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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 $$A(x) = {\delta F\over \delta ... 3 Whenever I have troubles with functional derivative things, I just do the replacement of a continuous variable x into a discrete index i. If I'm not mistaken this is what they call a "DeWitt notation". The hand waiving idea is that you can think of a functional F[f(x)] as of a "ordinary function" of many variables ... 3 You aren't doing anything wrong, the paper made a mistake. It probably doesn't affect the result at all, since it is only a complex conjugation difference. But you are working a little too hard. First note:$$ {\delta \Lambda(k) \over \delta \Lambda(x) } = {\delta\over\delta\Lambda(x)} \int e^{-ikx'} \Lambda(x') dx' = e^{-ikx} $$you could say by ... 3 One way to see that considering the dependence of \dot{x} on x is problematic is as follows: x(t) maps a real number t to another real number x. So \dot{x}=dx/dt is the derivative of that map, meaning we take$$\lim_{\Delta t \to 0} \frac{x(t+\Delta t) - x(t)}{\Delta t}$$So we can see that dx/dt is itself another map from a real number t ... 2 The physicist's derivative notation denotes the components of a Frechet derivative in the direction of the delta-function supported at y. This is one of those places where the habit of denoting the function f by its value f(x) gets confusing. It's somewhat clearer if you write \delta_y for the delta function at y, and$$ \frac{\delta F}{\delta ...

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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 ...

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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 ...

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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 ...

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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 ...

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