# Definition:Field of Directions/Functional

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## 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