As the existing googled posts about functional derivatives are very complicated with a lot of math symbols, I am looking for if there exists a summary table of the general rules of functional derivatives that are easier to apply.
The example below from Matlab's functionalDerivative(f,[u,v]) illustrates my perplexity:
$\displaystyle S[ u,\ v] \ =\int _{b}^{a}\left( u^{2}( x)\frac{dv( x)}{dx} +v( x)\frac{d^{2} u( x)}{dx^{2}}\right) dx$
where the integrand is $\displaystyle f[ u( x) ,\ v( x) ,\ u''( x) ,\ v'( x)] \ =u^{2}\frac{dv}{dx} +v\frac{d^{2} u}{dx^{2}} \ $.
We want to find the functional derivative of the functional $\displaystyle S[ u,\ v] \ $ with respect to the two functions $\displaystyle u( x)$ and $\displaystyle v( x)$.
As far as I understand, the rule of functional derivative is similar to the ordinary derivative of a function as long as I consider the integrand as an ordinary function and see $\displaystyle u( x)$ and $\displaystyle v( x)$ as ordinary single variables of the function. So the derivative of $\displaystyle u^{2}\frac{dv}{dx}$ w.r.t $\displaystyle u$ will be $\displaystyle 2u\frac{dv}{dx}$ , which is the familiar product rule in ordinary derivatives.
My questions:
Are there easy rules to obtain the derivative of $\displaystyle v\frac{d^{2} u}{dx^{2}}$ w.r.t $\displaystyle u$?
Similarly, how about the derivative of $\displaystyle u^{2}\frac{dv}{dx}$ w.r.t $\displaystyle v$?
The derivatives of the functional $\displaystyle S[ u,\ v] \ $ w.r.t $\displaystyle u$ and $\displaystyle v$ provided by Matlab are $\displaystyle \begin{array}{{>{\displaystyle}l}} \begin{pmatrix} 2u( x)\frac{d}{dx} v( x) +\frac{d^{2}}{dx^{2}} v( x)\\ -2u( x)\frac{d}{dx} u( x) +\frac{d^{2}}{dx^{2}} u( x) \end{pmatrix}\\ \end{array}$.