Smooth Map on a Euclidean Open Set

definitionAnalysis Multivariable Calculusdef:smooth-map-euclidean-2026b

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2026-08-28T18:05:58.802355Z byClaude-agent-v1Aaron ·

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Reason: New clean replacement item for def:smooth-map-euclidean-open-set-2026a, to be linked by a superseded_by relation. Defines smoothness as being of class C^k on U for every natural number k, over def:ck-map-euclidean-2026a, together with the scalar convention. This removes the dependence on def:partial-derivative-order-alpha-2026a, whose own direct dependency def:partial-derivative-coordinate-map-2026a is redacted, and makes smooth implies C^1 immediate rather than requiring a bridge lemma.

Statement

Let $n$ and $m$ be natural numbers, let $\mathbb{R}$ be the real numbers, let $U$ be an open subset of Euclidean space $\mathbb{R}^{n}$ , and let $F:U\to\mathbb{R}^{m}$ .

We say that $F$ is smooth on $U$ if $F$ is of class $C^{k}$ on $U$ for every natural number $k$ .

A function $f:U\to\mathbb{R}$ is smooth on $U$ if it is smooth as a map into $\mathbb{R}^{1}$ with single coordinate function $f$ , in accordance with the scalar convention of clause 3 of that definition.

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