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 ) {\displaystyle (M,d)} be a metric space. Let E ⊆ R {\displaystyle E\subseteq \mathbb {R} } have a limit point at t ∈ R {\displaystyle t\in \mathbb {R} } . Let γ : E → M {\displaystyle \gamma :E\to M} be a path. Then the metric derivative of γ {\displaystyle \gamma } at t {\displaystyle t} , denoted | γ ′ | ( t ) {\displaystyle |\gamma '|(t)} , is defined by
| γ ′ | ( t ) := lim s → 0 d ( γ ( t + s ) , γ ( t ) ) | s | , {\displaystyle |\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 ( γ ( s ) , γ ( t ) ) ≤ ∫ s t m ( τ ) d τ for all [ s , t ] ⊆ I {\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 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 R n {\displaystyle \mathbb {R} ^{n}} is equipped with its usual Euclidean norm ‖ − ‖ {\displaystyle \|-\|} , and γ ˙ : E → V ∗ {\displaystyle {\dot {\gamma }}:E\to V^{*}} is the usual Fréchet derivative with respect to time, then
| γ ′ | ( t ) = ‖ γ ˙ ( t ) ‖ , {\displaystyle |\gamma '|(t)=\|{\dot {\gamma }}(t)\|,}
where d ( x , y ) := ‖ x − y ‖ {\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.{{cite book}}: CS1 maint: multiple names: authors list (link)
