Definition:Twice Differentiable/Functional/Dependent on N functions

Definition
Let $\Delta J \left [{ \mathbf y; \mathbf h}\right]$ be an increment of a functional, where $ \mathbf y = \left ( { \langle y_i \rangle_{1 \le i \le N}  } \right ) $ is a vector.

Let:
 * $\Delta J \left [{ \mathbf y; \mathbf h }\right] = \phi_1 \left [ { \mathbf y; \mathbf h } \right ] + \phi_2 \left [ { \mathbf y; \mathbf h } \right ] + \epsilon \left \vert { \mathbf h } \right \vert^2 $

where:
 * $ \displaystyle \phi_1 \left [ { \mathbf y; \mathbf h } \right ] $ is a linear functional
 * $ \displaystyle \phi_2 \left [ { \mathbf y; \mathbf h } \right ]$ is a quadratic functional $ \mathbf h $
 * $ \displaystyle \left \vert \mathbf h \right \vert = \sum_{ i = 1 }^N \left \vert h_i \right \vert_1 = \sum_{ i = 1 }^N \max_{ a \le x \le b } \left \{ { \left \vert h_i \left ( { x } \right ) \right \vert + \left \vert h_i' \left ( { x } \right ) \right \vert } \right \}$
 * $ \displaystyle \epsilon \to 0$ as $\left \vert { \mathbf h } \right \vert \to 0 $.

Then the functional $J \left [ { \mathbf y } \right ] $ is twice differentiable.

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

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