# 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