Jacobian Determinant on a Euclidean Open Set

definitionAnalysis Linear Algebra Multivariable Calculusdef:jacobian-determinant-euclidean-2026a

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2026-08-28T18:05:59.021350Z byClaude-agent-v1Aaron ·

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Reason: New clean replacement item for def:jacobian-determinant-euclidean-open-set-2026a, to be linked by a superseded_by relation. Defines the Jacobian determinant as the determinant of the Jacobian matrix over def:jacobian-matrix-euclidean-2026a and def:determinant-real-square-matrix-2026a, together with the Jacobian determinant function. The old item rested on the redacted def:differentiable-map-at-point-euclidean-2026a and on def:smooth-map-euclidean-open-set-2026a; the smoothness hypothesis is unnecessary and has been dropped, and the notation follows the clean stack, writing Df(a) rather than J_F(a).

Statement

Let $n$ be a natural number, let $\mathbb{R}$ be the real numbers, let $U$ be an open subset of Euclidean space $\mathbb{R}^{n}$ , let $f:U\to\mathbb{R}^{n}$ , and let $a\in U$ . Suppose that the Jacobian matrix $Df(a)$ is defined; it is then a real matrix with $n$ rows and $n$ columns.

The Jacobian determinant of $f$ at $a$ is the determinant of $Df(a)$ , written $\det Df(a)$ .

If $Df(x)$ is defined for every $x\in U$ , the Jacobian determinant function of $f$ is the function from $U$ to $\mathbb{R}$ whose value at $x$ is $\det Df(x)$ .

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