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 be a metric space. Let have a limit point at . Let be a path. Then the metric derivative of at , denoted , is defined by

if this limit exists.

Properties

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

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 is equipped with its usual Euclidean norm , and is the usual Fréchet derivative with respect to time, then

where is the Euclidean metric.

References