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

${\displaystyle |\gamma '|(t):=\lim _{s\to 0}{\frac {d(\gamma (t+s),\gamma (t))}{|s|}},}$

if this limit exists.

Properties

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

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

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

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

where ${\displaystyle d(x,y):=\|x-y\|}$ 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.CS1 maint: multiple names: authors list (link)