Definition:Differentiable Functional

Definition
Let $\Delta J[y; h]=\phi[y; h]+\epsilon \left\vert h\right\vert$,

where $\phi[y; h]$ is a linear functional w.r.t. $h$ and $\epsilon\to 0$ as $\left\vert h\right\vert\to 0$.

Then the functional $J[y]$ is said to be differentiable, and the principal linear part of the increment is called the variation (or differential) of a functional, and is denoted by $\delta J[y; h]$. In other words, $\phi[y; h]=\delta J[y; h]$.