Metric derivative

Template:Short description 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 $E\subseteq ℝ$ have a limit point at $t\in ℝ$. Let $\gamma :E\to M$ be a path. Then the metric derivative of $\gamma$ at $t$, denoted $|{\gamma }^{\prime }|(t)$, is defined by

$|{\gamma }^{\prime }|(t):=\underset{s\to 0}{lim}\frac{d(\gamma (t+s),\gamma (t))}{|s|},$

if this limit exists.

Properties

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

$d(\gamma (s),\gamma (t))\le {\int }_s^{t}m(\tau )\mathrm{d}\tau \text{ for all }[s,t]\subseteq I$

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 ${ℝ}^n$ is equipped with its usual Euclidean norm $\Vert -\Vert$, and $\dot{\gamma }:E\to V^{\ast }$ is the usual Fréchet derivative with respect to time, then

$|{\gamma }^{\prime }|(t)=\Vert \dot{\gamma }(t)\Vert ,$

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

References

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