\begin{definition}[Definition:Field of Directions]
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.
\end{definition}