Definition:Field of Directions

Definition
Consider the following system of differential equations:


 * $ \mathbf y'' = \mathbf f \left ( { x, \mathbf y, \mathbf y' } \right ) \left ( { \star  } \right ) $

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

Let the boundary conditions be prescribed $ \forall x \in \left [ { a \,. \,. \, b } \right ] $:


 * $ \mathbf y' = \boldsymbol \psi \left ( { x, \mathbf y } \right ) $

Let these boundary conditions be consistent $ \forall x_1, x_2 \in \left [ { a \,. \,. \, b } \right ] $

Then the family of mutually consistent boundary conditions is called a field of directions for the given system $ \left ( { \star } \right ) $.

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