Gâteaux derivative

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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 of a function may be nonlinear.

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[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 dF(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

is continuous.

[edit] Properties

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.

However, for X and Y Banach spaces, if we suppose the Gâteaux derivative dF(u, ψ) of F is continuous and linear in ψ for all u ∈ U, and dF is a continuous mapping of metric spaces U → L(X, Y), then F is Fréchet differentiable. This is analogous to the criterion for the differentiability of a function from continuity of its partial derivatives.

If F is Fréchet differentiable, then it is also Gâteaux differentiable, and its Fréchet and Gâteaux derivatives agree.

[edit] Example

Let X be the Hilbert space of square-integrable functions on a Lebesgue measurable set Ω in the Euclidean space RN. The functional

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).\,

[edit] See also

[edit] References

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