Definition:Twice Differentiable/Functional/Dependent on N functions

Definition
Let $\Delta J\sqbrk{\mathbf y;\mathbf h}$ be an increment of a functional, where $\mathbf y=\paren{\sequence {y_i}_{1\le i\le N} }$ is a vector.

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

where:
 * $\displaystyle\phi_1\sqbrk{\mathbf y;\mathbf h}$ is a linear functional
 * $\displaystyle\phi_2\sqbrk{\mathbf y;\mathbf h}$ is a quadratic functional $\mathbf h$
 * $\displaystyle\size {\mathbf h}=\sum_{i=1}^N\size {h_i}_1=\sum_{i=1}^N \max_{a\le x\le b} \lbrace{ \size {\map {h_i} x}+\size {\map {h_i'} x} }\rbrace$
 * $\displaystyle\epsilon\to 0$ as $\size {\mathbf h}\to 0$.

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

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

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