# Definition:Differential/Functional

## 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.