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 $P\colon U \to G$ a continuous map. The derivative of $P$ at the point $f \in U$ in the direction $h \in F$ is the limit

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

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

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

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

A simple but nontrivial example is the operator

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

given by

$$P(f) \coloneqq f f'$$

with the derivative

$$D P(f) h = f' h + f h' .$$

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)

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

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

An analogue of the directional derivative and Faa di Bruno formula in the Goodwillie calculus are in

• Kristine Bauer, Brenda Johnson, Christina Osborne, Emily Riehl, Amelia Tebbe, Directional derivatives and higher order chain rules for abelian functor calculus (arxiv/1610.01930)