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Gâteaux derivative

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In mathematics, the Gâteaux derivative is a generalisation of the concept of directional derivative in differential calculus. It is named after René Gâteaux, a French mathematician who died young in World War I. Defined for locally convex topological vector spaces, it should be contrasted to the derivative on Banach spaces, the Fréchet derivative. Either derivative is often used to formalize the concept of the functional derivative that is frequently used in physics and in particular, in quantum field theory. Unlike other forms of derivatives, the Gâteaux derivative is not linear.

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[edit] Definition

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

<math>F:X\rightarrow Y.</math>

The Gâteaux derivative <math>dF(u,\psi)</math> of <math>F</math> at <math>u\in U</math> in the direction <math>\psi\in X</math> is defined as

<math>

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} </math>

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

One says that <math>F</math> is continuously differentiable in <math>U</math> if

<math>dF:U\times X \rightarrow Y</math>

is continuous.

[edit] Properties

If the Gâteaux derivative exists, it is unique.

For each <math>u\in U</math> the Gâteaux derivative is an operator

<math>dF(u,\cdot):X\rightarrow Y.</math>

This operator is homogeneous, so that

<math>dF(u,\alpha\psi)=\alpha dF(u,\psi)\,</math>,

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

[edit] Example

Let <math>X</math> be the Hilbert space of square-integrable functions on a Lebesgue measurable set <math>\Omega</math> in the Euclidean space RN. The functional

<math>E:X\rightarrow \mathbb{R}</math>

given by

<math> E(u)=\int_\Omega F\left( u(x) \right)dx </math>

where <math>F</math> is a real-valued function of a real variable with <math>F'=f\,</math> and <math>u</math> is defined on <math>\Omega</math> with real values, has Gâteaux derivative

<math>

dE(u,\psi)=(f(u),\psi)\,. </math>

Indeed,

<math>

\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) </math>

<math>

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

<math>

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

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

<math>\int_\Omega f(u(x))\psi(x) \,dx,</math>

that is, the inner product <math>(f(u),\psi).\,</math>

[edit] See also

[edit] References

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