Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: Literature-sourcedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Cylindrical coordinates have absolute Jacobian determinant rr on an injective compact box

Example

The cylindrical-coordinate map C(r,θ,z)=(rcos⁡θ,rsin⁡θ,z)C(r,\theta,z)=(r\cos\theta,r\sin\theta,z) is injective on [1,2]×[π/6,π/3]×[−1,1][1,2]\times[\pi/6,\pi/3]\times[-1,1] and has absolute Jacobian determinant rr there.

Facts & Assumptions

Given: The cylindrical map on the displayed parameter box.

[L1]

Sine and cosine have their standard derivatives and satisfy sin⁡2+cos⁡2=1\sin^2+\cos^2=1 (The derivatives of sine and cosine are cosine and minus sine, Parity and the Pythagorean identity for sine and cosine).

[L2]

Cosine is strictly decreasing on [0,π][0,\pi] (Signs, monotonicity intervals, and ranges of sine and cosine).

[L3]

Compact-Jordan change of variables applies on injective boxes with nonzero Jacobian (Change of variables for an injective C1C^1 map on a compact Jordan set).

Verification

technique · computation
1.1

The derivative matrix is block triangular over the polar block, and [L1] gives det⁡DC=det⁡(cos⁡θ−rsin⁡θ0sin⁡θrcos⁡θ0001)=r.\det DC=\det\begin{pmatrix}\cos\theta&-r\sin\theta&0\\\sin\theta&r\cos\theta&0\\0&0&1\end{pmatrix}=r.

L1
2.1

The third image coordinate recovers zz. The first two recover rr from their Euclidean norm and then θ\theta from strict cosine monotonicity [L2]. The same recovery works on (1/2,5/2)×(π/12,5π/12)×(−2,2)(1/2,5/2)\times(\pi/12,5\pi/12)\times(-2,2), where r>0r>0, so the map is injective with invertible derivative on an open neighborhood of the compact box.

L1L2step 1.1
3.1

Therefore [L3] applies on this seam-free compact box, and every transformed volume integral carries precisely the factor rr.

L3step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 135 results over 21 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources