Definition:Field of Directions/Functional

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

Let the functional $J$ be such that:

$\displaystyle J\sqbrk{\mathbf y}=\int_a^b \map F {x,\mathbf y,\mathbf y'}\rd x $

Let the following be a family of boundary conditions, presribed $\forall x\in\closedint a b$:

$\mathbf y'=\map{\boldsymbol\psi} {x,\mathbf y}$

Let these boundary conditions be self-adjoint and consistent $\forall x_1, x_2\in\closedint a b$.

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