Functional differentation My question is from the polymer field's famous literature: The Equilibrium Theory of Inhomogeneous Polymers by Glenn H.Fredrickson.
In its Appendix C, the C.2 Functional differentiation item, it says a Taylor-expanded form of a functional $F[f+\delta f]$ is:
$F[f+\delta f] = F[f] + \int_a^bdx\Gamma_1(x)\delta f(x)+\frac{1}{2!}\int_a^bdx\int_a^bdx'\Gamma_2(x,x')\delta f(x)\delta f(x') + \cdots$
where the functions $\Gamma_i$ represent Taylor expansion coefficents.
I want to know what the general item of the functional Taylor Expansion, and why can it be expanded like this, how to deduce the expansion ? And which area is foucus on the above calculus of Functionals ?
I will honestly appreciate any valued suggestion.
 A: In Dieudonne's book  Foundations of Modern Analysis you will find a description of Taylor expansion appropriate to your situation.   It involves certain multi-linear maps.  There is a famous theorem   due to L. Schwartz stating  that under certain assumptions multilinear  maps can be given integral descriptions involving  kernel functions such as the  functions $\Gamma_k$ i in your question.  (References for  the kernel theorem can be quite demanding. A good place is F. Treves, Topological vector spaces, distributions  and kernels. )  The story is   a  bit more involved than my explanation. It looks to me that you are not a mathematician by training (this is not a criticism),   and the above references may seem impenetrable. If this is the case, it would be better to get hold of a live mathematician who is willing to spend an hour with you. He/she   will be able to help you more efficiently than any written source. In any case the question you asked belongs to the overlap of two fields: calculus of variations and functional analysis.
A: You can probably derive the formula by expanding the function $t \mapsto F[f + t\delta f]$ into its Taylor series centered at $t = 0$ and then setting $t = 1$ in the Taylor series. It then comes down to computing the $k$-th derivative with respect to $t$ of $F[f + t\delta f]$. If the integrals involved all converge properly, then you can differentiate under the integral sign to obtain the appropriate formulas for these derivatives. Since you don't say anything abut what $F$ is, I can't provide any more details than that.
A: As pointed out by Liviu Nicolaescu, there is a Taylor theorem for functionals on infinite dimensional spaces which could be used here  but there are problems in applying it to your case.  Non linear functionals on function spaces occur regularly in applications and, since they can be very complicated, it is natural to seek a  tractable concrete representation and the one you display is very commonly used.  As a general principle, if such a (strictly speaking incorrect) formulaiton is useful, then there is probably a rigorous formulation which lies behind it
and it is usually useful to seek it out.  In your case, I think that the required result is a pair of theorems by the French mathematicians Fr\'echet and Gateaux (incidentally, the two names associated with the introduction of the concept of differentiability in infinite dimensional spaces), dating from before the first world war.
Before describing them, a couple of generalities.  The prototype of such results is the question of approximating continuous, one-dimensional functions (say, on the unit interval) by simpler ones, e.g., polynomials.  There are two natural ways to do this.  The first one is probably more familiar to non-mathematicians---the Taylor series---whereas the second one (the Weierstrass approximation theorem) tends to be known only to professional mathematicians.  On the surface, the first one seems preferable---the approximation is by an infinite power series whose coefficients can be explicitly described.  However, it has two very important disadvantages---it only applies to very smooth functionals and the explicit expression for the coefficients is not usually very useful for computations since it involves (potentially arbitrarily) high derivatives of the function.  The second one
is more cumbersome to state since it does not present a series approximation but simply a sequence of polynomials which converge to the given function but is usually the better version for applications.  Indeed, whole libraries have been written about methods of approximating functions by polynomials (Google under approximation theory or constructive function theory).
Of course, the same applies to your problem which is at the next metalevel---approximating functionals which depend on functions.  Again there are two methods.  Using Taylor-type expansions  has the same disadvantages, together with the one mentioned in the  above answer that the kernel functions involved in the representation need not, in general, be continuous functions, but could be something much more pathological (measures or distributions) and so not likely to be useful for computations.
This brings us finally to the result mentioned above.  This is a  complete analogue of the Weierstra\ss theorem and states that any continuous functional on $C([0,1])$, say, can be approximated by a sequence of functionals of the type in your formulation (with finite sums).
Since these results seem to be forgotten lore and  have not , to my knowledge, found their way into the secondary (English language) textbook literature, I can only refer to the original articles, "Sur les fonctionnelles continues" (F) and "Sur les fonctionneles continues et les fonctionnelles analytiques" (G), both of which are, fortunately, available online.  I think that a modern, accessible write-up in English would be a service to engineers.  I should remark that a modern analyst would be able to prove these results (in  more general form) in a couple of lines, using the abstract version of Weierstra\ss' theorem, the Stone-Weierstra\ss theorem.  However, the methods used by the above-mentioned authors could be used, together with a version of the Weierstra\ss theorem in higher dimensions,
to cobble together a version with an explicit expression for the approximating functionals (using, e.g., spline functions as did G, trigonometric approximations as did F and Bernstein polynomials.)
