Definition:Twice Differentiable/Functional

Definition
Let $\Delta J\sqbrk{y;h}$ be an increment of a functional.

Let:
 * $\Delta J\sqbrk{y;h}=\phi_1\sqbrk{y;h}+\phi_2\sqbrk{y;h}+\epsilon\size {h}^2$

where:
 * $\phi_1\sqbrk{y;h}$ is a linear functional
 * $\phi_2\sqbrk{y;h}$ is a quadratic functional $h$
 * $\epsilon\to 0$ as $\size h\to 0$.

Then the functional $J\sqbrk y$ is twice differentiable.

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

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