# 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

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 5.24$: Quadratic Functionals. The Second Variation of a Functional