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:
 * $\delta^2 J \sqbrk {y; h}$