# Definition:Field of Directions/Functional

## Definition

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 $J$.

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 6.32$: The Field of a Functional