# Definition:Differential/Functional

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## Contents

## Definition

Let $J \sqbrk y$ be a differentiable functional.

Let $h$ be an increment of the independent variable $y$.

Then the term linear with respect to $h$ is called the **differential** of the functional $J$, and is denoted by $\delta J \sqbrk {y; h}$.

## Notes

For a differentiable functional is holds that:

- $\Delta J \sqbrk {y; h} = \phi \sqbrk {y; h} + \epsilon \size h$

where $\phi$ is linear with respect to $h$.

Thus:

- $\delta J \sqbrk {y; h} = \phi \sqbrk {y; h}$

## Also known as

The **differential** $\delta J \sqbrk {y; h}$ is also known as the (**first**) **variation**.

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 1.3$: The Variation of a Functional. A Necessary Condition for an Extremum