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 $T(F)$.

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

Last revised on October 7, 2016 at 07:27:05.
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