Metric derivative explained

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\subseteqR

have a limit point at

t\inR

. Let

\gamma:E\toM

be a path. Then the metric derivative of

\gamma

, denoted

|\gamma'|(t)

, is defined by

|\gamma'|(t):=\lim_s

	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\left(\gamma(s),\gamma(t)\right)\leq

	t
\int
	s

m(\tau)d\tauforall[s,t]\subseteqI

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.

Rⁿ

is equipped with its usual Euclidean norm

\|-\|

, and

•

\gamma

:E\toV^*

is the usual Fréchet derivative with respect to time, then

|\gamma'|(t)=\|

•

\gamma

(t)\|,

where

d(x,y):=\|x-y\|

is the Euclidean metric.

References

Book: Ambrosio, L., Gigli, N. & Savaré, G. . Gradient Flows in Metric Spaces and in the Space of Probability Measures . ETH Zürich, Birkhäuser Verlag, Basel . 2005 . 3-7643-2428-7 . 24.