# 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 with respect to $h$
$\epsilon\to 0$ 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:

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