nLab
directional derivative

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 $F$ and $G$ be locally convex topological vector spaces, $U\subseteq F$ an open subspace and $P:U\to G$ a continuous map. The derivative of $P$ at the point $f\in U$ in the direction $h\in F$ is the limit

$$D{P}_{f}h=\underset{t\to 0}{\lim }\frac{1}{t}(P(f+th)-P(f)).$$D P_f h = \lim_{t \to 0} \frac{1}{t} (P(f + t h) - P(f)) .

If the limit exists for every $f\in U$ and every $h\in F$ then one can define a map

$$DP:U\times F\to G.$$D P\colon U \times F \to G .

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

A simple but nontrivial example is the operator

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

given by

$$P(f)≔ff\prime$$P(f) \coloneqq f f'

with the derivative

$$DP(f)h=f\prime h+fh\prime .$$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)

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

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

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