*}$ is the usual Fréchet derivative with respect to time, then

$|{\gamma }^{\prime }|(t)=\Vert \dot{\gamma }(t)\Vert ,$

where $d(x,y):=\Vert x-y\Vert$ 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.