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 $$E \subseteq \mathbb{R}$$ 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 $$| \gamma' | (t)$$, 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 $$\mathbb{R}^{n}$$ 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.