nLab
directional derivative

Directional derivatives

Idea

A directional derivative, or Gâteaux derivative, is a partial derivative of a function on a manifold along the direction given by a tangent vector.

Definitions

Let FF and GG be locally convex topological vector spaces, UFU \subseteq F an open subspace and P:UGP\colon U \to G a continuous map. The derivative of PP at the point fUf \in U in the direction hFh \in F is the limit

DP fh=lim t01t(P(f+th)P(f)). D P_f h = \lim_{t \to 0} \frac{1}{t} (P(f + t h) - P(f)) .

If the limit exists for every fUf \in U and every hFh \in F then one can define a map

DP:U×FG. D P\colon U \times F \to G .

If the limit exists and DPD P is continuous (jointly in both variables), we say that PP is continuously differentiable or C 1C^1.

A simple but nontrivial example is the operator

P:C [a,b]C [a,b] P\colon C^{\infty}[a, b] \to C^{\infty}[a, b]

given by

P(f)ff P(f) \coloneqq f f'

with the derivative

DP(f)h=fh+fh. D P(f) h = f' h + f h' .

In the context of a Fréchet space, it may be that the directional derivative in every direction exists but the Fréchet derivative does not; however the existence of Fréchet derivative implies the existence of directional derivatives in all directions.

The notion of directional derivatives extends to smooth manifolds (including infinite-dimensional ones based on Fréchet spaces) using local coordinates; the differentiability does not depend on the choice of a local chart. In this case we have (if everything is defined)

DP:T(U)G, D P\colon T(U) \to G ,

where T(U)T(U) is the tangent space of UU (an open subspace of T(F)T(F).

References

  • Wikipedia (English): Gâteaux derivative

  • eom: Gâteaux derivative, Gâteaux variation, René Gâteaux

  • R. Gâteaux, Sur les fonctionnelles continues et les fonctionnelles analytiques, C.R. Acad. Sci. Paris Sér. I Math. 157 (1913) pp. 325–327; Fonctions d’une infinités des variables indépendantes, Bull. Soc. Math. France 47 (1919) 70–96, numdam; Sur diverses questions du calcul fonctionnel, Bulletin de la Société Mathématique de France tome 50 (1922) 1–37, numdam

Revised on May 24, 2013 18:52:27 by Ingo Blechschmidt (79.219.135.215)