Definition:Field of Directions/Functional

Definition
Let $ \mathbf y $ be an N-dimensional vector.

Let the functional $ J $ be such that:


 * $ \displaystyle J \left [ { \mathbf y } \right ] = \int_a^b F \left ( { x, \mathbf y, \mathbf y' } \right ) \mathrm d x $

Let the following be a family of boundary conditions, presribed $ \forall x \in \left [ { a \,. \,. \, b } \right ] $:


 * $ \mathbf y' = \boldsymbol \psi \left ( { x, \mathbf y } \right ) $

Let these boundary conditions be self-adjoint and consistent $ \forall x_1, x_2 \in \left [ { a \,. \,. \, b } \right ] $.

Then these boundary conditions are called field of directions of the functional $ J $.