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.



Functional differentation
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.


