Metric derivative - Alchetron, The Free Social Encyclopedia

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 ⊆ R have a limit point at t ∈ R . Let γ : E → M be a path. Then the metric derivative of γ at t , denoted | γ ′ | ( t ) , is defined by

| γ ′ | ( t ) := lim s → 0 d ( γ ( t + s ) , γ ( t ) ) | s | ,

if this limit exists.

Properties

Recall that AC^p(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

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 ∥ − ∥ , and γ ˙ : E → V ∗ is the usual Fréchet derivative with respect to time, then

| γ ′ | ( t ) = ∥ γ ˙ ( t ) ∥ ,

where d ( x , y ) := ∥ x − y ∥ is the Euclidean metric.

References

Metric derivative Wikipedia

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Contents

Definition

Properties

References