# Definition:Central Field

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## Definition

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:

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

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 $\tuple {x, \mathbf y}$, 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**.

## Sources

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