Metric derivative

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))}, 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. ==References==