# Definition:Differential of Mapping/Functional

Jump to navigation
Jump to search

This article needs to be linked to other articles.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{MissingLinks}}` from the code. |

## 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

This page has been identified as a candidate for refactoring of medium complexity.In particular: Extract this into its own pageUntil this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Refactor}}` from the code. |

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