Definition:Increment/Functional

Definition
Let $J[y]$ be a functional defined on a normed linear space.

Then $\Delta J[y; h]=J[y+h]-J[y]$ is an increment of a functional.

The increment of an independent variable $y=y(x)$ is denoted by $h=h(x)$

Also defined as
For fixed y an increment of a functional is only a functional of h and is denoted by $\Delta J[h]$.