LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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The reference-triangle Duffy map has the stated image and Jacobian

Statement

For the map $D (r, s) = (r, r s)$ of The reference-triangle Duffy map from a vertex, the image of $[0, 1]^{2}$ is $T = {(x, y) : 0 \leq y \leq x \leq 1}$ . On $r > 0$ it has inverse $(x, y) \mapsto (x, y / x)$ and Jacobian determinant $r$ ; the edge $r = 0$ maps to $(0, 0)$ .

Facts & Assumptions

Given: The map $D (r, s) = (r, r s)$ on the square $[0, 1]^{2}$ .

[F1]

The reference-triangle Duffy map from a vertex: $D (r, s) = (r, r s)$ and $T = {(x, y) : 0 \leq y \leq x \leq 1}$ .

[F2]

Continuously differentiable maps, local inverses, and local diffeomorphisms: a map on an open Euclidean set is $C^{1}$ when all components are differentiable with continuous derivative entries; a local inverse is the inverse of a bijective restriction.

[F3]

The Jacobian determinant of a square-dimensional $C^{1}$ map is the determinant of its Jacobian matrix: the Jacobian determinant is the determinant of the derivative matrix.

Proof

1.1F1F2F3algebra

For an increment $(h, k)$ , $D (r + h, s + k) - D (r, s) = (h, s h + r k + h k)$ . The linear part $(h, s h + r k)$ has matrix $(\begin{array}{cc} 1 & 0 \\ s & r \end{array})$ , while the remainder has Euclidean norm $∣ h k ∣ \leq (h^{2} + k^{2}) / 2$ ; divided by $∥ (h, k) ∥_{2} = \sqrt{h^{2} + k^{2}}$ it tends to zero. Thus this is the total derivative. Its entries are continuous, so the polynomial extension is $C^{1}$ on $R^{2}$ by [F2]; its determinant is $r$ by [F3].

2.1F1step 1.1givenalgebra∎

If $0 \leq r, s \leq 1$ , then $0 \leq r s \leq r \leq 1$ , so $D (r, s) \in T$ . Conversely, for $(x, y) \in T$ with $x > 0$ , $r = x$ and $s = y / x$ lie in $[0, 1]$ and map to $(x, y)$ ; when $x = 0$ , $y = 0$ and every $(0, s)$ maps to the vertex. For $r > 0$ , the displayed inverse is unique. This proves the image, inverse, and collapsed-edge claims.

Depends on

Used by

The Duffy map turns a vertex singularity into an integrable radial weight Theorem

Cited to discharge well-definedness by The reference-triangle Duffy map from a vertex.

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Imperial College London, Numerical quadrature, Finite Element course notes, §1.6 (standard reference, not scraped)