Metric derivative

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).


Let (M,d) be a metric space. Let E \subseteq \mathbb{R} have a limit point at t \in \mathbb{R}. Let \gamma : E \to M be a path. Then the metric derivative of γ at t, denoted | γ' | (t), is defined by

| \gamma' | (t) := \lim_{s \to 0} \frac{d (\gamma(t + s), \gamma (t))}{| s |},

if this limit exists.


Recall that ACp(I; X) is the space of curves γ : IX such that

d \left( \gamma(s), \gamma(t) \right) \leq \int_{s}^{t} m(\tau) \, \mathrm{d} \tau \mbox{ for all } [s, t] \subseteq I

for some m in the Lp space Lp(I; R). For γ ∈ ACp(I; X), the metric derivative of γ exists for Lebesgue-almost all times in I, and the metric derivative is the smallest mLp(I; R) such that the above inequality holds.

If Euclidean space \mathbb{R}^{n} is equipped with its usual Euclidean norm \| - \|, and \dot{\gamma} : E \to V^{*} is the usual Fréchet derivative with respect to time, then

| \gamma' | (t) = \| \dot{\gamma} (t) \|,

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


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

Wikimedia Foundation. 2010.

См. также в других словарях:

  • Derivative (generalizations) — Derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, and geometry. Derivatives in analysis In real, complex, and functional… …   Wikipedia

  • Metric differential — In mathematical analysis, a metric differential is a generalization of a derivative for a Lipschitz continuous function defined on a Euclidean space and taking values in an arbitrary metric space. With this definition of a derivative, one can… …   Wikipedia

  • Metric tensor — In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold (such as a surface in space) which takes as input a pair of tangent vectors v and w and produces a real number (scalar) g(v,w) in a… …   Wikipedia

  • Metric connection — In mathematics, a metric connection is a connection in a vector bundle E equipped with a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve. Other common… …   Wikipedia

  • Metric Pixel Canvas — The Metric Pixel Canvas is a pixel based derivative of the ISO 216 standard developed by Dennis Pennekamp to unite pixel based raster graphics with vector graphics on a square root of two based metric canvas. Size Dimensions (pixels) Dimensions… …   Wikipedia

  • Metric compatibility — This article is about the concept in Riemannian geometry. For the typographic concept, see Typeface#Font metrics. In mathematics, given a metric tensor gab, a covariant derivative is said to be compatible with the metric if the following… …   Wikipedia

  • Covariant derivative — In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a… …   Wikipedia

  • Fréchet derivative — In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in mathematical analysis, especially functional… …   Wikipedia

  • Generalizations of the derivative — The derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, and geometry. Contents 1 Derivatives in analysis 1.1 Multivariable… …   Wikipedia

  • Glossary of Riemannian and metric geometry — This is a glossary of some terms used in Riemannian geometry and metric geometry mdash; it doesn t cover the terminology of differential topology. The following articles may also be useful. These either contain specialised vocabulary or provide… …   Wikipedia

Поделиться ссылкой на выделенное

Прямая ссылка:
Нажмите правой клавишей мыши и выберите «Копировать ссылку»