Definition:Differentiable Functional

Definition
Let $ \Delta J \left [ { y; h } \right ] $ be an increment of the functional $ J $ such that:


 * $ \displaystyle \Delta J \left [ { y; h } \right ] = \phi \left [ { y; h } \right ] + \epsilon \left \vert h \right \vert $,

where $ \phi \left [ { y; h } \right ] $ is a linear functional $ h $ and


 * $ \displaystyle \lim_{ \left\vert h \right\vert \to 0 } = 0 $.

Then the functional $ J \left [ { y } \right ]$ is said to be differentiable, and the principal linear part of the increment is called the variation (or differential) of a functional, and is denoted by $ \delta J \left [ { y; h } \right ] $. In other words, $ \phi \left [ { y; h } \right ] = \delta J \left [ { y; h } \right ] $.