Definition:Twice Differentiable/Functional

Definition
Let $\Delta J \left [{y; h}\right]$ be an increment of a functional.

Let:
 * $\Delta J \left [{y; h}\right] = \phi_1 \left [{y; h}\right] + \phi_2 \left [{y; h}\right] + \epsilon \left\vert{h}\right\vert^2$

where:
 * $\phi_1 \left [{y; h}\right]$ is a linear functional
 * $\phi_2 \left [{y; h}\right]$ is a quadratic functional $h$
 * $\epsilon \to 0$ as $\left \vert{h}\right \vert \to 0$.

Then the functional $J \left [{y}\right ]$ is twice differentiable.

The linear part $\phi_1$ is the first variation, denoted:
 * $\delta J \left [{y; h}\right]$

$\phi_2$ is called the second variation (or differential) of a functional, and is denoted:
 * $\delta^2 J \left[{y; h}\right]$