Definition:Twice Differentiable/Functional

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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:

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


Sources