Functional derivative

Let $𝑉$ be a function space over $𝕂$ and $𝐹 : 𝑉 \to 𝕂$ be functional. The functional derivative or Euler operator¹ at $𝑓$

𝛿 𝐹 𝛿 𝑓 \in 𝑉

is a function such that

𝛿 𝐹 [𝑓; 𝜂] = \int c o d 𝑉 𝛿 𝐹 𝛿 𝑓 𝜂 𝑑 𝑥

for all $𝜂$ satisfying certain boundary conditions. We informally identify $𝜂$ with $𝛿 𝑓$ to get a more intuitive expression.

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