====== Functional Derivative ======

It has also the name of variational derivative, and it is the equivalent to the derivative of a function. 


__** Intuition and working definition **__

We proceed with analogy to the ordinary derivative at x

$$
f(x) \to \left.\frac{d}{dy} f(y)\right|_{y=x} \equiv \lim_{\varepsilon\mapsto 0 } \frac{f(x+\varepsilon)-f(x)}{\varepsilon} = \frac{\partial}{\partial \varepsilon}f(x+\varepsilon)
$$
where the last expression is equivalent, and will help us in the next definition, $\varepsilon$ represents the variation around the point $x$.

For a functional, we will denote the differential with $\delta$, as representation of variation. As we know how to derive functions of "numbers", we will transform our functional of $q$ to a "normal" function for $\varepsilon$ with the map $F[q]\mapsto F[q+\varepsilon \phi]$ where $\phi$ is called a test function (analogous to a vector on multivariate calculus), then derive:

$$
S[q] \to \delta F[q|\phi] = \frac{\partial}{\partial \varepsilon}F[q+\varepsilon\phi]
$$

The summand $\varepsilon \phi$ is called //variation// of $q$.

=== Derivative of Integral Functionals ===

For a [[basic_tools:variational_calculus:functional:integral_functional| integral functional ]], what interests us is the derivative with respect a function. This is INCOMPLETE; WILL DO LATER


==== Formal Definitions ====

Frechet derivative on Banach spaces, and more generally Gatheaux derivative on locally convex spaces ... 



<tabbox Exercises>

1. 

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