# Definition:Field of Directions

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## Definition

Consider the following system of differential equations:

- $(1): \quad \mathbf y' = \map {\mathbf f} {x, \mathbf y, \mathbf y'}$

where $\mathbf y$ is an $n$-dimensional vector.

Let the boundary conditions be prescribed $\forall x \in \closedint a b$:

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

Let these boundary conditions be consistent for all $x_1, x_2 \in \closedint a b$.

Then the family of mutually consistent boundary conditions is called a **field of directions** for the given system $(1)$.

That is, the first-order system is valid in an interval instead of a countable set of points.

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 6.31$: Consistent Boundary Conditions. General Definition of a Field