# Functional Derivative¶

The functional derivative we talk about here is not the same as the variation used widely in Lagrangian formalism — although they look so much alike formally. To make sure you will not mistake them, we first illustrate what is the variation of a functional.

The variation of a functional $$F[\phi]$$ whose domain is a Banach space $$B$$ (a linear space where you can talk about the “length” — norm of its element) is usually implemented as Gâteaux derivative

$\frac{\delta F[\phi]}{\delta\phi} := (\forall \eta) :: \lim_{\varepsilon\rightarrow 0}\frac{F[\phi+\varepsilon\eta]-F[\phi]}{\varepsilon}$

which very much resembles the directional derivative in ordinary Calculus. Please notice that the $$\eta$$ here is just an arbitrary element of the Banach space, and in its application it is any physical field.

The variation is the “derivative” of the argument $$\phi$$ itself, without any designation of it value at some point $$\phi(x)$$; while functional derivative is defined differently. The functional derivative of $$F[\phi(x)]$$ with respect to value $$\phi(y)$$ is

$\frac{\delta F[\phi(x)]}{\delta\phi(y)} = \lim_{\varepsilon\rightarrow 0}\frac{F[\phi(x)+\varepsilon\delta(x;y)]-F[\phi(x)]}{\varepsilon}$

Therefore, you shall now see the differences between the variation and the functional derivative — the variation performs “derivative” on $$\phi$$ itself, while the functional derivative does so on the value $$\phi(y)$$. And thus a delta function has to be invoked to deal with the different points $$x$$ and $$y$$. You will see in chapter Quantum Theory that it is this delta function which brings the infinities into the S-matrix.