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This is a formal notation for the following general thing: $$F(f+\delta f) = F(f) + \int A(x) \delta f(x) $$ Where $\delta f$ is the infinitesimal change in f, and it is a smooth test function, and then on the right hand side, $A(x)$ is just a linear operator on the space of functions. The notation for the $A(x)$ is then $$ A(x) = {\delta F\over \delta ...

Whenever I have troubles with functional derivative things, I just do the replacement of a continuous variable $x$ into a discrete index $i$. If I'm not mistaken this is what they call a "DeWitt notation". The hand waiving idea is that you can think of a functional $F[f(x)]$ as of a "ordinary function" of many variables ...

You aren't doing anything wrong, the paper made a mistake. It probably doesn't affect the result at all, since it is only a complex conjugation difference. But you are working a little too hard. First note: $$ {\delta \Lambda(k) \over \delta \Lambda(x) } = {\delta\over\delta\Lambda(x)} \int e^{-ikx'} \Lambda(x') dx' = e^{-ikx} $$ you could say by ...

One way to see that considering the dependence of $\dot{x}$ on $x$ is problematic is as follows: $x(t)$ maps a real number $t$ to another real number $x$. So $\dot{x}=dx/dt$ is the derivative of that map, meaning we take $$\lim_{\Delta t \to 0} \frac{x(t+\Delta t) - x(t)}{\Delta t}$$ So we can see that $dx/dt$ is itself another map from a real number $t$ ...

The physicist's derivative notation denotes the components of a Frechet derivative in the direction of the delta-function supported at $y$. This is one of those places where the habit of denoting the function $f$ by its value $f(x)$ gets confusing. It's somewhat clearer if you write $\delta_y$ for the delta function at $y$, and $$ \frac{\delta F}{\delta ...

The least error-prone way for computing the functional derivative $df(M)/dM(x)$ by hand is the use of the formula $\int dx \frac{\partial f(M)}{\partial M(x)} N(x) = \frac{d}{dt} f(M+tN) |_{t=0}$, where $N$ is of the same type as $M$ (but c-valued if $M$ is an operator). The right hand side is easy to work out, and the result is a linear functional in ...

Have a look first at several chapters in Stone and Goldbart, "Mathematics for Physics" (the free preprint is here) before entering into more specific books. I think you may want to see chapters 1, and parts of 2 and 9. You may find some parts of what you want in classic books of the "Comprehensive Mathematical Methods for Physics" type, but they don't ...

1) Let us write the Wilson-line of a simple open curve $\gamma: [s_i,s_f]\to \mathbb{R}^4$ as $$\tag{1} U(s_f,s_i) ~=~ \mathcal{P}\exp \left[ i\int_{\gamma} A_{\mu}~ dx^{\mu} \right]. $$ 2) The path-ordering $\mathcal{P}$ becomes important if the gauge potential $$\tag{2}A_{\mu}~=~A^a_{\mu} T_a$$ is non-abelian. Here $T_a$ are the generators of the ...

The standard encyclopedic treatise of nonlinear functional analysis is the 5 volume opus of Eberhard Zeidler, "Nonlinear Functional Analysis and Its Applications". It covers a lot of material about variational calculus, for example, in volume III "Variational Methods and Optimization". The applications are usually applications from physics. If that is too ...