# 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 $F$ 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

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

$DP:U×F\to G.$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

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

given by

$P\left(f\right)≔ff\prime$P(f) \coloneqq f f'

with the derivative

$DP\left(f\right)h=f\prime h+fh\prime .$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 cooridnates; the differentiability does not depend on the choice of a local chart. In this case we have (if everything is defined)

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

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

## References

• Wikipedia (English): Gâteaux derivative

• 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 October 20, 2011 01:02:48 by Toby Bartels (64.89.53.218)