# Definition:Central Field

## Definition

Let $R$ be a simply connected region in $\paren{n+1}$-dimensional space.

Let $\paren{x,\mathbf y}$ be a point in $R$.

Let $c=\paren{\sequence{c_i}_{0\le i\le n} }$ be a point lying outside of $R$.

Let $J$ be a functional such that:

- $\displaystyle J\sqbrk{\mathbf y}=\int_a^b \map F {x,\mathbf y,\mathbf y'}\rd x$

whose extremals $\mathbf y$ are curves in $\paren{n+1}$-dimensional space.

Suppose that one and only one extremal of $J$ leaves $c$ and passes through $\paren{x,\mathbf y}$, thereby for every point in $ R $ defining a relation:

- $\map{\mathbf y'} x=\map{\boldsymbol\psi} {x,\mathbf y}\quad\paren{\star}$.

Then the field of directions $\paren{\star}$ is called a **central field**.

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 6.32$: The Field of a Functional