\begin{definition}[Definition:Central Field]
Let $R$ be a simply connected region in $\paren {n + 1}$-dimensional space.
Let $\tuple {x, \mathbf y}$ be a point in $R$.
Let $c = \tuple {\sequence {c_i}_{0 \mathop \le i \mathop \le n} }$ be a point lying outside of $R$.
Let $J$ be a functional such that:
:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
whose extremals $\mathbf y$ are curves in $\paren{n+1}$-dimensional space.
{{explain|Extremum or extremal, or are they the same thing? If they are, then an "also known as" section needs to be placed on the page Definition:Extremum of Functional. If not, then the terminology needs to be specifically amended and/or corrected.}}
Suppose that one and only one extremal of $J$ leaves $c$ and passes through $\tuple {x, \mathbf y}$, thereby for every point in $R$ defining a relation:
:$(1): \quad \map {\mathbf y'} x = \map {\boldsymbol \psi} {x, \mathbf y}$.
Then the field of directions $(1)$ is called a '''central field'''.
\end{definition}