Gâteaux derivative

In mathematics, the Gâteaux derivative is a generalisation of the concept of directional derivative in differential calculus. Named after René Gâteaux, a French mathematician who died young in World War I, it is defined for locally convex topological vector spaces, contrasting with the derivative on Banach spaces, the Fréchet derivative. Both derivatives are often used to formalize the functional derivative commonly used in physics, particularly quantum field theory. Unlike other forms of derivatives, the Gâteaux derivative is not linear.

[edit] Definition

Suppose $X$ and $Y$ are locally convex topological vector spaces (for example, Banach spaces), $U\subset X$ is open, and

$F:X\rightarrow Y.$

The Gâteaux derivative $d F (u,ψ)$ of $F$ at $u\in U$ in the direction $\psi\in X$ is defined as

$dF(u,\psi)=\lim_{\tau\rightarrow 0}\frac{F(u+\tau \psi)-F(u)}{\tau}=\left.\frac{d}{d\tau}F(u+\tau \psi)\right|_{\tau=0}$

if the limit exists. If the limit exists for all $\psi \in X$ , then one says that $F$ has Gâteaux derivative at $u\in U$ .

One says that $F$ is continuously differentiable in $U$ if

$dF:U\times X \rightarrow Y$

If the Gâteaux derivative exists, it is unique.

For each $u\in U$ the Gâteaux derivative is an operator

$dF(u,\cdot):X\rightarrow Y.$

This operator is homogeneous, so that

$dF(u,\alpha\psi)=\alpha dF(u,\psi)\,$ ,

but it is not additive in general case, and, hence, is not always linear, unlike the Fréchet derivative.

$E:X\rightarrow \mathbb{R}$

given by

$E(u)=\int_\Omega F\left( u(x) \right)dx$

where $F$ is a real-valued function of a real variable with $F'=f\,$ and $u$ is defined on $Ω$ with real values, has Gâteaux derivative

$dE(u,\psi)=(f(u),\psi)\,.$

Indeed,

$\frac{E(u+\tau\psi) - E(u)}{\tau} = \frac{1}{\tau} \left( \int_\Omega F(u+\tau\psi)dx - \int_\Omega F(u)dx \right)$

$\quad\quad =\frac{1}{\tau} \left( \int_\Omega\int_0^1 \frac{d}{ds} F(u+s\tau\psi) \,ds\,dx \right)$

$\quad \quad =\int_\Omega\int_0^1 f(u+s\tau\psi)\psi \,ds\,dx.$

Letting $\tau\rightarrow 0$ (and assuming that all integrals are well-defined) gives as answer for the Gâteaux derivative

$\int_\Omega f(u(x))\psi(x) \,dx,$

that is, the inner product $(f(u),\psi).\,$

R Gâteaux. Sur les fonctionnelles continues et les fonctionnelles analytiques. Comptes rendus de l'academie des sciences, Paris, Vol. 157 (1913). Retrieved on 30 July, 2006.