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:

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

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

This article, or a section of it, needs explaining. In particular: 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. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

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