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

:$, \gamma' , (t) := \lim_ \frac,$

if this limit exists.

Properties

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

:d \left( \gamma(s), \gamma(t) \right) \leq \int_^ m(\tau) \, \mathrm \tau \mbox , 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 $\mathbb^$ is equipped with its usual Euclidean norm $\, - \,$, and $\dot : E \to V^$ is the usual Fréchet derivative with respect to time, then

:$, \gamma' , (t) = \, \dot (t) \, ,$

where $d(x, y) := \, x - y \,$ is the Euclidean metric.

References

* {{cite book , author=Ambrosio, L., Gigli, N. & Savaré, G. , title=Gradient Flows in Metric Spaces and in the Space of Probability Measures , publisher=ETH Zürich, Birkhäuser Verlag, Basel , year=2005 , isbn=3-7643-2428-7 , page=24

Differential calculus
Metric geometry