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 but not direction.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 ACp is the space of curves γ : I → X such that
for some m in the Lp space Lp. For γ ∈ ACp, the metric derivative of γ exists for Lebesgue-almost all times in I, and the metric derivative is the smallest m ∈ Lp 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.
