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